Computational Fluid Dynamics Applications

Chapter 4

Abstract

Baver Okutmuştur

Lagrangian approaches.

1. Introduction

69

Eulerian coordinates, Lagrangian coordinates

them. The equations described by

Scalar Conservation Laws

We present a theoretical aspect of conservation laws by using simplest scalar models with essential properties. We start by rewriting the general scalar conservation law as a quasilinear partial differential equation and solve it by method of characteristics. Here we come across with the notion of strong and weak solutions depending on the initial value of the problem. Taking into account a special initial data for the left and right side of a discontinuity point, we get the related Riemann problem. An illustration of this problem is provided by some examples. In the remaining part of the chapter, we extend this analysis to the gas dynamics given in the Euler system of equations in one dimension. The transformations of this system into the Lagrangian coordinates follow by applying a suitable change of coordinates which is one of the main issues of this section. We next introduce a first-order Godunov finite volume scheme for scalar conservation laws which leads us to write Godunov schemes in both Eulerian and Lagrangian coordinates in one dimension where, in particular, the Lagrangian scheme is reformulated as a finite volume method. Finally, we end up the chapter by providing a comparison of Eulerian and

Keywords: conservation laws, Burgers' equation, shock and rarefaction waves, weak and strong solutions, Riemann problem, Euler system, Godunov schemes,

We present a general form of scalar conservation laws with further properties including some basic models and provide examples of computational methods for

in one dimension are known as scalar conservation laws where u ¼ u tð Þ ; x is the conserved quantity and f ¼ f uð Þ is the associated flux function depending on t and x. Whenever an initial condition uð Þ¼ 0; x u0ð Þ x is attached to Eq. (1), the problem is called the Cauchy problem the solution of which is a content of this chapter. The outlook of chapter is as follows. We introduce basic concepts and provide particular examples of scalar conservation laws in the first part. The equation of gas dynamics in Eulerian coordinates in one dimension is the main issue

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup><sup>x</sup> f uð Þ¼ 0, <sup>t</sup> . 0, <sup>x</sup><sup>∈</sup> <sup>R</sup> (1)

## Chapter 4 Scalar Conservation Laws

Baver Okutmuştur

### Abstract

We present a theoretical aspect of conservation laws by using simplest scalar models with essential properties. We start by rewriting the general scalar conservation law as a quasilinear partial differential equation and solve it by method of characteristics. Here we come across with the notion of strong and weak solutions depending on the initial value of the problem. Taking into account a special initial data for the left and right side of a discontinuity point, we get the related Riemann problem. An illustration of this problem is provided by some examples. In the remaining part of the chapter, we extend this analysis to the gas dynamics given in the Euler system of equations in one dimension. The transformations of this system into the Lagrangian coordinates follow by applying a suitable change of coordinates which is one of the main issues of this section. We next introduce a first-order Godunov finite volume scheme for scalar conservation laws which leads us to write Godunov schemes in both Eulerian and Lagrangian coordinates in one dimension where, in particular, the Lagrangian scheme is reformulated as a finite volume method. Finally, we end up the chapter by providing a comparison of Eulerian and Lagrangian approaches.

Keywords: conservation laws, Burgers' equation, shock and rarefaction waves, weak and strong solutions, Riemann problem, Euler system, Godunov schemes, Eulerian coordinates, Lagrangian coordinates

### 1. Introduction

We present a general form of scalar conservation laws with further properties including some basic models and provide examples of computational methods for them. The equations described by

$$
\partial\_t u + \partial\_x f(u) = \mathbf{0}, \quad t \ge \mathbf{0}, \boldsymbol{\omega} \in \mathbb{R} \tag{1}
$$

in one dimension are known as scalar conservation laws where u ¼ u tð Þ ; x is the conserved quantity and f ¼ f uð Þ is the associated flux function depending on t and x. Whenever an initial condition uð Þ¼ 0; x u0ð Þ x is attached to Eq. (1), the problem is called the Cauchy problem the solution of which is a content of this chapter. The outlook of chapter is as follows. We introduce basic concepts and provide particular examples of scalar conservation laws in the first part. The equation of gas dynamics in Eulerian coordinates in one dimension is the main issue

of the second part. After providing further instruction for these equations, we provide a transformation of the Eulerian equations in the Lagrangian coordinates. In the final part, we give as an example of computational methods for conservation laws, the Godunov schemes for the Eulerian, and the Lagrangian coordinates, respectively.

dt a tð Þ ; <sup>x</sup>; <sup>u</sup> <sup>¼</sup> dx

dc <sup>¼</sup> a tð Þ ; <sup>x</sup>; <sup>u</sup> , dx

ordinary differential equation (ODE):

DOI: http://dx.doi.org/10.5772/intechopen.83637

Scalar Conservation Laws

dt

u tð Þ ; x dx ¼

1.3 Strong (classical) solutions

Z <sup>x</sup><sup>1</sup> x0

of the flux functions between the points x<sup>0</sup> and x1:

<sup>∂</sup>tu <sup>þ</sup> <sup>f</sup> 0

0

We consider the initial value problem

to Eq. (7).

d dt <sup>Z</sup> <sup>x</sup><sup>1</sup> x0

<sup>u</sup>0ð Þ <sup>x</sup> <sup>∈</sup>C<sup>1</sup>

0

with λð Þ¼ u f

ð Þ¼ u tð Þ ; x tð Þ f

0

d dt x tðÞ¼ f

71

b tð Þ ; <sup>x</sup>; <sup>u</sup> <sup>¼</sup> du

By applying a parametrization of c, the relation (8) is transformed to a system of

dc <sup>¼</sup> b tð Þ ; <sup>x</sup>; <sup>u</sup> , du

In addition to these equations, if an initial condition u<sup>0</sup> ¼ u xð Þ<sup>0</sup> is also given, then by the existence theorem of ODE, there is a unique characteristic curve passing from each point ð Þ t0; x0; u<sup>0</sup> leading to an integral surface which is the solution

Observe that the scalar conservation law (1) is a particular example of Eq. (7) if

x0

This means, the quantity u tð Þ ; x is conserved so that it depends on the difference

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup>xð Þ¼ f uð Þ 0, <sup>t</sup> . 0, <sup>x</sup><sup>∈</sup> <sup>R</sup>

where the initial data is assumed to be continuously differentiable, that is,

where we define characteristic curves of Eq. (12) to be the solution of

u<sup>0</sup> be differentiable. Observe that (12) is equivalent to a quasilinear form:

We consider the characteristic curve passing through the point 0ð Þ ; x<sup>0</sup> :

is said to be a strong (or classical) solution if it satisfies Eq. (11), and it is continuously differentiable on a domain Ω ∈ R: Let u be a strong solution and the initial data

differential equation is transformed to a system of ordinary differential equations.

ð Þ R . Applying the chain rule to the relation (11), it follows that

ð Þ <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> 0, <sup>t</sup> . 0, <sup>x</sup><sup>∈</sup> <sup>R</sup>,

¼ ½inflow at the point x1� � ½ � outflow at the point x<sup>0</sup> :

f ut ð Þ ð Þ ; x <sup>x</sup> dx

<sup>u</sup>ð Þ¼ <sup>0</sup>; <sup>x</sup> <sup>u</sup>0ð Þ <sup>x</sup> , <sup>x</sup><sup>∈</sup> <sup>R</sup> (11)

<sup>u</sup>ð Þ¼ <sup>0</sup>; <sup>x</sup> <sup>u</sup>0ð Þ <sup>x</sup> , <sup>x</sup><sup>∈</sup> <sup>R</sup>, (12)

ð Þ u . Then a solution to the system (12) in a domain Ω ∈ R

ð Þ u . Applying the method of characteristics to Eq. (13), the partial

<sup>∂</sup>tu <sup>þ</sup> <sup>λ</sup>ð Þ <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> 0, (13)

we assign a tð Þ¼ ; x; u 1, b tð Þ ; x; u ux ¼ ð Þ f uð Þ <sup>x</sup>, and c tð Þ¼ ; x; u 0. The conserved quantity can be observed by integrating equation (1) over ½ � x0; x<sup>1</sup> . Indeed

<sup>∂</sup>tu tð Þ ; <sup>x</sup> dx ¼ � <sup>Z</sup> <sup>x</sup><sup>1</sup>

¼ f ut ð Þ� ð Þ ; x<sup>1</sup> f ut ð Þ ð Þ ; x<sup>0</sup>

c tð Þ ; <sup>x</sup>; <sup>u</sup> : (8)

dc <sup>¼</sup> c tð Þ ; <sup>x</sup>; <sup>u</sup> : (9)

(10)

#### 1.1 Conservation laws: integral form and differential form

We start by investigating the relation of the equations in gas dynamics with conservation laws. We take into account the equation of conservation of mass in one dimension. The density and the velocity are assumed to be constant in the tube where x is the distance and ρð Þ t; x is the density at the time t and at the point x. Then if we integrate the density on ½ � <sup>x</sup>1; <sup>x</sup><sup>2</sup> , we get total mass <sup>R</sup> <sup>x</sup><sup>2</sup> <sup>x</sup><sup>1</sup> ρð Þ t; x dx at time t. Assigning the velocity by u tð Þ ; x , then mass flux at becomes ρð Þ t; x u tð Þ ; x : It follows that the rate of change of the mass in ½ � x1; x<sup>2</sup> is

$$\frac{d}{dt}\int\_{\mathcal{X}\_1}^{\mathcal{X}\_2} \rho(t,\mathbf{x})d\mathbf{x} = \rho(t,\mathbf{x}\_1)u(t,\mathbf{x}\_1) - \rho(t,\mathbf{x}\_2)u(t,\mathbf{x}\_2). \tag{2}$$

The last equation is called integral form of conservation law. Integrating this expression in time from t<sup>1</sup> to t2, we get

$$\int\_{x\_1}^{x\_2} \rho(t\_2, \mathbf{x}) d\mathbf{x} - \int\_{x\_1}^{x\_2} \rho(t\_1, \mathbf{x}) d\mathbf{x} = \int\_{t\_1}^{t\_2} \rho(t, \mathbf{x}\_1) u(t, \mathbf{x}\_1) dt - \int\_{t\_1}^{t\_2} \rho(t, \mathbf{x}\_2) u(t, \mathbf{x}\_2) dt. \tag{3}$$

Using the fundamental theorem of calculus after reduction of Eq. (3), it follows that

$$
\rho(\mathbf{t}, \mathbf{x}\_2)u(\mathbf{t}, \mathbf{x}\_2) - \rho(\mathbf{t}, \mathbf{x}\_1)u(\mathbf{t}, \mathbf{x}\_1) = \int\_{\mathbf{x}\_1}^{\mathbf{x}\_2} \partial\_\mathbf{x} (\rho(\mathbf{t}, \mathbf{x})u(\mathbf{t}, \mathbf{x})) d\mathbf{x}.\tag{4}
$$

As a result, we get

$$\int\_{t\_1}^{t\_2} \int\_{x\_1}^{x\_2} \left\{ \partial\_t \rho(t, \mathbf{x}) + \partial\_\mathbf{x} \rho(t, \mathbf{x}) u(t, \mathbf{x}) \right\} d\mathbf{x} \, dt = \mathbf{0}.\tag{5}$$

Here the end points of the integrations are arbitrary; that is, for any ½ � x1; x<sup>2</sup> and ½ � t1; t<sup>2</sup> , the integrant must be zero. It follows that the conservation of mass yields

$$
\partial\_t \rho + \partial\_\mathbf{x} (u\rho) = \mathbf{0},\tag{6}
$$

which is said to be the differential form of the conservation law.

#### 1.2 A first-order quasilinear partial differential equations

A general solution to a quasilinear partial differential equation of the form

$$a(t, \mathfrak{x}, \mathfrak{u})\partial\_t \mathfrak{u} + b(t, \mathfrak{x}, \mathfrak{u})\partial\_{\mathfrak{x}}\mathfrak{u} = \mathfrak{c}(t, \mathfrak{x}, \mathfrak{u})\tag{7}$$

where a, b,c are non-zero and smooth on a given domain D ∈ R<sup>3</sup> follows by the characteristic method where the characteristic curves are defined by

Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

of the second part. After providing further instruction for these equations, we provide a transformation of the Eulerian equations in the Lagrangian coordinates. In the final part, we give as an example of computational methods for conservation laws, the Godunov schemes for the Eulerian, and the Lagrangian coordinates,

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

We start by investigating the relation of the equations in gas dynamics with conservation laws. We take into account the equation of conservation of mass in one dimension. The density and the velocity are assumed to be constant in the tube where x is the distance and ρð Þ t; x is the density at the time t and at the point x. Then

Assigning the velocity by u tð Þ ; x , then mass flux at becomes ρð Þ t; x u tð Þ ; x : It follows

The last equation is called integral form of conservation law. Integrating this

Using the fundamental theorem of calculus after reduction of Eq. (3), it follows

Here the end points of the integrations are arbitrary; that is, for any ½ � x1; x<sup>2</sup> and ½ � t1; t<sup>2</sup> , the integrant must be zero. It follows that the conservation of mass yields

which is said to be the differential form of the conservation law.

characteristic method where the characteristic curves are defined by

A general solution to a quasilinear partial differential equation of the form

where a, b,c are non-zero and smooth on a given domain D ∈ R<sup>3</sup> follows by the

1.2 A first-order quasilinear partial differential equations

Z <sup>t</sup><sup>2</sup> t1

ρð Þ t; x dx ¼ ρð Þ t; x<sup>1</sup> u tð Þ� ; x<sup>1</sup> ρð Þ t; x<sup>2</sup> u tð Þ ; x<sup>2</sup> : (2)

ρð Þ t; x<sup>1</sup> u tð Þ ; x<sup>1</sup> dt �

Z <sup>x</sup><sup>2</sup> x1

f g <sup>∂</sup>tρð Þþ <sup>t</sup>; <sup>x</sup> <sup>∂</sup>xρð Þ <sup>t</sup>; <sup>x</sup> u tð Þ ; <sup>x</sup> dx dt <sup>¼</sup> <sup>0</sup>: (5)

<sup>∂</sup>t<sup>ρ</sup> <sup>þ</sup> <sup>∂</sup>xð Þ¼ <sup>u</sup><sup>ρ</sup> 0, (6)

a tð Þ ; <sup>x</sup>; <sup>u</sup> <sup>∂</sup>tu <sup>þ</sup> b tð Þ ; <sup>x</sup>; <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> c tð Þ ; <sup>x</sup>; <sup>u</sup> (7)

<sup>x</sup><sup>1</sup> ρð Þ t; x dx at time t.

ρð Þ t; x<sup>2</sup> u tð Þ ; x<sup>2</sup> dt:

(3)

Z <sup>t</sup><sup>2</sup> t1

<sup>∂</sup>xð Þ <sup>ρ</sup>ð Þ <sup>t</sup>; <sup>x</sup> u tð Þ ; <sup>x</sup> dx: (4)

1.1 Conservation laws: integral form and differential form

if we integrate the density on ½ � <sup>x</sup>1; <sup>x</sup><sup>2</sup> , we get total mass <sup>R</sup> <sup>x</sup><sup>2</sup>

ρð Þ t1; x dx ¼

ρð Þ t; x<sup>2</sup> u tð Þ� ; x<sup>2</sup> ρð Þ t; x<sup>1</sup> u tð Þ¼ ; x<sup>1</sup>

that the rate of change of the mass in ½ � x1; x<sup>2</sup> is

Z <sup>x</sup><sup>2</sup> x1

d dt

expression in time from t<sup>1</sup> to t2, we get

Z <sup>x</sup><sup>2</sup> x1

Z <sup>t</sup><sup>2</sup> t1

Z <sup>x</sup><sup>2</sup> x1

ρð Þ t2; x dx �

As a result, we get

respectively.

Z <sup>x</sup><sup>2</sup> x1

that

70

$$\frac{dt}{a(t,\infty,u)} = \frac{dx}{b(t,\infty,u)} = \frac{du}{c(t,\infty,u)}.\tag{8}$$

By applying a parametrization of c, the relation (8) is transformed to a system of ordinary differential equation (ODE):

$$\frac{dt}{dc} = a(t, \boldsymbol{x}, \boldsymbol{u}), \quad \frac{d\boldsymbol{x}}{dc} = b(t, \boldsymbol{x}, \boldsymbol{u}), \quad \frac{d\boldsymbol{u}}{dc} = c(t, \boldsymbol{x}, \boldsymbol{u}).\tag{9}$$

In addition to these equations, if an initial condition u<sup>0</sup> ¼ u xð Þ<sup>0</sup> is also given, then by the existence theorem of ODE, there is a unique characteristic curve passing from each point ð Þ t0; x0; u<sup>0</sup> leading to an integral surface which is the solution to Eq. (7).

Observe that the scalar conservation law (1) is a particular example of Eq. (7) if we assign a tð Þ¼ ; x; u 1, b tð Þ ; x; u ux ¼ ð Þ f uð Þ <sup>x</sup>, and c tð Þ¼ ; x; u 0. The conserved quantity can be observed by integrating equation (1) over ½ � x0; x<sup>1</sup> . Indeed

$$\frac{d}{dt}\int\_{x\_0}^{x\_1} u(t,x) \, d\mathbf{x} = \int\_{x\_0}^{x\_1} \partial\_t u(t,x) \, d\mathbf{x} = -\int\_{x\_0}^{x\_1} f(u(t,x))\_{\mathbf{x}} \, d\mathbf{x}$$

$$= f(u(t, x\_1)) - f(u(t, x\_0)) \tag{10}$$

¼ ½inflow at the point x1� � ½ � outflow at the point x<sup>0</sup> :

This means, the quantity u tð Þ ; x is conserved so that it depends on the difference of the flux functions between the points x<sup>0</sup> and x1:

#### 1.3 Strong (classical) solutions

We consider the initial value problem

$$\begin{aligned} \partial\_t u + \partial\_x (f(u)) &= 0, \quad t \ge 0, \quad x \in \mathbb{R} \\ u(0, \boldsymbol{x}) &= u\_0(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathbb{R} \end{aligned} \tag{11}$$

where the initial data is assumed to be continuously differentiable, that is, <sup>u</sup>0ð Þ <sup>x</sup> <sup>∈</sup>C<sup>1</sup> ð Þ R . Applying the chain rule to the relation (11), it follows that

$$\begin{aligned} \partial\_t u + f'(u)\,\partial\_x u &= 0, \quad t > 0, \quad \varkappa \in \mathbb{R}, \\ u(0, \varkappa) &= u\_0(\varkappa), \quad \varkappa \in \mathbb{R}, \end{aligned} \tag{12}$$

where we define characteristic curves of Eq. (12) to be the solution of d dt x tðÞ¼ f 0 ð Þ¼ u tð Þ ; x tð Þ f 0 ð Þ u . Then a solution to the system (12) in a domain Ω ∈ R is said to be a strong (or classical) solution if it satisfies Eq. (11), and it is continuously differentiable on a domain Ω ∈ R: Let u be a strong solution and the initial data u<sup>0</sup> be differentiable. Observe that (12) is equivalent to a quasilinear form:

$$
\partial\_t u + \lambda(u)\partial\_x u = 0,\tag{13}
$$

with λð Þ¼ u f 0 ð Þ u . Applying the method of characteristics to Eq. (13), the partial differential equation is transformed to a system of ordinary differential equations. We consider the characteristic curve passing through the point 0ð Þ ; x<sup>0</sup> :

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{aligned} \partial\_t \mathfrak{x} &= \lambda(u(t, \mathfrak{x}(t))) \\ \mathfrak{x}(\mathbf{0}) &= \mathfrak{x}\_0. \end{aligned} \tag{14}$$

<sup>∂</sup>tu <sup>þ</sup> <sup>u</sup>∂xu <sup>¼</sup> <sup>ν</sup>∂xxu, (21)

ð Þ<sup>ξ</sup> , <sup>∂</sup>xxu <sup>¼</sup> <sup>g</sup>″ð Þ<sup>ξ</sup> : (23)

ð Þ� <sup>ξ</sup> <sup>ν</sup> <sup>g</sup>″ð Þ¼ <sup>ξ</sup> <sup>0</sup>: (24)

. Supposing that g1, g<sup>2</sup> are real implies g<sup>1</sup> . g2.

4ν ξ � � (27)

� � (28)

<sup>2</sup> ¼ λ, that is the wave

<sup>2</sup> tan <sup>h</sup> <sup>g</sup><sup>1</sup> � <sup>g</sup><sup>2</sup>

� �ð Þ <sup>x</sup> � <sup>λ</sup><sup>t</sup>

<sup>g</sup><sup>2</sup> � <sup>ν</sup> <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>C</sup>, <sup>C</sup> : constant: (25)

� � <sup>¼</sup> <sup>g</sup><sup>2</sup> � <sup>2</sup> <sup>λ</sup> <sup>g</sup> � <sup>2</sup><sup>C</sup> <sup>¼</sup> <sup>2</sup><sup>ν</sup> dg=dξ, (26)

<sup>2</sup> � <sup>g</sup><sup>1</sup> � <sup>g</sup><sup>2</sup>

<sup>4</sup><sup>ν</sup> <sup>g</sup><sup>1</sup> � <sup>g</sup><sup>2</sup>

ð Þξ , 0 for all ξ. This means the solution gð Þξ decreases

<sup>2</sup> is the wave speed. We can observe that lim<sup>ξ</sup>!�<sup>∞</sup> gð Þ¼ ξ g<sup>1</sup> and

u tð Þ¼ ; x gð Þ¼ ξ g xð Þ � λ t , with ξ ¼ x � λt, (22)

where ν∂xxu is the viscosity term. Equation (21) can be considered as a combination of nonlinear wave motion and linear diffusion term so that it is balance between time evolution, nonlinearity, and diffusion. The term u∂xu is a convection term that may have an effect to wave breaking, and the term ν∂xxu is a diffusion term that may cause to efface the wave breaking and to flatten discontinuities, and thus we expect to achieve a smooth solution. We try to find a traveling wave

where g and λ are to be determined. Applying the chain rule, we get

ð Þ<sup>ξ</sup> , <sup>∂</sup>xu <sup>¼</sup> <sup>g</sup><sup>0</sup>

ð Þþ ξ gð Þξ g<sup>0</sup>

solution of Eq. (21) of the form

Scalar Conservation Laws

DOI: http://dx.doi.org/10.5772/intechopen.83637

Rewriting Eq. (25) by

it follows that g1, <sup>2</sup> ¼ λ �

where <sup>λ</sup> <sup>¼</sup> <sup>g</sup>1þg<sup>2</sup>

lim<sup>ξ</sup>!<sup>∞</sup> gð Þ¼ ξ g<sup>2</sup> with g<sup>0</sup>

to the inviscid Burgers' equation.

73

<sup>∂</sup>tu ¼ �<sup>λ</sup> <sup>g</sup><sup>0</sup>

�λ g<sup>0</sup>

�λ g þ

1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>C</sup>

Using separation of variable and then integrating equation (26), we get

<sup>g</sup>1�g<sup>2</sup> ð Þ <sup>2</sup><sup>ν</sup> <sup>ξ</sup> <sup>¼</sup> <sup>g</sup><sup>1</sup> <sup>þ</sup> <sup>g</sup><sup>2</sup>

2

monotonically with <sup>ξ</sup> from the value <sup>g</sup><sup>1</sup> to <sup>g</sup>2. At <sup>ξ</sup> <sup>¼</sup> 0, <sup>u</sup> <sup>¼</sup> <sup>g</sup>1þg<sup>2</sup>

As a result the explicit form of traveling wave solution of Eq. (21) becomes

form gð Þξ travels from left to right with speed λ equal to the average value of its asymptotic values. The solution resembles to a shock form as it connects the asymptotic states g<sup>1</sup> and g2. Without the viscosity term, the solutions to Burgers equation allow shock forms which finally break. The diffusion term prevents incrementally deformation of the wave and its breaking by withstanding the nonlinearity. As a conclusion, there exists a balance between nonlinear advection term and the linear diffusion term. The wave form is notably affected by the diffusion coefficient ν. If ν is smaller, then the transition layer between two asymptotic values of solution is sharper. In the limit ν ! 0, the solutions converge to the step shock wave solutions

tan <sup>h</sup> <sup>1</sup>

p

<sup>g</sup>1�g<sup>2</sup> ð Þ <sup>2</sup><sup>ν</sup> <sup>ξ</sup>

Plugging these terms in Eq. (21), we get

Taking integration with respect to ξ gives

g � g<sup>1</sup> � � <sup>g</sup> � <sup>g</sup><sup>2</sup>

<sup>g</sup>ð Þ¼ <sup>ξ</sup> <sup>g</sup><sup>1</sup> <sup>þ</sup> <sup>g</sup>2<sup>e</sup>

1 þ e

u tð Þ¼ ; <sup>x</sup> <sup>λ</sup> � <sup>g</sup><sup>1</sup> � <sup>g</sup><sup>2</sup>

Along this characteristic curve,

$$
\partial\_t u(t, \mathbf{x}(t)) = \partial\_t u(t, \mathbf{x}(t)) + \partial\_t \mathbf{x} \partial\_\mathbf{x} u(t, \mathbf{x}(t)) = \partial\_t u + \lambda(u) \partial\_\mathbf{x} u = \mathbf{0} \tag{15}
$$

is satisfied, that is, u is constant. Hence, the characteristic curves are straight lines satisfying

$$
\boldsymbol{\mathfrak{x}} = \boldsymbol{\mathfrak{x}}\_0 + \lambda (\boldsymbol{\mathfrak{u}}\_0(\boldsymbol{\mathfrak{x}}\_0)) \boldsymbol{t} = \mathbf{0}.\tag{16}
$$

Hence we can define smooth solutions by u tð Þ¼ ; x u0ð Þ x<sup>0</sup> . If the slope of the characteristics is mchar <sup>¼</sup> <sup>1</sup> <sup>λ</sup>ð Þ <sup>u</sup>0ð Þ xi , then depending on the behavior of <sup>λ</sup>, the solution takes different forms. If λð Þ u0ð Þ x is increasing, then the slopes of the characteristics are decreasing. As a result, the characteristics do not intersect, and thus solution can be defined for all t which is greater than zero. On the other hand, if λð Þ u0ð Þ x is decreasing, then the slopes of the characteristics will be increasing which implies that the characteristics intersect at some point. But at the intersection point, solution cannot take both values u0ð Þ x<sup>1</sup> and u0ð Þ x<sup>2</sup> . Therefore, we cannot define the strong solution for all t . 0.

#### 1.4 Linear advection equation

The basic example of the scalar conservation law is the linear advection equation. It can be obtained by setting a tð Þ¼ ; x; u 1, b tð Þ¼ ; x; u λ, and c tð Þ¼ ; x; u 0 in Eq. (7). The flux function takes the form f uð Þ¼ λu where λ is a constant. Then the following quasilinear partial differential equation

$$
\partial\_t \mathfrak{u} + \lambda \partial\_\mathbf{x} \mathfrak{u} = \mathbf{0} \tag{17}
$$

is a linear advection equation. Similar to Eqs. (11) and (12), an initial value problem for linear advection equation is described by

$$\begin{aligned} \partial\_t u + \partial\_x f(u) &= 0, \quad -\infty \le x \le \infty, \quad t \ge 0, \\ u(0, \boldsymbol{x}) &= u\_0(\boldsymbol{x}) = f(\boldsymbol{x}\_0), \quad -\infty \le \boldsymbol{x} \le \infty. \end{aligned} \tag{18}$$

Applying the method of characteristics, it follows that dt <sup>1</sup> <sup>¼</sup> dx <sup>λ</sup> <sup>¼</sup> du <sup>0</sup> or equivalently

$$
\mu = c\_1, \quad \frac{d\mathbf{x}}{dt} = \lambda = c\_1, \quad \mathbf{x} = c\_1 t + c\_2,\tag{19}
$$

where c<sup>1</sup> and c<sup>2</sup> are constant and x � λt ¼ c2: As a conclusion, the solution is

$$u(t, \mathbf{x}) = u\_0(\mathbf{x} - \lambda t), \quad t \ge \mathbf{0}. \tag{20}$$

Here λ is the wave speed, and the characteristic lines x � λt ¼ c<sup>2</sup> are wavefronts which are constants.

#### 1.5 Burgers' equation

Burgers' equation is the simplest nonlinear partial differential equation and is the one of the most common models used in the scalar conservation laws and fluid dynamics. The classical Burgers' equation is described by

Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

<sup>∂</sup>tx <sup>¼</sup> <sup>λ</sup>ð Þ u tð Þ ; x tð Þ

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

is satisfied, that is, u is constant. Hence, the characteristic curves are straight

Hence we can define smooth solutions by u tð Þ¼ ; x u0ð Þ x<sup>0</sup> . If the slope of the

takes different forms. If λð Þ u0ð Þ x is increasing, then the slopes of the characteristics are decreasing. As a result, the characteristics do not intersect, and thus solution can be defined for all t which is greater than zero. On the other hand, if λð Þ u0ð Þ x is decreasing, then the slopes of the characteristics will be increasing which implies that the characteristics intersect at some point. But at the intersection point, solution cannot take both values u0ð Þ x<sup>1</sup> and u0ð Þ x<sup>2</sup> . Therefore, we cannot define the

The basic example of the scalar conservation law is the linear advection equation. It can be obtained by setting a tð Þ¼ ; x; u 1, b tð Þ¼ ; x; u λ, and c tð Þ¼ ; x; u 0 in Eq. (7). The flux function takes the form f uð Þ¼ λu where λ is a constant. Then the

is a linear advection equation. Similar to Eqs. (11) and (12), an initial value

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup>xf uð Þ¼ 0, � <sup>∞</sup> , <sup>x</sup> , <sup>∞</sup>, <sup>t</sup><sup>≥</sup> 0,

where c<sup>1</sup> and c<sup>2</sup> are constant and x � λt ¼ c2: As a conclusion, the solution is

Here λ is the wave speed, and the characteristic lines x � λt ¼ c<sup>2</sup> are wavefronts

Burgers' equation is the simplest nonlinear partial differential equation and is the one of the most common models used in the scalar conservation laws and fluid

<sup>u</sup>ð Þ¼ <sup>0</sup>; <sup>x</sup> <sup>u</sup>0ð Þ¼ <sup>x</sup> f xð Þ<sup>0</sup> , � <sup>∞</sup> , <sup>x</sup> , <sup>∞</sup>: (18)

<sup>∂</sup>tu tð Þ¼ ; x tð Þ <sup>∂</sup>tu tð Þþ ; x tð Þ <sup>∂</sup>tx∂xu tð Þ¼ ; x tð Þ <sup>∂</sup>tu <sup>þ</sup> <sup>λ</sup>ð Þ <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> 0 (15)

Along this characteristic curve,

lines satisfying

characteristics is mchar <sup>¼</sup> <sup>1</sup>

strong solution for all t . 0.

which are constants.

1.5 Burgers' equation

72

1.4 Linear advection equation

following quasilinear partial differential equation

problem for linear advection equation is described by

Applying the method of characteristics, it follows that dt

<sup>u</sup> <sup>¼</sup> <sup>c</sup>1, dx

dynamics. The classical Burgers' equation is described by

<sup>x</sup>ð Þ¼ <sup>0</sup> <sup>x</sup>0: (14)

x ¼ x<sup>0</sup> þ λð Þ u0ð Þ x<sup>0</sup> t ¼ 0: (16)

<sup>λ</sup>ð Þ <sup>u</sup>0ð Þ xi , then depending on the behavior of <sup>λ</sup>, the solution

<sup>∂</sup>tu <sup>þ</sup> <sup>λ</sup>∂xu <sup>¼</sup> <sup>0</sup> (17)

<sup>1</sup> <sup>¼</sup> dx

dt <sup>¼</sup> <sup>λ</sup> <sup>¼</sup> <sup>c</sup>1, <sup>x</sup> <sup>¼</sup> <sup>c</sup>1<sup>t</sup> <sup>þ</sup> <sup>c</sup>2, (19)

u tð Þ¼ ; x u0ð Þ x � λt , t≥0: (20)

<sup>λ</sup> <sup>¼</sup> du

<sup>0</sup> or equivalently

$$
\partial\_t \mathfrak{u} + \mathfrak{u} \partial\_\mathfrak{x} \mathfrak{u} = \nu \partial\_{\mathfrak{X}\mathfrak{X}} \mathfrak{u}, \tag{21}
$$

where ν∂xxu is the viscosity term. Equation (21) can be considered as a combination of nonlinear wave motion and linear diffusion term so that it is balance between time evolution, nonlinearity, and diffusion. The term u∂xu is a convection term that may have an effect to wave breaking, and the term ν∂xxu is a diffusion term that may cause to efface the wave breaking and to flatten discontinuities, and thus we expect to achieve a smooth solution. We try to find a traveling wave solution of Eq. (21) of the form

$$\mathfrak{u}(t,\mathfrak{x}) = \mathfrak{g}(\xi) = \mathfrak{g}(\mathfrak{x} - \lambda t), \quad \text{with} \quad \xi = \mathfrak{x} - \lambda t,\tag{22}$$

where g and λ are to be determined. Applying the chain rule, we get

$$
\partial \mathfrak{a} = -\lambda \mathfrak{g}'(\xi), \quad \partial\_{\mathfrak{x}} \mathfrak{u} = \mathfrak{g}'(\xi), \quad \partial\_{\mathfrak{x}\mathfrak{x}} \mathfrak{u} = \mathfrak{g}''(\xi). \tag{23}
$$

Plugging these terms in Eq. (21), we get

$$-\lambda \mathbf{g}'(\xi) + \mathbf{g}(\xi)\mathbf{g}'(\xi) - \nu \mathbf{g}''(\xi) = \mathbf{0}.\tag{24}$$

Taking integration with respect to ξ gives

$$-\lambda \text{g} + \frac{1}{2} \text{g}^2 - \nu \text{g}' = \text{C}, \quad \text{C}: constant. \tag{25}$$

Rewriting Eq. (25) by

$$(\mathbf{g} - \mathbf{g}\_1)(\mathbf{g} - \mathbf{g}\_2) = \mathbf{g}^2 - 2\lambda \mathbf{g} - 2\mathbf{C} = 2\nu \, d\mathbf{g}/d\xi,\tag{26}$$

it follows that g1, <sup>2</sup> ¼ λ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>C</sup> p . Supposing that g1, g<sup>2</sup> are real implies g<sup>1</sup> . g2. Using separation of variable and then integrating equation (26), we get

$$g(\xi) = \frac{\mathbf{g}\_1 + \mathbf{g}\_2 e^{\left(\frac{\mathbf{g}\_1 - \mathbf{g}\_2}{\omega}\right)\xi}}{1 + e^{\left(\frac{\mathbf{g}\_1 - \mathbf{g}\_2}{\omega}\right)\xi}} = \frac{\mathbf{g}\_1 + \mathbf{g}\_2}{2} - \frac{\mathbf{g}\_1 - \mathbf{g}\_2}{2} \tan h\left(\frac{\mathbf{g}\_1 - \mathbf{g}\_2}{4\nu}\xi\right) \tag{27}$$

As a result the explicit form of traveling wave solution of Eq. (21) becomes

$$u(t, \mathbf{x}) = \lambda - \frac{\mathbf{g}\_1 - \mathbf{g}\_2}{2} \tan h \left(\frac{\mathbf{1}}{4\nu} (\mathbf{g}\_1 - \mathbf{g}\_2)(\mathbf{x} - \lambda t)\right) \tag{28}$$

where <sup>λ</sup> <sup>¼</sup> <sup>g</sup>1þg<sup>2</sup> <sup>2</sup> is the wave speed. We can observe that lim<sup>ξ</sup>!�<sup>∞</sup> gð Þ¼ ξ g<sup>1</sup> and lim<sup>ξ</sup>!<sup>∞</sup> gð Þ¼ ξ g<sup>2</sup> with g<sup>0</sup> ð Þξ , 0 for all ξ. This means the solution gð Þξ decreases monotonically with <sup>ξ</sup> from the value <sup>g</sup><sup>1</sup> to <sup>g</sup>2. At <sup>ξ</sup> <sup>¼</sup> 0, <sup>u</sup> <sup>¼</sup> <sup>g</sup>1þg<sup>2</sup> <sup>2</sup> ¼ λ, that is the wave form gð Þξ travels from left to right with speed λ equal to the average value of its asymptotic values. The solution resembles to a shock form as it connects the asymptotic states g<sup>1</sup> and g2. Without the viscosity term, the solutions to Burgers equation allow shock forms which finally break. The diffusion term prevents incrementally deformation of the wave and its breaking by withstanding the nonlinearity. As a conclusion, there exists a balance between nonlinear advection term and the linear diffusion term. The wave form is notably affected by the diffusion coefficient ν. If ν is smaller, then the transition layer between two asymptotic values of solution is sharper. In the limit ν ! 0, the solutions converge to the step shock wave solutions to the inviscid Burgers' equation.

Remark. If the initial data is smooth and very small, then the uxx term is negligible compared to other terms before the beginning of wave breaking. As the wave breaking starts, the uxx term raises faster than ux term. After a while, the term uxx becomes comparable to the other terms so that it keeps the solution smooth, giving rise to avoid breakdown solutions.

#### 1.6 Inviscid Burgers' equation

Whenever ν ¼ 0, Eq. (21) is called the inviscid Burgers' equation. This equation can be obtained by substituting f uð Þ¼ <sup>u</sup>2=2 in the scalar conservation law (1), that is

$$
\partial\_t u + \partial\_\mathbf{x} \left( u^2 / 2 \right) = \partial\_t u + u \partial\_\mathbf{x} u = \mathbf{0}. \tag{29}
$$

is a simple example of discontinuous solution of the conservation law (11). If uL 6¼ uR, the relation (33) is called a shock wave connecting uL to uR with shock speed λ. As an example, if we take into account the characteristics of the inviscid

where u0ð Þ¼ x uð Þ 0; x and x<sup>0</sup> ¼ xð Þ 0 ; thus, the characteristics are straight lines. Depending on the behavior of these characteristics, we have two cases. If uL . uR, characteristics intersect, the solution will have an infinite slope, and the wave will break; as a result a shock is obtained. This is illustrated in Figure 1. On the other hand, if uR . uL, the characteristics do not intersect, and hence a region without characteristic will appear which is physically unacceptable. This is shown in

A rarefaction wave is a strong solution which is a union of characteristic lines. A rarefaction fan is a collection of rarefaction waves. These waves are constant on

> 0 ð Þ uL ; f 0 ð Þ uR

<sup>0</sup> � ��<sup>1</sup> x � x<sup>0</sup>

0

0

If, for instance, f is convex, then the rarefaction waves are increasing. If we consider again the inviscid Burgers' equation with the initial values, then the region without characteristics in Figure 2 will be covered by rarefaction solution which is

uL if x=t ≤ f

ð Þ x=t if f

uR if f

t

0 ð Þ uL ,

ð Þ uR ≤ x=t:

ð Þ uL ≤ x=t ≤ f

Figure 2. We get rid of this by introducing the rarefaction waves.

values of u at the edge of the rarefaction wave fan. If moreover f

f <sup>0</sup> � ��<sup>1</sup>

8 ><

>:

For the initial value uL . uR, characteristics, and shock wave.

For the initial value uR . uL, characteristics and rarefaction waves.

u xð Þ¼ ; t f

dt ¼ u tð Þ ; x , it follows that

x tðÞ¼ u0ð Þ x<sup>0</sup> t þ x<sup>0</sup> (34)

� � where uL and uR are the

� �: (35)

0 ð Þ uR ,

<sup>0</sup> is invertible, then

(36)

Burgers' equations which are of the form dx

DOI: http://dx.doi.org/10.5772/intechopen.83637

the characteristic line x � x<sup>0</sup> ¼ αt. Here α∈ f

u tð Þ¼ ; x

1.8 Rarefaction waves

Scalar Conservation Laws

described by

Figure 1.

Figure 2.

75

the solution u ¼ u tð Þ ; x satisfies

Observe that f uð Þ is a nonlinear function of u; thus, the inviscid Burgers' equation is a nonlinear equation. Equation (29) is now equivalent to Eq. (17) with λ ¼ u. We know the solution of Eq. (17); so, plugging λ ¼ u into the relation (20) implies that the solution of Eq. (29) is

$$u(t, \mathbf{x}) = f(\mathbf{x} - ut) = u\_0(\mathbf{x} - ut). \tag{30}$$

Recall that the characteristic speed λ is constant for linear advection equation; that is, the characteristic curves become parallel for Eq. (17). In contrast, for the inviscid Burgers' equation (29), the characteristic speed λ ¼ u depends on u. As a result the characteristic lines are not parallel. If we apply the implicit function theorem to Eq. (29), the solution can be written as a function of t and x as u<sup>0</sup> is differentiable. More particularly, differentiating Eq. (30) with respect to t, we get

$$
\partial\_t u = -u\_0'(u\_t t + u) \quad \Rightarrow \partial\_t u = -\frac{u\_0' u}{1 + u\_0' t}; \tag{31}
$$

and differentiating equation (30) with respect to x, we get

$$
\partial\_\mathbf{x} u = u\_0'(\mathbf{1} - u\_\mathbf{x} t) \quad \Rightarrow \partial\_\mathbf{x} u = \frac{u\_0'}{\mathbf{1} + u\_0' t}. \tag{32}
$$

Thus, substituting Eqs. (31) and (32) in (29), we can recover the inviscid Burgers' equation. Consequently, the relations (31) and (32) imply that the solutions of Eq. (1) and particularly of Eq. (29) depend on the initial value u0. It can be observed that whenever u<sup>0</sup> <sup>0</sup>ð Þ <sup>x</sup> . 0, then by Eq. (32), <sup>∂</sup>xu decreases in time because 1 þ u<sup>0</sup>0t . 0 for t . 0. In other words, the profile of the wave flattens as time increases. On the other hand, whenever u<sup>0</sup> <sup>0</sup>ð Þ <sup>x</sup> , 0, then <sup>∂</sup>xu increases in time as 1 þ u0t , 0: Hence ux in Eq. (32) tends to ∞ as 1 þ u<sup>0</sup> <sup>0</sup>t approaches to zero. As a result, wave profile become sharp after some time. For further details on the Burgers' equations, we refer the reader to [12, 13, 22] and the references therein.

#### 1.7 Shock waves

Let the constants uL and uR are given with a linear function, φðÞ¼ t λt. Then

$$u(t, \boldsymbol{x}) = \begin{cases} u\_R & \text{if } \qquad \boldsymbol{x} \ge \lambda t, \\ u\_L & \text{if } \qquad \boldsymbol{x} \le \lambda t, \end{cases} \tag{33}$$

#### Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

is a simple example of discontinuous solution of the conservation law (11). If uL 6¼ uR, the relation (33) is called a shock wave connecting uL to uR with shock speed λ. As an example, if we take into account the characteristics of the inviscid Burgers' equations which are of the form dx dt ¼ u tð Þ ; x , it follows that

$$\boldsymbol{\omega}(t) = \boldsymbol{\omega}\_0(\boldsymbol{\omega}\_0)t + \boldsymbol{\omega}\_0 \tag{34}$$

where u0ð Þ¼ x uð Þ 0; x and x<sup>0</sup> ¼ xð Þ 0 ; thus, the characteristics are straight lines. Depending on the behavior of these characteristics, we have two cases. If uL . uR, characteristics intersect, the solution will have an infinite slope, and the wave will break; as a result a shock is obtained. This is illustrated in Figure 1. On the other hand, if uR . uL, the characteristics do not intersect, and hence a region without characteristic will appear which is physically unacceptable. This is shown in Figure 2. We get rid of this by introducing the rarefaction waves.

#### 1.8 Rarefaction waves

Remark. If the initial data is smooth and very small, then the uxx term is negligible compared to other terms before the beginning of wave breaking. As the wave breaking starts, the uxx term raises faster than ux term. After a while, the term uxx becomes comparable to the other terms so that it keeps the solution smooth, giving

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Whenever ν ¼ 0, Eq. (21) is called the inviscid Burgers' equation. This equation

Observe that f uð Þ is a nonlinear function of u; thus, the inviscid Burgers' equation is a nonlinear equation. Equation (29) is now equivalent to Eq. (17) with λ ¼ u. We know the solution of Eq. (17); so, plugging λ ¼ u into the relation (20) implies

Recall that the characteristic speed λ is constant for linear advection equation; that is, the characteristic curves become parallel for Eq. (17). In contrast, for the inviscid Burgers' equation (29), the characteristic speed λ ¼ u depends on u. As a result the characteristic lines are not parallel. If we apply the implicit function theorem to Eq. (29), the solution can be written as a function of t and x as u<sup>0</sup> is differentiable. More particularly, differentiating Eq. (30) with respect to t, we get

<sup>0</sup>ð Þ) utt <sup>þ</sup> <sup>u</sup> <sup>∂</sup>tu ¼ � <sup>u</sup><sup>0</sup>

<sup>0</sup>ð Þ) <sup>1</sup> � uxt <sup>∂</sup>xu <sup>¼</sup> <sup>u</sup><sup>0</sup>

Thus, substituting Eqs. (31) and (32) in (29), we can recover the inviscid Burgers' equation. Consequently, the relations (31) and (32) imply that the solutions of Eq. (1) and particularly of Eq. (29) depend on the initial value u0. It can be

1 þ u<sup>0</sup>0t . 0 for t . 0. In other words, the profile of the wave flattens as time

result, wave profile become sharp after some time. For further details on the Burgers' equations, we refer the reader to [12, 13, 22] and the references therein.

Let the constants uL and uR are given with a linear function, φðÞ¼ t λt. Then

uR if x . λt, uL if x , λt,

<sup>=</sup><sup>2</sup> <sup>¼</sup> <sup>∂</sup>tu <sup>þ</sup> <sup>u</sup>∂xu <sup>¼</sup> <sup>0</sup>: (29)

u tð Þ¼ ; x f xð Þ¼ � ut u0ð Þ x � ut : (30)

0u 1 þ u<sup>0</sup> 0t

0 1 þ u<sup>0</sup> 0t

<sup>0</sup>ð Þ <sup>x</sup> . 0, then by Eq. (32), <sup>∂</sup>xu decreases in time because

<sup>0</sup>ð Þ <sup>x</sup> , 0, then <sup>∂</sup>xu increases in time as

<sup>0</sup>t approaches to zero. As a

(33)

; (31)

: (32)

can be obtained by substituting f uð Þ¼ <sup>u</sup>2=2 in the scalar conservation law (1),

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup><sup>x</sup> <sup>u</sup><sup>2</sup>

<sup>∂</sup>tu ¼ �u<sup>0</sup>

<sup>∂</sup>xu <sup>¼</sup> <sup>u</sup><sup>0</sup>

1 þ u0t , 0: Hence ux in Eq. (32) tends to ∞ as 1 þ u<sup>0</sup>

u tð Þ¼ ; x

and differentiating equation (30) with respect to x, we get

rise to avoid breakdown solutions.

1.6 Inviscid Burgers' equation

that the solution of Eq. (29) is

observed that whenever u<sup>0</sup>

1.7 Shock waves

74

increases. On the other hand, whenever u<sup>0</sup>

that is

A rarefaction wave is a strong solution which is a union of characteristic lines. A rarefaction fan is a collection of rarefaction waves. These waves are constant on the characteristic line x � x<sup>0</sup> ¼ αt. Here α∈ f 0 ð Þ uL ; f 0 ð Þ uR � � where uL and uR are the values of u at the edge of the rarefaction wave fan. If moreover f <sup>0</sup> is invertible, then the solution u ¼ u tð Þ ; x satisfies

$$
\mu(\mathbf{x}, t) = \left(f'\right)^{-1} \left(\frac{\mathbf{x} - \mathbf{x}\_0}{t}\right). \tag{35}
$$

If, for instance, f is convex, then the rarefaction waves are increasing. If we consider again the inviscid Burgers' equation with the initial values, then the region without characteristics in Figure 2 will be covered by rarefaction solution which is described by

$$u(t, \mathbf{x}) = \begin{cases} u\_L & \text{if} \quad \mathbf{x}/t \le f'(u\_L), \\ \left(f'\right)^{-1}(\mathbf{x}/t) & \text{if} \quad f'(u\_L) \le \mathbf{x}/t \le f'(u\_R), \\ u\_R & \text{if} \quad f'(u\_R) \le \mathbf{x}/t. \end{cases} \tag{36}$$

Figure 1.

For the initial value uL . uR, characteristics, and shock wave.

Figure 2. For the initial value uR . uL, characteristics and rarefaction waves.

An illustration of rarefaction waves and rarefaction fan in Eq. (36) is given in Figure 3.

that strong solutions are also weak solutions and a weak solution which is continu-

The Riemann problem is a Cauchy problem with a particular initial value which consists a conservation law together with piecewise constant data having a single discontinuity. We consider the Riemann problem for a convex flux described by

> uL if x , 0, uR if x . 0:

uL if x=t , λ, uR if x=t . λ,

uR�uL :

u tð Þ ; x dx ¼ �ð Þ f tð Þ� ; x<sup>2</sup> f tð Þ ; x<sup>1</sup> : (42)

d dt <sup>Z</sup> <sup>x</sup><sup>2</sup> ξð Þt

ðÞ¼ <sup>t</sup> f xð Þ� <sup>2</sup> f xð Þ<sup>1</sup> x<sup>2</sup> � x<sup>1</sup>

¼ 0: (41)

utð Þ t; x dx: (43)

: (44)

(39)

(40)

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup>xð Þ¼ f uð Þ 0, <sup>x</sup><sup>∈</sup> <sup>R</sup>, <sup>t</sup> <sup>∈</sup> <sup>R</sup>þ,

The solution is a set of shock and rarefaction waves depending on the relation

Case 1: ð Þ uL . uR A shock is obtained because the left-hand side wave moves

Case 2: (uL , uR) The solution given in Case 1 is also a solution for this case. In addition, we have rarefaction solutions of the form (36) illustrated by Figure 3.

A jump discontinuity along the characteristic line is controlled by the Rankine-Hugoniot jump condition. Integrating the scalar conservation law (1) in ½ � x1; x<sup>2</sup> , it

u tð Þ ; x dx þ f uð Þ

u<sup>0</sup> are continuous on the ½ Þ x1; ξð Þt and ð � ξð Þt ; x<sup>2</sup> , respectively. Suppose also that whenever x<sup>1</sup> ! ξð Þt � and x<sup>2</sup> ! ξð Þt <sup>þ</sup>, their limits exist. Next, Eq. (41) can be

By the fundamental theorem of calculus, the relations (41) and (42) yield

Taking the limit whenever x<sup>1</sup> ! ξð Þt � and x<sup>2</sup> ! ξð Þt <sup>þ</sup>, it follows that

Suppose that there is a discontinuity at the point x ¼ ξð Þt ∈ð Þ x1; x<sup>2</sup> where u and

� � � x2 x1

utð Þ t; x dx þ

(

�

ous and piecewise differentiable is also strong solution.

DOI: http://dx.doi.org/10.5772/intechopen.83637

uð Þ¼ 0; x

between uL and uR: There are two cases to investigate:

faster than the right-hand side one. Thus the solution

1.11 Rankine-Hugoniot jump condition

u tð Þ ; x dx þ

ðÞ�t u ξ<sup>þ</sup> ð Þ ; x ξ<sup>0</sup>

follows that

rewritten as

d dt <sup>Z</sup> <sup>ξ</sup>ð Þ<sup>t</sup> x1

u ξ� ð Þ ; x ξ<sup>0</sup>

ξ0

77

u tð Þ¼ ; x

is a shock wave satisfying the shock speed <sup>λ</sup> <sup>¼</sup> f uð Þ�<sup>R</sup> f uð Þ <sup>L</sup>

d dt <sup>Z</sup> <sup>x</sup><sup>2</sup> x1

> d dt <sup>Z</sup> <sup>x</sup><sup>2</sup> ξð Þt

> > ðÞþt d dt <sup>Z</sup> <sup>ξ</sup>ð Þ<sup>t</sup> x1

ð Þt ð Þ¼ x<sup>2</sup> � x<sup>1</sup> f xð Þ� <sup>2</sup> f xð Þ)<sup>1</sup> λ ¼ ξ<sup>0</sup>

1.10 Riemann problem

Scalar Conservation Laws

Remark. Whenever characteristics intersect, we may have multiple valued solution or no solution; but we have no more classical (strong) solution. To get rid of this situation, we introduce a more wide-ranging notion of solution, the weak solution, in the next part. By this arrangement, we may have non-differentiable and even discontinuous solutions.

#### 1.9 Weak solution

Weak solutions occur whenever there is no smooth (classical) solution. These solutions may not be differentiable or even not continuous. Considering ϕ : R � R<sup>þ</sup> ! R as a smooth test function with a compact support and multiplying the scalar conservation law (1) by this test function ϕ, it follows after integration by parts that

$$\begin{split} &\int\_{0}^{\infty} \int\_{-\infty}^{\infty} \phi \partial\_{t} u + \phi \partial\_{x} f(u) \, dx \, dt \\ &= \int\_{-\infty}^{\infty} \phi u \Big|\_{0}^{\infty} dx - \int\_{0}^{\infty} \int\_{-\infty}^{\infty} u \partial\_{t} \phi \, dx \, dt + \int\_{0}^{\infty} \phi f(u) \Big|\_{-\infty}^{\infty} dt - \int\_{0}^{\infty} \int\_{-\infty}^{\infty} f(u) \partial\_{x} \phi \, dx \, dt \\ &= - \int\_{0}^{\infty} \int\_{-\infty}^{\infty} u \partial\_{t} \phi \, dx \, dt - \int\_{0}^{\infty} \int\_{-\infty}^{\infty} f(u) \, \partial\_{x} \phi \, dx \, dt - \int\_{-\infty}^{\infty} u \phi \Big|\_{t=0} \, dx. \end{split} \tag{37}$$

Putting the initial condition u0ð Þ¼ x uð Þ 0; x to the above relation, it follows that

$$\int\_{0}^{\infty} \int\_{-\infty}^{\infty} u \phi\_t + f(u) \phi\_x dxdt + \int\_{-\infty}^{\infty} u(0, \infty) \phi(\infty) d\mathfrak{x} = 0. \tag{38}$$

Observe that there are no more derivatives of u and f which may lead less smoothness. In other words, the smoothness requirement is reduced for finding a solution. Thus, the function u tð Þ ; x is said to be the weak solution of the initial value problem (11) if the relation (38) satisfied for all test function ϕ: Here it is significant to note that u needs not be smooth or continuous to satisfy Eq. (38). Consequently, by weak solutions, we extend the solutions so that discontinuous solutions may also be covered. However, in general weak solutions are not unique. We can also notice

Figure 3. Rarefaction fan.

that strong solutions are also weak solutions and a weak solution which is continuous and piecewise differentiable is also strong solution.

#### 1.10 Riemann problem

An illustration of rarefaction waves and rarefaction fan in Eq. (36) is given in

Weak solutions occur whenever there is no smooth (classical) solution. These

ϕ : R � R<sup>þ</sup> ! R as a smooth test function with a compact support and multiplying the scalar conservation law (1) by this test function ϕ, it follows after integration by

> Z <sup>∞</sup> 0

f uð Þ∂xϕdxdt �

Putting the initial condition u0ð Þ¼ x uð Þ 0; x to the above relation, it follows that

Observe that there are no more derivatives of u and f which may lead less smoothness. In other words, the smoothness requirement is reduced for finding a solution. Thus, the function u tð Þ ; x is said to be the weak solution of the initial value problem (11) if the relation (38) satisfied for all test function ϕ: Here it is significant to note that u needs not be smooth or continuous to satisfy Eq. (38). Consequently, by weak solutions, we extend the solutions so that discontinuous solutions may also be covered. However, in general weak solutions are not unique. We can also notice

Z <sup>∞</sup> �∞

ϕf uð Þ � � � ∞ �∞dt �

> Z <sup>∞</sup> �∞ uϕ � � � � t¼0 dx:

Z <sup>∞</sup> 0

Z <sup>∞</sup> �∞

uð Þ 0; x ϕð Þ x dx ¼ 0: (38)

f uð Þ∂xϕdxdt

(37)

solutions may not be differentiable or even not continuous. Considering

<sup>u</sup>∂tϕdxdt <sup>þ</sup>

Z <sup>∞</sup> �∞

uϕ<sup>t</sup> þ f uð Þϕxdxdt þ

Remark. Whenever characteristics intersect, we may have multiple valued solution or no solution; but we have no more classical (strong) solution. To get rid of this situation, we introduce a more wide-ranging notion of solution, the weak solution, in the next part. By this arrangement, we may have non-differentiable and

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Figure 3.

parts that

Z <sup>∞</sup> 0

¼ Z <sup>∞</sup> �∞ ϕu � � � ∞ 0 dx �

¼ �

Figure 3. Rarefaction fan.

76

Z <sup>∞</sup> �∞

Z <sup>∞</sup> 0

Z <sup>∞</sup> �∞

even discontinuous solutions.

<sup>ϕ</sup>∂tu <sup>þ</sup> <sup>ϕ</sup>∂xf uð Þdxdt

Z <sup>∞</sup> 0

<sup>u</sup>∂tϕdxdt �

Z <sup>∞</sup> �∞

Z <sup>∞</sup> 0

Z <sup>∞</sup> �∞

Z <sup>∞</sup> 0

1.9 Weak solution

The Riemann problem is a Cauchy problem with a particular initial value which consists a conservation law together with piecewise constant data having a single discontinuity. We consider the Riemann problem for a convex flux described by

$$\begin{aligned} \partial\_t u + \partial\_x (f(u)) &= 0, \quad \boldsymbol{x} \in \mathbb{R}, \quad t \in \mathbb{R}\_+, \\ u(0, \boldsymbol{x}) &= \begin{cases} u\_L & \text{if } \quad \boldsymbol{x} \le \mathbf{0}, \\ u\_R & \text{if } \quad \boldsymbol{x} \ge \mathbf{0}. \end{cases} \end{aligned} \tag{39}$$

The solution is a set of shock and rarefaction waves depending on the relation between uL and uR: There are two cases to investigate:

Case 1: ð Þ uL . uR A shock is obtained because the left-hand side wave moves faster than the right-hand side one. Thus the solution

$$u(t, \mathbf{x}) = \begin{cases} u\_L & \text{if } \qquad \mathbf{x}/t \le \lambda, \\ u\_R & \text{if } \qquad \mathbf{x}/t \ge \lambda, \end{cases} \tag{40}$$

is a shock wave satisfying the shock speed <sup>λ</sup> <sup>¼</sup> f uð Þ�<sup>R</sup> f uð Þ <sup>L</sup> uR�uL :

Case 2: (uL , uR) The solution given in Case 1 is also a solution for this case. In addition, we have rarefaction solutions of the form (36) illustrated by Figure 3.

#### 1.11 Rankine-Hugoniot jump condition

A jump discontinuity along the characteristic line is controlled by the Rankine-Hugoniot jump condition. Integrating the scalar conservation law (1) in ½ � x1; x<sup>2</sup> , it follows that

$$\frac{d}{dt}\int\_{\mathbf{x}\_1}^{\mathbf{x}\_2} u(t, \mathbf{x})d\mathbf{x} + f(u)\Big|\_{\mathbf{x}\_1}^{\mathbf{x}\_2} = \mathbf{0}.\tag{41}$$

Suppose that there is a discontinuity at the point x ¼ ξð Þt ∈ð Þ x1; x<sup>2</sup> where u and u<sup>0</sup> are continuous on the ½ Þ x1; ξð Þt and ð � ξð Þt ; x<sup>2</sup> , respectively. Suppose also that whenever x<sup>1</sup> ! ξð Þt � and x<sup>2</sup> ! ξð Þt <sup>þ</sup>, their limits exist. Next, Eq. (41) can be rewritten as

$$\frac{d}{dt}\int\_{\infty}^{\xi(t)} u(t, \mathbf{x})d\mathbf{x} + \frac{d}{dt}\int\_{\xi(t)}^{\infty} u(t, \mathbf{x})d\mathbf{x} = -(f(t, \mathbf{x}\_2) - f(t, \mathbf{x}\_1)).\tag{42}$$

By the fundamental theorem of calculus, the relations (41) and (42) yield

$$u(\xi^-, \mathfrak{x})\xi'(t) - u(\xi^+, \mathfrak{x})\xi'(t) + \frac{d}{dt} \int\_{\mathfrak{x}\_1}^{\xi(t)} u\_t(t, \mathfrak{x}) d\mathfrak{x} + \frac{d}{dt} \int\_{\xi(t)}^{\chi\_2} u\_t(t, \mathfrak{x}) d\mathfrak{x}.\tag{43}$$

Taking the limit whenever x<sup>1</sup> ! ξð Þt � and x<sup>2</sup> ! ξð Þt <sup>þ</sup>, it follows that

$$f(t)(\mathbf{x}\_2 - \mathbf{x}\_1) = f(\mathbf{x}\_2) - f(\mathbf{x}\_1) \Rightarrow \lambda = \xi'(t) = \frac{f(\mathbf{x}\_2) - f(\mathbf{x}\_1)}{\mathbf{x}\_2 - \mathbf{x}\_1}.\tag{44}$$

The relation (44) is said to be the Rankine-Hugoniot jump condition. Geometrical meaning of the Rankine-Hugoniot jump condition is that the shock speed is the slope of the secant line through the points ð Þ uL; f uð Þ<sup>L</sup> and ð Þ uR; f uð Þ<sup>R</sup> on the graph of f.

#### 1.12 Entropy functions

Entropy and entropy flux are defined for attaining physically meaningful solutions. If u is the smooth solution of the conservation law (1), then the relation

$$
\partial\_t G(u) + \partial\_x F(u) = \mathbf{0} \tag{45}
$$

inviscid Burgers' equation is λ<sup>1</sup> ¼ ð Þ uL þ uR =2; however for Eq. (49), we have

Applying the method of characteristics for t . 0, it follows that

dt <sup>¼</sup> 0, dx

� � �. That is <sup>λ</sup><sup>1</sup> 6¼ <sup>λ</sup><sup>2</sup> whenever uL 6¼ uR, and thus these two equations

Example 1.2. We first consider the initial value problem for uL . uR given by

�

Next if we integrate Eq. (51) with respect to t, we get the characteristic curves

<sup>x</sup> <sup>¼</sup> <sup>t</sup> � <sup>c</sup> if <sup>x</sup><sup>≤</sup> 0, <sup>b</sup> if <sup>x</sup> . 0, �

where c . 0 and b are constants. Due to the discontinuity at the point x ¼ 0,

1 if <sup>x</sup>

0 if <sup>x</sup>

Example 1.3. We now interchange the roles of uL and uR of the Example 1.2 so

<sup>=</sup><sup>2</sup> � � <sup>¼</sup> 0, <sup>u</sup><sup>0</sup> <sup>¼</sup> 0 if <sup>x</sup>≤0,

0 if <sup>x</sup>

8 ><

>:

1 if <sup>x</sup>

t ≤1

t . 1

�

1 if x . 0:

t ≤0:5

t . 0:5 ,

there is no strong (classical) solution. The speed of propagation is <sup>λ</sup> <sup>¼</sup> uLþuR

8 ><

>:

which satisfies both the jump condition and the entropy condition as uL ¼ 1 . uR ¼ 0. The characteristic curves can be observed in Figure 4.

<sup>=</sup><sup>2</sup> � � <sup>¼</sup> 0, <sup>u</sup><sup>0</sup> <sup>¼</sup> 1 if <sup>x</sup>≤0,

�

dt <sup>¼</sup> 1 if <sup>x</sup>≤0, 0 if x . 0:

0 if x . 0:

(50)

(51)

(52)

<sup>2</sup> ¼ 0:5:

(53)

(54)

(55)

<sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> 3 u3 L�u<sup>3</sup> R u2 L�u<sup>2</sup> R

Figure 4.

79

have different weak solutions.

DOI: http://dx.doi.org/10.5772/intechopen.83637

Scalar Conservation Laws

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup><sup>x</sup> <sup>u</sup><sup>2</sup>

du

Moreover, the weak solution for t≤λ ¼ 0:5 becomes

that uL , uR to get an initial value problem:

For initial value uL . uR, the characteristic solutions.

<sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup><sup>x</sup> <sup>u</sup><sup>2</sup>

By the method of characteristics, we obtain a solution

u1ð Þ¼ t; x

u tð Þ¼ ; x

is satisfied for continuously differentiable functions G and F where the pair ð Þ G; F is called as entropy pair so that G is entropy and F is entropy flux. If in addition u is smooth, then Eq. (45) becomes

$$G'(u)\partial\_t u + F'(u)\partial\_x u = 0 \tag{46}$$

which looks like to the scalar conservation law (1). Indeed, if we multiply Eq. (1) by G<sup>0</sup> ð Þ u , it follows that

$$G'(u)\partial\_t u + G'(u)f'(u)\partial\_x u = 0.\tag{47}$$

It follows that Eqs. (46) and (47) are equivalent with F<sup>0</sup> ð Þ¼ u G<sup>0</sup> ð Þ u f 0 ð Þ u : Here the function u tð Þ ; x is said to be the entropy solution of Eq. (1) if

$$
\partial\_t G(u) + \partial\_x F(u) \le 0
$$

holds for all convex entropy pairs ð Þ G uð Þ; F uð Þ .

#### 1.13 Entropy condition

Weak solutions to conservation laws may contain discontinuities as a result of a discontinuity in the initial data or of characteristics that cross each other or because of the jump conditions which are satisfied across the discontinuities. Although the Rankine-Hugoniot jump condition is satisfied, the uniqueness of the solution may always not be guaranteed. In order to eliminate the nonphysical solutions among the weak solutions, we need an additional condition, so-called entropy condition. It is described by the following: A discontinuity propagating with the characteristic speed λ given by the Rankine-Hugoniot jump condition satisfies the entropy condition if holds.

$$f'(u\_L) \succeq \lambda \rhd f'(u\_R) \tag{48}$$

Example 1.1. The weak solutions to conservation laws need not be unique. If we write the inviscid Burgers' equation in quasilinear form and multiply by 2u, we obtain 2u∂tu <sup>þ</sup> <sup>2</sup>u<sup>2</sup>∂xu <sup>¼</sup> 0. In conservative form it becomes

$$
\partial\_t \left( u^2 \right) + \partial\_x \left( \frac{2}{3} u^3 \right) = 0, \quad \text{with } f\left( u^2 \right) = \frac{2}{3} \left( u^2 \right)^{3/2}. \tag{49}
$$

The inviscid Burgers' equation and Eq. (49) have exactly the same smooth solutions. But their weak solutions are different. A shock traveling speed for the

The relation (44) is said to be the Rankine-Hugoniot jump condition. Geometrical meaning of the Rankine-Hugoniot jump condition is that the shock speed is the slope of the secant line through the points ð Þ uL; f uð Þ<sup>L</sup> and ð Þ uR; f uð Þ<sup>R</sup> on the

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

Entropy and entropy flux are defined for attaining physically meaningful solu-

is satisfied for continuously differentiable functions G and F where the pair ð Þ G; F is called as entropy pair so that G is entropy and F is entropy flux. If in

which looks like to the scalar conservation law (1). Indeed, if we multiply Eq. (1)

ð Þ u f 0

<sup>∂</sup>tG uð Þþ <sup>∂</sup>xF uð Þ≤<sup>0</sup>

Weak solutions to conservation laws may contain discontinuities as a result of a discontinuity in the initial data or of characteristics that cross each other or because of the jump conditions which are satisfied across the discontinuities. Although the Rankine-Hugoniot jump condition is satisfied, the uniqueness of the solution may always not be guaranteed. In order to eliminate the nonphysical solutions among the weak solutions, we need an additional condition, so-called entropy condition. It is described by the following: A discontinuity propagating with the characteristic speed λ given by the Rankine-Hugoniot jump condition satisfies the entropy condi-

<sup>∂</sup>tG uð Þþ <sup>∂</sup>xF uð Þ¼ <sup>0</sup> (45)

ð Þ <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> <sup>0</sup> (46)

ð Þ <sup>u</sup> <sup>∂</sup>xu <sup>¼</sup> <sup>0</sup>: (47)

ð Þ¼ u G<sup>0</sup>

ð Þ uR (48)

<sup>3</sup> <sup>u</sup><sup>2</sup> <sup>3</sup>=<sup>2</sup>

: (49)

ð Þ u f 0 ð Þ u : Here

tions. If u is the smooth solution of the conservation law (1), then the relation

ð Þ <sup>u</sup> <sup>∂</sup>tu <sup>þ</sup> <sup>F</sup><sup>0</sup>

ð Þ <sup>u</sup> <sup>∂</sup>tu <sup>þ</sup> <sup>G</sup><sup>0</sup>

graph of f.

by G<sup>0</sup>

1.12 Entropy functions

ð Þ u , it follows that

1.13 Entropy condition

tion if holds.

78

addition u is smooth, then Eq. (45) becomes

G0

It follows that Eqs. (46) and (47) are equivalent with F<sup>0</sup>

the function u tð Þ ; x is said to be the entropy solution of Eq. (1) if

f 0

obtain 2u∂tu <sup>þ</sup> <sup>2</sup>u<sup>2</sup>∂xu <sup>¼</sup> 0. In conservative form it becomes

2 3 u3 

<sup>∂</sup><sup>t</sup> <sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>x</sup>

ð Þ uL . λ . f

write the inviscid Burgers' equation in quasilinear form and multiply by 2u, we

The inviscid Burgers' equation and Eq. (49) have exactly the same smooth solutions. But their weak solutions are different. A shock traveling speed for the

0

<sup>¼</sup> 0, with f u<sup>2</sup> <sup>¼</sup> <sup>2</sup>

Example 1.1. The weak solutions to conservation laws need not be unique. If we

G0

holds for all convex entropy pairs ð Þ G uð Þ; F uð Þ .

inviscid Burgers' equation is λ<sup>1</sup> ¼ ð Þ uL þ uR =2; however for Eq. (49), we have <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> 3 u3 L�u<sup>3</sup> R u2 L�u<sup>2</sup> R � � �. That is <sup>λ</sup><sup>1</sup> 6¼ <sup>λ</sup><sup>2</sup> whenever uL 6¼ uR, and thus these two equations have different weak solutions.

Example 1.2. We first consider the initial value problem for uL . uR given by

$$
\partial\_t u + \partial\_x (u^2/2) = 0, \quad u\_0 = \begin{cases} 1 & \text{if } \quad x \le 0, \\ 0 & \text{if } \quad x \ge 0. \end{cases} \tag{50}
$$

Applying the method of characteristics for t . 0, it follows that

$$\frac{du}{dt} = 0, \quad \frac{d\boldsymbol{x}}{dt} = \begin{cases} 1 & \text{if} \quad \boldsymbol{x} \le \mathbf{0}, \\ 0 & \text{if} \quad \boldsymbol{x} \ge \mathbf{0}. \end{cases} \tag{51}$$

Next if we integrate Eq. (51) with respect to t, we get the characteristic curves

$$\mathfrak{x} = \begin{cases} t - c & \text{if} \quad \mathfrak{x} \le \mathbf{0}, \\\ b & \text{if} \quad \mathfrak{x} \ge \mathbf{0}, \end{cases} \tag{52}$$

where c . 0 and b are constants. Due to the discontinuity at the point x ¼ 0, there is no strong (classical) solution. The speed of propagation is <sup>λ</sup> <sup>¼</sup> uLþuR <sup>2</sup> ¼ 0:5: Moreover, the weak solution for t≤λ ¼ 0:5 becomes

$$u(t, \mathbf{x}) = \begin{cases} 1 & \text{if} \quad \frac{\mathbf{x}}{t} \le \mathbf{0}.5\\ 0 & \text{if} \quad \frac{\mathbf{x}}{t} > \mathbf{0}.5 \end{cases},\tag{53}$$

which satisfies both the jump condition and the entropy condition as uL ¼ 1 . uR ¼ 0. The characteristic curves can be observed in Figure 4.

Example 1.3. We now interchange the roles of uL and uR of the Example 1.2 so that uL , uR to get an initial value problem:

$$
\partial\_t u + \partial\_x \left( u^2 / 2 \right) = 0, \quad u\_0 = \begin{cases} 0 & \text{if} \quad x \le 0, \\ 1 & \text{if} \quad x \ge 0. \end{cases} \tag{54}
$$

By the method of characteristics, we obtain a solution

Figure 4. For initial value uL . uR, the characteristic solutions.

Figure 5. For initial value uL , uR, characteristic solutions u1ð Þ t; x and u2ð Þ x; t with rarefaction fan.

which is a classical (strong) solution on both sides of the characteristic line <sup>x</sup> <sup>t</sup> ¼ 1. Since it satisfies the Rankine-Hugoniot jump condition along the discontinuity curve, it is a weak solution. However, the entropy condition is not satisfied. It yields an empty region between the characteristic lines shown in Figure 4. In order to cover this empty state, we consider another solution described by

$$u\_2(t, \mathbf{x}) = \begin{cases} 0 & \text{if} \quad x \le 0, \\ \frac{\mathcal{X}}{t} & \text{if} \quad 0 \le \frac{\mathcal{X}}{t} \le \mathbf{1}, \\ 1 & \text{if} \quad \frac{\mathcal{X}}{t} \ge \mathbf{1} \end{cases} \tag{56}$$

where ρ is density, p is pressure, u is velocity, and e is the specific internal

If we do not neglect the heat conduction, then the U and F terms in Eq. (59)

ρu <sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>p</sup> ð Þ E þ p u

<sup>∂</sup>tU <sup>þ</sup> A Uð Þ∂xU <sup>¼</sup> 0, (61)

0

B@

1

1 u þ a H þ ua 1

CA (62)

CA, (60)

ð Þ <sup>δ</sup>�<sup>1</sup> <sup>ρ</sup>, and for perfect gases

ffiffiffiffi δp ρ q .

(63)

(64)

0

B@

<sup>2</sup> <sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>ρ</sup>e, <sup>e</sup> <sup>¼</sup> <sup>p</sup>

<sup>∂</sup><sup>U</sup> is the Jacobian matrix. The eigenvalues of A Uð Þ then are

CA, <sup>E</sup>ð Þ<sup>3</sup> <sup>¼</sup>

CA and <sup>F</sup> <sup>¼</sup>

λ<sup>1</sup> ¼ u, λ<sup>2</sup> ¼ u � a, λ<sup>3</sup> ¼ u þ a where a is the sound speed given by a ¼

0

B@

1 u � a H � ua 1

which are real, and the eigenvectors are linearly independent implying that the

Using the results in the previous part, the Rankine-Hugoniot jump conditions for

<sup>þ</sup> <sup>p</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> 1 ρ1

� p1,

UL if x , 0, UR if x . 0:

s ρ<sup>1</sup> � ρ<sup>2</sup> ð Þ¼ m<sup>2</sup> � m1,

sð Þ¼ ρ2E<sup>2</sup> � ρ1E<sup>1</sup> m2H<sup>2</sup> � H1m1,

where the indices 1 and 2 refer to the left and right of the shock, respectively,

The Riemann problem for the one-dimensional Euler equation (57) is

The reader is addressed to the references [18, 24] for further details.

<sup>∂</sup>tU <sup>þ</sup> <sup>∂</sup>xð Þ¼ F Uð Þ 0, <sup>x</sup><sup>∈</sup> <sup>R</sup>, <sup>t</sup> . 0,

(

m<sup>2</sup> 2 ρ2

δ ¼ cp=cv is the ratio of specific heats. Rewriting Eq. (59) in quasilinear form, we get

energy.

Scalar Conservation Laws

become

2.2 Hyperbolicity of the Euler system

DOI: http://dx.doi.org/10.5772/intechopen.83637

U ¼

where <sup>E</sup> is total energy such that <sup>E</sup> <sup>¼</sup> <sup>1</sup>

Moreover the corresponding eigenvectors are

1

Euler equations for perfect gases are hyperbolic.

CCA, <sup>E</sup>ð Þ<sup>2</sup> <sup>¼</sup>

2.3 Rankine-Hugoniot conditions for the Euler system

s mð Þ¼ <sup>2</sup> � m<sup>1</sup>

Uð Þ¼ 0; x U0ð Þ¼ x

1 u 1 2 u2

0

BB@

the Euler system will be of the form

and s denotes the wave speed.

represented by

81

2.4 Riemann problem for the Euler system

where A Uð Þ¼ <sup>∂</sup><sup>F</sup>

<sup>E</sup>ð Þ<sup>1</sup> <sup>¼</sup>

ρ ρu E

1

0

B@

which satisfies both jump and entropy conditions. Here we can observe the rarefaction fan arising on the interval 0 ≤ <sup>x</sup> <sup>t</sup> ≤ 1. An illustration of this solution is supplied in Figure 5.

#### 2. The gas dynamic equations in one dimension

The equation of fluid dynamics can be represented in Eulerian and Lagrangian forms. Eulerian coordinates are related to the coordinates of a fixed observer. On the other hand, Lagrangian coordinates are in usual related to the local flow velocity. That is, due to the velocity taking different values in different parts of the fluid, the change of coordinates is different from one point to another one.

#### 2.1 Eulerian coordinates

The equations of gas dynamics in Eulerian coordinates can be written in the following conservative forms:

$$\begin{cases} \partial\_t(\rho) + \partial\_x(\rho u) = \mathbf{0}, \\ \partial\_t(\rho u) + \partial\_x(\rho u^2 + p) = \mathbf{0}, \\ \partial\_t(\rho e) + \partial\_x((\rho e + p)u) = \mathbf{0} \end{cases} \tag{57}$$

where we ignored the heat conduction. If we denote

$$U = \begin{pmatrix} \rho \\ \rho u \\ \rho \end{pmatrix}, \quad F(U) = \begin{pmatrix} \rho \mathbf{e} \\ \rho u^2 + p \\ \rho \mathbf{e}u + pu \end{pmatrix}, \tag{58}$$

then Eq. (57) can be written by

$$
\partial\_t U + \partial\_x F(U) = 0 \tag{59}
$$

where ρ is density, p is pressure, u is velocity, and e is the specific internal energy.

#### 2.2 Hyperbolicity of the Euler system

which is a classical (strong) solution on both sides of the characteristic line <sup>x</sup>

0 if x≤0,

if 0 ≤

1 if <sup>x</sup> t ≥1

which satisfies both jump and entropy conditions. Here we can observe the

The equation of fluid dynamics can be represented in Eulerian and Lagrangian forms. Eulerian coordinates are related to the coordinates of a fixed observer. On the other hand, Lagrangian coordinates are in usual related to the local flow velocity. That is, due to the velocity taking different values in different parts of the fluid,

The equations of gas dynamics in Eulerian coordinates can be written in the

<sup>∂</sup>tð Þþ <sup>ρ</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> <sup>ρ</sup><sup>u</sup> ð Þ¼ <sup>2</sup> <sup>þ</sup> <sup>p</sup> 0, <sup>∂</sup>tð Þþ <sup>ρ</sup><sup>e</sup> <sup>∂</sup>xð Þ¼ ð Þ <sup>ρ</sup><sup>e</sup> <sup>þ</sup> <sup>p</sup> <sup>u</sup> <sup>0</sup>

> ρe <sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>p</sup> ρeu þ pu

1

<sup>∂</sup>tU <sup>þ</sup> <sup>∂</sup>xF Uð Þ¼ <sup>0</sup> (59)

CA, (58)

0

B@

<sup>∂</sup>tð Þþ <sup>ρ</sup> <sup>∂</sup>xð Þ¼ <sup>ρ</sup><sup>u</sup> 0,

CA, F Uð Þ¼

x t ≤1,

<sup>t</sup> ≤ 1. An illustration of this solution is

Since it satisfies the Rankine-Hugoniot jump condition along the discontinuity curve, it is a weak solution. However, the entropy condition is not satisfied. It yields an empty region between the characteristic lines shown in Figure 4. In order to

> x t

8 >>><

>>>:

cover this empty state, we consider another solution described by

For initial value uL , uR, characteristic solutions u1ð Þ t; x and u2ð Þ x; t with rarefaction fan.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

u2ð Þ¼ t; x

rarefaction fan arising on the interval 0 ≤ <sup>x</sup>

2. The gas dynamic equations in one dimension

8 ><

>:

where we ignored the heat conduction. If we denote

0

B@

ρ ρu ρe

1

U ¼

then Eq. (57) can be written by

80

the change of coordinates is different from one point to another one.

supplied in Figure 5.

Figure 5.

2.1 Eulerian coordinates

following conservative forms:

<sup>t</sup> ¼ 1.

(56)

(57)

If we do not neglect the heat conduction, then the U and F terms in Eq. (59) become

$$U = \begin{pmatrix} \rho \\ \rho u \\ E \end{pmatrix} \qquad \text{and} \qquad F = \begin{pmatrix} \rho u \\ \rho u^2 + p \\ (E+p)u \end{pmatrix}, \tag{60}$$

where <sup>E</sup> is total energy such that <sup>E</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>ρ</sup>e, <sup>e</sup> <sup>¼</sup> <sup>p</sup> ð Þ <sup>δ</sup>�<sup>1</sup> <sup>ρ</sup>, and for perfect gases δ ¼ cp=cv is the ratio of specific heats. Rewriting Eq. (59) in quasilinear form, we get

$$
\partial\_t U + A(U)\partial\_x U = 0,\tag{61}
$$

where A Uð Þ¼ <sup>∂</sup><sup>F</sup> <sup>∂</sup><sup>U</sup> is the Jacobian matrix. The eigenvalues of A Uð Þ then are λ<sup>1</sup> ¼ u, λ<sup>2</sup> ¼ u � a, λ<sup>3</sup> ¼ u þ a where a is the sound speed given by a ¼ ffiffiffiffi δp ρ q . Moreover the corresponding eigenvectors are

$$E^{(1)} = \begin{pmatrix} 1 \\ u \\ \frac{1}{2}u^2 \end{pmatrix}, \quad E^{(2)} = \begin{pmatrix} 1 \\ u - a \\ H - ua \end{pmatrix}, \quad E^{(3)} = \begin{pmatrix} 1 \\ u + a \\ H + ua \end{pmatrix} \tag{62}$$

which are real, and the eigenvectors are linearly independent implying that the Euler equations for perfect gases are hyperbolic.

#### 2.3 Rankine-Hugoniot conditions for the Euler system

Using the results in the previous part, the Rankine-Hugoniot jump conditions for the Euler system will be of the form

$$\begin{aligned} s(\rho\_1 - \rho\_2) &= m\_2 - m\_1, \\ s(m\_2 - m\_1) &= \frac{m\_2^2}{\rho\_2} + p\_2 - \frac{m\_1^2}{\rho\_1} - p\_1, \\ s(\rho\_2 E\_2 - \rho\_1 E\_1) &= m\_2 H\_2 - H\_1 m\_1, \end{aligned} \tag{63}$$

where the indices 1 and 2 refer to the left and right of the shock, respectively, and s denotes the wave speed.

#### 2.4 Riemann problem for the Euler system

The Riemann problem for the one-dimensional Euler equation (57) is represented by

$$\begin{aligned} \partial\_t U + \partial\_x (F(U)) &= \mathbf{0}, \quad \boldsymbol{\varkappa} \in \mathbb{R}, \quad t > 0, \\ U(\mathbf{0}, \boldsymbol{\varkappa}) = U\_0(\boldsymbol{\varkappa}) &= \begin{cases} U\_L & \text{if } \qquad \boldsymbol{\varkappa} \le \mathbf{0}, \\ U\_R & \text{if } \qquad \boldsymbol{\varkappa} > \mathbf{0}. \end{cases} \end{aligned} \tag{64}$$

The reader is addressed to the references [18, 24] for further details.

#### 2.5 Lagrangian coordinates

We aim to transform the equations of gas dynamics (57) given in the Eulerian coordinates into the Lagrangian coordinates for one-dimensional case. We start denoting by u ¼ uð Þ t; x the velocity field of the fluid flow and consider the differential system

$$\frac{d\mathbf{x}}{dt} = \mathbf{u}(t, \mathbf{x}).\tag{65}$$

Moreover, we define a mass variable m by

DOI: http://dx.doi.org/10.5772/intechopen.83637

Lagrangian coordinates with the mass variable in the form

where <sup>p</sup> <sup>¼</sup> <sup>p</sup>ð Þ¼ <sup>τ</sup>; <sup>ξ</sup> <sup>p</sup> <sup>τ</sup>;<sup>e</sup> � <sup>u</sup><sup>2</sup> ð Þ <sup>=</sup><sup>2</sup> . If we set <sup>V</sup> <sup>¼</sup>

of the flux calculated with respect to the variables ð Þ τ; u;e

0

B@

2.6 Rankine-Hugoniot conditions for the Lagrangian system

details we cite these works with references therein.

Lagrangian system (79) are of the form

relations (see [9] for further detail).

nates. Then the system of equations

8 ><

>:

<sup>τ</sup> . 0, <sup>u</sup><sup>∈</sup> <sup>R</sup>,<sup>e</sup> � <sup>u</sup><sup>2</sup>=<sup>2</sup> . 0, we obtain a scalar conservation law of the form

<sup>2</sup> <sup>u</sup>2. The eigenvalues are <sup>σ</sup><sup>1</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p so that they are all distinct, and thus the system is strictly hyperbolic. In fact there are different versions of the gas dynamics in Lagrangian coordinates. In this part we followed the approaches stated in [9, 10, 12]. For further

Similarly as in the Euler system, the Rankine-Hugoniot jump conditions for the

σ τð Þ¼� <sup>1</sup> � τ<sup>0</sup> ð Þ u<sup>1</sup> � u<sup>0</sup> , σð Þ¼ u<sup>1</sup> � u<sup>0</sup> p<sup>1</sup> � p0, σð Þ¼ e<sup>1</sup> � e<sup>0</sup> p1u<sup>1</sup> � p0u0,

where σ denotes the speed of propagation of the discontinuity with respect to

Remark. The Eulerian and Lagrangian Rankine-Hugoniot relations are equivalent. Moreover, Eulerian entropy relations are equivalent to all Lagrangian entropy

Example 2.1. For simplicity of notation, we take ð Þ t; x as the Lagrangian coordi-

<sup>∂</sup>t<sup>τ</sup> � <sup>∂</sup>xu <sup>¼</sup> 0, <sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup>xpð Þ¼ <sup>τ</sup> 0, �

which is strictly hyperbolic. This can be verified by checking the Jacobian matrix

0 �1 0 p<sup>τ</sup> �up<sup>ε</sup> p<sup>ε</sup> up<sup>τ</sup> <sup>p</sup> � <sup>u</sup><sup>2</sup>p<sup>ε</sup> up<sup>ε</sup>

ρð Þ 0; y dy, or equivalently, dm ¼ ρð Þ 0; ξ dξ ¼ ρ0dξ: (74)

τ u e

1

<sup>∂</sup>tV <sup>þ</sup> <sup>∂</sup>mF Vð Þ¼ <sup>0</sup> (76)

CA, F Vð Þ¼

0

B@

1

p<sup>τ</sup> � pp<sup>ε</sup>

(75)

�u p pu

1

CA with

(78)

(79)

0

B@

CA (77)

<sup>p</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> , <sup>σ</sup><sup>3</sup> <sup>¼</sup>

Finally, using Eqs. (69) and (73), the Euler system (57) can be written in

<sup>∂</sup>t<sup>τ</sup> � <sup>∂</sup>mu <sup>¼</sup> 0, <sup>∂</sup>tu <sup>þ</sup> <sup>∂</sup>mp <sup>¼</sup> 0, <sup>∂</sup>te <sup>þ</sup> <sup>∂</sup>mð Þ¼ pu 0,

mð Þ¼ ξ

with <sup>e</sup> <sup>¼</sup> <sup>ε</sup> <sup>þ</sup> <sup>1</sup>

the mass variable.

83

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p<sup>τ</sup> � pp<sup>ε</sup> Z <sup>ξ</sup> 0

Scalar Conservation Laws

We set the following change of coordinates from Euler coordinates to Lagrange coordinates for space and time as ð Þ! t; x t <sup>0</sup> ð Þ ; <sup>ξ</sup> where <sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup>1; <sup>ξ</sup>2; <sup>ξ</sup><sup>3</sup> ð Þ<sup>∈</sup> <sup>R</sup><sup>3</sup> so that

$$t' = t, \quad \frac{\partial \mathbf{x}(t', \xi)}{\partial t'} = \mathbf{u}(t', \mathbf{x}(t', \xi)), \quad \mathbf{x}\_0 = \mathbf{x}(0, \xi) = \xi. \tag{66}$$

It follows that t <sup>0</sup> ð Þ¼ ; ξ t; ξ1; ξ2; ξ<sup>3</sup> ð Þ ð Þ are the Lagrangian coordinates associated with the velocity field u. We set

$$J(t,\xi) = \det \left(\frac{\partial \mathbf{x}\_i}{\partial \xi\_j}(t,\xi)\right),\tag{67}$$

which gives

$$\frac{\partial f}{\partial t}(t,\xi) = f(t,\xi)(\mathbf{div}\mathbf{u})(t,\varkappa(t,\xi)), \qquad \text{where,} \qquad \mathbf{div}\,\mathbf{u} = \sum\_{j=1}^{3} \frac{\partial u\_j}{\partial \mathbf{x}\_j}.\tag{68}$$

It follows by some algebraic manipulations that the gas dynamic equations become

$$\begin{cases} \partial\_t(\rho f) = 0, \quad \text{(Conservation of mass)},\\ \partial\_t(\rho uf) + \partial\_\xi(p) = 0, \quad \text{(Conservation of momentum)},\\ \partial\_t(\rho ef) + \partial\_\xi(pu) = 0, \quad \text{(Conservation of energy)}. \end{cases} \tag{69}$$

In order to derive a more convenient form of the system (69), we derive firstly the equation of conservation of mass:

$$
\rho \mathbf{J} = \rho\_0 = \rho(\mathbf{0}, \xi) \tag{70}
$$

where ρ0ð Þ¼ ξ ρð Þ 0; ξ : Assuming that ρ . 0, we introduce the specific volume τ ¼ 1=ρ, and by using Eq. (68) we get

$$J = \rho\_0 \pi,\quad\text{and}\quad\partial\_t f = f \partial\_\mathbf{x} u = \partial\_\xi u \tag{71}$$

which yields

$$
\rho\_0 \partial\_t \mathfrak{x} - \partial\_{\xi} \mathfrak{u} = \mathbf{0}.\tag{72}
$$

Hence the second and third equations of Eq. (69) become

$$\begin{aligned} \rho\_0 \partial\_t u + \partial\_{\tilde{\xi}} p &= 0, \quad \text{(Conservation of momentum)},\\ \rho\_0 \partial\_t e + \partial\_{\tilde{\xi}} (p u) &= 0, \quad \text{(Conservation of energy)}. \end{aligned} \tag{73}$$

2.5 Lagrangian coordinates

coordinates for space and time as ð Þ! t; x t

<sup>0</sup> <sup>¼</sup> <sup>t</sup>, <sup>∂</sup>x t<sup>0</sup> ð Þ ; <sup>ξ</sup>

t

with the velocity field u. We set

It follows that t

which gives

8 ><

>:

which yields

82

the equation of conservation of mass:

τ ¼ 1=ρ, and by using Eq. (68) we get

∂J ∂t

become

ential system

We aim to transform the equations of gas dynamics (57) given in the Eulerian coordinates into the Lagrangian coordinates for one-dimensional case. We start denoting by u ¼ uð Þ t; x the velocity field of the fluid flow and consider the differ-

We set the following change of coordinates from Euler coordinates to Lagrange

<sup>0</sup> ð Þ¼ ; ξ t; ξ1; ξ2; ξ<sup>3</sup> ð Þ ð Þ are the Lagrangian coordinates associated

dt <sup>¼</sup> <sup>u</sup>ð Þ <sup>t</sup>; <sup>x</sup> : (65)

; x t<sup>0</sup> ð Þ ð Þ ; ξ , x<sup>0</sup> ¼ xð Þ¼ 0; ξ ξ: (66)

<sup>0</sup> ð Þ ; <sup>ξ</sup> where <sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup>1; <sup>ξ</sup>2; <sup>ξ</sup><sup>3</sup> ð Þ<sup>∈</sup> <sup>R</sup><sup>3</sup> so that

, (67)

∂uj ∂xj

: (68)

(69)

3 j¼1

ρJ ¼ ρ<sup>0</sup> ¼ ρð Þ 0; ξ (70)

<sup>ρ</sup>0∂t<sup>τ</sup> � <sup>∂</sup>ξ<sup>u</sup> <sup>¼</sup> <sup>0</sup>: (72)

<sup>J</sup> <sup>¼</sup> <sup>ρ</sup>0τ, and <sup>∂</sup><sup>t</sup> <sup>J</sup> <sup>¼</sup> <sup>J</sup>∂xu <sup>¼</sup> <sup>∂</sup>ξ<sup>u</sup> (71)

dx

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

0

J tð Þ¼ ; <sup>ξ</sup> det <sup>∂</sup>xi

ð Þ¼ t; ξ J tð Þ ; ξ ð Þ divu ð Þ t; x tð Þ ; ξ , where, div u ¼ ∑

<sup>∂</sup>tð Þ¼ <sup>ρ</sup><sup>J</sup> 0, Conservation of mass ð Þ,

Hence the second and third equations of Eq. (69) become

It follows by some algebraic manipulations that the gas dynamic equations

<sup>∂</sup>tð Þþ <sup>ρ</sup>uJ <sup>∂</sup>ξð Þ¼ <sup>p</sup> 0, Conservation of momentum ð Þ, <sup>∂</sup>tð Þþ <sup>ρ</sup>eJ <sup>∂</sup>ξð Þ¼ pu 0, Conservation of energy � �:

In order to derive a more convenient form of the system (69), we derive firstly

where ρ0ð Þ¼ ξ ρð Þ 0; ξ : Assuming that ρ . 0, we introduce the specific volume

<sup>ρ</sup>0∂tu <sup>þ</sup> <sup>∂</sup>ξ<sup>p</sup> <sup>¼</sup> 0, Conservation of momentum ð Þ,

<sup>ρ</sup>0∂te <sup>þ</sup> <sup>∂</sup>ξð Þ¼ pu 0, Conservation of energy � �: (73)

∂ξj ð Þ t; ξ !

<sup>∂</sup>t<sup>0</sup> <sup>¼</sup> <sup>u</sup> <sup>t</sup>

Moreover, we define a mass variable m by

$$m(\xi) = \int\_0^{\xi} \rho(0, y) dy,\quad \text{or equivalently,}\qquad dm = \rho(0, \xi) d\xi = \rho\_0 d\xi. \tag{74}$$

Finally, using Eqs. (69) and (73), the Euler system (57) can be written in Lagrangian coordinates with the mass variable in the form

$$\begin{cases} \partial\_t \tau - \partial\_m u = 0, \\ \partial\_t u + \partial\_m p = 0, \\ \partial\_t e + \partial\_m (p u) = 0, \end{cases} \tag{75}$$

$$\text{where } p = p(\tau, \xi) = p(\tau, e - u^2/2). \text{ If we set } V = \begin{pmatrix} \tau \\ u \\ e \end{pmatrix}, \quad F(V) = \begin{pmatrix} -u \\ p \\ pu \end{pmatrix} \text{ with }$$

<sup>τ</sup> . 0, <sup>u</sup><sup>∈</sup> <sup>R</sup>,<sup>e</sup> � <sup>u</sup><sup>2</sup>=<sup>2</sup> . 0, we obtain a scalar conservation law of the form

$$
\partial\_t V + \partial\_m F(V) = 0 \tag{76}
$$

which is strictly hyperbolic. This can be verified by checking the Jacobian matrix of the flux calculated with respect to the variables ð Þ τ; u;e

$$
\begin{pmatrix}
0 & -1 & 0 \\
p\_\tau & -up\_\varepsilon & p\_\varepsilon \\
up\_\tau & p - u^2 p\_\varepsilon & up\_\varepsilon
\end{pmatrix}
\tag{77}
$$

with <sup>e</sup> <sup>¼</sup> <sup>ε</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>u</sup>2. The eigenvalues are <sup>σ</sup><sup>1</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p<sup>τ</sup> � pp<sup>ε</sup> <sup>p</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> , <sup>σ</sup><sup>3</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p<sup>τ</sup> � pp<sup>ε</sup> p so that they are all distinct, and thus the system is strictly hyperbolic.

In fact there are different versions of the gas dynamics in Lagrangian coordinates. In this part we followed the approaches stated in [9, 10, 12]. For further details we cite these works with references therein.

#### 2.6 Rankine-Hugoniot conditions for the Lagrangian system

Similarly as in the Euler system, the Rankine-Hugoniot jump conditions for the Lagrangian system (79) are of the form

$$\begin{aligned} \sigma(\tau\_1 - \tau\_0) &= -(u\_1 - u\_0), \\ \sigma(u\_1 - u\_0) &= p\_1 - p\_0, \\ \sigma(\varepsilon\_1 - \varepsilon\_0) &= p\_1 u\_1 - p\_0 u\_0, \end{aligned} \tag{78}$$

where σ denotes the speed of propagation of the discontinuity with respect to the mass variable.

Remark. The Eulerian and Lagrangian Rankine-Hugoniot relations are equivalent. Moreover, Eulerian entropy relations are equivalent to all Lagrangian entropy relations (see [9] for further detail).

Example 2.1. For simplicity of notation, we take ð Þ t; x as the Lagrangian coordinates. Then the system of equations

$$\begin{cases} \partial\_t \tau - \partial\_\mathbf{x} \mu = \mathbf{0}, \\ \partial\_t \mu + \partial\_\mathbf{x} p(\tau) = \mathbf{0}, \end{cases} \tag{79}$$

is a one-dimensional isentropic gas dynamics in Lagrangian coordinates which is also known as p-system. It is the simplest nontrivial example of a nonlinear system of conservation laws. Here τ is the specific volume, u is the velocity, and the pressure p ¼ pð Þτ is given as a function of τ by

$$p(\tau) = \kappa \tau^{-\gamma}, \quad \gamma > 0, \quad \kappa = \frac{\left(\chi - 1\right)^2}{4\chi}. \tag{80}$$

uLð Þ¼ x

DOI: http://dx.doi.org/10.5772/intechopen.83637

Scalar Conservation Laws

mate the solution at the next time step t

Z xjþ1=<sup>2</sup> xj�1=<sup>2</sup>

by <sup>u</sup><sup>∗</sup> <sup>ð</sup>U<sup>n</sup>

become

85

<sup>j</sup>�<sup>1</sup>, U<sup>n</sup>

u~<sup>n</sup> x; t

end points xj�1=<sup>2</sup> and xjþ1<sup>=</sup>2, we get

<sup>n</sup>þ<sup>1</sup> � �dx �

unþ<sup>1</sup> <sup>j</sup> <sup>¼</sup> un

<sup>j</sup> <sup>Þ</sup> and <sup>u</sup><sup>∗</sup> <sup>ð</sup>U<sup>n</sup>

f u~<sup>n</sup> j�1=2

f u~<sup>n</sup> jþ1=2

U<sup>n</sup>þ<sup>1</sup> <sup>j</sup> <sup>¼</sup> <sup>U</sup><sup>n</sup>

an alternative form. Assigning the constant value of un

<sup>j</sup> , U<sup>n</sup>

� � <sup>¼</sup> f u<sup>∗</sup> <sup>U</sup><sup>n</sup>

� � <sup>¼</sup> f u<sup>∗</sup> <sup>U</sup><sup>n</sup>

Therefore, a first-order Godunov method takes the form

<sup>j</sup> � <sup>Δ</sup><sup>t</sup> Δx

are picked as the numerical fluxes at the grid boundary.

lem for the conservation law (1). If we suppose that u<sup>n</sup>

and Fð Þ¼ u; u fð Þ u . For the stability, CFL condition requires that

sup <sup>x</sup><sup>∈</sup> <sup>R</sup>, <sup>t</sup> . <sup>0</sup> ∣f 0 ð Þ u tð Þ ; x ∣

un

un

<sup>j</sup>�<sup>1</sup> ; <sup>x</sup> , 0,

solution u t ~ð Þ ; x . Once establishing the solution over the mesh t

Unþ<sup>1</sup> <sup>j</sup> <sup>¼</sup> <sup>1</sup> Δx

Proceeding this process, we define the solution u x ~ ; t

¼ Z <sup>t</sup> nþ1

Dividing both parts by Δx and using the fact that u x ~ ; t

Z xjþ1=<sup>2</sup> xj�1=<sup>2</sup>

tn

<sup>j</sup> � <sup>Δ</sup><sup>t</sup>

<sup>j</sup> ; <sup>x</sup> . 0, uRð Þ¼ <sup>x</sup>

Z xjþ1=<sup>2</sup> xj�1=<sup>2</sup>

can be calculated by using the integral form of the conservation law (1) in the following way: We integrate (1) for u tð Þ ; x over each grid cell ½ �� tn; tnþ<sup>1</sup> Ij :

> u~<sup>n</sup> x; t <sup>n</sup> ð Þdx

f u~<sup>n</sup> j�1=2 � �dt �

<sup>Δ</sup><sup>x</sup> <sup>f</sup> <sup>u</sup>~<sup>n</sup>

j�1=2 � � � <sup>f</sup> <sup>u</sup>~<sup>n</sup>

Thus, Godunov method is a conservative numerical scheme. It can be restated in

<sup>j</sup>�<sup>1</sup>; <sup>U</sup><sup>n</sup> j � � � � <sup>¼</sup> <sup>F</sup> <sup>U</sup><sup>n</sup>

<sup>j</sup> ; U<sup>n</sup> jþ1 � � � � <sup>¼</sup> <sup>F</sup> <sup>U</sup><sup>n</sup>

Here the constant value of un~ depends on the initial data. In other words, the Godunov method considers the Riemann problem as constant in each grid interval Ii. It follows that, at the subsequent time stage, the exact solutions of the problem

The Godunov method is consistent with the exact solution of the Riemann prob-

Δt Δx

F U<sup>n</sup> <sup>j</sup> ; U<sup>n</sup> jþ1 � � � <sup>F</sup> <sup>U</sup><sup>n</sup>

<sup>j</sup>þ<sup>1</sup> ; <sup>x</sup> . 0, ( (

u x ~ ; t

respectively. These two solutions to the Riemann problem will be the numerical

un

un

<sup>n</sup>þ<sup>1</sup> by the average value

Z <sup>t</sup> nþ1

tn

f u~<sup>n</sup> jþ1=2 � �dt:

<sup>n</sup> ð Þ¼ un

jþ1=2 � � � � : (86)

> <sup>j</sup>�<sup>1</sup>; <sup>U</sup><sup>n</sup> j � �,

> <sup>j</sup> ; U<sup>n</sup> jþ1 � �:

<sup>j</sup>�<sup>1</sup>; <sup>U</sup><sup>n</sup> j � � � � : (88)

<sup>j</sup> <sup>¼</sup> unþ<sup>1</sup>

<sup>j</sup> <sup>¼</sup> <sup>u</sup>, then <sup>u</sup>~<sup>n</sup>

≤1 (89)

<sup>j</sup>þ<sup>1</sup>Þ, respectively, the numerical flux functions

<sup>j</sup> ; x , 0,

<sup>n</sup>; t

<sup>n</sup>þ<sup>1</sup> � �dx: (84)

<sup>n</sup>þ<sup>1</sup> ð Þ iteratively. Then <sup>U</sup>nþ<sup>1</sup>

<sup>n</sup>þ<sup>1</sup> ½ �, we approxi-

(83)

j

(85)

(87)

<sup>j</sup>þ1=<sup>2</sup> ¼ u

<sup>j</sup> is constant at the

<sup>j</sup> at the points xj�1=<sup>2</sup> and xjþ1=<sup>2</sup>

The system (79) is equivalent to

$$
\partial\_t V + \partial\_x f(V) = 0, \qquad \text{with} \qquad V = \begin{pmatrix} \tau \\ u \end{pmatrix}, \quad f(V) = \begin{pmatrix} -u \\ p(\tau) \end{pmatrix}, \tag{81}
$$

where <sup>τ</sup> . 0 and ð Þ <sup>τ</sup>; <sup>u</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> : If we assume that p<sup>0</sup> ð Þτ , 0, it follows that the Jacobian matrix of f

$$J(f) = \begin{pmatrix} \mathbf{0} & -\mathbf{1} \\ p'(\mathbf{r}) & \mathbf{0} \end{pmatrix} \tag{82}$$

has two real distinct eigenvalues <sup>σ</sup><sup>1</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �p<sup>0</sup> <sup>ð</sup> ð Þ<sup>τ</sup> <sup>p</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi �p<sup>0</sup> ð Þ<sup>τ</sup> <sup>p</sup> . In other words, the system (81) is strictly hyperbolic. On the other hand, for the case p0 ð Þτ . 0, it becomes elliptic. Moreover, one can verify that the solutions of the p-system (79) and the Euler system (57) are equivalent.

#### 3. Godunov schemes

The Godunov scheme deals with solving the Riemann problem forward in time for each grid cell and then taking the mean value over these cells. The Riemann problem is solved per mesh point at each time step iteratively. If there are no strong shock discontinuities, this process may cost much and will not be effective. To get rid of such a situation, we establish approximate Riemann solvers that are easier to implement and also low cost to use. Eulerian and Lagrangian Godunov schemes are current Godunov scheme in literature. Both have advantages and disadvantages depending on the structure of the problem. A brief comparison of the method for these two approaches is presented in the last part of the chapter. In this work we will not go further in numerical examples and details of these methods; instead, we aim to present a general form of Godunov schemes for gas dynamics in Eulerian and Lagrangian coordinate. Before introducing these, we present a first-order Godunov scheme for scalar conservation laws.

#### 3.1 First-order Godunov scheme

Consider the scalar conservation law (1). Godunov scheme is a numerical scheme which takes advantage of analytical solutions of the Riemann problem for the conservation law (1). The numerical flux functions are evaluated at the spatial steps xj�1=<sup>2</sup> and xjþ1=<sup>2</sup> by handling the solutions of the Riemann problem. On each grid cell I<sup>i</sup> ¼ xj�1=2; xjþ1=<sup>2</sup> � �, we have a piecewise constant function. The Riemann problem for (1) for the left and right sides of I<sup>i</sup> are described by

Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

is a one-dimensional isentropic gas dynamics in Lagrangian coordinates which is also known as p-system. It is the simplest nontrivial example of a nonlinear system of conservation laws. Here τ is the specific volume, u is the velocity, and the

, <sup>γ</sup> . 0, <sup>κ</sup> <sup>¼</sup> ð Þ <sup>γ</sup> � <sup>1</sup> <sup>2</sup>

u � �

: If we assume that p<sup>0</sup>

J fð Þ¼ <sup>0</sup> �<sup>1</sup> p0 ð Þτ 0 � �

words, the system (81) is strictly hyperbolic. On the other hand, for the case

ð Þτ . 0, it becomes elliptic. Moreover, one can verify that the solutions of the

The Godunov scheme deals with solving the Riemann problem forward in time for each grid cell and then taking the mean value over these cells. The Riemann problem is solved per mesh point at each time step iteratively. If there are no strong shock discontinuities, this process may cost much and will not be effective. To get rid of such a situation, we establish approximate Riemann solvers that are easier to implement and also low cost to use. Eulerian and Lagrangian Godunov schemes are current Godunov scheme in literature. Both have advantages and disadvantages depending on the structure of the problem. A brief comparison of the method for these two approaches is presented in the last part of the chapter. In this work we will not go further in numerical examples and details of these methods; instead, we aim to present a general form of Godunov schemes for gas dynamics in Eulerian and Lagrangian coordinate. Before introducing these, we present a first-order Godunov

Consider the scalar conservation law (1). Godunov scheme is a numerical scheme which takes advantage of analytical solutions of the Riemann problem for the conservation law (1). The numerical flux functions are evaluated at the spatial steps xj�1=<sup>2</sup> and xjþ1=<sup>2</sup> by handling the solutions of the Riemann problem.

function. The Riemann problem for (1) for the left and right sides of I<sup>i</sup> are

� �, we have a piecewise constant

, f Vð Þ¼ �<sup>u</sup>

�p<sup>0</sup> <sup>ð</sup> ð Þ<sup>τ</sup> <sup>p</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>4</sup><sup>γ</sup> : (80)

, (81)

(82)

pð Þτ � �

ð Þτ , 0, it follows that the

�p<sup>0</sup>

ð Þ<sup>τ</sup> <sup>p</sup> . In other

pressure p ¼ pð Þτ is given as a function of τ by

The system (79) is equivalent to

where <sup>τ</sup> . 0 and ð Þ <sup>τ</sup>; <sup>u</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>

Jacobian matrix of f

3. Godunov schemes

scheme for scalar conservation laws.

3.1 First-order Godunov scheme

On each grid cell I<sup>i</sup> ¼ xj�1=2; xjþ1=<sup>2</sup>

described by

84

p0

<sup>p</sup>ð Þ¼ <sup>τ</sup> κτ�<sup>γ</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

<sup>∂</sup>tV <sup>þ</sup> <sup>∂</sup><sup>x</sup> f Vð Þ¼ 0, with <sup>V</sup> <sup>¼</sup> <sup>τ</sup>

has two real distinct eigenvalues <sup>σ</sup><sup>1</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p-system (79) and the Euler system (57) are equivalent.

$$u\_L(\mathbf{x}) = \begin{cases} u\_{j-1}^n & ; \mathbf{x} \le \mathbf{0}, \\\ u\_j^n & ; \mathbf{x} > \mathbf{0}, \end{cases} \qquad u\_R(\mathbf{x}) = \begin{cases} u\_j^n & ; \mathbf{x} \le \mathbf{0}, \\\ u\_{j+1}^n & ; \mathbf{x} > \mathbf{0}, \end{cases} \tag{83}$$

respectively. These two solutions to the Riemann problem will be the numerical solution u t ~ð Þ ; x . Once establishing the solution over the mesh t <sup>n</sup>; t <sup>n</sup>þ<sup>1</sup> ½ �, we approximate the solution at the next time step t <sup>n</sup>þ<sup>1</sup> by the average value

$$U\_j^{n+1} = \frac{1}{\Delta \mathbf{x}} \int\_{\mathbf{x}\_{j-1/2}}^{\mathbf{x}\_{j+1/2}} \tilde{u}\left(\mathbf{x}, t^{n+1}\right) d\mathbf{x}.\tag{84}$$

Proceeding this process, we define the solution u x ~ ; t <sup>n</sup>þ<sup>1</sup> ð Þ iteratively. Then <sup>U</sup>nþ<sup>1</sup> j can be calculated by using the integral form of the conservation law (1) in the following way: We integrate (1) for u tð Þ ; x over each grid cell ½ �� tn; tnþ<sup>1</sup> Ij :

$$\begin{split} \int\_{x\_{j-1/2}}^{x\_{j+1/2}} \bar{u}^n(\mathbf{x}, t^{n+1}) \, d\mathbf{x} - \int\_{x\_{j-1/2}}^{x\_{j+1/2}} \bar{u}^n(\mathbf{x}, t^n) \, d\mathbf{x} \\ = \int\_{t^n}^{t^{n+1}} f\left(\tilde{u}^n\_{j-1/2}\right) dt - \int\_{t^n}^{t^{n+1}} f\left(\tilde{u}^n\_{j+1/2}\right) dt. \end{split} \tag{85}$$

Dividing both parts by Δx and using the fact that u x ~ ; t <sup>n</sup> ð Þ¼ un <sup>j</sup> is constant at the end points xj�1=<sup>2</sup> and xjþ1<sup>=</sup>2, we get

$$u\_{j}^{n+1} = u\_{j}^{n} - \frac{\Delta t}{\Delta \mathbf{x}} \left( f\left(\tilde{u}\_{j-1/2}^{n}\right) - f\left(\tilde{u}\_{j+1/2}^{n}\right) \right). \tag{86}$$

Thus, Godunov method is a conservative numerical scheme. It can be restated in an alternative form. Assigning the constant value of un <sup>j</sup> at the points xj�1=<sup>2</sup> and xjþ1=<sup>2</sup> by <sup>u</sup><sup>∗</sup> <sup>ð</sup>U<sup>n</sup> <sup>j</sup>�<sup>1</sup>, U<sup>n</sup> <sup>j</sup> <sup>Þ</sup> and <sup>u</sup><sup>∗</sup> <sup>ð</sup>U<sup>n</sup> <sup>j</sup> , U<sup>n</sup> <sup>j</sup>þ<sup>1</sup>Þ, respectively, the numerical flux functions become

$$\begin{aligned} f\left(\bar{\boldsymbol{u}}\_{j-1/2}^{n}\right) &= f\left(\boldsymbol{u}^\*\left(\boldsymbol{U}\_{j-1}^{n}, \boldsymbol{U}\_{j}^{n}\right)\right) = \mathcal{F}\left(\boldsymbol{U}\_{j-1}^{n}, \boldsymbol{U}\_{j}^{n}\right), \\ f\left(\bar{\boldsymbol{u}}\_{j+1/2}^{n}\right) &= f\left(\boldsymbol{u}^\*\left(\boldsymbol{U}\_{j}^{n}, \boldsymbol{U}\_{j+1}^{n}\right)\right) = \mathcal{F}\left(\boldsymbol{U}\_{j}^{n}, \boldsymbol{U}\_{j+1}^{n}\right). \end{aligned} \tag{87}$$

Therefore, a first-order Godunov method takes the form

$$U\_j^{n+1} = U\_j^n - \frac{\Delta t}{\Delta \mathbf{x}} \left( \mathcal{F} \left( U\_j^n, U\_{j+1}^n \right) - \mathcal{F} \left( U\_{j-1}^n, U\_j^n \right) \right). \tag{88}$$

Here the constant value of un~ depends on the initial data. In other words, the Godunov method considers the Riemann problem as constant in each grid interval Ii. It follows that, at the subsequent time stage, the exact solutions of the problem are picked as the numerical fluxes at the grid boundary.

The Godunov method is consistent with the exact solution of the Riemann problem for the conservation law (1). If we suppose that u<sup>n</sup> <sup>j</sup> <sup>¼</sup> unþ<sup>1</sup> <sup>j</sup> <sup>¼</sup> <sup>u</sup>, then <sup>u</sup>~<sup>n</sup> <sup>j</sup>þ1=<sup>2</sup> ¼ u and Fð Þ¼ u; u fð Þ u . For the stability, CFL condition requires that

$$\sup\_{\lambda \in \mathcal{R}, t \ge 0} |f'(u(t, \infty))| \frac{\Delta t}{\Delta \infty} \le 1 \tag{89}$$

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

for each un <sup>j</sup> . Next, if assigning u<sup>∗</sup> as the intermediate value over the grid I<sup>i</sup> in the Riemann solution, it implies that

$$u^\*\left(u\_L, u\_R\right) = \begin{cases} u\_L, & \lambda > 0, \\ u\_R, & \lambda < 0, \end{cases} \tag{90}$$

where

Scalar Conservation Laws

<sup>Δ</sup>mi <sup>¼</sup> <sup>ρ</sup><sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83637

at tn is upgraded with respect to

Next we deduce

with

that is

scheme:

87

<sup>i</sup> Δξi, pn

xnþ<sup>1</sup> <sup>i</sup>þ1=<sup>2</sup> <sup>¼</sup> xn

ρn <sup>i</sup> xn

8 < :

ρ<sup>n</sup>þ<sup>1</sup>

8 >>>>>>><

u<sup>n</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>u</sup><sup>n</sup>

If we integrate these equations on ξ<sup>i</sup>�1=2; ξ<sup>i</sup>�1=<sup>2</sup>

d dt <sup>Z</sup> xiþ1=<sup>2</sup> xi�1=<sup>2</sup>

Δx<sup>n</sup>þ<sup>1</sup> <sup>i</sup> φ<sup>n</sup>þ<sup>1</sup>

φ is constant in each cell ξ<sup>i</sup>�1=2; ξ<sup>i</sup>�1=<sup>2</sup>

e nþ1 <sup>i</sup> <sup>¼</sup> <sup>e</sup><sup>n</sup>

>>>>>>>:

<sup>i</sup> <sup>¼</sup> <sup>x</sup><sup>n</sup>þ<sup>1</sup>

<sup>i</sup> <sup>¼</sup> <sup>p</sup> <sup>τ</sup><sup>n</sup>

If we now consider the moving coordinates, Godunov scheme can also be derived equivalently by the following. Setting xiþ1=<sup>2</sup> ¼ ξiþ1=<sup>2</sup> with the approximation of u ¼ dx=dt, it follows that the Eulerian coordinate xiþ1=<sup>2</sup> of the interface ξiþ1=<sup>2</sup>

<sup>i</sup>þ1=<sup>2</sup> � xn

<sup>Δ</sup>mi <sup>¼</sup> <sup>ρ</sup><sup>0</sup>

xnþ<sup>1</sup> <sup>i</sup>þ1=<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>n</sup>

<sup>i</sup> ; ε<sup>n</sup> i � �, ε<sup>n</sup>

<sup>i</sup>þ1=<sup>2</sup> <sup>þ</sup> <sup>Δ</sup>tu<sup>n</sup>

i�1=2

by a simple induction process. Hence the Lagrangian Godunov schemes become

<sup>i</sup>þ1=<sup>2</sup> � <sup>x</sup><sup>0</sup>

<sup>i</sup>þ1=<sup>2</sup> <sup>þ</sup> <sup>Δ</sup>tu<sup>n</sup>

� �

i�1=2

iþ1=2

Δmi

i�1=2

i�1=2

<sup>∂</sup>tð Þþ <sup>φ</sup><sup>J</sup> <sup>∂</sup><sup>ξ</sup> <sup>f</sup> <sup>¼</sup> <sup>0</sup>: (101)

� � <sup>¼</sup> <sup>0</sup>: (102)

<sup>i</sup>þ1=<sup>2</sup> � ð Þ pu <sup>n</sup>

� � it follows that

� �, it follows by an explicit one-step method

<sup>i</sup>þ1=<sup>2</sup> � <sup>f</sup>

n i�1=2

� �: (103)

� �

<sup>i</sup> x<sup>0</sup>

<sup>i</sup>þ1=<sup>2</sup> � xnþ<sup>1</sup>

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δmi

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δmi

volume method. Equation (100) can be written in conservative form:

<sup>i</sup> <sup>¼</sup> <sup>Δ</sup>x<sup>n</sup>

<sup>i</sup> φ<sup>n</sup>

i�1=2 � ��<sup>1</sup>

pn

ð Þ pu <sup>n</sup>

Notice that the Lagrangian Godunov schemes can be reformulated as a finite

<sup>φ</sup>d<sup>ξ</sup> <sup>þ</sup> fiþ1=<sup>2</sup> � fi�1=<sup>2</sup>

Here we omit the dependency of f, φ and x on t. Moreover, if we suppose that

<sup>i</sup> � <sup>Δ</sup>t f <sup>n</sup>

Moreover, if ð Þ ρ; u;e are constant in each cell with v ¼ u, we get the Godunov

<sup>i</sup>þ1=<sup>2</sup> � pn

� �:

<sup>i</sup> ¼ e n <sup>i</sup> � un i � �<sup>2</sup>

� � <sup>¼</sup> <sup>Δ</sup>mi (98)

<sup>2</sup> : (96)

(99)

(100)

<sup>i</sup>þ1=2: (97)

where λ is the wave propagation speed. Hence the numerical flux for Godunov's method can be generalized by

$$f(u\_L, u\_R) = \begin{cases} \min\_{\substack{u\_L \le u \le u\_R \\ u\_L \ge u\_R}} f(u), & \text{if} \quad u\_L \le u\_R, \\\max\_{\substack{u\_L \ge u \ge u\_R}} f(u), & \text{if} \quad u\_R < u\_L. \end{cases} \tag{91}$$

For numerical illustration of Godunov schemes, we cite the articles [14, 20, 27].

#### 3.2 Godunov method in Eulerian coordinates

We consider Eq. (59) with (60). The eigenvalues of F<sup>0</sup> ð Þ U are σ<sup>1</sup> ¼ u � c , σ<sup>2</sup> ¼ u , σ<sup>3</sup> ¼ u þ c. Then the Riemann problem at the point xiþ1=<sup>2</sup> between the states Ui and Uiþ<sup>1</sup> which is solved by the Godunov scheme can be written by

$$\begin{cases} \rho\_i^{n+1} = \rho\_i^n - \frac{\Delta t}{\Delta \mathbf{x}\_i} \left( (\rho u)\_{i+1/2}^n - (\rho u)\_{i-1/2}^n \right) \\ \quad \left( \rho u \right)\_i^{n+1} = \left( \rho u \right)\_i^n - \frac{\Delta t}{\Delta \mathbf{x}\_i} \left( \left( \rho u^2 + p \right)\_{i+1/2}^n - \left( \rho u^2 + p \right)\_{i-1/2}^n \right). \end{cases} \tag{92}$$
 
$$\left( \rho e \right)\_i^{n+1} = \left( \rho e \right)\_i^n - \frac{\Delta t}{\Delta \mathbf{x}\_i} \left( \left( (\rho e + p) u \right)\_{i+1/2}^n - \left( (\rho e + p) u \right)\_{i-1/2}^n \right)$$

#### 3.3 Godunov method in Lagrangian coordinates

Consider the initial condition for a quantity v given by the mean value

$$v\_i^0 = \frac{1}{\Delta \xi\_i} \int\_{\xi\_{i-1/2}}^{\xi\_{i+1/2}} v(\xi, \mathbf{0}) d\xi. \tag{93}$$

The eigenvalues satisfy <sup>σ</sup><sup>1</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> , <sup>σ</sup>3: Setting uiþ1=<sup>2</sup> and piþ1=<sup>2</sup> as the values of u and p at the contact discontinuity between V<sup>n</sup> <sup>i</sup> and V<sup>n</sup> <sup>i</sup>þ1, it follows that

$$F(\left(w\_R(0;V\_i^n,V\_{i+1}^n)\right)) = \left(-u\_{i+j/2}^n, p\_{i+j/2}^n, \left(pu\right)\_{i+j/2}^n\right)^T.\tag{94}$$

Then Godunov scheme for the Lagrangian coordinates takes the form

$$\begin{cases} \mathbf{r}\_{i}^{n+1} = \mathbf{r}\_{i}^{n} + \frac{\Delta t}{\Delta m\_{i}} \left( u\_{i+1/2}^{n} - u\_{i-1/2}^{n} \right) \\\ u\_{i}^{n+1} = u\_{i}^{n} - \frac{\Delta t}{\Delta m\_{i}} \left( p\_{i+1/2}^{n} - p\_{i-1/2}^{n} \right) \\\ e\_{i}^{n+1} = e\_{i}^{n} - \frac{\Delta t}{\Delta m\_{i}} \left( (\boldsymbol{p} \, u)\_{i+1/2}^{n} - (\boldsymbol{p} \, u)\_{i-1/2}^{n} \right) \end{cases} \tag{95}$$

Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

where

for each un

written by

86

ρ<sup>n</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>ρ</sup><sup>n</sup>

8 >>>>>><

>>>>>>:

ð Þ <sup>ρ</sup><sup>u</sup> <sup>n</sup>þ<sup>1</sup>

ð Þ <sup>ρ</sup><sup>e</sup> <sup>n</sup>þ<sup>1</sup>

Riemann solution, it implies that

method can be generalized by

f uð Þ¼ <sup>L</sup>; uR

3.2 Godunov method in Eulerian coordinates

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δxi

<sup>i</sup> <sup>¼</sup> ð Þ <sup>ρ</sup><sup>u</sup> <sup>n</sup>

<sup>i</sup> <sup>¼</sup> ð Þ <sup>ρ</sup><sup>e</sup> <sup>n</sup>

<sup>j</sup> . Next, if assigning u<sup>∗</sup> as the intermediate value over the grid I<sup>i</sup> in the

uL, λ . 0, uR, λ , 0,

f uð Þ, if uL ≤uR,

f uð Þ, if uR , uL:

ð Þ U are

i�1=2

:

i�1=2

vð Þ ξ; 0 dξ: (93)

<sup>i</sup>þ1, it follows that

: (94)

(95)

iþj=2

(90)

(91)

(92)

�

where λ is the wave propagation speed. Hence the numerical flux for Godunov's

For numerical illustration of Godunov schemes, we cite the articles [14, 20, 27].

<sup>u</sup><sup>∗</sup> ð Þ¼ uL; uR

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

8 < :

We consider Eq. (59) with (60). The eigenvalues of F<sup>0</sup>

ð Þ <sup>ρ</sup><sup>u</sup> <sup>n</sup>

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δxi

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δxi

> v0 <sup>i</sup> <sup>¼</sup> <sup>1</sup> Δξ<sup>i</sup>

<sup>i</sup> ;V<sup>n</sup> iþ1 � � � � � � ¼ �u<sup>n</sup>

<sup>i</sup> þ

3.3 Godunov method in Lagrangian coordinates

u and p at the contact discontinuity between V<sup>n</sup>

τ<sup>n</sup>þ<sup>1</sup> <sup>i</sup> <sup>¼</sup> <sup>τ</sup><sup>n</sup>

unþ<sup>1</sup> <sup>i</sup> <sup>¼</sup> un

e nþ1 <sup>i</sup> <sup>¼</sup> en

>>>>>>:

F wR 0;V<sup>n</sup>

8 >>>>>>< min uL <sup>≤</sup><sup>u</sup> <sup>≤</sup>uR

max uL <sup>≥</sup><sup>u</sup> <sup>≥</sup>uR

σ<sup>1</sup> ¼ u � c , σ<sup>2</sup> ¼ u , σ<sup>3</sup> ¼ u þ c. Then the Riemann problem at the point xiþ1=<sup>2</sup> between the states Ui and Uiþ<sup>1</sup> which is solved by the Godunov scheme can be

<sup>i</sup>þ1=<sup>2</sup> � ð Þ <sup>ρ</sup><sup>u</sup> <sup>n</sup>

<sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>p</sup> � �<sup>n</sup>

ð Þ ð Þ <sup>ρ</sup><sup>e</sup> <sup>þ</sup> <sup>p</sup> <sup>u</sup> <sup>n</sup>

Consider the initial condition for a quantity v given by the mean value

Then Godunov scheme for the Lagrangian coordinates takes the form

un

pn

ð Þ pu <sup>n</sup>

Δt Δmi

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δmi

<sup>i</sup> � <sup>Δ</sup><sup>t</sup> Δmi

Z <sup>ξ</sup>iþ1=<sup>2</sup> ξi�1=<sup>2</sup>

The eigenvalues satisfy <sup>σ</sup><sup>1</sup> , <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> , <sup>σ</sup>3: Setting uiþ1=<sup>2</sup> and piþ1=<sup>2</sup> as the values of

� �

i�1=2

<sup>i</sup>þ1=<sup>2</sup> � <sup>ρ</sup>u<sup>2</sup> <sup>þ</sup> <sup>p</sup> � �<sup>n</sup>

<sup>i</sup>þ1=<sup>2</sup> � ð Þ ð Þ <sup>ρ</sup><sup>e</sup> <sup>þ</sup> <sup>p</sup> <sup>u</sup> <sup>n</sup>

� �

� �

<sup>i</sup> and V<sup>n</sup>

<sup>i</sup>þj=2;ð Þ pu <sup>n</sup>

� �<sup>T</sup>

i�1=2

i�1=2

<sup>i</sup>þ1=<sup>2</sup> � ð Þ pu <sup>n</sup>

� �

i�1=2

<sup>i</sup>þj=2; pn

<sup>i</sup>þ1=<sup>2</sup> � <sup>u</sup><sup>n</sup>

<sup>i</sup>þ1=<sup>2</sup> � pn

� �

� �

$$
\Delta m\_i = \rho\_i^0 \Delta \xi\_i, \qquad p\_i^n = p\left(\tau\_i^n, \varepsilon\_i^n\right), \quad \varepsilon\_i^n = e\_i^n - \frac{\left(u\_i^n\right)^2}{2}. \tag{96}
$$

If we now consider the moving coordinates, Godunov scheme can also be derived equivalently by the following. Setting xiþ1=<sup>2</sup> ¼ ξiþ1=<sup>2</sup> with the approximation of u ¼ dx=dt, it follows that the Eulerian coordinate xiþ1=<sup>2</sup> of the interface ξiþ1=<sup>2</sup> at tn is upgraded with respect to

$$
\varkappa\_{i+1/2}^{n+1} = \varkappa\_{i+1/2}^{n} + \Delta t u\_{i+1/2}^{n}.\tag{97}
$$

Next we deduce

$$
\rho\_i^n \left( \mathbf{x}\_{i+1/2}^n - \mathbf{x}\_{i-1/2}^n \right) = \Delta m\_i \tag{98}
$$

by a simple induction process. Hence the Lagrangian Godunov schemes become

$$\begin{cases} \Delta m\_i = \rho\_i^0 \left( \mathbf{x}\_{i+1/2}^0 - \mathbf{x}\_{i-1/2}^0 \right) \\\\ \mathbf{x}\_{i+1/2}^{n+1} = \mathbf{x}\_{i+1/2}^n + \Delta t \mathbf{u}\_{i+1/2}^n \end{cases} \tag{99}$$

with

$$\begin{cases} \rho\_i^{n+1} = \left(\mathbf{x}\_{i+1/2}^{n+1} - \mathbf{x}\_{i-1/2}^{n+1}\right)^{-1} \Delta m\_i \\\\ u\_i^{n+1} = u\_i^n - \frac{\Delta t}{\Delta m\_i} \left(p\_{i+1/2}^n - p\_{i-1/2}^n\right). \\\\ e\_i^{n+1} = e\_i^n - \frac{\Delta t}{\Delta m\_i} \left(\left(p\,u\right)\_{i+1/2}^n - \left(p\,u\right)\_{i-1/2}^n\right) \end{cases} \tag{100}$$

Notice that the Lagrangian Godunov schemes can be reformulated as a finite volume method. Equation (100) can be written in conservative form:

$$
\partial\_t(q\mathcal{J}) + \partial\_{\xi}\mathcal{J} = \mathbf{0}.\tag{101}
$$

If we integrate these equations on ξ<sup>i</sup>�1=2; ξ<sup>i</sup>�1=<sup>2</sup> � � it follows that

$$\frac{d}{dt} \int\_{\mathbf{x}\_{i-1/2}}^{\mathbf{x}\_{i+1/2}} \rho d\xi + \left(f\_{i+1/2} - f\_{i-1/2}\right) = \mathbf{0}.\tag{102}$$

Here we omit the dependency of f, φ and x on t. Moreover, if we suppose that φ is constant in each cell ξ<sup>i</sup>�1=2; ξ<sup>i</sup>�1=<sup>2</sup> � �, it follows by an explicit one-step method that is

$$
\Delta \mathfrak{x}\_i^{n+1} \varrho\_i^{n+1} = \Delta \mathfrak{x}\_i^n \varrho\_i^n - \Delta t \left( f\_{i+1/2}^n - f\_{i-1/2}^n \right). \tag{103}
$$

Moreover, if ð Þ ρ; u;e are constant in each cell with v ¼ u, we get the Godunov scheme:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{cases} \Delta x\_i^n \rho\_i^n = \Delta m\_i \\ \Delta m\_i^{n+1} u\_i^{n+1} = \Delta m\_i^n u\_i^n - \Delta t \left( p\_{i+1/2}^n - p\_{i-1/2}^n \right) \\ \Delta m\_i^{n+1} e\_i^{n+1} = \Delta m\_i^n e\_i^n - \Delta t \left( (\boldsymbol{p}\boldsymbol{u})\_{i+1/2}^n - (\boldsymbol{p}\boldsymbol{u})\_{i-1/2}^n \right) \end{cases} \tag{104}$$

subsonic character of the flow makes the transformation much easier than in Eulerian schemes. Lagrangian schemes consider the implementation in a grid that moves with the flow which is an advantage for the problems like the transport equations since the advective and diffusion terms can separately be examined. Apart from the two main approaches, there is another method which is a combination of both, so-called Eulerian-Lagrangian methods. It combines the advantages and eliminates disadvantages of both approaches to get a more efficient method. For further details we address the reader to the reference in the next part.

We have tried to present only the theoretical aspects of scalar conservation laws with some basic models and provide some examples of computational methods for the scalar models. There are plenty of contributors to the subject; however, we just cite some important of these and the references therein. Scalar conservation laws are thoroughly studied in particular in [12]; for a more general introduction including systems, see [13, 15, 18, 19, 22] and the references therein. There are some important works related to the concept of entropy provided by [7, 15, 16]. A more precise study of the shock and rarefaction waves can be found in [23]. A simple analysis for inviscid Burgers' equation is done by [21]. The readers who are deeply interested in systems of conservation laws and the Riemann problem should see [8, 13, 15, 22, 24]. A well-ordered work of the propagation and the interaction of nonlinear waves are provided by [26]. We refer the reader to the papers [1, 17] for the theory of hyperbolic conservation laws on spacetime geometries and finite volume analysis with different aspects. A widely introductory material for finite difference and finite volume schemes to scalar conservation laws can be found in [18]. In this chapter we have studied the one-dimensional gas dynamics on the Eulerian and Lagrangian coordinates. For the detail on the Lagrangian conservation laws, we refer [10]; moreover for both Eulerian and Lagrangian conservation laws, we cite [11]. The proof of the equivalency of the Euler and Lagrangian equations for weak solutions is given in [25]. There are several numerical works for Lagrangian approach; some of the basic works on Lagrangian schemes are given in [2–6]. We refer the reader to the book [7] for a detailed analysis of the mathematical standpoint of compressible flows. Moreover Godunov-type schemes are precisely analyzed in [14, 27]; whereas, Lagrangian Godunov schemes can be found in [2, 12, 20]. As a last word, we must cite [9] as a recent and more general book consisting of scalar and system approaches of both Eulerian and Lagrangian conservation laws with theoretical and numerical parts which can be a basic source

Notes

Scalar Conservation Laws

DOI: http://dx.doi.org/10.5772/intechopen.83637

for the curious readers.

89

provided un <sup>i</sup>þ1=2; pn iþ1=2 � � are determined by the solution of the Riemann problem, which is the desired result.

#### 3.4 Comparison of Eulerian and Lagrangian schemes

In the literature there are two types of Godunov schemes: the Eulerian and Lagrangian. To compare one with the other, both have advantages and disadvantages. These are briefly listed in the following:

#### 3.4.1 Eulerian approach

It is more nature; that is the properties of a flow field are described as functions of the coordinates which are in the natural physical space and time. The flow is determined by examining the behavior of the functions. Eulerian coordinates correspond to the coordinates of a fixed observer. This approach is ease of implementation and computation. The computational grids derived from the geometry constraints are generated in advance. The computational cells are fixed in space, and the fluid particles move across the cell interfaces. Since the Eulerian schemes consider the implementation at the nodes of a fixed grid, this may lead to spurious oscillations for the problems like diffusion-dominated transport equations. By adding artificial diffusion, one can get rid of these oscillations; however the nature of the problem may differ from the original one. Besides, refining the grids may also lead to remove numerical oscillations, but this process may augment the computation cost. Besides, while refining the grids, it may cause restriction of the size of time step which is limited by CFL condition. This restriction does not occur in Lagrangian case.

#### 3.4.2 Lagrangian approach

It is based on the notion of mass coordinate denoted by mð Þξ . An important feature of the mass coordinate is that two segments have the same length if the mass contained in these segments is the same. This leads to face with a disadvantage; that is, at each iteration time step, the problem has to be converted from the natural coordinate system to the mass coordinate system. Once the solution at the next step is known, it has to be remapped into the natural coordinate system. As a result, this process raises the cost of the computation. Lagrangian coordinates are associated to the local flow velocity. In other words, as the velocity has different values in different parts of the fluid, then the change of coordinates is different from one point to another one in Lagrangian coordinates. Thus Lagrangian coordinates are equivalent to the Eulerian coordinates at another time. Lagrangian description states the motions and properties of the given fluid particles as they travel to different locations. Hence the computational grid points are precisely fluid particles. Since the particle paths in steady flow coincide with the streamlines, no fluid particles will cross the streamlines. Hence, there is no convective flux across cell boundaries, and the numerical diffusion is minimized. As a result, Godunov method in a Lagrangian grid is easier to handle. Moreover, in the case of higher schemes, the

#### Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

subsonic character of the flow makes the transformation much easier than in Eulerian schemes. Lagrangian schemes consider the implementation in a grid that moves with the flow which is an advantage for the problems like the transport equations since the advective and diffusion terms can separately be examined.

Apart from the two main approaches, there is another method which is a combination of both, so-called Eulerian-Lagrangian methods. It combines the advantages and eliminates disadvantages of both approaches to get a more efficient method. For further details we address the reader to the reference in the next part.

#### Notes

Δ xn <sup>i</sup> ρ<sup>n</sup>

8 >>><

>>>:

provided un

3.4.1 Eulerian approach

Lagrangian case.

88

3.4.2 Lagrangian approach

Δmnþ<sup>1</sup> <sup>i</sup> unþ<sup>1</sup>

Δmnþ<sup>1</sup> <sup>i</sup> e nþ1 <sup>i</sup> <sup>¼</sup> <sup>Δ</sup>m<sup>n</sup>

lem, which is the desired result.

<sup>i</sup>þ1=2; pn

<sup>i</sup> ¼ Δmi

iþ1=2 � �

tages. These are briefly listed in the following:

<sup>i</sup> <sup>¼</sup> <sup>Δ</sup>m<sup>n</sup>

3.4 Comparison of Eulerian and Lagrangian schemes

<sup>i</sup> un

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

i en

<sup>i</sup> � <sup>Δ</sup>t p<sup>n</sup>

<sup>i</sup> � <sup>Δ</sup>t pu ð Þ<sup>n</sup>

In the literature there are two types of Godunov schemes: the Eulerian and Lagrangian. To compare one with the other, both have advantages and disadvan-

It is more nature; that is the properties of a flow field are described as functions of the coordinates which are in the natural physical space and time. The flow is determined by examining the behavior of the functions. Eulerian coordinates correspond to the coordinates of a fixed observer. This approach is ease of implementation and computation. The computational grids derived from the geometry constraints are generated in advance. The computational cells are fixed in space, and the fluid particles move across the cell interfaces. Since the Eulerian schemes consider the implementation at the nodes of a fixed grid, this may lead to spurious oscillations for the problems like diffusion-dominated transport equations. By adding artificial diffusion, one can get rid of these oscillations; however the nature of the problem may differ from the original one. Besides, refining the grids may also lead to remove numerical oscillations, but this process may augment the computation cost. Besides, while refining the grids, it may cause restriction of the size of time step which is limited by CFL condition. This restriction does not occur in

It is based on the notion of mass coordinate denoted by mð Þξ . An important feature of the mass coordinate is that two segments have the same length if the mass contained in these segments is the same. This leads to face with a disadvantage; that is, at each iteration time step, the problem has to be converted from the natural coordinate system to the mass coordinate system. Once the solution at the next step is known, it has to be remapped into the natural coordinate system. As a result, this process raises the cost of the computation. Lagrangian coordinates are associated to the local flow velocity. In other words, as the velocity has different values in different parts of the fluid, then the change of coordinates is different from one point to another one in Lagrangian coordinates. Thus Lagrangian coordinates are equivalent to the Eulerian coordinates at another time. Lagrangian description states the motions and properties of the given fluid particles as they travel to different locations. Hence the computational grid points are precisely fluid particles. Since the particle paths in steady flow coincide with the streamlines, no fluid particles will cross the streamlines. Hence, there is no convective flux across cell boundaries, and the numerical diffusion is minimized. As a result, Godunov method in a Lagrangian grid is easier to handle. Moreover, in the case of higher schemes, the

<sup>i</sup>þ1=<sup>2</sup> � <sup>p</sup><sup>n</sup>

� �

i�1=2

are determined by the solution of the Riemann prob-

i�1=2

(104)

<sup>i</sup>þ1=<sup>2</sup> � ð Þ pu <sup>n</sup>

� �

We have tried to present only the theoretical aspects of scalar conservation laws with some basic models and provide some examples of computational methods for the scalar models. There are plenty of contributors to the subject; however, we just cite some important of these and the references therein. Scalar conservation laws are thoroughly studied in particular in [12]; for a more general introduction including systems, see [13, 15, 18, 19, 22] and the references therein. There are some important works related to the concept of entropy provided by [7, 15, 16]. A more precise study of the shock and rarefaction waves can be found in [23]. A simple analysis for inviscid Burgers' equation is done by [21]. The readers who are deeply interested in systems of conservation laws and the Riemann problem should see [8, 13, 15, 22, 24]. A well-ordered work of the propagation and the interaction of nonlinear waves are provided by [26]. We refer the reader to the papers [1, 17] for the theory of hyperbolic conservation laws on spacetime geometries and finite volume analysis with different aspects. A widely introductory material for finite difference and finite volume schemes to scalar conservation laws can be found in [18]. In this chapter we have studied the one-dimensional gas dynamics on the Eulerian and Lagrangian coordinates. For the detail on the Lagrangian conservation laws, we refer [10]; moreover for both Eulerian and Lagrangian conservation laws, we cite [11]. The proof of the equivalency of the Euler and Lagrangian equations for weak solutions is given in [25]. There are several numerical works for Lagrangian approach; some of the basic works on Lagrangian schemes are given in [2–6]. We refer the reader to the book [7] for a detailed analysis of the mathematical standpoint of compressible flows. Moreover Godunov-type schemes are precisely analyzed in [14, 27]; whereas, Lagrangian Godunov schemes can be found in [2, 12, 20]. As a last word, we must cite [9] as a recent and more general book consisting of scalar and system approaches of both Eulerian and Lagrangian conservation laws with theoretical and numerical parts which can be a basic source for the curious readers.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

References

Scalar Conservation Laws

1059-1086

5160-5183

261-262:56-65

4324-4354

91

Publishers; 1948

2011;46:133-136

[1] Amorim P, LeFloch PG, Okutmustur B. Finite volume schemes on Lorentzian

DOI: http://dx.doi.org/10.5772/intechopen.83637

325. Berlin Heidelberg: Springer-Verlag;

[9] Despres B. Numerical Methods for Eulerian and Lagrangian Conservation

[10] Despres B. Lagrangian systems of conservation laws. Numerische Mathematik. 2001;89:99-134

[11] Despres B. Lois de Conservation Euleriennes et Lagrangiennes,

Mathematiques et Applications. Berlin Heidelberg: Springer-Verlag; 2009

[12] Godlevski E, Raviart PA. Numerical Approximation of Hyperbolic Systems

[13] Godlevski E, Raviart PA. Hyperbolic Systems of Conservation Laws. Paris,

[14] Guinot V. Godunov–Type Schemes: An Introduction for Engineers. 1st ed. Amsterdam, Netherlands: Elsevier; 2003

[15] Lax PD. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. In: Conf. Board. Math. Sci. Regional Conferences series in Applied Math., vol. 11. SIAM,

[16] LeFloch PG. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. ETH Lecture Notes Series, Birkhauser;

[17] LeFloch PG, Okutmustur B. Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms. Far East Journal of Mathematical Sciences. 2008;

of Conservation Laws, Applied Mathematical Sciences. Vol. 118. New

York: Springer-Verlag; 1996

France: Ellipse; 1991

Philadelphia; 1972

2002

31:49-83

Laws. Birkhauser; 2017

2000

[2] Barlow AJ, Roe PL. A cell centred Lagrangian Godunov scheme for shock hydrodynamics. Computers and Fluids.

[3] Carre G, Del Pino S, Despres B, Labourasse E. A cell-centered

[4] Cheng J, Shu CW, Zeng Q. A conservative Lagrangian scheme for solving compressible fluid flows with multiple internal energy equations. Communications in Computational Physics. 2012;12(5):1307-1328

[5] Clair G, Despres B, Labourasse E. A new method to introduce constraints in cell-centered Lagrangian schemes. Computer Methods in Applied Mechanics and Engineering. 2013;

[6] Claisse A, Despres B, Labourasse E, Ledoux F. A new exceptional points method with application to cellcentered Lagrangian schemes and

Computational Physics. 2012;231(11):

[7] Courant R, Friedrich KO. Supersonic Flow and Shock Waves. New York: Springer-Verlag, Interscience

mathematischen Wissenschaften. Vol.

curved meshes. Journal of

[8] Dafermos CM. Hyperbolic Conservation Laws in Continuum

Physics, Grundlehren der

Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. Journal of Computational Physics. 2009;228:

manifolds. Communications in Mathematical Sciences. 2008;6(4):

#### Author details

Baver Okutmuştur Middle East Technical University, Ankara, Turkey

\*Address all correspondence to: baver@metu.edu.tr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Scalar Conservation Laws DOI: http://dx.doi.org/10.5772/intechopen.83637

#### References

[1] Amorim P, LeFloch PG, Okutmustur B. Finite volume schemes on Lorentzian manifolds. Communications in Mathematical Sciences. 2008;6(4): 1059-1086

[2] Barlow AJ, Roe PL. A cell centred Lagrangian Godunov scheme for shock hydrodynamics. Computers and Fluids. 2011;46:133-136

[3] Carre G, Del Pino S, Despres B, Labourasse E. A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. Journal of Computational Physics. 2009;228: 5160-5183

[4] Cheng J, Shu CW, Zeng Q. A conservative Lagrangian scheme for solving compressible fluid flows with multiple internal energy equations. Communications in Computational Physics. 2012;12(5):1307-1328

[5] Clair G, Despres B, Labourasse E. A new method to introduce constraints in cell-centered Lagrangian schemes. Computer Methods in Applied Mechanics and Engineering. 2013; 261-262:56-65

[6] Claisse A, Despres B, Labourasse E, Ledoux F. A new exceptional points method with application to cellcentered Lagrangian schemes and curved meshes. Journal of Computational Physics. 2012;231(11): 4324-4354

[7] Courant R, Friedrich KO. Supersonic Flow and Shock Waves. New York: Springer-Verlag, Interscience Publishers; 1948

[8] Dafermos CM. Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften. Vol. 325. Berlin Heidelberg: Springer-Verlag; 2000

[9] Despres B. Numerical Methods for Eulerian and Lagrangian Conservation Laws. Birkhauser; 2017

[10] Despres B. Lagrangian systems of conservation laws. Numerische Mathematik. 2001;89:99-134

[11] Despres B. Lois de Conservation Euleriennes et Lagrangiennes, Mathematiques et Applications. Berlin Heidelberg: Springer-Verlag; 2009

[12] Godlevski E, Raviart PA. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences. Vol. 118. New York: Springer-Verlag; 1996

[13] Godlevski E, Raviart PA. Hyperbolic Systems of Conservation Laws. Paris, France: Ellipse; 1991

[14] Guinot V. Godunov–Type Schemes: An Introduction for Engineers. 1st ed. Amsterdam, Netherlands: Elsevier; 2003

[15] Lax PD. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. In: Conf. Board. Math. Sci. Regional Conferences series in Applied Math., vol. 11. SIAM, Philadelphia; 1972

[16] LeFloch PG. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. ETH Lecture Notes Series, Birkhauser; 2002

[17] LeFloch PG, Okutmustur B. Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms. Far East Journal of Mathematical Sciences. 2008; 31:49-83

Author details

Baver Okutmuştur

90

Middle East Technical University, Ankara, Turkey

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

\*Address all correspondence to: baver@metu.edu.tr

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Chapter 5

Abstract

1. Introduction

93

Unsteady CFD with Heat and Mass

Transfer Simulation of Solar

Wahiba Yaïci and Evgueniy Entchev

systems and for predicting their performance.

Adsorption Cooling System for

Optimal Design and Performance

The purpose of the work described here was to investigate the effects of design and operating parameters on the performance of an adsorption cooling system. An unsteady Computational Fluid Dynamics (CFD) coupled with heat a mass transfer model was created for predicting the flow behaviour, pressure, temperature, and water adsorption distributions. Silica gel and zeolite 13X were both considered as possible adsorbents, though the study included silica gel given the lower working temperature range required for operation, which makes it more appropriate for residential cooling applications powered by solar heat. Validation of the unsteady computation results with experimental data found in the literature has shown a good agreement. Different computation cases during the desorption process were simulated in a parametric study that considered adsorbent bed thickness (lbed), heat exchanger tube thickness (b), heat transfer fluid (HTF) velocity (v), and adsorbent particle diameter (dp), to systematically analyse the effects of key geometrical and operating parameters on the system performance. The CFD results revealed the importance of v, lbed and dp while b had relatively insignificant changes in the system performance. Moreover, the coupled CFD with heat and mass transfer model is suitable as a valuable tool for simulating and optimising adsorption cooling

Keywords: CFD analysis, adsorption cooling, solar thermally activated chiller, silica gel, zeolite, fluid flow, heat transfer, mass transfer, design, performance

Conventional vapour compression cooling systems are major consumers of electricity. In addition, these systems use non-natural refrigerants, which have high global warming as well as ozone layer depletion potentials and are responsible for the emission of CO2 and other greenhouse gases such as chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs). From this perspective, interest in adsorption systems powered by solar energy or waste heat has been increased as they do not use ozone-depleting substances as the working fluid nor do they need electricity or fossil fuels as driving sources. Furthermore, adsorption cooling

[18] LeVeque RJ. Finite Volume Methods for Hyperbolic Problems. 1st ed. Cambridge, England: Cambridge University Press; 2002

[19] Liu TP. Nonlinear stability of shock waves for viscous conservation laws. AMS Memoirs, 328, Providence; 1985

[20] Munz D. On Godunov-type schemes for Lagrangian gas dynamics. SIAM Journal on Numerical Analysis. 1994;31(1):17-42

[21] Oyar N. Inviscid Burgers equation and its numerical solutions [Master thesis]. Ankara, Turkey: METU; 2017

[22] Serre D. Systems of Conservation Laws 1–2. Cambridge, England: Cambridge University Press; 1999

[23] Smoller J. Shock Waves and Reaction–Diffusion Equations. New York: Springer-Verlag; 1967

[24] Toro EF. Riemann Solvers and Numerical Methods in Fluid Dynamics. A Practical Introduction. 3rd ed. Berlin Heidelberg: Springer-Verlag; 2009

[25] Wagner DH. Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. The Journal of Differential Equations. 1987; 68:118-136

[26] Whitham G. Linear and Nonlinear Waves. New York: Wiley-Interscience; 1974

[27] Van Leer B. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. SIAM Journal on Scientific and Statistical Computing. 2012;5:1-20

#### Chapter 5

[18] LeVeque RJ. Finite Volume Methods

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

[19] Liu TP. Nonlinear stability of shock waves for viscous conservation laws. AMS Memoirs, 328, Providence; 1985

[21] Oyar N. Inviscid Burgers equation and its numerical solutions [Master thesis]. Ankara, Turkey: METU; 2017

[22] Serre D. Systems of Conservation Laws 1–2. Cambridge, England: Cambridge University Press; 1999

[23] Smoller J. Shock Waves and Reaction–Diffusion Equations. New York: Springer-Verlag; 1967

[24] Toro EF. Riemann Solvers and Numerical Methods in Fluid Dynamics. A Practical Introduction. 3rd ed. Berlin Heidelberg: Springer-Verlag; 2009

[25] Wagner DH. Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. The Journal of Differential Equations. 1987;

[26] Whitham G. Linear and Nonlinear Waves. New York: Wiley-Interscience;

[27] Van Leer B. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. SIAM Journal on Scientific and Statistical Computing. 2012;5:1-20

68:118-136

1974

92

for Hyperbolic Problems. 1st ed. Cambridge, England: Cambridge

[20] Munz D. On Godunov-type schemes for Lagrangian gas dynamics. SIAM Journal on Numerical Analysis.

University Press; 2002

1994;31(1):17-42

## Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System for Optimal Design and Performance

Wahiba Yaïci and Evgueniy Entchev

### Abstract

The purpose of the work described here was to investigate the effects of design and operating parameters on the performance of an adsorption cooling system. An unsteady Computational Fluid Dynamics (CFD) coupled with heat a mass transfer model was created for predicting the flow behaviour, pressure, temperature, and water adsorption distributions. Silica gel and zeolite 13X were both considered as possible adsorbents, though the study included silica gel given the lower working temperature range required for operation, which makes it more appropriate for residential cooling applications powered by solar heat. Validation of the unsteady computation results with experimental data found in the literature has shown a good agreement. Different computation cases during the desorption process were simulated in a parametric study that considered adsorbent bed thickness (lbed), heat exchanger tube thickness (b), heat transfer fluid (HTF) velocity (v), and adsorbent particle diameter (dp), to systematically analyse the effects of key geometrical and operating parameters on the system performance. The CFD results revealed the importance of v, lbed and dp while b had relatively insignificant changes in the system performance. Moreover, the coupled CFD with heat and mass transfer model is suitable as a valuable tool for simulating and optimising adsorption cooling systems and for predicting their performance.

Keywords: CFD analysis, adsorption cooling, solar thermally activated chiller, silica gel, zeolite, fluid flow, heat transfer, mass transfer, design, performance

#### 1. Introduction

Conventional vapour compression cooling systems are major consumers of electricity. In addition, these systems use non-natural refrigerants, which have high global warming as well as ozone layer depletion potentials and are responsible for the emission of CO2 and other greenhouse gases such as chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs). From this perspective, interest in adsorption systems powered by solar energy or waste heat has been increased as they do not use ozone-depleting substances as the working fluid nor do they need electricity or fossil fuels as driving sources. Furthermore, adsorption cooling

systems have many other advantages, such as simple construction, no solution pumps, powered directly by solar energy or waste heat and no need for electricity [1–3].

cycles. The models were sorted into three main groups: lumped parameter, thermodynamics, and heat and mass transfer models. Among the various models existing in the literature, Computational Fluid Dynamics (CFD) models based on heat and mass transfer are especially important, as they provide understanding into the operation dynamics of the adsorber in the related cooling system. A heat and mass transfer model solving the problem in the form of partial differential equations is featured, with the temperature or the mass content of adsorbate varies with space and time. Based on the geometry of the adsorption cooling system, the models can be clustered under one-dimensional (1D), two-dimensional (2D) and threedimensional (3D) models. In general, heat and mass transfer processes are not taken into account in thermodynamic models. While heat transfer is reflected in the lumped parameters model, the temperature variation with space is not considered in it (zero-dimensional (0D) model), which is considered in the heat and mass transfer model. The distinction between the diverse models applied to simulate adsorptive cooling systems, usually relates with the variations in the simplifying assumptions, numerical solution methods, design and utilisation of the modelled

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

DOI: http://dx.doi.org/10.5772/intechopen.81144

Some of the previous models, which may be categorised either as uniformtemperature models [16–18] or as uniform-pressure models [19–21], considered only heat transfer while neglecting mass transfer in the adsorbent. Hajji and Worek [22] suggested a model allowing for only convection term for heat flow to assess the effect of design and operating parameters on the system performance of a zeolite heat pump system. Alam et al. [23] investigated the design parameters of the fluid side on the system performance by developing a two-dimensional heat equation for both the fluid and adsorbent sides. They used the model to explore the effect of heat exchanger design parameters on the system performance of a two-bed silica gel/water adsorption cooling unit, as well as the effect of switching speed on the

It is only lately that a number of numerical studies with consideration of using CFD and coupled heat and mass transfer have been presented. Using Darcy's law to account for the mass transfer resistance among adsorbent pellets, Sun et al. [24] studied the momentum and heat transfer in an adsorption cooling system with two adsorbent-adsorbate pairs. Their findings suggested that for low-density adsorbates, like water or methanol, the operation dynamics of the adsorber may possibly be critically controlled by mass transfer resistance within the adsorption unit if its size is bulky. Their model is 1D. Amar et al. [25] analysed a 2D model, which similarly took into account the combined heat and mass transfers in the adsorber to investigate the impacts of different functional parameters on the performance of a thermal wave regenerative heat pump. A three-dimensional model was investigated by Zhang and Wang [11] and Zhang [12] to study the effect of coupled heat and mass transfer in adsorbent beds on the performance of a waste heat adsorption cooling unit. They also studied the effect of reactor configuration on the performance. Solmus et al. [26, 27] presented numerical studies of heat and mass transfer within the adsorbent bed of a silica gel/water adsorptive cooling system by means of the local volume averaging method. They utilised a transient 1D local thermal nonequilibrium model, which taken into account both internal and external mass transfer resistances. Their results showed the significance of spatial temperature and pressure gradients clearly indicated that external mass transfer resistance and heat transfer were important. Caglar et al. [28] developed a 2D mathematical model of the heat and mass transfer inside a cylindrical adsorption bed for a thermal wave adsorption cycle, with a heat transfer fluid flowing through an inner tube and the adsorbent in the annulus. He investigated the effect of determining factors that enhance the heat and mass transfer inside the adsorbent bed. Çağlar [29] used a 2D

system [14, 15].

system performance.

95

Adsorption is the adhesion of atoms, ions, molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. The desorption is the reverse of adsorption. It is a surface phenomenon. The adsorption process is usually considered as physisorption, specific of weak van der Waals forces or chemisorption, specific of covalent bonding. Adsorption is generally stated by way of isotherms, which represent the quantity of adsorbate (vapour or liquid phase) on the adsorbent (solid phase) as a function of its concentration (if liquid) or pressure (if gas) at constant temperature. For convenience, the adsorbed phase is normalised by the amount of the adsorbent in order to facilitate comparison of various adsorbent-adsorbate pairs [4–6].

An adsorption cooling system consists of adsorbing material (adsorbent) packed in a vessel (adsorber) and an evacuated vessel (the evaporator). The working fluid, generally water, is the adsorbate. The working principle of the system consists in adsorbing the vapour produced in the evaporator, generating a cooling effect. Water constantly evaporates at low pressure, cooling the process air while the heat produced simultaneously in the adsorption process is removed by the cooling water, from the adsorber. At the end of the adsorption process, the desorption stage begins by heating the adsorber, using hot water, synthetic oil or any appropriate means. The extracted vapour is directed to the condenser and eventually returns as liquid to the evaporator. The thermodynamic cycle of the complete process is therefore ended. As an energy cost-effective solution, hot water/oil can be heated by free solar energy or waste heat. The two adsorption/desorption chambers of the adsorption cooling systems operate alternatively so that to generate continuous cooling power [7].

Numerous heat-pumping, refrigeration and desalination applications have been studied using various adsorbent and adsorbate pairs. Most of the cycles need medium and/or high temperature heat sources to work as the powering sources. But adsorption cycles using the silica gel/water and zeolite/water (adsorbent-adsorbate) pairs, exhibit a distinctive benefit above other systems in their capability to be driven by heat of quite low, near-ambient temperatures, so that heat from solar panels or waste heat below 100°C can be recovered, which is highly desirable, especially if flat plat collectors are used [8–11].

Nevertheless, conventional vapour compression systems still dominate in practically all applications, because adsorption cooling has some drawbacks, which require to be improved. The recognised limitation of adsorption cycles is that the heat and mass transfer coefficients of the bed are relatively small due to low conductivity of adsorbent pellets/particles and high contact resistance between particles and metal tubes/fins. The performance of an adsorbent bed is affected adversely by the heat and mass transfer limitations inside the bed, such as reduced thermal conductivity of the solid adsorbent, and internal (intra-particle) and external (inter-particle) mass transfer resistances. The other adverse consequences on the performance are: (a) lengthy adsorption/desorption cycle time; (b) low coefficient of performance (COP), resulting to enlarged energy consumption and expenditure; and (c) small specific cooling power (SCP), resulting to a bulky and outsized system [11–13].

To simulate and optimise the performance of adsorptive heat pump/cooling systems, various numerical models and several approaches have been proposed and reported in the literature.

For example, Yong and Sumathy [14] reviewed various categories of mathematical models, used to predict the functioning and effectiveness of adsorption cooling

#### Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

cycles. The models were sorted into three main groups: lumped parameter, thermodynamics, and heat and mass transfer models. Among the various models existing in the literature, Computational Fluid Dynamics (CFD) models based on heat and mass transfer are especially important, as they provide understanding into the operation dynamics of the adsorber in the related cooling system. A heat and mass transfer model solving the problem in the form of partial differential equations is featured, with the temperature or the mass content of adsorbate varies with space and time. Based on the geometry of the adsorption cooling system, the models can be clustered under one-dimensional (1D), two-dimensional (2D) and threedimensional (3D) models. In general, heat and mass transfer processes are not taken into account in thermodynamic models. While heat transfer is reflected in the lumped parameters model, the temperature variation with space is not considered in it (zero-dimensional (0D) model), which is considered in the heat and mass transfer model. The distinction between the diverse models applied to simulate adsorptive cooling systems, usually relates with the variations in the simplifying assumptions, numerical solution methods, design and utilisation of the modelled system [14, 15].

Some of the previous models, which may be categorised either as uniformtemperature models [16–18] or as uniform-pressure models [19–21], considered only heat transfer while neglecting mass transfer in the adsorbent. Hajji and Worek [22] suggested a model allowing for only convection term for heat flow to assess the effect of design and operating parameters on the system performance of a zeolite heat pump system. Alam et al. [23] investigated the design parameters of the fluid side on the system performance by developing a two-dimensional heat equation for both the fluid and adsorbent sides. They used the model to explore the effect of heat exchanger design parameters on the system performance of a two-bed silica gel/water adsorption cooling unit, as well as the effect of switching speed on the system performance.

It is only lately that a number of numerical studies with consideration of using CFD and coupled heat and mass transfer have been presented. Using Darcy's law to account for the mass transfer resistance among adsorbent pellets, Sun et al. [24] studied the momentum and heat transfer in an adsorption cooling system with two adsorbent-adsorbate pairs. Their findings suggested that for low-density adsorbates, like water or methanol, the operation dynamics of the adsorber may possibly be critically controlled by mass transfer resistance within the adsorption unit if its size is bulky. Their model is 1D. Amar et al. [25] analysed a 2D model, which similarly took into account the combined heat and mass transfers in the adsorber to investigate the impacts of different functional parameters on the performance of a thermal wave regenerative heat pump. A three-dimensional model was investigated by Zhang and Wang [11] and Zhang [12] to study the effect of coupled heat and mass transfer in adsorbent beds on the performance of a waste heat adsorption cooling unit. They also studied the effect of reactor configuration on the performance. Solmus et al. [26, 27] presented numerical studies of heat and mass transfer within the adsorbent bed of a silica gel/water adsorptive cooling system by means of the local volume averaging method. They utilised a transient 1D local thermal nonequilibrium model, which taken into account both internal and external mass transfer resistances. Their results showed the significance of spatial temperature and pressure gradients clearly indicated that external mass transfer resistance and heat transfer were important. Caglar et al. [28] developed a 2D mathematical model of the heat and mass transfer inside a cylindrical adsorption bed for a thermal wave adsorption cycle, with a heat transfer fluid flowing through an inner tube and the adsorbent in the annulus. He investigated the effect of determining factors that enhance the heat and mass transfer inside the adsorbent bed. Çağlar [29] used a 2D

systems have many other advantages, such as simple construction, no solution pumps, powered directly by solar energy or waste heat and no need for

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

comparison of various adsorbent-adsorbate pairs [4–6].

especially if flat plat collectors are used [8–11].

Adsorption is the adhesion of atoms, ions, molecules of gas, liquid, or dissolved solids to a surface. This process creates a film of the adsorbate on the surface of the adsorbent. The desorption is the reverse of adsorption. It is a surface phenomenon. The adsorption process is usually considered as physisorption, specific of weak van der Waals forces or chemisorption, specific of covalent bonding. Adsorption is generally stated by way of isotherms, which represent the quantity of adsorbate (vapour or liquid phase) on the adsorbent (solid phase) as a function of its concentration (if liquid) or pressure (if gas) at constant temperature. For convenience, the adsorbed phase is normalised by the amount of the adsorbent in order to facilitate

An adsorption cooling system consists of adsorbing material (adsorbent) packed in a vessel (adsorber) and an evacuated vessel (the evaporator). The working fluid, generally water, is the adsorbate. The working principle of the system consists in adsorbing the vapour produced in the evaporator, generating a cooling effect. Water constantly evaporates at low pressure, cooling the process air while the heat produced simultaneously in the adsorption process is removed by the cooling water, from the adsorber. At the end of the adsorption process, the desorption stage begins by heating the adsorber, using hot water, synthetic oil or any appropriate means. The extracted vapour is directed to the condenser and eventually returns as liquid to the evaporator. The thermodynamic cycle of the complete process is therefore ended. As an energy cost-effective solution, hot water/oil can be heated by free solar energy or waste heat. The two adsorption/desorption chambers of the adsorption cooling systems operate alternatively so that to generate continuous cooling

Numerous heat-pumping, refrigeration and desalination applications have been

Nevertheless, conventional vapour compression systems still dominate in practically all applications, because adsorption cooling has some drawbacks, which require to be improved. The recognised limitation of adsorption cycles is that the heat and mass transfer coefficients of the bed are relatively small due to low conductivity of adsorbent pellets/particles and high contact resistance between particles and metal tubes/fins. The performance of an adsorbent bed is affected adversely by the heat and mass transfer limitations inside the bed, such as reduced thermal conductivity of the solid adsorbent, and internal (intra-particle) and external (inter-particle) mass transfer resistances. The other adverse consequences on the performance are: (a) lengthy adsorption/desorption cycle time; (b) low coefficient of performance (COP), resulting to enlarged energy consumption and expenditure; and (c) small specific cooling power (SCP), resulting to a bulky and outsized

To simulate and optimise the performance of adsorptive heat pump/cooling systems, various numerical models and several approaches have been proposed and

For example, Yong and Sumathy [14] reviewed various categories of mathematical models, used to predict the functioning and effectiveness of adsorption cooling

studied using various adsorbent and adsorbate pairs. Most of the cycles need medium and/or high temperature heat sources to work as the powering sources. But adsorption cycles using the silica gel/water and zeolite/water (adsorbent-adsorbate) pairs, exhibit a distinctive benefit above other systems in their capability to be driven by heat of quite low, near-ambient temperatures, so that heat from solar panels or waste heat below 100°C can be recovered, which is highly desirable,

electricity [1–3].

power [7].

system [11–13].

94

reported in the literature.

coupled heat and mass transfer model to analyse both finless and finned tube-type adsorbent bed for a thermal wave adsorption cooling cycle with silica gel/water as the working pair. He showed that a significant enhancement in the heat transfer is obtained using a finned tube such that the temperature of the adsorbent in the finned tube adsorbent bed.

Despite the continuous research effort about the dynamic behaviour of heat and mass transfers inside the adsorbent bed and attempts to enhance the overall system performance, there remains comprehensive research effort to be made for the accurate design and performance prediction of adsorption cooling systems. Although the limited number of studies have dealt recently with multi-dimensional effects, most of the modelling efforts have focused on a one-dimensional description of the adsorption process for its simplicity, either on adsorption or desorption processes. However, 0D or 1D model cannot describe the flow structure, the dynamic behaviour and interactions of heat and mass transfers inside the adsorbent bed and the heat exchanger. In addition, in the analyses, by assuming an equilibrium adsorption state, the internal mass transfer resistance is not taken into account, which can be very limiting for performance enhancement when the cycle time is small.

In the present work, the configuration and operating conditions influence on the performance of a solar heat driven adsorption cooling system operating in desorption mode is simulated. A 3D/2D unsteady CFD coupled with heat and mass transfer model using silica gel/water or zeolite/water pairs is created, and validated from literature data. Effects of the adsorption bed with a finned tube heat exchanger geometry as well as the operating conditions on the system performance are then fully investigated in detail. Distinct characteristics of significance to optimum design and operation that have effects on the adsorption cooling system are demonstrated and analysed. The CFD model developed in this study may be useful to design and optimise a new and more efficient adsorption cooling bed. It also provides a tool for optimisation of adsorption cooling systems driven by solar heat or low-grade/waste heat.

For this purpose, the rest of the paper is organised as follows. In Section 2, in depth steps taken for developing the CFD coupled with heat and mass transfer model of a solar adsorption cooling system, including the physical model, the governing equations, boundary conditions, and the numerical procedure are thoroughly described. The validation of the model is provided in Section 3. Section 4 discusses in detail the base case simulation and the parametric study results of the solar-driven adsorption cooling system. Finally, Section 5 summarises the main conclusions of the work.

> with fins extending off of the tube. The HTF flows in though the bottom of the tube and exits through the top to control the temperature of the unit. Around the tube is the adsorbent material that is secured between the copper fins. There are a total of 38 fins in the full geometry. The system is symmetrical, thus the computational domain is modelled with symmetric boundary conditions; the geometry was reduced a 2D axisymmetric face of the 3D geometry, as visualised in Figure 2a. Figure 2b depicts the schematic of the model with fully labelled geometric lengths,

Simulated finned tube adsorber: (a) geometric selection and domains and (b) schematic of the geometry and

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

Unsteady flow models in desorption mode were built in the commercial software Multiphysics COMSOL [29]. It was used as the grid generator and as the CFD solver. In order to better predict the different field characteristics, the optimised solution-adaptive mesh refinement is used. More cells were added at locations where significant phenomena changes are expected, for example near the adsorber/heat exchanger walls and inlet/outlet ports. The resulting mesh thus enabled the features of the different fields to be better resolved. The symmetric

which will be investigated in the parametric study.

A simple annular-finned tube used in adsorption cooling system [15].

DOI: http://dx.doi.org/10.5772/intechopen.81144

Figure 1.

Figure 2.

97

nomenclature.

#### 2. CFD model details

#### 2.1 Physical model

A heat exchanger in adsorption cooling system is a device that is in thermal contact with the adsorbent (solid phase) and helps to heat and cool the adsorbent throughout the desorption and adsorption periods, respectively. The plate-finned bed can take various configurations, such as the finned tube type, the fin plate type, and the flat-pipe type [15, 30]. A basic design of a finned tube heat exchanger is displayed in Figure 1.

From the possible adsorber bed geometries, the finned tube was selected to study. Figure 2a and b present the geometric model of the finned tube adsorption cooling unit. Figure 2a shows the full geometry consisting of a single copper tube

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

Figure 1. A simple annular-finned tube used in adsorption cooling system [15].

Figure 2.

coupled heat and mass transfer model to analyse both finless and finned tube-type adsorbent bed for a thermal wave adsorption cooling cycle with silica gel/water as the working pair. He showed that a significant enhancement in the heat transfer is obtained using a finned tube such that the temperature of the adsorbent in the

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

Despite the continuous research effort about the dynamic behaviour of heat and mass transfers inside the adsorbent bed and attempts to enhance the overall system performance, there remains comprehensive research effort to be made for the accurate design and performance prediction of adsorption cooling systems.

Although the limited number of studies have dealt recently with multi-dimensional effects, most of the modelling efforts have focused on a one-dimensional description of the adsorption process for its simplicity, either on adsorption or desorption processes. However, 0D or 1D model cannot describe the flow structure, the dynamic behaviour and interactions of heat and mass transfers inside the adsorbent bed and the heat exchanger. In addition, in the analyses, by assuming an equilibrium adsorption state, the internal mass transfer resistance is not taken into account, which can be very limiting for performance enhancement when the cycle

In the present work, the configuration and operating conditions influence on the performance of a solar heat driven adsorption cooling system operating in desorption mode is simulated. A 3D/2D unsteady CFD coupled with heat and mass transfer model using silica gel/water or zeolite/water pairs is created, and validated from literature data. Effects of the adsorption bed with a finned tube heat exchanger geometry as well as the operating conditions on the system performance are then fully investigated in detail. Distinct characteristics of significance to optimum design and operation that have effects on the adsorption cooling system are demonstrated and analysed. The CFD model developed in this study may be useful to design and optimise a new and more efficient adsorption cooling bed. It also provides a tool for optimisation of adsorption cooling systems driven by solar heat or

For this purpose, the rest of the paper is organised as follows. In Section 2, in depth steps taken for developing the CFD coupled with heat and mass transfer model of a solar adsorption cooling system, including the physical model, the governing equations, boundary conditions, and the numerical procedure are thoroughly described. The validation of the model is provided in Section 3. Section 4 discusses in detail the base case simulation and the parametric study results of the solar-driven adsorption cooling system. Finally, Section 5 summarises the main

A heat exchanger in adsorption cooling system is a device that is in thermal contact with the adsorbent (solid phase) and helps to heat and cool the adsorbent throughout the desorption and adsorption periods, respectively. The plate-finned bed can take various configurations, such as the finned tube type, the fin plate type, and the flat-pipe type [15, 30]. A basic design of a finned tube heat exchanger is

From the possible adsorber bed geometries, the finned tube was selected to study. Figure 2a and b present the geometric model of the finned tube adsorption cooling unit. Figure 2a shows the full geometry consisting of a single copper tube

finned tube adsorbent bed.

time is small.

low-grade/waste heat.

conclusions of the work.

2. CFD model details

2.1 Physical model

displayed in Figure 1.

96

Simulated finned tube adsorber: (a) geometric selection and domains and (b) schematic of the geometry and nomenclature.

with fins extending off of the tube. The HTF flows in though the bottom of the tube and exits through the top to control the temperature of the unit. Around the tube is the adsorbent material that is secured between the copper fins. There are a total of 38 fins in the full geometry. The system is symmetrical, thus the computational domain is modelled with symmetric boundary conditions; the geometry was reduced a 2D axisymmetric face of the 3D geometry, as visualised in Figure 2a. Figure 2b depicts the schematic of the model with fully labelled geometric lengths, which will be investigated in the parametric study.

Unsteady flow models in desorption mode were built in the commercial software Multiphysics COMSOL [29]. It was used as the grid generator and as the CFD solver. In order to better predict the different field characteristics, the optimised solution-adaptive mesh refinement is used. More cells were added at locations where significant phenomena changes are expected, for example near the adsorber/heat exchanger walls and inlet/outlet ports. The resulting mesh thus enabled the features of the different fields to be better resolved. The symmetric

solver selected here accounts for the three-dimensional effects. Mixed topology of unstructured grids was utilised, and the final mesh was composed of about 35,305–85,000 elements depending on the volume of the finned tube adsorption bed considered. Details on the grid system and selected mesh elements can be found in Section 2.3.

Conservation of momentum:

Conservation of energy:

Heat transfer in the HTF:

Heat transfer in the metal:

Heat transfer in the adsorbent:

adsorbed to the adsorbent [13]:

represented:

99

ð Þ 1 � ε ρscs

velocity and the pressure are relatively low.

ρfcf ∂Tf

<sup>∂</sup><sup>t</sup> � kfΔ<sup>2</sup>

ρtct ∂Tt <sup>∂</sup><sup>t</sup> � ktΔ<sup>2</sup>

<sup>∂</sup><sup>t</sup> � <sup>ε</sup>ρvcp, <sup>v</sup>

<sup>∂</sup><sup>t</sup> � ksΔ<sup>2</sup>

The heat of adsorption (kJ/kg) changes linearly with the amount of water

The velocity of the water vapour is calculated using Darcy's equation, which defines the external mass transfer resistance in the water adsorption process. Darcy's equation is valid in this condition since the adsorbent is porous and both the

> , v ¼ � <sup>κ</sup> μ ∂P ∂z

The internal mass transfer resistance for the water adsorption process is defined

, w ¼ � <sup>κ</sup> μ ∂P ∂z

<sup>∂</sup><sup>t</sup> <sup>¼</sup> kmð Þ Xe � <sup>X</sup> (10)

The 2D cylindrical and 3D Cartesian coordinate expressions are both

, v ¼ � <sup>κ</sup> μ ∂P ∂y

where X represents the amount adsorbed and Xe is the equilibrium adsorption capacity of the adsorbent-adsorbate pair under study, provided in Eqs. (13)–(18).

<sup>u</sup> ¼ � <sup>κ</sup> μ ∂P ∂r

∂X

<sup>u</sup> ¼ � <sup>κ</sup> μ ∂P ∂x

by means of a linear driving force expression as follows:

∂Ts ∂t

þ ð Þ 1 � ε ρsXcp,l

Ts þ ρvcp, <sup>v</sup>uΔTs ¼ 0

ΔH ¼ 2950 � 1400X (7)

∂Ts

�ð Þ <sup>1</sup> � <sup>ε</sup> <sup>ρ</sup><sup>s</sup> j j <sup>Δ</sup><sup>H</sup> <sup>∂</sup><sup>X</sup>

process.

ρ ∂u ∂t

DOI: http://dx.doi.org/10.5772/intechopen.81144

The momentum balance expresses the motion of the HTF within the tube.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

þ ρð Þ u � ∇ u ¼ ∇ � �pI þ μ ∇u þ ð Þ ∇u

The energy transfer within the system can be described with respect to the three discrete domains involved: the HTF, the metal tube and fins, and the adsorbent. The adsorbent domain is assumed to be under local thermal equilibrium between the adsorbed and vapour phase. Therefore, the equation has three additional terms to take into account for the change of temperature due to the water vapour, water adsorbed and the heat of adsorption associated with the adsorption

<sup>2</sup> h i � � (2)

Tf þ ρfcf vf ΔTf ¼ 0 (4)

Tt ¼ 0 (5)

∂Ts ∂t

(6)

(8)

(9)

ρ∇ � ð Þ¼ u 0 (3)

#### 2.2 Governing equations and boundary conditions

The Navier-Stokes and the mass and energy equations in three-dimensional form were used to solve for the transient fluid dynamics with the coupled heat and mass transfer fields inside the finned tube adsorption bed system with silica gel/water or zeolite/water as the adsorbent/adsorbate working pairs. The mass balance describes the rate of adsorption or desorption within the adsorbent bed. The models were created in the COMSOL Multiphysics software package including the CFD, Heat Transfer and Chemical Reaction Engineering modules [31].

The governing equations of mass, momentum and energy conservation were solved by using the finite element method, based on the following assumptions:


Therefore, the resulting governing equations can be stated as follows. For conciseness, the basic variables defined in the following equations can be found in [12–14, 24, 26, 28].

Conservation of mass:

$$
\varepsilon \frac{\partial \rho\_v}{\partial t} + (\mathbf{1} - \varepsilon)\rho\_s \frac{\partial \mathbf{X}}{\partial t} - D\_m \Delta^2 \rho\_v + \Delta(\mathbf{u}\rho\_v) = \mathbf{0} \tag{1}
$$

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

Conservation of momentum:

solver selected here accounts for the three-dimensional effects. Mixed topology of unstructured grids was utilised, and the final mesh was composed of about 35,305–85,000 elements depending on the volume of the finned tube adsorption bed considered. Details on the grid system and selected mesh elements can be

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

The Navier-Stokes and the mass and energy equations in three-dimensional form were used to solve for the transient fluid dynamics with the coupled heat and mass transfer fields inside the finned tube adsorption bed system with silica gel/water or zeolite/water as the adsorbent/adsorbate working pairs. The mass balance describes the rate of adsorption or desorption within the adsorbent bed. The models were created in the COMSOL Multiphysics software package including the CFD, Heat Transfer and Chemical Reaction Engineering modules [31]. The governing equations of mass, momentum and energy conservation were solved by using the finite element method, based on the following assumptions:

• the finned tube adsorption bed operates under unsteady-state, non-

• the surface porosity is considered to be equal to the total porosity;

• the adsorbate's water vapour phase is assumed to be an ideal gas;

• the adsorbate's water phase is assumed to be a liquid;

• the volume fraction of the gas phase is assumed to be equal to the total

• the absorbent particles are assumed to have uniform size, shape and porosity;

• the adsorbed phase is considered to be a liquid, while the gas phase is assumed

• the working fluid is water at high velocity is assumed to be Newtonian, incompressible flow inside the tubes, resulting in a constant tube-surface temperature; the viscous dissipation and viscous work are neglected; there are

• the work done by pressure changes, radiative heat transfer and viscous

Therefore, the resulting governing equations can be stated as follows. For conciseness, the basic variables defined in the following equations can be found in

<sup>∂</sup><sup>t</sup> � DmΔ<sup>2</sup>

ρ<sup>v</sup> þ Δ uρ<sup>v</sup> ð Þ¼ 0 (1)

∂X

• the wall thickness of the vacuum tube is assumed to be very thin and hence, its

found in Section 2.3.

equilibrium conditions;

porosity;

to be an ideal gas;

no body forces;

[12–14, 24, 26, 28].

98

Conservation of mass:

dissipation is neglected; and

thermal resistance was neglected.

ε ∂ρv ∂t

þ ð Þ 1 � ε ρ<sup>s</sup>

2.2 Governing equations and boundary conditions

The momentum balance expresses the motion of the HTF within the tube.

$$
\rho \frac{\partial \mathfrak{u}}{\partial t} + \rho (\mathfrak{u} \cdot \nabla) \mathfrak{u} = \nabla \cdot \left[ -pI + \mu \left( \nabla \mathfrak{u} + (\nabla \mathfrak{u})^2 \right) \right] \tag{2}
$$

$$
\rho \nabla \cdot (\mathfrak{u}) = \mathbf{0} \tag{3}
$$

Conservation of energy:

The energy transfer within the system can be described with respect to the three discrete domains involved: the HTF, the metal tube and fins, and the adsorbent. The adsorbent domain is assumed to be under local thermal equilibrium between the adsorbed and vapour phase. Therefore, the equation has three additional terms to take into account for the change of temperature due to the water vapour, water adsorbed and the heat of adsorption associated with the adsorption process.

Heat transfer in the HTF:

$$
\rho\_f c\_f \frac{\partial T\_f}{\partial t} - k\_f \Delta^2 T\_f + \rho\_f c\_f v\_f \Delta T\_f = 0 \tag{4}
$$

Heat transfer in the metal:

$$
\rho\_t c\_t \frac{\partial T\_t}{\partial t} - k\_t \Delta^2 T\_t = \mathbf{0} \tag{5}
$$

Heat transfer in the adsorbent:

$$\begin{aligned} \rho(\mathbf{1} - \boldsymbol{\varepsilon})\rho\_s c\_s \, \frac{\partial T\_s}{\partial t} - \boldsymbol{\varepsilon}\rho\_v c\_{p,v} \frac{\partial T\_s}{\partial t} + (\mathbf{1} - \boldsymbol{\varepsilon})\rho\_s \mathbf{X} c\_{p,l} \frac{\partial T\_s}{\partial t} \\ - (\mathbf{1} - \boldsymbol{\varepsilon})\rho\_s \, |\Delta H| \frac{\partial \mathbf{X}}{\partial t} - k\_s \Delta^2 T\_s + \rho\_v c\_{p,v} \mu \Delta T\_s = \mathbf{0} \end{aligned} \tag{6}$$

The heat of adsorption (kJ/kg) changes linearly with the amount of water adsorbed to the adsorbent [13]:

$$
\Delta H = 2950 - 1400X \tag{7}
$$

The velocity of the water vapour is calculated using Darcy's equation, which defines the external mass transfer resistance in the water adsorption process. Darcy's equation is valid in this condition since the adsorbent is porous and both the velocity and the pressure are relatively low.

The 2D cylindrical and 3D Cartesian coordinate expressions are both represented:

$$u = -\frac{\kappa}{\mu} \frac{\partial P}{\partial r}, v = -\frac{\kappa}{\mu} \frac{\partial P}{\partial \mathbf{z}}\tag{8}$$

$$u = -\frac{\kappa}{\mu} \frac{\partial P}{\partial \mathbf{x}}, v = -\frac{\kappa}{\mu} \frac{\partial P}{\partial \mathbf{y}}, w = -\frac{\kappa}{\mu} \frac{\partial P}{\partial \mathbf{z}} \tag{9}$$

The internal mass transfer resistance for the water adsorption process is defined by means of a linear driving force expression as follows:

$$\frac{\partial X}{\partial t} = k\_m (X\_\epsilon - X) \tag{10}$$

where X represents the amount adsorbed and Xe is the equilibrium adsorption capacity of the adsorbent-adsorbate pair under study, provided in Eqs. (13)–(18).

The temperature dependent mass transfer coefficient and diffusion coefficient are calculated as follows [25]:

$$k\_m = \frac{15}{r\_p^2} D\_m \tag{11}$$

Category Description Variable Unit Validation

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

Geometry Bed thickness lbed mm 13 12

Adsorption initial concentration

DOI: http://dx.doi.org/10.5772/intechopen.81144

Desorption initial concentration

Mass Reference diffusivity Do m<sup>2</sup>

Activation energy of surface diffusion

Convective heat transfer coefficient: tube and adsorbent

Convective heat transfer coefficient: tube and fluid

Temperature pressure

Thermal properties

101

zeolite 13X/water

Xa<sup>0</sup> kg/kg — 0.024

Xd<sup>0</sup> kg/kg — 0.72

/s 5.8e-9 2.54e4

Ea J/mole 1.0e5 3.36e4

ho W/m<sup>2</sup> K 100 100

hi W/m<sup>2</sup> K 100,000 100,000

Hot temperature Thot K 473 338 Cold temperature Tcold K 313 298 Condenser pressure Pc kPa 4.247 4.246 Evaporator pressure Pe kPa n/a 1.228

Fin distance d mm 10 15 Tube thickness (copper) b mm 1 1 Adsorbent particle diameter dp mm 1.25 3 Absorber length L mm 500 600 Tube radius rt mm 13 13 Tube thickness bt mm 1 1 Void fraction ε — 0.635 0.635 Permeability <sup>κ</sup> <sup>m</sup><sup>2</sup> 3.04e-7 3.4e<sup>9</sup>

Fluid density ρ<sup>f</sup> kg/m<sup>3</sup> 914 f(T) Absorbent density ρ<sup>s</sup> kg/m<sup>3</sup> 1000 670 Tube density ρ<sup>t</sup> kg/m<sup>3</sup> 8700 8700 Dynamic viscosity μ kg/m s 1.0e-5 1.5e5

Fluid velocity v m/s 0.001 0.001

Thermal conductivity of bed ks W/m K 1 0.3 Thermal conductivity of fluid kf W/m K 0.155 f(T) Thermal conductivity of tube kt W/m K 400 400

Specific heat of fluid cf J/kg K 1930 f(T) Specific heat of particle liquid cpl J/kg K 4180 4180 Specific heat of particle vapour cpv J/kg K 1880 1880 Specific heat of adsorbent solid cs J/kg K 837 f(T) Specific heat of tube ct J/kg K 385 f(T)

> D-A constant B — 5.36 — D-A constant n — 1.73 —

Kinetics Water adsorbed reference Xo kgw/kgs 0.261 —

Base case silica gel/water

$$D\_m = D\_o \exp\left(-\frac{E\_a}{RT\_S}\right) \tag{12}$$

where Do is reference diffusivity and rp is the particle radius.

2.2.1 Kinetic expression for regular density (RD) silica gel

Regular density (RD) silica gel is the most widespread type of silica gel and is obtainable from any supplier that retails silica gel. Because of its capacity for high moisture uptake in the low RH (relative humidity) range, it is a very effective desiccant.

The equilibrium adsorption capacity characterises the theoretical maximum capacity that the adsorbent bed can adsorb for a given pressure and temperature [32].

For RD silica gel, the following pressure dependent isotherm is used as follows:

$$X\_{\epsilon} = \mathbf{a} \left(\frac{\mathbf{P}}{\mathbf{P}\_{\text{sat}}}\right)^{\mathsf{b}} \tag{13}$$

The values of a and b are temperatures dependent are provided in the next section, Table 1 and given as follows [33]:

$$\mathfrak{a} = \mathfrak{a}\_0 + \mathfrak{a}\_1 T\_s + \mathfrak{a}\_2 T\_s^2 + \mathfrak{a}\_3 T\_s^3 \tag{14}$$

$$b = b\_0 + b\_1 T\_s + b\_2 T\_s^2 + b\_3 T\_s^3 \tag{15}$$

The saturated water pressure (kPa) for the adsorbate is provided by Antoine's equation [34].

$$P\_{sat} = 0.1333 \cdot 10^{8.07131 - \frac{1790 \text{bs}}{T\_t - 39.724}} \tag{16}$$

#### 2.2.2 Kinetic expression for zeolite 13X

The equilibrium adsorption capacity for zeolite 13X is determined with the following Dubinin-Astakhov (D-A) equation as the adsorption isotherm where the values of B and n are constants [35]:

$$X\_{\varepsilon} = X\_o \exp\left[-B\left(\frac{T\_s}{T\_{sat}} - \mathbf{1}\right)^n\right] \tag{17}$$

The saturated water temperature (K) for the adsorbate is given by Antoine's equation [34] as follows:

$$T\_{sat} = 39.724 + \frac{1730.63}{8.07131 - \log 10 \left(7.500638 \cdot 10^{-3} \cdot P\right)}\tag{18}$$

Assuming symmetry conditions, the boundary conditions in a view of the 2D axisymmetric finned adsorbent, used in this work are depicted Figure 3. At the upstream or inlet boundary, Dirichlet boundary conditions, uniform flow with


Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

The temperature dependent mass transfer coefficient and diffusion coefficient

Dm (11)

(12)

(13)

(17)

<sup>s</sup> (14)

<sup>s</sup> (15)

km <sup>¼</sup> <sup>15</sup> r2 p

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

Dm <sup>¼</sup> Do exp � Ea

Regular density (RD) silica gel is the most widespread type of silica gel and is obtainable from any supplier that retails silica gel. Because of its capacity for high moisture uptake in the low RH (relative humidity) range, it is a very effective

The equilibrium adsorption capacity characterises the theoretical maximum capacity that the adsorbent bed can adsorb for a given pressure and temperature [32]. For RD silica gel, the following pressure dependent isotherm is used as follows:

> P Psat <sup>b</sup>

> > <sup>s</sup> <sup>þ</sup> <sup>a</sup>3T<sup>3</sup>

<sup>s</sup> <sup>þ</sup> <sup>b</sup>3T<sup>3</sup>

Psat <sup>¼</sup> <sup>0</sup>:<sup>1333</sup> � <sup>10</sup><sup>8</sup>:07131� <sup>1730</sup>:<sup>63</sup> Ts�39:<sup>724</sup> (16)

Xe ¼ a

<sup>a</sup> <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup> <sup>a</sup>1Ts <sup>þ</sup> <sup>a</sup>2T<sup>2</sup>

<sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>0</sup> <sup>þ</sup> <sup>b</sup>1Ts <sup>þ</sup> <sup>b</sup>2T<sup>2</sup>

The values of a and b are temperatures dependent are provided in the next

The saturated water pressure (kPa) for the adsorbate is provided by Antoine's

The equilibrium adsorption capacity for zeolite 13X is determined with the following Dubinin-Astakhov (D-A) equation as the adsorption isotherm where the

The saturated water temperature (K) for the adsorbate is given by Antoine's

Assuming symmetry conditions, the boundary conditions in a view of the 2D axisymmetric finned adsorbent, used in this work are depicted Figure 3. At the upstream or inlet boundary, Dirichlet boundary conditions, uniform flow with

Tsat � 1 <sup>n</sup>

1730:63

<sup>8</sup>:<sup>07131</sup> � log 10 7:<sup>500638</sup> � <sup>10</sup>�<sup>3</sup> � <sup>P</sup> (18)

Xe <sup>¼</sup> Xo exp �<sup>B</sup> Ts

where Do is reference diffusivity and rp is the particle radius.

2.2.1 Kinetic expression for regular density (RD) silica gel

section, Table 1 and given as follows [33]:

2.2.2 Kinetic expression for zeolite 13X

values of B and n are constants [35]:

Tsat ¼ 39:724 þ

equation [34] as follows:

100

RTS 

are calculated as follows [25]:

desiccant.

equation [34].

#### Advanced Computational Fluid Dynamics for Emerging Engineering Processes...


For energy transfer, symmetry, inlet and outlet of the metal tube, have the same definitions as in the momentum transfer physics. However, the metal tube surface with the fins shown in green and purple lines, define the heat transfer between the three domains. The convective heat transfer boundary conditions between the heat

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

qi ¼ hi Tt � Tf

where hi is the convective heat transfer coefficient between tube and fluid.

where h<sup>0</sup> is the convective heat transfer coefficient between tube and adsorbent.

The governing equations of mass, momentum and energy conservation are solved by using the finite element method, based on the assumptions listed in Section 2.2. The governing equations are discretised on the computational domain, linearised in an implicit manner and solved by the finite element method using a pressure-based coupled solver (PBCS). This latter solves pressure and momentum

(19)

qo ¼ hoð Þ Ts � Tt (20)

transfer fluid, metal tube and the adsorbent are defined as follows:

HTF to metal tube:

DOI: http://dx.doi.org/10.5772/intechopen.81144

Metal tube to adsorbent:

2.3 Numerical procedure

Figure 4.

103

Overview of the pressure-based coupled solver algorithm.

#### Table 1.

Parameter settings for the validation and parametric studies.

Figure 3. View of the 2D axisymmetric finned tube adsorber: computational domain with boundary conditions.

constant velocity and constant temperature are assumed. At the downstream end of the computational domain or outlet, the Neumann boundary condition is used, i.e. stream wise gradients for all the variables are set to zero. No-slip boundary condition is used at the adsorbent fin surfaces. These surfaces are assumed to be solid walls with no slip wall boundary condition; the velocity of the fluid at the wall is zero and constant wall temperature is presumed. This sweeping statement relates to isothermal wall boundary condition. The fins and tube are presumed to be made of copper. As copper is a rather high thermal conductivity material, constant wall temperature boundary condition can be confidently supposed throughout the shells. In the right side of the figure, is shown the adsorbent bed under vacuum, where there is zero flux for mass transfer.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

For energy transfer, symmetry, inlet and outlet of the metal tube, have the same definitions as in the momentum transfer physics. However, the metal tube surface with the fins shown in green and purple lines, define the heat transfer between the three domains. The convective heat transfer boundary conditions between the heat transfer fluid, metal tube and the adsorbent are defined as follows:

HTF to metal tube:

$$q\_i = h\_i \left( T\_t - T\_f \right) \tag{19}$$

where hi is the convective heat transfer coefficient between tube and fluid. Metal tube to adsorbent:

$$q\_o = h\_o(T\_s - T\_t) \tag{20}$$

where h<sup>0</sup> is the convective heat transfer coefficient between tube and adsorbent.

#### 2.3 Numerical procedure

The governing equations of mass, momentum and energy conservation are solved by using the finite element method, based on the assumptions listed in Section 2.2. The governing equations are discretised on the computational domain, linearised in an implicit manner and solved by the finite element method using a pressure-based coupled solver (PBCS). This latter solves pressure and momentum

Figure 4. Overview of the pressure-based coupled solver algorithm.

constant velocity and constant temperature are assumed. At the downstream end of the computational domain or outlet, the Neumann boundary condition is used, i.e. stream wise gradients for all the variables are set to zero. No-slip boundary condition is used at the adsorbent fin surfaces. These surfaces are assumed to be solid walls with no slip wall boundary condition; the velocity of the fluid at the wall is zero and constant wall temperature is presumed. This sweeping statement relates to isothermal wall boundary condition. The fins and tube are presumed to be made of copper. As copper is a rather high thermal conductivity material, constant wall temperature boundary condition can be confidently supposed throughout the shells. In the right side of the figure, is shown the adsorbent bed under vacuum, where

View of the 2D axisymmetric finned tube adsorber: computational domain with boundary conditions.

Category Description Variable Unit Validation

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

Misc. Universal gas constant R J/mole

Ideal gas constant for water vapour

Parameter settings for the validation and parametric studies.

Table 1.

Figure 3.

102

zeolite 13X/water

8.314 8.314

Kinetic constant a<sup>0</sup> — — 6.5314 Kinetic constant a<sup>1</sup> — — 0.072452 Kinetic constant a<sup>2</sup> — — 0.00023951 Kinetic constant a<sup>3</sup> — — 2.5493e7 Kinetic constant b<sup>0</sup> — — 15.587 Kinetic constant b<sup>1</sup> — — 0.15915 Kinetic constant b<sup>2</sup> — — 0.00050612 Kinetic constant b<sup>3</sup> — — 5.329e7

K

Rv J/kg K 461.5 461.5

Base case silica gel/water

there is zero flux for mass transfer.

simultaneously. Figure 4 represents an overview of the PBCS algorithm. SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm is applied for the pressure-velocity coupling; the second order upwind discretisation scheme is used for the convection terms and each governing equation is solved using QUICK (Quadratic Upwind Interpolation) scheme [36].

shows the results of the simulated temperature profile in COMSOL Multiphysics along r = 0.02 m of the unit. It can be seen that the experimental data for the desorption mode compared to the CFD simulation results provide good agreement between the each other with deviation of no more than 5.8%. The difference between the experimental and numerical results is higher at the inlet of the unit. This may be due to the experimental uncertainties due to the temperature measurement errors because of lower heat transfer resulted from flow maldistribution or various losses. However, the observed uncertainties are well within the uncertainties of sensor measurements. This CFD model of the adsorption cooling system was therefore utilised for further transient analysis with supporting reliability of the

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

The selected adsorption chiller applied a finned tube geometry with a silica gel/water working pair. A base line case on desorption mode is initially simulated and studied to determine the intrinsic behaviour of the system. The parameter values for the base case and for the parametric study are presented in Table 1.

Figure 6 presents a 3D qualitative assessment of the pressure, temperature and

Pressure, temperature and adsorbed water distributions at t = 200, 600 and 1200 s of the 3D geometry for the

water desorption distributions at operation times of 200, 600 and 1200 s. The temperature profile captures the heat transfer characteristics of the system throughout the entire system while the pressure and adsorbed water only capture phenomena in the adsorbent bed. Desorption is the mode of operation highlighted in this study. The HTF inlet temperature is 338 K while the bed temperature is 298 K. The HTF enters the tube at z = 0 and exits at z = 0.6 m. A white set of arrows

computation.

4.1 Base case

Figure 6.

105

baseline case under desorption mode.

4. Results and discussion

DOI: http://dx.doi.org/10.5772/intechopen.81144

It is required to carry out independency verification of the grid system before CFD computation. The mesh independence study is examined by utilising three different mesh sizes (normal, fine and finest meshes) of 23,083, 35,305, and 84,992 for the finned tube adsorption cooling bed are adopted for computation for the baseline case. The relative error compared to the fine mesh, in the average bed temperature and the total water adsorbed as a function of time, are 2.85 and 1.84%, and 0.13 and 0.07%, respectively, before settling to a fine mesh for the geometry of the computational adsorption cooling bed cases. Computations were then run for a geometry comprising about 35,305 meshes, which was considered satisfactory in terms of accuracy and efficiency.

Furthermore, the solution is iterated until convergence is achieved, that is, residual for each equation achieves values less than 10<sup>6</sup> , and variations in energy, mass and temperature, respectively become negligible. A workstation with 2 (R) Xeon processors and a 2 core 2.4 GHz CPU with an installed memory of 32 GB (RAM), which took between 4 and 8 hours of CPU time depending on the computation case, was utilised to execute the necessary task.

#### 3. Model validation

Experimental data of a finned tube adsorption bed with zeolite/water (adsorbent/adsorbate) as the working pair, zeolite as the adsorbent material by Çağlar et al. [28] have been used to validate the CFD model in the present study. The finned tube heat exchanger geometry was modified to have the identical geometric dimensions and operating conditions as found in [27]. Table 1 shows the parameter settings associated with the experimental operation with particular settings associated with the experimental tests for t, k, v, lbed and r shown in Figure 5. In Figure 4

#### Figure 5.

Comparison of simulation results with experimental results of Çağlar et al. [28] in finned tube adsorption cooling system using zeolite/water for validation study.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

shows the results of the simulated temperature profile in COMSOL Multiphysics along r = 0.02 m of the unit. It can be seen that the experimental data for the desorption mode compared to the CFD simulation results provide good agreement between the each other with deviation of no more than 5.8%. The difference between the experimental and numerical results is higher at the inlet of the unit. This may be due to the experimental uncertainties due to the temperature measurement errors because of lower heat transfer resulted from flow maldistribution or various losses. However, the observed uncertainties are well within the uncertainties of sensor measurements. This CFD model of the adsorption cooling system was therefore utilised for further transient analysis with supporting reliability of the computation.

#### 4. Results and discussion

The selected adsorption chiller applied a finned tube geometry with a silica gel/water working pair. A base line case on desorption mode is initially simulated and studied to determine the intrinsic behaviour of the system. The parameter values for the base case and for the parametric study are presented in Table 1.

#### 4.1 Base case

simultaneously. Figure 4 represents an overview of the PBCS algorithm. SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm is applied for the pressure-velocity coupling; the second order upwind discretisation scheme is used for the convection terms and each governing equation is solved

It is required to carry out independency verification of the grid system before CFD computation. The mesh independence study is examined by utilising three different mesh sizes (normal, fine and finest meshes) of 23,083, 35,305, and 84,992 for the finned tube adsorption cooling bed are adopted for computation for the baseline case. The relative error compared to the fine mesh, in the average bed temperature and the total water adsorbed as a function of time, are 2.85 and 1.84%, and 0.13 and 0.07%, respectively, before settling to a fine mesh for the geometry of the computational adsorption cooling bed cases. Computations were then run for a geometry comprising about 35,305 meshes, which was considered satisfactory in

Furthermore, the solution is iterated until convergence is achieved, that is,

mass and temperature, respectively become negligible. A workstation with 2 (R) Xeon processors and a 2 core 2.4 GHz CPU with an installed memory of 32 GB (RAM), which took between 4 and 8 hours of CPU time depending on the compu-

Experimental data of a finned tube adsorption bed with zeolite/water (adsorbent/adsorbate) as the working pair, zeolite as the adsorbent material by Çağlar et al. [28] have been used to validate the CFD model in the present study. The finned tube heat exchanger geometry was modified to have the identical geometric dimensions and operating conditions as found in [27]. Table 1 shows the parameter settings associated with the experimental operation with particular settings associated with the experimental tests for t, k, v, lbed and r shown in Figure 5. In Figure 4

Comparison of simulation results with experimental results of Çağlar et al. [28] in finned tube adsorption

, and variations in energy,

using QUICK (Quadratic Upwind Interpolation) scheme [36].

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

residual for each equation achieves values less than 10<sup>6</sup>

tation case, was utilised to execute the necessary task.

terms of accuracy and efficiency.

3. Model validation

Figure 5.

104

cooling system using zeolite/water for validation study.

Figure 6 presents a 3D qualitative assessment of the pressure, temperature and water desorption distributions at operation times of 200, 600 and 1200 s. The temperature profile captures the heat transfer characteristics of the system throughout the entire system while the pressure and adsorbed water only capture phenomena in the adsorbent bed. Desorption is the mode of operation highlighted in this study. The HTF inlet temperature is 338 K while the bed temperature is 298 K. The HTF enters the tube at z = 0 and exits at z = 0.6 m. A white set of arrows

#### Figure 6.

Pressure, temperature and adsorbed water distributions at t = 200, 600 and 1200 s of the 3D geometry for the baseline case under desorption mode.

show the direction of the fluid flow. The kinetics of the adsorbed water is slow enough that there exists a time delay between reaching the thermodynamic temperature and establishing the equilibrium. The adsorbent is fully saturated after 2000 s, but this is not illustrated on the figure for brevity. For desorption the initial pressure of the system increases to 4.246 kPa and the initial adsorbed water begins at 0.72 kg/kg and reaches 0.08 kg/kg upon completion. The pressure of the system closely follows the behaviour of the temperature profile and affects the equilibrium water adsorption.

Figure 7 presents a 2D qualitative assessment of the pressure, temperature and desorbed water distributions. The same layout and results are chosen to be displayed as in Figure 6 for an alternative, but more insightful, assessment of the simulation results. From this visual perspective, the profiles for all simulated measures may be better compared in the parametric study.

#### 4.2 Parametric study

A parametric study applying the developed transient CFD coupled with heat and mass transfer model, was conducted in an effort to predict the influence of various parameters on the design and performance of an adsorption cooling system during the desorption process. The effect of these parameters on the governing independent parameters influencing the fluid flow, the heat and mass transfer on the adsorber performance, are the geometrical, particle size, physicalchemical, thermodynamic and thermal property parameters of the adsorber. The parameters investigated in this study are the adsorbent bed thickness (lbed), the heat exchanger tube thickness (b), the HTF velocity (v), and the adsorbent particle diameter (dp).

The basic aim of the analysis is to better understand the optimum design of the adsorption cooling system with heat exchanger configurations using CFD approach and, thereby, learn how to successfully maximise the overall heat and mass transfer

9 5

Case description Geometry Fluid flow and particle size

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

Tube thickness

HTF velocity

lbed (mm) b (mm) v (m/s) dp (mm)

12 1 0.001 2

thickness

1 1 Base 12 1 0.001 3 2 2 Bed thickness 30 1 0.001 3

3 4 Tube thickness 12 1.5 0.001 3

4 6 Fluid velocity 12 1 0.005 3 7 0.015

Case description Bed

5 2

3 45

DOI: http://dx.doi.org/10.5772/intechopen.81144

diameter

properties

Adsorbent particle diameter

Table 2 depicts the four-parameter settings, which were selected in the study.

The effect of adsorbent bed thickness on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is investigated for adsorbent bed thicknesses of 12 (baseline case), 30 and 45 mm. Figure 8 presents the effect of bed thickness on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 9 shows

The results reveals that the thickness of adsorbent bed has a substantial effect on the desorption performance. As the bed thickness increases, the bed temperature takes more time to increase. Therefore, the rate of the pressure drop is lower and the water adsorption takes a greater extent of time. The pressure, temperature and the amount of water adsorbed distributions inside the bed reaches equilibrium after about 1500, 2500 and 3000 s for bed thicknesses of 12, 30 and 45 mm, respectively. The desorption bed produced decreased average pressure values of 0.8 and 1.6%, for adsorption bed thicknesses of 30 and 40 mm relative to the baseline case having a bed thickness of 12 mm, respectively. Alternatively, under the same conditions, the desorption bed produced the same amounts of decreased average temperature

the simulated temperature and water adsorbed distributions at t = 600 s.

and decreased adsorbed water values of 15.3 and 35.0%, respectively.

As adsorbent bed thickness increases, more adsorbent is used, external mass transfer resistances increase with increasing mass of adsorbent. Thus, more heat is expected to be required to increase the temperature of the entire unit; the thermal resistances across the adsorbent bed are substantially increased. Fin temperature is reduced and cannot generate a sufficient heat transfer since the contact resistances between fins and adsorbent material increase. This causes reduced mass transfer, smaller amounts of adsorption capacities and increased cycle times. This same

performance of the system such as temperature and desorption rates whilst

Summary of the simulation matrix with the input data of the adsorption cooling system.

minimising the pressure drop.

5 8 Adsorbent particle

Case Case no.

Table 2.

107

Eight cases were investigated.

4.2.1 Effect of adsorbent bed thickness (lbed)

#### Figure 7.

Pressure, temperature and adsorbed water distributions at t = 200, 600 and 1200 s of the 2D geometry for the baseline case under desorption mode.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144


#### Table 2.

show the direction of the fluid flow. The kinetics of the adsorbed water is slow enough that there exists a time delay between reaching the thermodynamic temperature and establishing the equilibrium. The adsorbent is fully saturated after 2000 s, but this is not illustrated on the figure for brevity. For desorption the initial pressure of the system increases to 4.246 kPa and the initial adsorbed water begins at 0.72 kg/kg and reaches 0.08 kg/kg upon completion. The pressure of the system closely follows the behaviour of the temperature profile and affects the equilibrium

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

Figure 7 presents a 2D qualitative assessment of the pressure, temperature and

A parametric study applying the developed transient CFD coupled with heat and mass transfer model, was conducted in an effort to predict the influence of various parameters on the design and performance of an adsorption cooling system during the desorption process. The effect of these parameters on the governing independent parameters influencing the fluid flow, the heat and mass transfer on the adsorber performance, are the geometrical, particle size, physicalchemical, thermodynamic and thermal property parameters of the adsorber. The parameters investigated in this study are the adsorbent bed thickness (lbed), the heat exchanger tube thickness (b), the HTF velocity (v), and the adsorbent particle

Pressure, temperature and adsorbed water distributions at t = 200, 600 and 1200 s of the 2D geometry for the

desorbed water distributions. The same layout and results are chosen to be displayed as in Figure 6 for an alternative, but more insightful, assessment of the simulation results. From this visual perspective, the profiles for all simulated mea-

sures may be better compared in the parametric study.

water adsorption.

4.2 Parametric study

diameter (dp).

Figure 7.

106

baseline case under desorption mode.

Summary of the simulation matrix with the input data of the adsorption cooling system.

The basic aim of the analysis is to better understand the optimum design of the adsorption cooling system with heat exchanger configurations using CFD approach and, thereby, learn how to successfully maximise the overall heat and mass transfer performance of the system such as temperature and desorption rates whilst minimising the pressure drop.

Table 2 depicts the four-parameter settings, which were selected in the study. Eight cases were investigated.

#### 4.2.1 Effect of adsorbent bed thickness (lbed)

The effect of adsorbent bed thickness on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is investigated for adsorbent bed thicknesses of 12 (baseline case), 30 and 45 mm. Figure 8 presents the effect of bed thickness on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 9 shows the simulated temperature and water adsorbed distributions at t = 600 s.

The results reveals that the thickness of adsorbent bed has a substantial effect on the desorption performance. As the bed thickness increases, the bed temperature takes more time to increase. Therefore, the rate of the pressure drop is lower and the water adsorption takes a greater extent of time. The pressure, temperature and the amount of water adsorbed distributions inside the bed reaches equilibrium after about 1500, 2500 and 3000 s for bed thicknesses of 12, 30 and 45 mm, respectively. The desorption bed produced decreased average pressure values of 0.8 and 1.6%, for adsorption bed thicknesses of 30 and 40 mm relative to the baseline case having a bed thickness of 12 mm, respectively. Alternatively, under the same conditions, the desorption bed produced the same amounts of decreased average temperature and decreased adsorbed water values of 15.3 and 35.0%, respectively.

As adsorbent bed thickness increases, more adsorbent is used, external mass transfer resistances increase with increasing mass of adsorbent. Thus, more heat is expected to be required to increase the temperature of the entire unit; the thermal resistances across the adsorbent bed are substantially increased. Fin temperature is reduced and cannot generate a sufficient heat transfer since the contact resistances between fins and adsorbent material increase. This causes reduced mass transfer, smaller amounts of adsorption capacities and increased cycle times. This same

#### Figure 8.

Effect of bed thickness (lbed) on desorption 1D profile as a function of time for average bed pressure drop, average bed temperature and adsorption of water.

Figure 9.

Effect of bed thickness (lbed) on desorption 2D temperature and adsorbed water distributions at t = 600 s.

The data in the figures disclose that as the tube thickness increases, neither the bed temperature, pressure or water adsorption shows any significant variance from the baseline case. It is expected that as tube thickness increases, a greater amount of heat should be transferred to the adsorbent bed, and thus should have a positive impact on the overall performance. However, the limited process is controlled by the heat and mass transfer phenomena in the adsorption cooling system. Heat and mass transfer coefficients of the bed are quite small due to low conductivity of adsorbent particles, and high contact resistance between particles and metal tubes/ fins. Therefore, the performance of the adsorbent bed is affected unfavourably by the heat and mass transfer constraints inside the bed, with reduced thermal conductivity of the solid adsorbent, and internal (intra-particle) and external (inter-

Effect of tube thickness (b) on desorption 2D temperature and adsorbed water distributions at t = 600 s.

Effect of tube thickness (b) on desorption 1D profiles as a function of time for average bed pressure drop,

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

The effect of HTF velocity on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is evaluated for velocity of 0.001 (baseline case), 0.005 and 0.015 m/s. Figure 12 shows the effects of HTF velocity on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 13 presents the simulated tempera-

particle) mass transfer resistances.

ture and water adsorbed distributions at t = 600 s.

4.2.3 Effect of HTF velocity (v)

Figure 10.

Figure 11.

109

average bed temperature and adsorption of water.

DOI: http://dx.doi.org/10.5772/intechopen.81144

outcome has been documented by previous researches in heat exchangers in general. Heat transfer in the radial direction is improved by reducing the adsorbent bed thickness. For design considerations, a reduced adsorbent bed thickness will produce a reduced cycle time and an enhanced specific cooling power. Increasing bed thickness has then an adverse effect on the heat transfer within the bed. There is a trade-off to consider; decreasing bed thickness, results in an improvement in the heat transfer across the adsorbent bed at the expense of a reduction in the mass of adsorbent and hence, of the adsorption capacity.

#### 4.2.2 Effect of tube thickness (b)

The effect of tube thickness on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is examined for tube thickness of 1 (baseline case), 1.5 and 2 mm. Figure 10 shows the effects of tube thickness on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 11 presents the simulated temperature and water adsorbed distributions at t = 600 s.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

Figure 10.

Effect of tube thickness (b) on desorption 1D profiles as a function of time for average bed pressure drop, average bed temperature and adsorption of water.

Figure 11. Effect of tube thickness (b) on desorption 2D temperature and adsorbed water distributions at t = 600 s.

The data in the figures disclose that as the tube thickness increases, neither the bed temperature, pressure or water adsorption shows any significant variance from the baseline case. It is expected that as tube thickness increases, a greater amount of heat should be transferred to the adsorbent bed, and thus should have a positive impact on the overall performance. However, the limited process is controlled by the heat and mass transfer phenomena in the adsorption cooling system. Heat and mass transfer coefficients of the bed are quite small due to low conductivity of adsorbent particles, and high contact resistance between particles and metal tubes/ fins. Therefore, the performance of the adsorbent bed is affected unfavourably by the heat and mass transfer constraints inside the bed, with reduced thermal conductivity of the solid adsorbent, and internal (intra-particle) and external (interparticle) mass transfer resistances.

#### 4.2.3 Effect of HTF velocity (v)

The effect of HTF velocity on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is evaluated for velocity of 0.001 (baseline case), 0.005 and 0.015 m/s. Figure 12 shows the effects of HTF velocity on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 13 presents the simulated temperature and water adsorbed distributions at t = 600 s.

outcome has been documented by previous researches in heat exchangers in general. Heat transfer in the radial direction is improved by reducing the adsorbent bed thickness. For design considerations, a reduced adsorbent bed thickness will produce a reduced cycle time and an enhanced specific cooling power. Increasing bed thickness has then an adverse effect on the heat transfer within the bed. There is a trade-off to consider; decreasing bed thickness, results in an improvement in the heat transfer across the adsorbent bed at the expense of a reduction in the mass of

Effect of bed thickness (lbed) on desorption 2D temperature and adsorbed water distributions at t = 600 s.

Effect of bed thickness (lbed) on desorption 1D profile as a function of time for average bed pressure drop,

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

The effect of tube thickness on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is examined for tube thickness of 1 (baseline case), 1.5 and 2 mm. Figure 10 shows the effects of tube thickness on simulated transient average bed pressure, average bed temperature and adsorption of water profiles. Figure 11 presents the simulated temperature

adsorbent and hence, of the adsorption capacity.

and water adsorbed distributions at t = 600 s.

4.2.2 Effect of tube thickness (b)

Figure 8.

Figure 9.

108

average bed temperature and adsorption of water.

#### Figure 12.

Effect of HTF velocity (v) on desorption 1D profiles as a function of time for average bed pressure drop, average bed temperature and adsorption of water.

The results reveal that as the HTF velocity increases, the bed temperature and pressure response increase substantially, as well as the water desorbed. With an increase of HTF velocity from 0.001 to 0.005 m/s the water desorption response almost decreases an upper threshold. Alike, for the fluid flow and thermal properties, as fluid velocity of the tube increases, the temperature profile evolution should rise. The desorption bed produced increased average pressure and temperature values of 1.3 and 1.7% for HTF velocity of 0.055 and 0.015 relative to the baseline case having a HTF velocity of 0.001 m/s, respectively. On the other hand, under the same conditions, the desorption bed produced decreased average adsorbed water values of 18 and 21%, respectively. The adsorbent bed reached a steady state uniform temperature profile at the cycle maximum temperature of 338 K at 500, 700 and 1500 s, respectively. The cycle time increases with the decrease of the HTF velocity in the heat exchanger. An optimal velocity value corresponding to maximum overall performance of the adsorption cooling system, such as a high COP and specific cooling power would be in the range between 0.005 and 0.015 m/s. The HTF velocity should be well selected in order to obtain a good heat transfer efficiency, but also a positive effect on the mass transfer inside the adsorbent bed and on the overall system performance. A too slow HTF will increase cycle time and decrease specific cooling power. A faster velocity will reduce thermal gradient, but will need more pumping energy.

the effects of particle diameter on simulated transient adsorption of water profile. Figure 15 shows the water adsorbed distributions at t = 600 s. The temperature and

Effect of particle diameter (dp) on desorption 2D adsorbed water distributions at t = 600 s.

Effect of particle diameter (dp) on desorption 1D profiles as a function of time for adsorption of water.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

DOI: http://dx.doi.org/10.5772/intechopen.81144

The desorption bed produced decreased/increased average temperature and pressure values of about 1.6% for adsorbent particle diameter of 2 mm and 5 mm relative to the baseline case having a particle diameter of 3 mm, respectively. Alternatively, the desorption bed produced decreased average adsorbed water value of 6% for adsorbent particle diameter of 2 mm, and increased value of 22% for adsorbent particle diameter of 5 mm, relative to the baseline case having a particle diameter of 3 mm. The adsorbent bed achieved a steady state uniform temperature profile at the cycle maximum temperature of 338 K at 1000, 1500 and 2000 s, respectively. The smaller the particle diameter the faster the water adsorption response. The kinetics is directly dependent upon the particle packing as seen by the kinetic expressions in Section 2. As the adsorbent particle diameter decreases, the specific area of adsorbent increases, the internal mass transfer resistances decrease. Therefore, the bed temperature, the pressure and the desorption rate increase. The

In this work, a transient CFD coupled with heat and mass transfer model has been created for a solar adsorption cooling system. Transient simulations have been

pressure profiles are not shown because of limited space.

particle diameter should be kept as smaller as possible.

5. Conclusions

111

Figure 14.

Figure 15.

#### 4.2.4 Effect of adsorbent particle diameter (dp)

The effect of adsorbent particle diameter on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is assessed for particle diameter of 2, 3 (baseline case) and 5 mm. Figure 14 presents

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

#### Figure 14.

The results reveal that as the HTF velocity increases, the bed temperature and pressure response increase substantially, as well as the water desorbed. With an increase of HTF velocity from 0.001 to 0.005 m/s the water desorption response almost decreases an upper threshold. Alike, for the fluid flow and thermal properties, as fluid velocity of the tube increases, the temperature profile evolution should rise. The desorption bed produced increased average pressure and temperature values of 1.3 and 1.7% for HTF velocity of 0.055 and 0.015 relative to the baseline case having a HTF velocity of 0.001 m/s, respectively. On the other hand, under the same conditions, the desorption bed produced decreased average adsorbed water values of 18 and 21%, respectively. The adsorbent bed reached a steady state uniform temperature profile at the cycle maximum temperature of 338 K at 500, 700 and 1500 s, respectively. The

Effect of HTF velocity on desorption 2D temperature and adsorption of water distributions at t = 600 s.

Effect of HTF velocity (v) on desorption 1D profiles as a function of time for average bed pressure drop, average

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

cycle time increases with the decrease of the HTF velocity in the heat

dient, but will need more pumping energy.

Figure 12.

Figure 13.

110

bed temperature and adsorption of water.

4.2.4 Effect of adsorbent particle diameter (dp)

exchanger. An optimal velocity value corresponding to maximum overall performance of the adsorption cooling system, such as a high COP and specific cooling power would be in the range between 0.005 and 0.015 m/s. The HTF velocity should be well selected in order to obtain a good heat transfer efficiency, but also a positive effect on the mass transfer inside the adsorbent bed and on the overall system performance. A too slow HTF will increase cycle time and decrease specific cooling power. A faster velocity will reduce thermal gra-

The effect of adsorbent particle diameter on the dependent variables pressure, temperature and amount of water adsorbed within the adsorption cooling system is assessed for particle diameter of 2, 3 (baseline case) and 5 mm. Figure 14 presents

Effect of particle diameter (dp) on desorption 1D profiles as a function of time for adsorption of water.

the effects of particle diameter on simulated transient adsorption of water profile. Figure 15 shows the water adsorbed distributions at t = 600 s. The temperature and pressure profiles are not shown because of limited space.

The desorption bed produced decreased/increased average temperature and pressure values of about 1.6% for adsorbent particle diameter of 2 mm and 5 mm relative to the baseline case having a particle diameter of 3 mm, respectively. Alternatively, the desorption bed produced decreased average adsorbed water value of 6% for adsorbent particle diameter of 2 mm, and increased value of 22% for adsorbent particle diameter of 5 mm, relative to the baseline case having a particle diameter of 3 mm. The adsorbent bed achieved a steady state uniform temperature profile at the cycle maximum temperature of 338 K at 1000, 1500 and 2000 s, respectively. The smaller the particle diameter the faster the water adsorption response. The kinetics is directly dependent upon the particle packing as seen by the kinetic expressions in Section 2. As the adsorbent particle diameter decreases, the specific area of adsorbent increases, the internal mass transfer resistances decrease. Therefore, the bed temperature, the pressure and the desorption rate increase. The particle diameter should be kept as smaller as possible.

#### 5. Conclusions

In this work, a transient CFD coupled with heat and mass transfer model has been created for a solar adsorption cooling system. Transient simulations have been carried out to investigate the influence of several design and operating parameters during the desorption process. Silica gel and zeolite 13X are both investigated as possible adsorbents, though the study incorporated the working pair silica gel/water given the lower working temperature range required for operation which makes it more suitable for residential cooling applications powered by solar heat. Flow behaviour, heat and mass transfer performances have been analysed in detail. The CFD model was validated against the experimental data using zeolite 13X/water pair available in the literature. Good agreement with experimental results was obtained, which demonstrates the effectiveness of the model. Therefore, the CFD model was utilised in the present study with confidence that it can predict the pressure, temperature and water adsorption of the system accurately.

Nomenclature

a kinetic constant b kinetic constant b tube thickness, m bt tube thickness, m

d fin distance, m

I identity tensor

lbed bed thickness, m P pressure, kPa rp particle radius

RD regular density

T temperature, K u velocity vector, m/s v fluid velocity, m/s

Greek letters

Subscripts

113

D-A Dubinin-Astakhov

C specific heat capacity, J/kg K COP coefficient of performance

DOI: http://dx.doi.org/10.5772/intechopen.81144

/s

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

/s

hi convective heat transfer coefficient: tube and fluid, W/m<sup>2</sup> K ho convective heat transfer coefficient: tube and adsorbent, W/m<sup>2</sup> K

Do reference diffusivity, m<sup>2</sup>

Dm molecular diffusivity, m<sup>2</sup>

ΔH heat of adsorption, kJ/kg

k thermal conductivity, W/m K km mass transfer coefficient, 1/s L adsorbent bed total length, m

R universal gas constant, J/mol K

SPC specific cooling power, W/kg

X adsorption of water, kgw/kgs

Δ gradient operator Δ<sup>2</sup> Laplacian operator ε void fraction κ permeability, m<sup>2</sup>

ρ density, kg/m<sup>3</sup>

0 initial condition a adsorption b adsorbent bed c condenser

d desorption

cold low temperature setting

e equilibrium or evaporator

μ dynamic viscosity, kg/m s

Rv ideal gas constant for water vapour, J/kg K

dp adsorbent particle diameter, m

Ea activation energy of surface diffusion, J/mol

The key parameters considered in this work were geometrical and operating factors such as bed thickness (lbed), heat exchanger tube thickness (b) and HTF velocity (v), and adsorbent particle diameter (dp). The results have not only confirmed some observations reported in earlier researches, but also provided an envelope for the optimum design parameters when the operating conditions and geometrical factors were varied for performance enhancement. In order of highest impact on the system, the parameters are listed as follows: v>dp>lbed>b. The results disclosed that from the four parameters; only the first three stated are the most influential factors of performance and significantly change the cycle time. In the design phase, the most important geometric parameter to consider is the bed thickness. As the bed thickness increases, the amount of water adsorbed increases but so does the cycle time. While selecting the adsorbent material, a smaller particle diameter is desired to minimise the cycle time. A lower threshold of 2 mm was identified. During operation, the fluid velocity should be operated at a higher velocity to minimise the cycle time. A upper threshold of 0.005 m/s was identified.

It is concluded that the present modelling approach provides a useful means of identifying significant features, which influence the levels of the pressure drops, temperature and water adsorption from adsorption cooling systems and for assessing the performance characteristic of proposed adsorption chiller configurations. Moreover, the coupled CFD with heat and mass transfer model is a useful tool to simulate and optimise adsorption cooling systems and detect the parameters in the adsorber that are responsible for excessive pressure drops and low performance levels. The effort to alleviate these problems can be directly evaluated. In addition, the influence of any modification that is made to help improve performance characteristics on other operating or geometrical parameters is easily evaluated.

As a final point, this work should lead to accurate design and optimisation of solar/heat-driven adsorption cooling systems in terms of better control of the heat and mass transfer levels over the entire adsorption/desorption operating range and improved overall system performance.

#### Acknowledgements

Funding for this work was provided by Natural Resources Canada through the Program of Energy Research and Development.

#### Conflict of interest

The authors declare no conflict of interest.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

#### Nomenclature

carried out to investigate the influence of several design and operating parameters during the desorption process. Silica gel and zeolite 13X are both investigated as possible adsorbents, though the study incorporated the working pair silica gel/water given the lower working temperature range required for operation which makes it more suitable for residential cooling applications powered by solar heat. Flow behaviour, heat and mass transfer performances have been analysed in detail. The CFD model was validated against the experimental data using zeolite 13X/water pair available in the literature. Good agreement with experimental results was obtained, which demonstrates the effectiveness of the model. Therefore, the CFD model was utilised in the present study with confidence that it can predict the

The key parameters considered in this work were geometrical and operating factors such as bed thickness (lbed), heat exchanger tube thickness (b) and HTF velocity (v), and adsorbent particle diameter (dp). The results have not only confirmed some observations reported in earlier researches, but also provided an envelope for the optimum design parameters when the operating conditions and geometrical factors were varied for performance enhancement. In order of highest impact on the system, the parameters are listed as follows: v>dp>lbed>b. The results disclosed that from the four parameters; only the first three stated are the most influential factors of performance and significantly change the cycle time. In the design phase, the most important geometric parameter to consider is the bed thickness. As the bed thickness increases, the amount of water adsorbed increases but so does the cycle time. While selecting the adsorbent material, a smaller particle diameter is desired to minimise the cycle time. A lower threshold of 2 mm was identified. During operation, the fluid velocity should be operated at a higher velocity to minimise the cycle time. A upper threshold of 0.005 m/s was

It is concluded that the present modelling approach provides a useful means of identifying significant features, which influence the levels of the pressure drops, temperature and water adsorption from adsorption cooling systems and for assessing the performance characteristic of proposed adsorption chiller configurations. Moreover, the coupled CFD with heat and mass transfer model is a useful tool to simulate and optimise adsorption cooling systems and detect the parameters in the adsorber that are responsible for excessive pressure drops and low performance levels. The effort to alleviate these problems can be directly evaluated. In addition, the influence of any modification that is made to help improve performance characteristics on other operating or geometrical parameters is easily evaluated.

As a final point, this work should lead to accurate design and optimisation of solar/heat-driven adsorption cooling systems in terms of better control of the heat and mass transfer levels over the entire adsorption/desorption operating range and

Funding for this work was provided by Natural Resources Canada through the

pressure, temperature and water adsorption of the system accurately.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

identified.

improved overall system performance.

Program of Energy Research and Development.

The authors declare no conflict of interest.

Acknowledgements

Conflict of interest

112


#### Greek letters


#### Subscripts


Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

References

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DOI: http://dx.doi.org/10.5772/intechopen.81144

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[12] Zhang LZ. A three-dimensional non-equilibrium model for an

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10.1016/j.ijrefrig.2011.12.006

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intermittent adsorption cooling system. Solar Energy. 2000;69:27-35. DOI:

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0038-092X/00

ijrefrig.2005.06.001

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System…

445-456. DOI: 10.1016/j.

rser.2015.05.035

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[3] Sah RP, Choudhury B, Das RK. A review on adsorption cooling systems with silica gel and carbon as adsorbents. Renewable and Sustainable Energy Reviews. 2015;45:123-134. DOI: 10.1016/j.rser.2015.01.039

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[6] Teng Y, Wang RZ, Wu JY. Study of the fundamentals of adsorption systems. Applied Thermal Engineering. 1997;17: 327-338. DOI: 10.1016/S1359-4311(96)

[7] Pons M, Meunier F, Cacciola G, Critoph R, Groll M, Puigjaner L, et al. Thermodynamic based comparison of sorption systems for cooling and heat pumping. International Journal of Refrigeration. 1999;22:5-17. DOI:

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S0140-7007(98)00048-6

0890-4332(93)90051-V

00039-7

115


### Author details

Wahiba Yaïci\* and Evgueniy Entchev CanmetENERGY Research Centre, Natural Resources Canada, Ottawa, Ontario, Canada

\*Address all correspondence to: wahiba.yaici@canada.ca

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

#### References

f fluid

v vapour t tube

Author details

Canada

114

Wahiba Yaïci\* and Evgueniy Entchev

provided the original work is properly cited.

\*Address all correspondence to: wahiba.yaici@canada.ca

CanmetENERGY Research Centre, Natural Resources Canada, Ottawa, Ontario,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

pl particle liquid pv particle vapour s solid adsorbent

hot high temperature setting

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

[1] Saha BB, Akisawa A, Kashiwagi T. Solar/waste heat driven two-stage adsorption chiller: The prototype. Renewable Energy. 2001;23:93-101. DOI: 10.1016/S0960-1481(00)00107-5

[2] Tso CY, Chao CYH. Activated carbon, silica-gel and calcium chloride composite adsorbents for energy efficient solar adsorption cooling and dehumidification systems. International Journal of Refrigeration. 2012;35: 1626-1638. DOI: 10.1016/j. ijrefrig.2012.05.007

[3] Sah RP, Choudhury B, Das RK. A review on adsorption cooling systems with silica gel and carbon as adsorbents. Renewable and Sustainable Energy Reviews. 2015;45:123-134. DOI: 10.1016/j.rser.2015.01.039

[4] Meunier F. Solid sorption: An alternative to CFCs. Heat Recovery CHP. 1993;13:289-295. DOI: 10.1016/ 0890-4332(93)90051-V

[5] Meunier F. Solid sorption heat powered cycles for cooling and heat pumping applications. Applied Thermal Engineering. 1998;18:715-729. DOI: 10.1016/S1359-4311(97)00122-1

[6] Teng Y, Wang RZ, Wu JY. Study of the fundamentals of adsorption systems. Applied Thermal Engineering. 1997;17: 327-338. DOI: 10.1016/S1359-4311(96) 00039-7

[7] Pons M, Meunier F, Cacciola G, Critoph R, Groll M, Puigjaner L, et al. Thermodynamic based comparison of sorption systems for cooling and heat pumping. International Journal of Refrigeration. 1999;22:5-17. DOI: S0140-7007(98)00048-6

[8] Hamamoto Y, Alam KC, Saha BB, Koyama S, Akisawa A, Kashiwagi T. Study on adsorption refrigeration cycle utilizing activated carbon fibers. Part 2: Cycle performance evaluation. International Journal of Refrigeration. 2006;29:315-327. DOI: 10.1016/j. ijrefrig.2005.06.001

[9] Shmroukh AN, Ali AHH, Ookawara S. Adsorption working pairs for adsorption cooling chillers: A review based on adsorption capacity and environmental impact. Renewable and Sustainable Energy Reviews. 2015;50: 445-456. DOI: 10.1016/j. rser.2015.05.035

[10] Kim AS, Lee HS, Moon DS, Kim HJ. Performance control modeling on adsorption desalination using initial time lag (ITL) of individual beds. Desalination. 2016;369:1-16. DOI: 10.1016/j.desal.2016.05.004

[11] Zhang LZ, Wang L. Effects of coupled heat and mass transfers in adsorbent on the performance of a waste heat adsorption cooling unit. Applied Thermal Engineering. 1999;19: 195-215. DOI: 1359-4311/99

[12] Zhang LZ. A three-dimensional non-equilibrium model for an intermittent adsorption cooling system. Solar Energy. 2000;69:27-35. DOI: 0038-092X/00

[13] Solmuş İ, Rees DAS, Yamalı C, Baker D. A two-energy equation model for dynamic heat and mass transfer in an adsorbent bed using silica gel/water pair. International Journal of Heat and Mass Transfer. 2012;55:5275-5288. DOI: 10.1016/j.ijrefrig.2011.12.006

[14] Yong L, Sumathy K. Review of mathematical investigation on the closed adsorption heat pump and cooling systems. Renewable and Sustainable Energy Reviews. 2002;6:305-338. DOI: 10.1016/S1364-0321(02)00010-2

[15] Teng WS, Leong KC, Chakraborty A. Revisiting adsorption cooling cycle

from mathematical modelling to system development. Renewable and Sustainable Energy Reviews. 2016;63: 315-332. DOI: 10.1016/j.rser.2016.05.059

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[18] Cacciola G, Restuccia G. Reversible heat pump: A thermodynamic model. International Journal of Refrigeration. 1995;18:100-106. DOI: 10.1016/ 0140-7007(94)00005-I0007/95/ 0140-7

[19] Guilleminot JJ, Meunier F, Pakleza J. Heat and mass transfer in a nonisothermal fixed bed solid adsorbent external reactor: A uniform pressure– non uniform temperature case. International Journal of Heat and Mass Transfer. 1987;30:1595-1606. DOI: 10.1016/0017-9310(87)90304-8

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[22] Hajji A, Worek WM. Simulation of a regenerative, closed-cycle adsorption cooling/heating system. Energy. 1991; 16:643-654. DOI: 10.1016/0360-5442 (91)90035

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cooling—Novel beds and advanced cycles. Energy Conversion and Management. 2015;94:221-232. DOI: 10.1016/j.enconman.2015.01.076

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[25] Amar NB, Sun LM, Meunier F. Numerical analysis of adsorptive temperature wave regenerative heat pump. Applied Thermal Engineering. 1996;16:405-418. DOI: 10.1016/ 1359-4311(95)00045-3

[26] Solmuş İ, Rees DAS, Yamalı C, Baker D, Kaftanoğlu B. Numerical investigation of coupled heat and mass transfer inside the adsorbent bed of an adsorption cooling unit. International Journal of Refrigeration. 2012;35: 652-662. DOI: 10.1016/j.ijrefrig.2011. 12.006

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[30] Li XH, Hou XH, Zhang X, Yuan ZX. A review on development of adsorption

Unsteady CFD with Heat and Mass Transfer Simulation of Solar Adsorption Cooling System… DOI: http://dx.doi.org/10.5772/intechopen.81144

cooling—Novel beds and advanced cycles. Energy Conversion and Management. 2015;94:221-232. DOI: 10.1016/j.enconman.2015.01.076

from mathematical modelling to system

Advanced Computational Fluid Dynamics for Emerging Engineering Processes...

performance of silica gel adsorption refrigeration systems. International Journal of Heat and Mass Transfer. 2000;43:4419-4431. DOI: 10.1016/

[24] Sun LM, Amar NB, Meunier F. Numerical study on coupled heat and mass transfer in an adsorber with external fluid heating. Heat Recovery Systems & CHP. 1995;15:19-29. DOI:

[25] Amar NB, Sun LM, Meunier F. Numerical analysis of adsorptive temperature wave regenerative heat pump. Applied Thermal Engineering. 1996;16:405-418. DOI: 10.1016/

[26] Solmuş İ, Rees DAS, Yamalı C, Baker D, Kaftanoğlu B. Numerical investigation of coupled heat and mass transfer inside the adsorbent bed of an adsorption cooling unit. International Journal of Refrigeration. 2012;35: 652-662. DOI: 10.1016/j.ijrefrig.2011.

[27] Solmuş İ, Yamalı C, Yıldırım C, Bilen K. Transient behavior of a cylindrical adsorbent bed during the adsorption process. Applied Energy. 2015;142:115-124. DOI: 10.1016/j.

[28] Çağlar A, Yamalı C, Baker DK. Two dimensional transient coupled analysis of a finned tube adsorbent bed for a thermal wave cycle. International Journal of Thermal Sciences. 2013;73: 58-68. DOI: 10.1016/j.ijthermalsci.2013.

[29] Çağlar A. The effect of fin design parameters on the heat transfer

enhancement in the adsorbent bed of a thermal wave cycle. Applied Thermal Engineering. 2016;104:386-393. DOI: 10.1016/j.applthermaleng.2016.05.092

[30] Li XH, Hou XH, Zhang X, Yuan ZX. A review on development of adsorption

S0017-9310(00)00072-7

0890-4332(94)E0011-8

1359-4311(95)00045-3

apenergy.2014.12.080

12.006

06.009

Sustainable Energy Reviews. 2016;63: 315-332. DOI: 10.1016/j.rser.2016.05.059

[16] Sakoda A, Suzuki M. Fundamental study on solar powered adsorption cooling system. Journal of Chemical Engineering of Japan. 1984;17:52-57.

[17] Cho SH, Kim JN. Modeling of a silica gel/water adsorption cooling system. Energy. 1992;17:829-839. DOI: 10.1016/

[18] Cacciola G, Restuccia G. Reversible heat pump: A thermodynamic model. International Journal of Refrigeration. 1995;18:100-106. DOI: 10.1016/ 0140-7007(94)00005-I0007/95/

[19] Guilleminot JJ, Meunier F, Pakleza J.

International Journal of Heat and Mass Transfer. 1987;30:1595-1606. DOI: 10.1016/0017-9310(87)90304-8

[21] Shelton SV, Wepfer WJ, Miles DJ. External fluid heating of a porous bed. Chemical Engineering Communications.

[22] Hajji A, Worek WM. Simulation of a regenerative, closed-cycle adsorption cooling/heating system. Energy. 1991; 16:643-654. DOI: 10.1016/0360-5442

[23] Alam KCA, Saha B, Kang YT, Akisawa A, Kashiwagi T. Heat

exchanger design effect on the system

Heat and mass transfer in a nonisothermal fixed bed solid adsorbent external reactor: A uniform pressure– non uniform temperature case.

[20] Aittomaki A, Harkonen M. Modelling of zeolite/methanol adsorption heat pump process. Heat Recovery Systems CHP. 1988;8:475-482. DOI: 10.1016/0890-4332(88)90053-1

1988;71:39-52. DOI: 10.1080/

00986448808940413

(91)90035

116

development. Renewable and

DOI: 10.1252/jcej.17.52

0360-5442(92)90101-5

0140-7

[31] COMSOL Inc. COMSOL Multiphysics Software Package Including the CFD, Heat Transfer and Chemical Reaction Engineering Modules. Version 5.2. 2017

[32] Aristov Y, Tokarev M, Freni A, Glaznev I. Kinetics of water adsorption on silica Fuji Davison RD. Microporous and Mesoporous Materials. 2006;96: 65-71. DOI: 10.1016/j. micromeso.2006.06.008

[33] Chihara K, Suzuki M. Air drying by pressure swing adsorption. Journal of Chemical Engineering of Japan. 1983;16: 293-299. DOI: 10.1252/jcej.16.293

[34] Reid RC, Prausnitz JM, Sherwood TK. Properties of Gases and Liquids. 3rd ed. New York, DC: McGraw-Hill; 1977. ISBN 978-007051790-5

[35] Solmuş İ, Yamalı C, Kaftanoğlu B, Baker D, Çağlar A. Adsorption properties of a natural zeolite–water pair for use in adsorption cooling cycles. Applied Energy. 2010;87:2062-2067. DOI: 10.1016/j.apenergy.2009.11.027

[36] Patankar SV. Numerical Heat Transfer and Fluid Flow. 1st ed. New York, DC: Hemisphere Publishing Corporation; 1980. 210 p. DOI: 10.1002/ cite.330530323

Chapter 6

Abstract

Modeling of Fluid-Solid

Zhenhua Huang and Cheng-Hsien Lee

included at the end of the chapter.

continuum model, OpenFOAM®

imental data for model validation.

119

1. Introduction

Two-Phase Geophysical Flows

Fluid-solid two-phase flows are frequently encountered in geophysical flow problems such as sediment transport and submarine landslides. It is still a challenge to the current experiment techniques to provide information such as detailed flow and pressure fields of each phase, which however is easily obtainable through numerical simulations using fluid-solid two-phase flow models. This chapter focuses on the Eulerian-Eulerian approach to two-phase geophysical flows. Brief derivations of the governing equations and some closure models are provided, and the numerical implementation in the finite-volume framework of OpenFOAM® is described. Two applications in sediment transport and submarine landslides are also

Keywords: granular flows, submarine landslides, sediment transport, scour,

Fluid-solid two-phase flows are important in many geophysical problems such as sediment erosion, transport and deposition in rivers or coastal environment, debris flows, scour at river or marine structures, and submarine landslides. Behaviors of fluid-solid two-phase flows are very different from those of liquid-gas two-phase flows where bubbles are dispersed in the liquid or droplets dispersed in the gas. Vast numbers of experiments on various scales have been carried out for different applications of fluid-solid two-phase flows; these experiments have advanced our understanding of bulk behaviors of some important flow characteristics. However, development of measurement techniques suitable for collecting data that contribute to understanding important physics involved in fluid-solid two-phase flows is a still-evolving science. With the modern computer technology, many data that are not obtainable currently in the experiment can be easily produced by performing time-dependent, multidimensional numerical simulations. Of course, empirical closure models required to close the governing equations still need high-quality exper-

Numerical approaches to two-phase flows include Eulerian-Eulerian approach, direct numerical simulations (DNS) based on Eulerian-Lagrangian formulations (Lagrangian point-particle approach), and fully resolved DNS approach [1]. Fully resolved DNS can resolve all important scales of the fluid and particles, but these simulations are currently limited to about 10 k uniform-size spheres on a Cray XE6 with 2048 cores [2], and it is not practical to use this method to model large-scale

#### Chapter 6

## Modeling of Fluid-Solid Two-Phase Geophysical Flows

Zhenhua Huang and Cheng-Hsien Lee

#### Abstract

Fluid-solid two-phase flows are frequently encountered in geophysical flow problems such as sediment transport and submarine landslides. It is still a challenge to the current experiment techniques to provide information such as detailed flow and pressure fields of each phase, which however is easily obtainable through numerical simulations using fluid-solid two-phase flow models. This chapter focuses on the Eulerian-Eulerian approach to two-phase geophysical flows. Brief derivations of the governing equations and some closure models are provided, and the numerical implementation in the finite-volume framework of OpenFOAM® is described. Two applications in sediment transport and submarine landslides are also included at the end of the chapter.

Keywords: granular flows, submarine landslides, sediment transport, scour, continuum model, OpenFOAM®

#### 1. Introduction

Fluid-solid two-phase flows are important in many geophysical problems such as sediment erosion, transport and deposition in rivers or coastal environment, debris flows, scour at river or marine structures, and submarine landslides. Behaviors of fluid-solid two-phase flows are very different from those of liquid-gas two-phase flows where bubbles are dispersed in the liquid or droplets dispersed in the gas. Vast numbers of experiments on various scales have been carried out for different applications of fluid-solid two-phase flows; these experiments have advanced our understanding of bulk behaviors of some important flow characteristics. However, development of measurement techniques suitable for collecting data that contribute to understanding important physics involved in fluid-solid two-phase flows is a still-evolving science. With the modern computer technology, many data that are not obtainable currently in the experiment can be easily produced by performing time-dependent, multidimensional numerical simulations. Of course, empirical closure models required to close the governing equations still need high-quality experimental data for model validation.

Numerical approaches to two-phase flows include Eulerian-Eulerian approach, direct numerical simulations (DNS) based on Eulerian-Lagrangian formulations (Lagrangian point-particle approach), and fully resolved DNS approach [1]. Fully resolved DNS can resolve all important scales of the fluid and particles, but these simulations are currently limited to about 10 k uniform-size spheres on a Cray XE6 with 2048 cores [2], and it is not practical to use this method to model large-scale

geophysical flow problems in the foreseeable future [1]. Lagrangian point-particle approach uses Eulerian formulation for the fluid phase and Lagrangian formulation for tracking the instantaneous positions of the particles. Lagrangian point-particle simulations make use of semiempirical relationships to provide both hydrodynamic force and torque acting on each particle and thus avoid modeling processes on scales smaller than Kolmogorov scale [1], making it possible to include more particles and run in a domain larger than that for fully resolved DNS. The application of Lagrangian point-particle approach is crucially dependent on the availability and accuracy of such semiempirical relationships. A recent study shows that good results can be obtained for about 100k uniform-size spherical particles in a vertical channel flow [3]; however, using this approach to investigate large-scale two-phase flow problems is still beyond the current computing capacity. Two-phase Eulerian-Eulerian approach treats both the fluid and particle phases as continuum media and is suitable for solving large-scale two-phase flow problems.

In the Eulerian-Eulerian approach to two-phase flows, it is assumed that the equations governing the motion of phase k (for the fluid phase k ¼ f and for the solid phase k ¼ s) at the microscopic scale are the following equations for the

where ρ<sup>k</sup> is the density, u<sup>k</sup> is the velocity, and g is the acceleration due to gravity.

Because the fluid phase and the solid phase are immiscible, at any time t, a point in space x can be occupied only by one phase, not both. This fact can be described

> 1, if the point x is occupied by phase k 0, if the point x is not occupied by phase k

There are several methods to derive the ensemble averaged equations governing the motion of phase k. This chapter treats the phase function as a general function and uses it to define the derivatives of the phase function ck with respect to time and space and the equation governing the evolution of ck. As stated in Drew [8], the phase function ck can be treated as a generalized function whose derivative can be defined in terms of a set of test functions. These test functions must be sufficiently smooth and have compact support so that the integration of a derivative of the phase function, weighed with the test function, is finite. The equation describing

where pk is the microscopic pressure and τ<sup>k</sup> is the microscopic stress tensor.

The volumetric concentration of phase k is directly related to the probability of occurrence of phase k at a given location x at the time t and can be obtained by taking ensemble average of ck. Using the phase function given in Eq. (4), the volumetric concentration of phase k is obtained by taking the ensemble average of ck, denoted by h i ck . The operator h i ⋯ means taking an ensemble average of its

mathematically by the following phase function ckð Þ x; t for phase k:

∂ck ∂t

where u<sup>i</sup> is the velocity of the interface between the region occupied by the fluid phase and the region occupied by the solid phase. It is stressed here that ∇ck is zero except at the interface between two phases where ∇ck behaves like a delta-

The ensemble averaged equations governing the motion of phase k are obtained by multiplying Eqs. (1) and (2) with ck and performing an ensemble average operation on every term in the resulting equations. When performing ensemble average operations, Reynolds' rules for algebraic operations, Leibniz' rule for time derivatives, Gauss' rule for spatial derivatives, and the following two

þ ∇ � u<sup>k</sup> ¼ 0, (1)

þ ∇ � ð Þ¼ ρkuku<sup>k</sup> ∇ � T<sup>k</sup> þ ρg, (2)

T<sup>k</sup> ¼ �pkI þ τ<sup>k</sup> (3)

<sup>þ</sup> <sup>u</sup><sup>i</sup> � <sup>∇</sup>ck <sup>¼</sup> <sup>0</sup>, (5)

:

(4)

∂ρk ∂t

conservation of mass and momentum [8, 10]:

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

> ∂ρu<sup>k</sup> ∂t

The stress tensor T<sup>k</sup> includes two components:

ckð Þ¼ x; t

and

argument.

the evolution of ck is

function [8].

identities are used:

121

Eulerian-Eulerian two-phase flow models based on large-eddy-simulations solve a separate set of equations describing conservation of mass, momentum, and kinetic energy for each phase [4–7] and thus have the potential to consider all important processes involved in the interactions between the two phases through parameterization of particle-scale processes. This chapter introduces the basics of Eulerian-Eulerian two-phase flow modeling, its implementation in the finite-volume framework of OpenFOAM®, and two applications in geophysical flow problems.

#### 2. Governing equations for fluid-solid two-phase flows

Let us consider a mixture of fluid and solid particles. Fluid can be gas, water, or a mixture of water and gas. In DNS and Lagrangian point-particle approaches to twophase flows, the flow field is solved by solving the Navier-Stokes equations, and the motion of each particle is determined by the Newton's equation of motion. In Eulerian-Eulerian two-phase flow approaches, however, the motions of individual particles are not of the interest, and the focus is on the macroscopic motion of the fluid and solid particles instead. For this purpose, the solid particles are modeled as a continuum mass through an ensemble averaging operation, which is based on the existence of possible equivalent realizations. After taking ensemble average, the mixture of fluid and particles consists of two continuous phases: the fluid (water, gas, or a mixture of water and gas) is the fluid phase, and the solid particle is the solid phase. Both phases are incompressible. The motions of the fluid and solid phases are governed by their own equations, which are obtained by taking ensemble average of the microscopic governing equations for each phase [8]. Even though some aspects of fluid-solid interaction can be considered through the ensemble average, the ensemble averaging operation itself, however, does not explicitly introduce any turbulent dispersion in the resulting equations. To consider the turbulent dispersion in the Eulerian-Eulerian description of the fluid-solid two-phase flows, another averaging operation (usually a Favre average) is needed to consider the correlations of turbulent components [5, 9].

#### 2.1 Ensemble averaged equations

At the microscopic scale, the fluid-solid mixture is a discrete system. The purpose of performing an ensemble averaging operation is to derive a set of equations describing this discrete system as a continuous system at the macroscopic scale, where the typical length scale should be much larger than one particle diameter.

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

In the Eulerian-Eulerian approach to two-phase flows, it is assumed that the equations governing the motion of phase k (for the fluid phase k ¼ f and for the solid phase k ¼ s) at the microscopic scale are the following equations for the conservation of mass and momentum [8, 10]:

$$\frac{\partial \rho\_k}{\partial t} + \nabla \cdot \mathbf{u}\_k = \mathbf{0},\tag{1}$$

and

geophysical flow problems in the foreseeable future [1]. Lagrangian point-particle approach uses Eulerian formulation for the fluid phase and Lagrangian formulation for tracking the instantaneous positions of the particles. Lagrangian point-particle simulations make use of semiempirical relationships to provide both hydrodynamic force and torque acting on each particle and thus avoid modeling processes on scales smaller than Kolmogorov scale [1], making it possible to include more particles and

Eulerian-Eulerian two-phase flow models based on large-eddy-simulations solve a separate set of equations describing conservation of mass, momentum, and kinetic energy for each phase [4–7] and thus have the potential to consider all important processes involved in the interactions between the two phases through parameterization of particle-scale processes. This chapter introduces the basics of Eulerian-Eulerian two-phase flow modeling, its implementation in the finite-volume frame-

Let us consider a mixture of fluid and solid particles. Fluid can be gas, water, or a mixture of water and gas. In DNS and Lagrangian point-particle approaches to twophase flows, the flow field is solved by solving the Navier-Stokes equations, and the motion of each particle is determined by the Newton's equation of motion. In Eulerian-Eulerian two-phase flow approaches, however, the motions of individual particles are not of the interest, and the focus is on the macroscopic motion of the fluid and solid particles instead. For this purpose, the solid particles are modeled as a continuum mass through an ensemble averaging operation, which is based on the existence of possible equivalent realizations. After taking ensemble average, the mixture of fluid and particles consists of two continuous phases: the fluid (water, gas, or a mixture of water and gas) is the fluid phase, and the solid particle is the solid phase. Both phases are incompressible. The motions of the fluid and solid phases are governed by their own equations, which are obtained by taking ensemble average of the microscopic governing equations for each phase [8]. Even though some aspects of fluid-solid interaction can be considered through the ensemble average, the ensemble averaging operation itself, however, does not explicitly introduce any turbulent dispersion in the resulting equations. To consider the turbulent dispersion in the Eulerian-Eulerian description of the fluid-solid two-phase flows, another averaging operation (usually a Favre average) is needed to consider

At the microscopic scale, the fluid-solid mixture is a discrete system. The purpose of performing an ensemble averaging operation is to derive a set of equations describing this discrete system as a continuous system at the macroscopic scale, where the typical length scale should be much larger than one particle diameter.

run in a domain larger than that for fully resolved DNS. The application of Lagrangian point-particle approach is crucially dependent on the availability and accuracy of such semiempirical relationships. A recent study shows that good results can be obtained for about 100k uniform-size spherical particles in a vertical channel flow [3]; however, using this approach to investigate large-scale two-phase flow problems is still beyond the current computing capacity. Two-phase Eulerian-Eulerian approach treats both the fluid and particle phases as continuum media and

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

work of OpenFOAM®, and two applications in geophysical flow problems.

2. Governing equations for fluid-solid two-phase flows

the correlations of turbulent components [5, 9].

2.1 Ensemble averaged equations

120

is suitable for solving large-scale two-phase flow problems.

$$\frac{\partial \rho \mathbf{u}\_k}{\partial t} + \nabla \cdot (\rho\_k \mathbf{u}\_k \mathbf{u}\_k) = \nabla \cdot \mathbf{T}\_k + \rho \mathbf{g},\tag{2}$$

where ρ<sup>k</sup> is the density, u<sup>k</sup> is the velocity, and g is the acceleration due to gravity. The stress tensor T<sup>k</sup> includes two components:

$$\mathbf{T}\_k = -p\_k \mathbf{I} + \mathbf{r}\_k \tag{3}$$

where pk is the microscopic pressure and τ<sup>k</sup> is the microscopic stress tensor.

Because the fluid phase and the solid phase are immiscible, at any time t, a point in space x can be occupied only by one phase, not both. This fact can be described mathematically by the following phase function ckð Þ x; t for phase k:

$$c\_k(\mathbf{x}, t) = \begin{cases} \mathbf{1}, & \text{if the point } \mathbf{x} \text{ is occupied by phase } k\\ \mathbf{0}, & \text{if the point } \mathbf{x} \text{ is not occupied by phase } k \end{cases} \tag{4}$$

The volumetric concentration of phase k is directly related to the probability of occurrence of phase k at a given location x at the time t and can be obtained by taking ensemble average of ck. Using the phase function given in Eq. (4), the volumetric concentration of phase k is obtained by taking the ensemble average of ck, denoted by h i ck . The operator h i ⋯ means taking an ensemble average of its argument.

There are several methods to derive the ensemble averaged equations governing the motion of phase k. This chapter treats the phase function as a general function and uses it to define the derivatives of the phase function ck with respect to time and space and the equation governing the evolution of ck. As stated in Drew [8], the phase function ck can be treated as a generalized function whose derivative can be defined in terms of a set of test functions. These test functions must be sufficiently smooth and have compact support so that the integration of a derivative of the phase function, weighed with the test function, is finite. The equation describing the evolution of ck is

$$\frac{\partial c\_k}{\partial t} + \mathbf{u}^i \cdot \nabla c\_k = \mathbf{0},\tag{5}$$

where u<sup>i</sup> is the velocity of the interface between the region occupied by the fluid phase and the region occupied by the solid phase. It is stressed here that ∇ck is zero except at the interface between two phases where ∇ck behaves like a deltafunction [8].

The ensemble averaged equations governing the motion of phase k are obtained by multiplying Eqs. (1) and (2) with ck and performing an ensemble average operation on every term in the resulting equations. When performing ensemble average operations, Reynolds' rules for algebraic operations, Leibniz' rule for time derivatives, Gauss' rule for spatial derivatives, and the following two identities are used:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$
\mathcal{L}\_k \frac{\partial \phi\_k}{\partial t} = \frac{\partial \mathcal{c}\_k \phi\_k}{\partial t} - \phi\_k \frac{\partial \mathcal{c}\_k}{\partial t} = \frac{\partial \mathcal{c}\_k \phi\_k}{\partial t} + \phi\_k \mathbf{u}^i \cdot \nabla \mathcal{c}\_k,\tag{6}
$$

~τ0

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

> <sup>~</sup>ρ<sup>k</sup> <sup>¼</sup> ckρ<sup>k</sup> h i h i ck

> > 0 ¼ 1 � ~cs

materials

reduce to

for the fluid phase, and

for the solid phase.

written as p~<sup>s</sup> ¼ p~<sup>f</sup> þ p�<sup>s</sup>

<sup>∂</sup>ð Þ <sup>1</sup> � <sup>c</sup> <sup>ρ</sup>fu<sup>f</sup> ∂t

and

123

ensemble averaged pressure.

follows that

<sup>k</sup> ¼ � ckρku<sup>0</sup>

ku0 k � � h i ck

For compressible materials ~ρ<sup>k</sup> is not a constant. However, for incompressible

Now we examine the limiting case where the fluid-solid system is at its static state. Because the phase functions for the two phases satisfy cf þ cs ¼ 1, both phases are not moving, and m~ <sup>f</sup> þ m~ <sup>s</sup> ¼ 0, the governing equations

� h

0 ¼ ~cs ~ρs

0 ¼ ~cs ~ρs

<sup>∂</sup>ð Þ <sup>1</sup> � <sup>c</sup> <sup>ρ</sup><sup>f</sup> ∂t

þ∇ � ð Þ 1 � c ρfufu<sup>f</sup> h i , u<sup>0</sup>

� <sup>ρ</sup>k, <sup>u</sup>^<sup>k</sup> <sup>¼</sup> h i ckρku<sup>k</sup>

Þ~ρ<sup>f</sup> g � ∇ 1 � ~cs

g � ∇ ~cs

Because p~<sup>f</sup> is the hydrostatic pressure in this case, i.e., ∇p~<sup>f</sup> ¼ ~ρ<sup>f</sup> g, it then

which, physically, is the buoyancy acting on the solid phase. Now Eq. (18) becomes

which states that the weight of the solid particles is supported by the buoyancy and the interparticle forces. Therefore, the ensemble pressure of the solid phase can be

the contributions from other factors such as collision and enduring contact to the

1 � c and drop the symbols representing the ensemble averages hereinafter. The ensemble averaged equations governing the motion of the fluid phase are

¼ ð Þ 1 � c ρ<sup>f</sup> g þ ∇ � ð Þ� 1 � c pfI þ τ<sup>f</sup>

For brevity of the presentation, we shall denote simply cs by c as well cf by

þ ∇ � ð Þ 1 � c ρfu<sup>f</sup>

p~s

, with p~<sup>f</sup> being the total fluid pressure and p�<sup>s</sup> accounting for

g � ∇ ~cs

p~s � þ m~ <sup>s</sup>

ckρ<sup>k</sup> h i <sup>¼</sup> h i cku<sup>k</sup>

Þp~f � i � <sup>m</sup><sup>~</sup> <sup>s</sup>

h i ck

,

, � (18)

m~ <sup>s</sup> ¼ p~f∇~cs (19)

� � <sup>þ</sup> <sup>p</sup>~<sup>f</sup> <sup>∇</sup>~cs (20)

h i <sup>¼</sup> <sup>0</sup>, (21)

f h i � <sup>m</sup>:

(22)

h i � � <sup>þ</sup> <sup>∇</sup> � ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>0</sup>

<sup>k</sup> ¼ u<sup>k</sup> � u^<sup>k</sup> (15)

(16)

(17)

and

$$
\mathcal{L}\_k \nabla \cdot (\rho\_k \mathbf{u}\_k) = \nabla \cdot (\mathbf{c}\_k \rho\_k \mathbf{u}\_k) - (\rho\_k \mathbf{u}\_k) \cdot \nabla \mathbf{c}\_k. \tag{7}
$$

The resulting equations governing the ensemble average motion of phase k are [8]

$$\frac{\partial \langle \mathbf{c}\_k \rho\_k \rangle}{\partial t} + \nabla \cdot \langle \mathbf{c}\_k \rho\_k \mathbf{u}\_k \rangle = \langle \rho\_k \left( \mathbf{u}\_k - \mathbf{u}^i \right) \cdot \nabla \mathbf{c}\_k \rangle,\tag{8}$$

and

$$\frac{\partial \langle c\_k \rho\_k \mathbf{u}\_k \rangle}{\partial t} + \nabla \cdot \langle c\_k \rho\_k \mathbf{u}\_k \mathbf{u}\_k \rangle = \nabla \cdot \langle c\_k \mathbf{T}\_k \rangle + \langle c\_k \rho\_k \mathbf{g} \rangle + \tilde{\mathbf{m}}\_k \tag{9}$$

with

$$
\tilde{\mathbf{m}}\_k = \langle \rho\_k \mathbf{u}\_k \left( \mathbf{u}\_k - \mathbf{u}^i \right) - \mathbf{T}\_k \cdot \nabla c\_k \rangle,\tag{10}
$$

Note that ∇ck is not zero only on the interface of the region occupied by phase k (grain boundary). For the fluid-solid two-phase flows, the interface of phase k must satisfy the no-slip and no-flux conditions; therefore, <sup>u</sup><sup>k</sup> � <sup>u</sup><sup>i</sup> <sup>¼</sup> 0. As a result, the right-hand side of Eq. (8) is zero and

$$
\tilde{\mathbf{m}}\_k = - \langle \mathbf{T}\_k \cdot \nabla c\_k \rangle,\tag{11}
$$

which is the density of the interfacial force [8]. Physically, <sup>T</sup><sup>k</sup> � <sup>∇</sup>ck is the microscopic density of the force acting on a surface whose normal direction is defined by ∇ck.

After using Eq. (3) for T<sup>k</sup> in Eq. (9), the ensemble averaged equations can be further written in terms of the ensemble averaged qualities describing the motion of phase k as

$$\frac{\partial \tilde{\boldsymbol{c}} \tilde{\rho}\_k}{\partial t} + \nabla \cdot \left[ \tilde{\boldsymbol{c}} \tilde{\rho}\_k \hat{\mathbf{u}}\_k \right] = \mathbf{0} \tag{12}$$

and

$$\frac{\partial \tilde{\boldsymbol{\varepsilon}}\_{k} \tilde{\rho}\_{k} \hat{\mathbf{u}}\_{k}}{\partial t} + \nabla \cdot \left[ \tilde{\boldsymbol{\varepsilon}} \tilde{\rho}\_{k} \hat{\mathbf{u}}\_{k} \hat{\mathbf{u}}\_{k} \right] = \tilde{\boldsymbol{\varepsilon}} \tilde{\rho}\_{k} \mathbf{g} + \nabla \cdot \left[ \tilde{\boldsymbol{\varepsilon}} \left( \tilde{p}\_{k} \mathbf{I} + \tilde{\boldsymbol{\tau}}\_{k} \right) \right] + \nabla \cdot \left[ \tilde{\boldsymbol{\varepsilon}} \tilde{\boldsymbol{\tau}}\_{k}' \right] + \tilde{\mathbf{m}}\_{k}, \tag{13}$$

where ~ck ¼ h i ck is the volumetric concentration of phase k. Other ensemble averaged quantities used in Eqs. (12) and (13) to describe the motion of phase k at the macroscopic scale are density ~ρk, pressure p~k, stress tensor ~τk, and velocity u^k, defined by

$$\tilde{\rho}\_k = \frac{\langle c\_k \rho\_k \rangle}{\langle c\_k \rangle}, \quad \tilde{p}\_k = \frac{\langle c\_k p\_k \rangle}{\langle c\_k \rangle}, \quad \tilde{\tau}\_k = \frac{\langle c\_k \tau\_k \rangle}{\langle c\_k \rangle}, \quad \hat{\mathbf{u}}\_k = \frac{\langle c\_k \rho\_k \mathbf{u}\_k \rangle}{\langle c\_k \rho\_k \rangle} \tag{14}$$

and ~t 0 <sup>k</sup> represents the c-weighted ensemble average of microscopic momentum flux associated with the fluctuation of the velocity u<sup>k</sup> around the ensemble averaged velocity u^<sup>k</sup>

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

$$\tilde{\mathbf{r}}'\_{k} = -\frac{\left<\boldsymbol{c}\_{k}\rho\_{k}\mathbf{u}'\_{k}\mathbf{u}'\_{k}\right>}{\left<\boldsymbol{c}\_{k}\right>}, \quad \mathbf{u}'\_{k} = \mathbf{u}\_{k} - \hat{\mathbf{u}}\_{k} \tag{15}$$

For compressible materials ~ρ<sup>k</sup> is not a constant. However, for incompressible materials

$$\rho\_k = \frac{\langle c\_k \rho\_k \rangle}{\langle c\_k \rangle} \equiv \rho\_k, \quad \hat{\mathbf{u}}\_k = \frac{\langle c\_k \rho\_k \mathbf{u}\_k \rangle}{\langle c\_k \rho\_k \rangle} = \frac{\langle c\_k \mathbf{u}\_k \rangle}{\langle c\_k \rangle} \tag{16}$$

Now we examine the limiting case where the fluid-solid system is at its static state. Because the phase functions for the two phases satisfy cf þ cs ¼ 1, both phases are not moving, and m~ <sup>f</sup> þ m~ <sup>s</sup> ¼ 0, the governing equations reduce to

$$\mathbf{0} = \left(\mathbf{1} - \tilde{c}\_s\right)\tilde{\rho}\_f \mathbf{g} - \nabla \left[ \left(\mathbf{1} - \tilde{c}\_s\right)\tilde{p}\_f \right] - \mathbf{\tilde{m}}\_{\circ} \tag{17}$$

for the fluid phase, and

ck ∂ϕ<sup>k</sup>

<sup>∂</sup> ckρ<sup>k</sup> h i ∂t

<sup>∂</sup>h i ckρku<sup>k</sup> ∂t

the right-hand side of Eq. (8) is zero and

þ ∇ � ~c~ρku^ku^<sup>k</sup>

<sup>~</sup>ρ<sup>k</sup> <sup>¼</sup> ckρ<sup>k</sup> h i h i ck

and

are [8]

and

with

by ∇ck.

phase k as

<sup>∂</sup>~ck~ρku^<sup>k</sup> ∂t

defined by

and ~t 0

122

velocity u^<sup>k</sup>

and

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup>ckϕ<sup>k</sup>

<sup>∂</sup><sup>t</sup> � <sup>ϕ</sup><sup>k</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

∂ck

The resulting equations governing the ensemble average motion of phase k

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup>ckϕ<sup>k</sup> ∂t

<sup>þ</sup> <sup>∇</sup> � h i ckρku<sup>k</sup> <sup>¼</sup> <sup>ρ</sup><sup>k</sup> <sup>u</sup><sup>k</sup> � <sup>u</sup><sup>i</sup> � <sup>∇</sup>ck

<sup>m</sup><sup>~</sup> <sup>k</sup> <sup>¼</sup> <sup>ρ</sup>ku<sup>k</sup> <sup>u</sup><sup>k</sup> � <sup>u</sup><sup>i</sup> � <sup>T</sup><sup>k</sup> � <sup>∇</sup>ck

Note that ∇ck is not zero only on the interface of the region occupied by phase k (grain boundary). For the fluid-solid two-phase flows, the interface of phase k must satisfy the no-slip and no-flux conditions; therefore, <sup>u</sup><sup>k</sup> � <sup>u</sup><sup>i</sup> <sup>¼</sup> 0. As a result,

which is the density of the interfacial force [8]. Physically, <sup>T</sup><sup>k</sup> � <sup>∇</sup>ck is the microscopic density of the force acting on a surface whose normal direction is defined

> <sup>∂</sup>~c~ρ<sup>k</sup> ∂t

, <sup>p</sup>~<sup>k</sup> <sup>¼</sup> ckpk

 h i ck

After using Eq. (3) for T<sup>k</sup> in Eq. (9), the ensemble averaged equations can be further written in terms of the ensemble averaged qualities describing the motion of

<sup>¼</sup> <sup>~</sup>c~ρk<sup>g</sup> <sup>þ</sup> <sup>∇</sup> � <sup>~</sup><sup>c</sup> <sup>p</sup>~k<sup>I</sup> <sup>þ</sup> <sup>~</sup>τk<sup>Þ</sup> <sup>þ</sup> <sup>∇</sup> � <sup>~</sup>c~τ<sup>0</sup>

where ~ck ¼ h i ck is the volumetric concentration of phase k. Other ensemble averaged quantities used in Eqs. (12) and (13) to describe the motion of phase k at the macroscopic scale are density ~ρk, pressure p~k, stress tensor ~τk, and velocity u^k,

> , <sup>~</sup>τ<sup>k</sup> <sup>¼</sup> h i ckτ<sup>k</sup> h i ck

<sup>k</sup> represents the c-weighted ensemble average of microscopic momentum flux

associated with the fluctuation of the velocity u<sup>k</sup> around the ensemble averaged

ck∇ � ð Þ¼ ρku<sup>k</sup> ∇ � ð Þ� ckρku<sup>k</sup> ð Þ� ρku<sup>k</sup> ∇ck: (7)

<sup>þ</sup> <sup>∇</sup> � h i ckρkuku<sup>k</sup> <sup>¼</sup> <sup>∇</sup> � h i ckT<sup>k</sup> <sup>þ</sup> ckρk<sup>g</sup> <sup>þ</sup> <sup>m</sup><sup>~</sup> <sup>k</sup> (9)

, (10)

m~ <sup>k</sup> ¼ �h i T<sup>k</sup> � ∇ck , (11)

<sup>þ</sup> <sup>∇</sup> � <sup>~</sup>c~ρku^k� ¼ <sup>0</sup> (12)

k <sup>þ</sup> <sup>m</sup><sup>~</sup> k, (13)

, <sup>u</sup>^<sup>k</sup> <sup>¼</sup> h i ckρku<sup>k</sup>

ckρ<sup>k</sup> h i (14)

<sup>þ</sup> <sup>ϕ</sup>ku<sup>i</sup> � <sup>∇</sup>ck, (6)

, (8)

$$\mathbf{0} = \tilde{c}\_s \tilde{\rho}\_s \mathbf{g} - \nabla \left[ \tilde{c}\_s \tilde{p}\_s \right] + \mathbf{\tilde{m}}\_s,\tag{18}$$

for the solid phase.

Because p~<sup>f</sup> is the hydrostatic pressure in this case, i.e., ∇p~<sup>f</sup> ¼ ~ρ<sup>f</sup> g, it then follows that

$$
\tilde{\mathbf{m}}\_s = \tilde{p}\_f \nabla \tilde{c}\_s \tag{19}
$$

which, physically, is the buoyancy acting on the solid phase. Now Eq. (18) becomes

$$\mathbf{0} = \tilde{c}\_s \tilde{\rho}\_s \mathbf{g} - \nabla \left[ \tilde{c}\_s \tilde{p}\_s \right] + \tilde{p}\_f \nabla \tilde{c}\_s \tag{20}$$

which states that the weight of the solid particles is supported by the buoyancy and the interparticle forces. Therefore, the ensemble pressure of the solid phase can be written as p~<sup>s</sup> ¼ p~<sup>f</sup> þ p�<sup>s</sup> , with p~<sup>f</sup> being the total fluid pressure and p�<sup>s</sup> accounting for the contributions from other factors such as collision and enduring contact to the ensemble averaged pressure.

For brevity of the presentation, we shall denote simply cs by c as well cf by 1 � c and drop the symbols representing the ensemble averages hereinafter. The ensemble averaged equations governing the motion of the fluid phase are

$$\frac{\partial(\mathbf{1}-\boldsymbol{c})\rho\_f}{\partial t} + \nabla \cdot \left[ (\mathbf{1}-\boldsymbol{c})\rho\_f \mathbf{u}\_f \right] = \mathbf{0},\tag{21}$$

and

$$\begin{split} \frac{\partial(\mathbf{1}-\boldsymbol{c})\rho\_{f}\mathbf{u}\_{f}}{\partial t} &\quad + \nabla \cdot \left[ (\mathbf{1}-\boldsymbol{c})\rho\_{f}\mathbf{u}\_{f}\mathbf{u}\_{f} \right] \\ &= (\mathbf{1}-\boldsymbol{c})\rho\_{f}\mathbf{g} + \nabla \cdot \left[ (\mathbf{1}-\boldsymbol{c}) \left( -p\_{f}\mathbf{I} + \tau\_{f} \right) \right] + \nabla \cdot \left[ (\mathbf{1}-\boldsymbol{c})\tau\_{f}' \right] - \mathbf{m}. \end{split} \tag{22}$$

The ensemble averaged equations governing the motion of the solid phase are

$$\frac{\partial \varepsilon \rho\_s}{\partial t} + \nabla \cdot (c \rho\_s \mathbf{u}\_s) = \mathbf{0},\tag{23}$$

The averaged equations for the mean flow fields of the two phases are obtained by taking the following steps: (i) substituting Eq. (25) with Eq. (26) in Eqs. (22) and (24), (ii) substituting Eq. (27) in the equations obtained at step (i), and (iii) taking average of the equations obtained at step (ii) to obtain the following

þ ∇ � ρfð Þ 1 � c u<sup>f</sup>

<sup>f</sup> þ τ<sup>00</sup> f � � � <sup>c</sup>ρ<sup>s</sup>

<sup>f</sup> ¼ �ρfu<sup>00</sup>

þ∇ � ρscusu<sup>s</sup> ½ �¼ ρscg � c∇pf � c00∇p<sup>00</sup>

<sup>s</sup> ¼ �ρsu<sup>00</sup>

It is remarked here that the terms 1ð Þ � ~c ∇p~<sup>f</sup> in Eq. (30) and ~c∇p~<sup>f</sup> in Eq. (33) have

In order to close these averaged equations, closure models are required for the

can be neglected based on an analysis of their orders of magnitude by Drew [12].

<sup>f</sup> ¼ � <sup>ν</sup>ft σc

where νft is the eddy viscosity and σ<sup>c</sup> is the Schmidt number, which represents the ratio of the eddy viscosity of the fluid phase to the eddy diffusivity of the solid

> <sup>f</sup> þ τ<sup>00</sup> f

For brevity of the presentation, the symbols representing Favre averages are dropped hereinafter, and the final equations governing the conservation of mass

cu<sup>00</sup>

phase. Furthermore, the following approximations are introduced:

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> ∂t

<sup>s</sup> u<sup>00</sup>

<sup>f</sup> þ τ<sup>00</sup> f � �, <sup>c</sup>u<sup>00</sup>

<sup>f</sup> is approximated by the following gradient transport hypotheses:

<sup>s</sup> þ τ<sup>00</sup> s � � <sup>þ</sup> <sup>c</sup>ρ<sup>s</sup>

<sup>s</sup> being defined by

τ00

<sup>f</sup> being defined by

τ00

∂ρsc ∂t

þ∇ � c τ<sup>s</sup> þ τ<sup>0</sup>

been obtained by using the expression for m given in Eq. (25).

<sup>s</sup> þ τ<sup>00</sup> s � �, ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>f</sup> <sup>þ</sup> <sup>τ</sup><sup>0</sup>

h i <sup>¼</sup> <sup>ρ</sup>fð Þ <sup>1</sup> � <sup>c</sup> <sup>g</sup> � ð Þ <sup>1</sup> � <sup>c</sup> <sup>∇</sup>pf <sup>þ</sup> <sup>c</sup>00∇p<sup>00</sup>

<sup>f</sup> u<sup>00</sup>

u<sup>f</sup> � u<sup>s</sup> τp

h i <sup>¼</sup> <sup>0</sup>, (29)

� ρs τp

þ ∇ � ρscu<sup>s</sup> ½ �¼ 0, (32)

þ ρs τp

u<sup>f</sup> � u<sup>s</sup> τp

cu<sup>00</sup> f � �,

<sup>f</sup> , (31)

<sup>f</sup> � ∇cps

cu<sup>00</sup> f � �,

<sup>s</sup> (34)

<sup>f</sup> , and c00∇p<sup>00</sup>

∇c (35)

� � <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>f</sup> , cps <sup>¼</sup> cps (36)

þ ∇ � ρfð Þ 1 � c u<sup>f</sup> ¼ 0, (37)

f

(30)

(33)

<sup>f</sup> . The last term

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> ∂t

þ∇ � ρfð Þ 1 � c ufu<sup>f</sup>

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

þ∇ � ð Þ 1 � c τ<sup>f</sup> þ τ<sup>0</sup>

equations:

and

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> <sup>u</sup><sup>f</sup> ∂t

for the fluid phase, with τ<sup>00</sup>

for the solid phase, with τ<sup>00</sup>

following terms: c τ<sup>s</sup> þ τ<sup>0</sup>

The term cu<sup>00</sup>

c τ<sup>s</sup> þ τ<sup>0</sup>

125

<sup>s</sup> þ τ<sup>00</sup> s

and momentum of each phase are

� � <sup>¼</sup> <sup>c</sup>τs, ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>f</sup> <sup>þ</sup> <sup>τ</sup><sup>0</sup>

∂ρscu<sup>s</sup> ∂t

and

$$\frac{\partial \mathcal{L} \rho\_s \mathbf{u}\_s}{\partial t} + \nabla \cdot \left[ c \rho\_f \mathbf{u}\_s \mathbf{u}\_s \right] = \rho\_s \varepsilon \mathbf{g} + \nabla \cdot \left[ c \left( -p\_f \mathbf{I} - p\_s \mathbf{I} + \boldsymbol{\tau}\_s \right) \right] + \nabla \cdot \left( c \boldsymbol{\tau}\_s' \right) + \mathbf{m}.\tag{24}$$

where ps denotes the contributions from interparticle interactions such as collision and enduring contact to the ensemble averaged pressure of the solid phase.

To close the equations for the fluid and solid phases, closure models are needed for τ<sup>0</sup> s , τ<sup>0</sup> <sup>f</sup> , τs, τ<sup>f</sup> , ps , and m.

It is remarked here that the definitions of the ensemble averages given in Eq. (14) do not consider the contribution from the correlations between the fluctuations of the velocities and the fluctuations of phase functions at microscopic scale; therefore, the effects of turbulent dispersion are not directly included in the ensemble averaged equations describing the motion of the each phase. In the literature, two approaches have been used to consider the turbulent dispersion: (i) considering the correlation between the fluctuations of h i ck and u <sup>f</sup> associated with the turbulent flow [9] and (ii) including a term in the model for m to account for the turbulent dispersion [8]. This chapter considers the turbulent dispersion using the first approach in the next section by taking another Favre averaging operation.

In the absence of the turbulent dispersion from m, the interphase force m should include the so-called general buoyancy pf∇c and a component f which includes drag force, inertial force, and lift force

$$\mathbf{m} = \mathbf{f} + p\_f \nabla c \equiv \mathbf{f} - c \nabla p\_f + \nabla \left( c p\_f \right). \tag{25}$$

This expression for m has been derived by [11] using a control volume/surface approach. For most fluid-solid two-phase geophysical flows, the drag force dominates f [9] and thus f can be modeled by

$$\mathbf{f} = c\rho\_s \frac{\mathbf{u}\_f - \mathbf{u}\_s}{\tau\_p},\tag{26}$$

where τ<sup>p</sup> is the so-called particle response time (i.e., a relaxation time of the particle to respond the surrounding flow). As expected, the particle response time should be related to drag coefficient and grain Reynolds number.

#### 2.2 Favre averaged equations

The volumetric concentration and the velocities can be written as

$$\mathcal{L} = \overline{\mathfrak{c}} + \mathfrak{c}'', \quad p\_f = \overline{p}\_f + p\_f'', \quad \mathbf{u}\_f = \overline{\mathfrak{u}}\_f + \mathbf{u}\_f'', \quad \mathbf{u}\_t = \overline{\mathfrak{u}}\_t + \mathbf{u}\_t'',\tag{27}$$

where the Favre averages are defined as

$$\overline{\rho\_s} = \frac{\overline{c\rho\_s}}{\overline{c}}, \quad \overline{\rho\_f} = \frac{\overline{(1-c)\rho\_f}}{1-\overline{c}}, \\ \overline{\mathbf{u}\_t} = \frac{\overline{c\rho\_s \mathbf{u}\_t}}{\overline{c\rho\_s}}, \\ \overline{\mathbf{u}\_f} = \frac{\overline{(1-c)\rho\_f \mathbf{u}\_f}}{\overline{(1-c)\rho\_f}}, \tag{28}$$

and the overline stands for an integration with respect to time over a time scale longer than small-scale turbulent fluctuations but shorter than the variation of the mean flow field.

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

The averaged equations for the mean flow fields of the two phases are obtained by taking the following steps: (i) substituting Eq. (25) with Eq. (26) in Eqs. (22) and (24), (ii) substituting Eq. (27) in the equations obtained at step (i), and (iii) taking average of the equations obtained at step (ii) to obtain the following equations:

$$\frac{\partial \overline{\rho}\_f(\mathbf{1} - \overline{\mathbf{c}})}{\partial t} + \nabla \cdot \left[ \overline{\rho}\_f(\mathbf{1} - \overline{\mathbf{c}}) \overline{\mathbf{u}}\_f \right] = \mathbf{0},\tag{29}$$

$$\begin{split} \frac{\partial \overline{\rho}\_{f} (\mathbf{1} - \overline{c}) \overline{\mathbf{u}}\_{f}}{\partial t} &+ \nabla \cdot \left[ \overline{\rho}\_{f} (\mathbf{1} - \overline{c}) \overline{\mathbf{u}}\_{f} \overline{\mathbf{u}}\_{f} \right] = \overline{\rho}\_{f} (\mathbf{1} - \overline{c}) \mathbf{g} - (\mathbf{1} - \overline{c}) \nabla \overline{p}\_{f} + \overline{c'' \nabla p\_{f}''} \\ &+ \nabla \cdot \left( \overline{\mathbf{1} - c} \right) \left( \tau\_{f} + \tau\_{f}' + \tau\_{f}'' \right) - \overline{c} \overline{\rho}\_{s} \frac{\overline{\mathbf{u}}\_{f} - \overline{\mathbf{u}}\_{s}}{\tau\_{p}} - \frac{\overline{\rho}\_{s}}{\tau\_{p}} \left( \overline{c \mathbf{u}\_{f}''} \right), \end{split} \tag{30}$$

for the fluid phase, with τ<sup>00</sup> <sup>f</sup> being defined by

$$
\pi\_f'' = -\rho\_f \mathbf{u}\_f'' \mathbf{u}\_f'',\tag{31}
$$

and

The ensemble averaged equations governing the motion of the solid phase are

þ ∇ � ð Þ¼ cρsu<sup>s</sup> 0, (23)

þ ∇ � cτ<sup>0</sup> s

� � <sup>þ</sup> <sup>m</sup>: (24)

I þ τ<sup>s</sup>

� �

: (25)

<sup>s</sup> , (27)

, (28)

, (26)

<sup>f</sup> , u<sup>s</sup> ¼ u<sup>s</sup> þ u<sup>00</sup>

, <sup>u</sup><sup>f</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>ρ</sup>fu<sup>f</sup> ð Þ 1 � c ρ<sup>f</sup>

h i � �

∂cρ<sup>s</sup> ∂t

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

¼ ρscg þ ∇ � c �pfI � ps

where ps denotes the contributions from interparticle interactions such as collision and enduring contact to the ensemble averaged pressure of the solid phase.

It is remarked here that the definitions of the ensemble averages given in Eq. (14) do not consider the contribution from the correlations between the fluctuations of the velocities and the fluctuations of phase functions at microscopic scale; therefore, the effects of turbulent dispersion are not directly included in the ensemble averaged equations describing the motion of the each phase. In the literature, two approaches have been used to consider the turbulent dispersion: (i) considering the correlation between the fluctuations of h i ck and u <sup>f</sup> associated with the turbulent flow [9] and (ii) including a term in the model for m to account for the turbulent dispersion [8]. This chapter considers the turbulent dispersion using the first approach in the next section by taking another Favre averaging

In the absence of the turbulent dispersion from m, the interphase force m

m ¼ f þ pf∇c � f � c∇pf þ ∇ cpf

f ¼ cρ<sup>s</sup>

should be related to drag coefficient and grain Reynolds number.

<sup>00</sup>, pf ¼ pf þ p<sup>00</sup>

<sup>c</sup> , <sup>ρ</sup><sup>f</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>ρ</sup><sup>f</sup>

The volumetric concentration and the velocities can be written as

This expression for m has been derived by [11] using a control volume/surface approach. For most fluid-solid two-phase geophysical flows, the drag force domi-

where τ<sup>p</sup> is the so-called particle response time (i.e., a relaxation time of the particle to respond the surrounding flow). As expected, the particle response time

u<sup>f</sup> � u<sup>s</sup> τp

<sup>f</sup> , u<sup>f</sup> ¼ u<sup>f</sup> þ u<sup>00</sup>

cρs

<sup>1</sup> � <sup>c</sup> , <sup>u</sup><sup>s</sup> <sup>¼</sup> <sup>c</sup>ρsu<sup>s</sup>

and the overline stands for an integration with respect to time over a time scale longer than small-scale turbulent fluctuations but shorter than the variation of the

should include the so-called general buoyancy pf∇c and a component f

which includes drag force, inertial force, and lift force

nates f [9] and thus f can be modeled by

2.2 Favre averaged equations

c ¼ c þ c

<sup>ρ</sup><sup>s</sup> <sup>¼</sup> <sup>c</sup>ρ<sup>s</sup>

mean flow field.

124

where the Favre averages are defined as

To close the equations for the fluid and solid phases, closure models are needed

and

for τ<sup>0</sup> s , τ<sup>0</sup>

operation.

∂cρsu<sup>s</sup> ∂t

þ ∇ � cρfusu<sup>s</sup> h i

, and m.

<sup>f</sup> , τs, τ<sup>f</sup> , ps

$$\frac{\partial \overline{\rho}\_{\iota} \overline{\mathcal{L}}}{\partial t} + \nabla \cdot [\overline{\rho}\_{\iota} \overline{\epsilon} \overline{\mathbf{u}}\_{\iota}] = \mathbf{0},\tag{32}$$

$$\begin{split} \frac{\partial \overline{\rho\_{s}} \overline{c \mathbf{u}\_{s}}}{\partial t} &+ \nabla \cdot [\overline{\rho\_{s}} \overline{c \mathbf{u}\_{s}} \overline{\mathbf{u}\_{s}}] = \overline{\rho\_{s}} \overline{c} \mathbf{g} - \overline{c} \nabla \overline{p\_{f}} - \overline{c'' \nabla p\_{f}''} - \nabla \overline{c \mathbf{p}\_{s}} \\ &+ \nabla \cdot \overline{c \left(\tau\_{s} + \tau\_{s}' + \tau\_{s}''\right)} + \overline{c \rho\_{s}} \frac{\overline{\mathbf{u}\_{f}} - \overline{\mathbf{u}}\_{s}}{\tau\_{p}} + \frac{\overline{\rho}\_{s}}{\tau\_{p}} \left(\overline{c \mathbf{u}\_{f}''}\right), \end{split} \tag{33}$$

for the solid phase, with τ<sup>00</sup> <sup>s</sup> being defined by

$$\mathbf{r}\_{\mathbf{s}}^{\prime\prime} = -\rho\_{\mathbf{s}} \mathbf{u}\_{\mathbf{s}}^{\prime\prime} \mathbf{u}\_{\mathbf{s}}^{\prime\prime} \tag{34}$$

It is remarked here that the terms 1ð Þ � ~c ∇p~<sup>f</sup> in Eq. (30) and ~c∇p~<sup>f</sup> in Eq. (33) have been obtained by using the expression for m given in Eq. (25).

In order to close these averaged equations, closure models are required for the following terms: c τ<sup>s</sup> þ τ<sup>0</sup> <sup>s</sup> þ τ<sup>00</sup> s � �, ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>f</sup> <sup>þ</sup> <sup>τ</sup><sup>0</sup> <sup>f</sup> þ τ<sup>00</sup> f � �, <sup>c</sup>u<sup>00</sup> <sup>f</sup> , and c00∇p<sup>00</sup> <sup>f</sup> . The last term can be neglected based on an analysis of their orders of magnitude by Drew [12]. The term cu<sup>00</sup> <sup>f</sup> is approximated by the following gradient transport hypotheses:

$$
\overline{\boldsymbol{\sigma} \mathbf{u}\_f''} = -\frac{\nu\_{\boldsymbol{\mathcal{H}}}}{\sigma\_\varepsilon} \nabla \boldsymbol{\mathcal{E}} \tag{35}
$$

where νft is the eddy viscosity and σ<sup>c</sup> is the Schmidt number, which represents the ratio of the eddy viscosity of the fluid phase to the eddy diffusivity of the solid phase. Furthermore, the following approximations are introduced:

$$
\overline{c\left(\tau\_s + \tau\_s' + \tau\_s''\right)} = \overline{c\tau\_s}, \quad \overline{(1-c)\left(\tau\_f + \tau\_f' + \tau\_f''\right)} = (1-\overline{c})\overline{\tau}\_f, \quad \overline{c\overline{p}\_s} = \overline{c\overline{p}\_s} \tag{36}
$$

For brevity of the presentation, the symbols representing Favre averages are dropped hereinafter, and the final equations governing the conservation of mass and momentum of each phase are

$$\frac{\partial \rho\_f(\mathbf{1} - \mathbf{c})}{\partial t} + \nabla \cdot \rho\_f(\mathbf{1} - \mathbf{c}) \mathbf{u}\_f = \mathbf{0},\tag{37}$$

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\begin{split} \frac{\partial \rho\_f(\mathbf{1} - c)\mathbf{u}\_f}{\partial t} &\quad + \nabla \cdot \left[\rho\_f(\mathbf{1} - c)\mathbf{u}\_f \mathbf{u}\_f\right] \\ &= \rho\_f(\mathbf{1} - c)\mathbf{g} - (\mathbf{1} - c)\nabla p\_f + \nabla \cdot (\mathbf{1} - c)\tau\_f \\ &\quad - \left\{c\rho\_s \frac{\mathbf{u}\_f - \mathbf{u}\_s}{\tau\_p} + \frac{\rho\_s}{\tau\_p} \frac{\nu\_{\text{ft}}}{\sigma\_c} \nabla c\right\}, \end{split} \tag{38}$$

for the fluid phase and

$$\frac{\partial \rho\_s \mathbf{c}}{\partial t} + \nabla \cdot \rho\_s \mathbf{c} \mathbf{u}\_s = \mathbf{0},\tag{39}$$

with ϵ being the turbulent dissipation of the fluid phase to be provide by solving the <sup>k</sup> � <sup>ϵ</sup> equation. The coefficient <sup>f</sup> <sup>μ</sup> <sup>¼</sup> exp �3:4=ð Þ <sup>1</sup> <sup>þ</sup> Ret=<sup>50</sup> <sup>2</sup> h i represents the low-

The equations governing k and ϵ are similar to those for clear water [15]

h i <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>t</sup><sup>f</sup> : <sup>∇</sup>u<sup>f</sup> � <sup>ρ</sup>fð Þ <sup>1</sup> � <sup>c</sup> <sup>ϵ</sup>

� ρ<sup>s</sup> � ρ<sup>f</sup> � � <sup>ν</sup><sup>t</sup>

f σc

∇c � g þ

<sup>k</sup> <sup>C</sup>ϵ<sup>1</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>1</sup> � <sup>c</sup> <sup>τ</sup><sup>f</sup> : <sup>∇</sup>u<sup>f</sup> � <sup>C</sup>ϵ<sup>2</sup> <sup>f</sup> <sup>2</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> <sup>ϵ</sup> h i

> f f σc

∇c � g þ

( )

=ν<sup>f</sup> ϵ. The coefficient C<sup>μ</sup> is usually

2ρscð Þ 1 � α k τp

,

2ρscð Þ 1 � α k τp

, (47)

(45)

9 = ;,

(46)

(48)

Reynolds-number correction with Re<sup>t</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

þ ∇ � ρfð Þ 1 � c u<sup>f</sup> k

νt f σc

þ ∇ ρfð Þ 1 � c u<sup>f</sup> ϵ

νt f σϵ

ð Þ 1 � c k " #

h i <sup>¼</sup> <sup>ϵ</sup>

ð Þ 1 � c ϵ " #

� ϵ k

<sup>α</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>τ</sup><sup>p</sup>

<sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>c</sup>rcp c � �<sup>1</sup> 3 � 1 � �<sup>d</sup> <sup>ρ</sup><sup>s</sup>

mean free dispersion distance to the diameter of the solid particle.

min τ<sup>l</sup> ð Þ ; τ<sup>c</sup> � ��<sup>1</sup>

> ps � �<sup>1</sup>=<sup>2</sup>

where τ<sup>l</sup> ¼ 0:165k=ϵ is a time scale for the turbulent flow and τ<sup>c</sup> is a time scale for

with crcp being the random close packing fraction and d being the particle diameter. <sup>c</sup>rcp is 0.634 for spheres [17]. The term <sup>c</sup>rcp=<sup>c</sup> � �<sup>1</sup>=<sup>3</sup> � 1 is related to the ratio of the

It is remarked here that the presence of solid particles in the turbulent flow may either enhance (for large particles) or reduce (for small particles) the turbulence [18]. The k � ϵ model given here can only reflect the reduction of turbulence and thus is not suitable for problems with large particles. Other turbulence models [7, 18] include a term describing the enhancement of turbulence; however, including that term in the present model may induce numerical instability in some cases.

Cϵ<sup>3</sup> ρ<sup>s</sup> � ρ<sup>f</sup>

8 < :

where coefficients Cϵ1, Cϵ2, σϵ, σk, and f <sup>2</sup> are model parameters, whose values can be taken the same as those in the k � ϵ model for clear fluid under low-Reynoldsnumber conditions [15]. There are two terms inside the curly brackets, and both terms account for the turbulence modulation by the presence of particles: the first term is associated with the general buoyancy, and the second term is due to the correlation of the fluctuating velocities of solid and fluid phases. Cϵ<sup>3</sup> ¼ 1 is usually adopted in the literature [28]; however, it is remarked that the value of Cϵ<sup>3</sup> is not well understood at the present and a sensitivity test to understand how the value of this Cϵ<sup>3</sup> on the simulation results is recommended. The parameter α reflects the correlation between the solid-phase and fluid-phase turbulent motions and is

� � ν

þ∇ � ρ<sup>f</sup>

þ∇ � ρ<sup>f</sup>

particle collisions given by [16]

assumed to be a constant.

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> <sup>k</sup> ∂t

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> <sup>ϵ</sup> ∂t

and

given by

127

$$\begin{split} \frac{\partial \rho\_{s} c \mathbf{u}\_{s}}{\partial t} &\quad + \nabla \cdot (\rho\_{s} c \mathbf{u}\_{s} \mathbf{u}\_{s}) \\ &= \rho\_{s} c \mathbf{g} - c \nabla p\_{f} - \nabla \left( c p\_{s} \right) + \nabla \cdot c \mathbf{\tau}\_{s} \\ &\quad + \left\{ c \rho\_{s} \frac{\left( \mathbf{u}\_{f} - \mathbf{u}\_{s} \right)}{\tau\_{p}} - \frac{\rho\_{s}}{\tau\_{p}} \frac{\nu\_{f}}{\sigma\_{c}} \nabla c \right\}, \end{split} \tag{40}$$

for the solid phase.

#### 3. Closure models

#### 3.1 Stresses for the fluid phase

The stress tensor for the fluid phase τ<sup>f</sup> includes two parts: a part for the viscous stress, τ<sup>v</sup> <sup>f</sup> , and the other part for the turbulent Reynolds stress, τ<sup>t</sup> f

$$
\pi\_f = \pi\_f^\nu + \pi\_f^t \tag{41}
$$

The viscous stress tensor τ<sup>v</sup> <sup>f</sup> is usually computed by

$$\boldsymbol{\sigma}\_{f}^{\boldsymbol{\nu}} = -\rho\_{f} \left(\frac{2}{3} \boldsymbol{\nu}\_{f} \nabla \cdot \mathbf{u}\_{f}\right) \mathbf{I} + 2\rho\_{f} \boldsymbol{\nu}\_{f} \mathbf{D}\_{f} \tag{42}$$

where ν<sup>f</sup> is the kinematic viscosity of the fluid phase and D<sup>f</sup> ¼ ∇u<sup>f</sup> þ ∇u<sup>f</sup> � �<sup>T</sup> h i=2, where the superscript T denotes a transpose. Some studies [13] suggested modifying ν<sup>f</sup> to consider the effect of the solid phase; other studies [14], however, obtained satisfactory results even without considering this effect.

The stress tensor τ<sup>t</sup> <sup>f</sup> is related to the turbulent characteristics, which need to be provided by solving a turbulent closure model such as k � ϵ or k � ω model. For a <sup>k</sup> � <sup>ϵ</sup> model with low-Reynolds-number correction [15], <sup>τ</sup><sup>t</sup> <sup>f</sup> can be computed by

$$\sigma\_f^t = -\rho\_f \left(\frac{2}{3}k + \frac{2}{3}\nu\_f^t \nabla \cdot \mathbf{u}\_f\right) \mathbf{I} + 2\rho\_f \nu\_f^t \mathbf{D}\_f \tag{43}$$

where k is the turbulence kinetic energy and ν<sup>t</sup> <sup>f</sup> is the eddy viscosity of the fluid phase, given by

$$
\nu\_f^t = f\_\mu \mathbf{C}\_\mu \mathbf{k}^2 / \mathbf{e} \tag{44}
$$

with ϵ being the turbulent dissipation of the fluid phase to be provide by solving the <sup>k</sup> � <sup>ϵ</sup> equation. The coefficient <sup>f</sup> <sup>μ</sup> <sup>¼</sup> exp �3:4=ð Þ <sup>1</sup> <sup>þ</sup> Ret=<sup>50</sup> <sup>2</sup> h i represents the low-Reynolds-number correction with Re<sup>t</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> =ν<sup>f</sup> ϵ. The coefficient C<sup>μ</sup> is usually assumed to be a constant.

The equations governing k and ϵ are similar to those for clear water [15]

$$\begin{split} \frac{d\rho\_f(\mathbf{1}-c)k}{\partial t} + \nabla \cdot \left[\rho\_f(\mathbf{1}-c)\mathbf{u}\_f k\right] &= (\mathbf{1}-c)\mathbf{t}\_f : \nabla \mathbf{u}\_f - \rho\_f(\mathbf{1}-c)\mathbf{c} \\ + \nabla \cdot \left[\rho\_f \frac{\nu\_f^t}{\sigma\_c}(\mathbf{1}-c)k\right] &- \left\{ \left(\rho\_s - \rho\_f\right) \frac{\nu\_f^t}{\sigma\_c} \nabla c \cdot \mathbf{g} + \frac{2\rho\_c c (\mathbf{1}-a)k}{\tau\_p} \right\}, \end{split} \tag{45}$$

and

<sup>∂</sup>ρfð Þ <sup>1</sup> � <sup>c</sup> <sup>u</sup><sup>f</sup> ∂t

for the fluid phase and

for the solid phase.

3. Closure models

stress, τ<sup>v</sup>

3.1 Stresses for the fluid phase

The viscous stress tensor τ<sup>v</sup>

The stress tensor τ<sup>t</sup>

phase, given by

126

þ∇ � ρfð Þ 1 � c ufu<sup>f</sup> h i

> u<sup>f</sup> � u<sup>s</sup> τp

� cρ<sup>s</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

∂ρsc ∂t

þ cρ<sup>s</sup>

þ ∇ � ρ<sup>s</sup> ð Þ cusu<sup>s</sup>

¼ ρscg � c∇pf � ∇ cps

u<sup>f</sup> � u<sup>s</sup> � � τp

� ρs τp νft σc ∇c

� �

The stress tensor for the fluid phase τ<sup>f</sup> includes two parts: a part for the viscous

<sup>f</sup> <sup>þ</sup> <sup>τ</sup><sup>t</sup>

<sup>f</sup> , and the other part for the turbulent Reynolds stress, τ<sup>t</sup>

τv <sup>f</sup> ¼ �ρ<sup>f</sup>

satisfactory results even without considering this effect.

<sup>k</sup> � <sup>ϵ</sup> model with low-Reynolds-number correction [15], <sup>τ</sup><sup>t</sup>

2 3 k þ 2 3 νt <sup>f</sup>∇ � u<sup>f</sup> � �

νt

τt <sup>f</sup> ¼ �ρ<sup>f</sup>

where k is the turbulence kinetic energy and ν<sup>t</sup>

<sup>τ</sup><sup>f</sup> <sup>¼</sup> <sup>τ</sup><sup>v</sup>

<sup>f</sup> is usually computed by

νf∇ � u<sup>f</sup> � �

where the superscript T denotes a transpose. Some studies [13] suggested modifying ν<sup>f</sup> to consider the effect of the solid phase; other studies [14], however, obtained

provided by solving a turbulent closure model such as k � ϵ or k � ω model. For a

<sup>f</sup> <sup>¼</sup> <sup>f</sup> <sup>μ</sup>Cμk<sup>2</sup>

<sup>f</sup> is related to the turbulent characteristics, which need to be

<sup>I</sup> <sup>þ</sup> <sup>2</sup>ρ<sup>f</sup> <sup>ν</sup><sup>t</sup>

2 3

where ν<sup>f</sup> is the kinematic viscosity of the fluid phase and D<sup>f</sup> ¼ ∇u<sup>f</sup> þ ∇u<sup>f</sup>

∂ρscu<sup>s</sup> ∂t

¼ ρfð Þ 1 � c g � ð Þ 1 � c ∇pf þ ∇ � ð Þ 1 � c τ<sup>f</sup>

,

� � <sup>þ</sup> <sup>∇</sup> � <sup>c</sup>τ<sup>s</sup>

þ ∇ � ρscu<sup>s</sup> ¼ 0, (39)

,

f

<sup>f</sup> (41)

I þ 2ρ<sup>f</sup> νfD<sup>f</sup> (42)

� �<sup>T</sup> h i

<sup>f</sup> can be computed by

<sup>f</sup>D<sup>f</sup> (43)

<sup>f</sup> is the eddy viscosity of the fluid

=ϵ (44)

=2,

(38)

(40)

þ ρs τp νft σc ∇c

� �

$$\begin{split} \frac{\partial \rho\_{\mathcal{f}}(\mathbf{1} - c)\mathbf{c}}{\partial t} + \nabla \left[\rho\_{\mathcal{f}}(\mathbf{1} - c)\mathbf{u}\_{\mathcal{f}}\mathbf{c}\right] &= \frac{\mathbf{c}}{k} \left[\mathbf{C}\_{\epsilon1}f\_{1}(\mathbf{1} - c)\boldsymbol{\tau}\_{\mathcal{f}} : \nabla \mathbf{u}\_{\mathcal{f}} - \mathbf{C}\_{\epsilon2}f\_{2}\rho\_{\mathcal{f}}(\mathbf{1} - c)\mathbf{c}\right] \\ &+ \nabla \cdot \left[\rho\_{\mathcal{f}}\frac{\nu\_{\mathcal{f}}^{\prime}}{\sigma\_{\epsilon}}(\mathbf{1} - c)\mathbf{c}\right] - \frac{\epsilon}{k}\mathbf{C}\_{\epsilon3}\left\{ \left(\rho\_{\mathcal{s}} - \rho\_{\mathcal{f}}\right)\frac{\nu\_{\mathcal{f}}^{\prime}}{\sigma\_{\mathcal{c}}}\nabla \mathbf{c} \cdot \mathbf{g} + \frac{2\rho\_{\mathcal{s}}\mathbf{c}(\mathbf{1} - a)\mathbf{k}}{\tau\_{\mathcal{p}}} \right\}, \end{split} \tag{45}$$

where coefficients Cϵ1, Cϵ2, σϵ, σk, and f <sup>2</sup> are model parameters, whose values can be taken the same as those in the k � ϵ model for clear fluid under low-Reynoldsnumber conditions [15]. There are two terms inside the curly brackets, and both terms account for the turbulence modulation by the presence of particles: the first term is associated with the general buoyancy, and the second term is due to the correlation of the fluctuating velocities of solid and fluid phases. Cϵ<sup>3</sup> ¼ 1 is usually adopted in the literature [28]; however, it is remarked that the value of Cϵ<sup>3</sup> is not well understood at the present and a sensitivity test to understand how the value of this Cϵ<sup>3</sup> on the simulation results is recommended. The parameter α reflects the correlation between the solid-phase and fluid-phase turbulent motions and is given by

$$a = \left(1 + \frac{\tau\_p}{\min(\tau\_l, \tau\_c)}\right)^{-1},\tag{47}$$

where τ<sup>l</sup> ¼ 0:165k=ϵ is a time scale for the turbulent flow and τ<sup>c</sup> is a time scale for particle collisions given by [16]

$$\pi\_c = \left[ \left( \frac{c\_{\rm rep}}{c} \right)^{\dagger} - 1 \right] d \left( \frac{\rho\_s}{p\_s} \right)^{1/2} \tag{48}$$

with crcp being the random close packing fraction and d being the particle diameter. <sup>c</sup>rcp is 0.634 for spheres [17]. The term <sup>c</sup>rcp=<sup>c</sup> � �<sup>1</sup>=<sup>3</sup> � 1 is related to the ratio of the mean free dispersion distance to the diameter of the solid particle.

It is remarked here that the presence of solid particles in the turbulent flow may either enhance (for large particles) or reduce (for small particles) the turbulence [18]. The k � ϵ model given here can only reflect the reduction of turbulence and thus is not suitable for problems with large particles. Other turbulence models [7, 18] include a term describing the enhancement of turbulence; however, including that term in the present model may induce numerical instability in some cases.

#### 3.2 Stresses for the solid phase

The closure models for ps and τ<sup>s</sup> used in Lee et al. [16] will be described here. In order to cover flow regimes with different solid-phase concentrations (dilute flows, dense flows, and compact beds), Lee et al. [16] suggested the following model for ps :

$$p\_s = p\_s^t + p\_s^r + p\_s^e,\tag{49}$$

ratio of the viscous stress to the quasi-static shear stress associated with the weight

stress. The relative importance of the inertial number to the viscous number can be

2

<sup>c</sup> <sup>¼</sup> cc

where cc is a critical concentration and b is a model parameter. Trulsson et al. [21]

<sup>η</sup> <sup>¼</sup> <sup>η</sup><sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> � <sup>η</sup><sup>1</sup>

where η<sup>1</sup> ¼ tan θ<sup>s</sup> with θ<sup>s</sup> being the angle of repose and η<sup>2</sup> and Io are constants.

Based on Eqs. (56) and (55), the following expressions for pr

<sup>s</sup> <sup>¼</sup> <sup>2</sup>b<sup>2</sup>

3.3.1 A model based on the particle sedimentation in still water

�ρfð Þ 1 � c g � ð Þ 1 � c

�ρscg � c

∂pf ∂z þ

pr

3.3 Closure models for particle response time

In this case, Eqs. (38) and (40) reduce to

νv <sup>s</sup> <sup>¼</sup> pr

c ð Þ cc � c

where b is a constant. In Lee et al. [7], a ¼ 0:11 and b ¼ 1 were taken.

which considers the solid phase in its static state as a very viscous fluid and

1 þ Io=I

<sup>s</sup> <sup>þ</sup> <sup>p</sup><sup>e</sup> s � �η 2ρsDs

<sup>2</sup> <sup>ρ</sup><sup>f</sup> <sup>ν</sup><sup>f</sup> <sup>þ</sup> <sup>2</sup>aρsd<sup>2</sup>

The drag force between the two phases is modeled through the particle response time τp. Three representative models for particle response time are introduced in

The first model is based on particle sedimentation in still water, which can be simplified as a one-dimensional problem, where the steady sedimentation assures that there are no stresses in both the solid and fluid phases in the vertical direction z.

∂pf

<sup>∂</sup><sup>z</sup> � <sup>c</sup>ρ<sup>s</sup> wf � ws

cρ<sup>s</sup> wf � ws � � τp

� � τp

Ds

� �Ds, (58)

p , which describes the ratio of the inertial stress to the quasi-static

<sup>i</sup> =Iν. Some formulas have been proposed in

<sup>1</sup> <sup>þ</sup> bI1=<sup>2</sup> (55)

<sup>1</sup>=<sup>2</sup> , (56)

, (57)

<sup>s</sup> can be

¼ 0, (59)

¼ 0, (60)

<sup>s</sup> and ν<sup>v</sup>

=ps with T<sup>s</sup> being

(resulting from the enduring contact). The inertial number is defined by

the literature to describe <sup>c</sup> � <sup>I</sup> and <sup>η</sup> � <sup>I</sup> relationships, where <sup>η</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup>

Following the work of Boyer et al. [22], Lee et al. [16] assumed

Ii <sup>¼</sup> <sup>2</sup>dD<sup>s</sup>

proposed

derived [16]:

this section.

and

129

= ffiffiffiffiffiffiffiffiffiffiffiffi cps =ρ<sup>s</sup>

the second invariant of τs.

measured by the Stokes number st<sup>v</sup> ¼ I

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

where p<sup>t</sup> <sup>s</sup> accounts for the turbulent motion of solid particles (important for dilute flows); pr <sup>s</sup> reflects the rheological characteristics of dense flows and includes the effects such as fluid viscosity, enduring contact, and particle inertial; pe <sup>s</sup> accounts for the elastic effect, which is important when the particles are in their static state in a compact bed.

For solid particles in a compact bed, the formula proposed by Hsu et al. [19] can be used to compute p<sup>e</sup> s

$$p\_s^\epsilon = K[\max(c - c\_o, 0)]^\mathbb{X} \left\{ \mathbf{1} + \sin \left[ \max \left( \frac{c - c\_o}{c\_{\rm rep} - c\_o}, 0 \right) \pi - \frac{\pi}{2} \right] \right\},\tag{50}$$

where co is random loose packing fraction and coefficients K and χ are model parameters. For spheres, co ranges from 0.54 to 0.634, depending on the friction [17]. The coefficient K is associated with the Young's modulus of the compact bed, and the other terms are related to material deformation.

The closure models for pr <sup>s</sup> and p<sup>t</sup> <sup>s</sup> are closely related to the stress tensor and the visco-plastic rheological characteristics for the solid phase. The stress tensor for the solid phase can be computed by

$$\mathbf{t}\_s = -\left(\frac{2}{3}\rho\_s\nu\_s\nabla \cdot \mathbf{u}\_s\right) + 2\rho\_s\nu\_s\mathbf{D}\_{\sigma} \tag{51}$$

The kinematic viscosity of the solid phase ν<sup>s</sup> is computed by the sum of two terms:

$$
\boldsymbol{\nu}\_{\boldsymbol{s}} = \boldsymbol{\nu}\_{\boldsymbol{s}}^{\boldsymbol{p}} + \boldsymbol{\nu}\_{\boldsymbol{s}}^{\boldsymbol{f}}.\tag{52}
$$

where ν<sup>v</sup> <sup>s</sup> and ν<sup>t</sup> <sup>s</sup> represent the visco-plastic and turbulence effects, respectively. This model for ν<sup>s</sup> can consider both the turbulence behavior (for dilute flows) and the visco-plastic behavior (for dense flows and compact beds).

Based on an analysis of heavy and small particles in homogeneous steady turbulent flows, Hinze [20] suggests that pt <sup>s</sup> and ν<sup>t</sup> <sup>s</sup> can be computed by

$$p\_s^t = \frac{2}{3} \rho\_s a k\_s \tag{53}$$

and

$$
\omega\_s^t = a \omega\_f^t. \tag{54}
$$

where the coefficient α is the same as that in Eqs.(45) and (46).

For dense fluid-solid two-phase flows, the visco-plastic rheological characteristics depend on a dimensionless parameter <sup>I</sup> <sup>¼</sup> Iv <sup>þ</sup> aI<sup>2</sup> <sup>i</sup> , where Iv is the viscous number, Ii is the inertial number, and a is a constant [21]. The viscous number is defined by Iv <sup>¼</sup> <sup>2</sup>ρ<sup>f</sup> <sup>ν</sup>fDs =cps where ν<sup>f</sup> is the kinematic viscosity of the fluid and D<sup>s</sup> is the second invariant of the strain rate. Physically, the viscous number describes the Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

3.2 Stresses for the solid phase

where p<sup>t</sup>

flows); pr

compact bed.

be used to compute p<sup>e</sup>

The closure models for pr

solid phase can be computed by

pe

terms:

where ν<sup>v</sup>

and

128

<sup>s</sup> and ν<sup>t</sup>

defined by Iv <sup>¼</sup> <sup>2</sup>ρ<sup>f</sup> <sup>ν</sup>fDs

lent flows, Hinze [20] suggests that pt

s

The closure models for ps and τ<sup>s</sup> used in Lee et al. [16] will be described here. In order to cover flow regimes with different solid-phase concentrations (dilute flows, dense flows, and compact beds), Lee et al. [16] suggested the following model for ps

<sup>s</sup> <sup>þ</sup> pr

the elastic effect, which is important when the particles are in their static state in a

where co is random loose packing fraction and coefficients K and χ are model parameters. For spheres, co ranges from 0.54 to 0.634, depending on the friction [17]. The coefficient K is associated with the Young's modulus of the compact bed,

visco-plastic rheological characteristics for the solid phase. The stress tensor for the

ρsνs∇ � u<sup>s</sup> 

The kinematic viscosity of the solid phase ν<sup>s</sup> is computed by the sum of two

<sup>s</sup> <sup>þ</sup> <sup>ν</sup><sup>t</sup> s

Based on an analysis of heavy and small particles in homogeneous steady turbu-

For dense fluid-solid two-phase flows, the visco-plastic rheological characteris-

number, Ii is the inertial number, and a is a constant [21]. The viscous number is

the second invariant of the strain rate. Physically, the viscous number describes the

<sup>s</sup> represent the visco-plastic and turbulence effects, respectively. This

<sup>s</sup> can be computed by

=cps where ν<sup>f</sup> is the kinematic viscosity of the fluid and D<sup>s</sup> is

<sup>ν</sup><sup>s</sup> <sup>¼</sup> <sup>ν</sup><sup>v</sup>

model for ν<sup>s</sup> can consider both the turbulence behavior (for dilute flows) and the

<sup>s</sup> and ν<sup>t</sup>

pt <sup>s</sup> <sup>¼</sup> <sup>2</sup> 3

> νt <sup>s</sup> <sup>¼</sup> αν<sup>t</sup>

where the coefficient α is the same as that in Eqs.(45) and (46).

tics depend on a dimensionless parameter <sup>I</sup> <sup>¼</sup> Iv <sup>þ</sup> aI<sup>2</sup>

For solid particles in a compact bed, the formula proposed by Hsu et al. [19] can

crcp � co

; 0 

<sup>s</sup> are closely related to the stress tensor and the

<sup>π</sup> � <sup>π</sup> 2

þ 2ρsνsDs, (51)

, (52)

ρsαk, (53)

<sup>f</sup> : (54)

<sup>i</sup> , where Iv is the viscous

<sup>s</sup> <sup>þ</sup> pe

<sup>s</sup> accounts for the turbulent motion of solid particles (important for dilute

<sup>s</sup> reflects the rheological characteristics of dense flows and includes the

<sup>s</sup>, (49)

ps <sup>¼</sup> pt

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

effects such as fluid viscosity, enduring contact, and particle inertial; pe

<sup>s</sup> <sup>¼</sup> <sup>K</sup>½ � maxð Þ <sup>c</sup> � co; <sup>0</sup> <sup>χ</sup> <sup>1</sup> <sup>þ</sup> sin max <sup>c</sup> � co

<sup>s</sup> and p<sup>t</sup>

<sup>t</sup><sup>s</sup> ¼ � <sup>2</sup> 3

visco-plastic behavior (for dense flows and compact beds).

and the other terms are related to material deformation.

:

<sup>s</sup> accounts for

, (50)

ratio of the viscous stress to the quasi-static shear stress associated with the weight (resulting from the enduring contact). The inertial number is defined by Ii <sup>¼</sup> <sup>2</sup>dD<sup>s</sup> = ffiffiffiffiffiffiffiffiffiffiffiffi cps =ρ<sup>s</sup> p , which describes the ratio of the inertial stress to the quasi-static stress. The relative importance of the inertial number to the viscous number can be measured by the Stokes number st<sup>v</sup> ¼ I 2 <sup>i</sup> =Iν. Some formulas have been proposed in the literature to describe <sup>c</sup> � <sup>I</sup> and <sup>η</sup> � <sup>I</sup> relationships, where <sup>η</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup> =ps with T<sup>s</sup> being the second invariant of τs.

Following the work of Boyer et al. [22], Lee et al. [16] assumed

$$\mathcal{L} = \frac{\mathcal{L}\_c}{\mathbf{1} + bI^{1/2}} \tag{55}$$

where cc is a critical concentration and b is a model parameter. Trulsson et al. [21] proposed

$$
\eta = \eta\_1 + \frac{\eta\_2 - \eta\_1}{1 + I\_o/I^{1/2}},
\tag{56}
$$

where η<sup>1</sup> ¼ tan θ<sup>s</sup> with θ<sup>s</sup> being the angle of repose and η<sup>2</sup> and Io are constants. Based on Eqs. (56) and (55), the following expressions for pr <sup>s</sup> and ν<sup>v</sup> <sup>s</sup> can be derived [16]:

$$\nu\_s^p = \frac{(p\_s^r + p\_s^e)\eta}{2\rho\_s D\_s},\tag{57}$$

which considers the solid phase in its static state as a very viscous fluid and

$$p\_s^r = \frac{2b^2c}{\left(c\_c - c\right)^2} \left(\rho\_f \nu\_f + 2a\rho\_s d^2 D\_s\right) D\_\nu \tag{58}$$

where b is a constant. In Lee et al. [7], a ¼ 0:11 and b ¼ 1 were taken.

#### 3.3 Closure models for particle response time

The drag force between the two phases is modeled through the particle response time τp. Three representative models for particle response time are introduced in this section.

#### 3.3.1 A model based on the particle sedimentation in still water

The first model is based on particle sedimentation in still water, which can be simplified as a one-dimensional problem, where the steady sedimentation assures that there are no stresses in both the solid and fluid phases in the vertical direction z. In this case, Eqs. (38) and (40) reduce to

$$-\rho\_f(\mathbf{1}-c)\mathbf{g} - (\mathbf{1}-c)\frac{\partial p\_f}{\partial \mathbf{z}} - \frac{c\rho\_s(w\_f - w\_s)}{\tau\_p} = \mathbf{0},\tag{59}$$

and

$$-\rho\_s c \mathbf{g} - c \frac{\partial p\_f}{\partial \mathbf{z}} + \frac{c \rho\_s (w\_f - w\_s)}{\tau\_p} = \mathbf{0},\tag{60}$$

where wf and ws are the vertical velocities of the fluid and solid phases, respectively.

Because net volume flux through any horizontal plane must be zero, we have

$$(\mathbf{1} - \mathbf{c})w\_f + cw\_s = \mathbf{0}.\tag{61}$$

3.3.2 A model based on the pressure drop in steady flows through a homogeneous porous

Another model for particle response time can be derived by examining the pressure drop in the steady flow through a porous media. For a one-dimensional problem of a horizontal, steady flow through porous media, the terms containing

> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>c</sup>ρsuf ð Þ 1 � c τ<sup>p</sup>

where the horizontal coordinate x points in the direction of the flow and u is the

<sup>∂</sup><sup>x</sup> <sup>¼</sup> aFρfð Þ <sup>1</sup> � <sup>c</sup> uf <sup>þ</sup> bFρfð Þ <sup>1</sup> � <sup>c</sup>

where aF and bF are two model parameters. Several formulas for computing aF and bF can be found in previous studies. The following two expressions for aF and bF suggested by Engelund [25] are recommended for the applications presented at the

<sup>d</sup><sup>2</sup> , bF <sup>¼</sup> bEc

1 aEc<sup>2</sup> þ bERe<sup>p</sup>

where aE and bE are two model parameters depending on the composition of the solid phase. The parameter aE is associated with kp as will be shown later. For <sup>d</sup>≈<sup>2</sup> � <sup>10</sup>�<sup>4</sup> m, kp <sup>≈</sup> <sup>10</sup>�<sup>10</sup> � <sup>10</sup>�11m2 [30], which gives aE <sup>≈</sup>1:<sup>6</sup> � <sup>10</sup><sup>3</sup> � <sup>1</sup>:<sup>6</sup> � <sup>10</sup><sup>4</sup>

For flow in a porous media, the particle response time can also be related to its permeability κp. According to Darcy's law for seepage [29], the pressure gradient

> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>ρ</sup><sup>f</sup> <sup>ν</sup>fð Þ <sup>1</sup> � <sup>c</sup> uf κp

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>c</sup>ρsκ<sup>p</sup> ð Þ 1 � c 2 ρ<sup>f</sup> ν<sup>f</sup>

aE <sup>¼</sup> <sup>d</sup><sup>2</sup>

which means that the particle response time can be related to the permeability.

kpð Þ 1 � c

When the flow is very slow, Eqs. (70), (71), and (73) suggest that

<sup>g</sup>ð Þ <sup>1</sup> � <sup>c</sup> <sup>3</sup>

d

, (69)

<sup>f</sup> , (70)

, (71)

, (72)

, (73)

<sup>2</sup> , (75)

(74)

2 u2

the stresses of the fluid phase disappear, and Eq. (38) reduces to

� ∂pf

media

velocity component in x-direction.

end of this chapter:

can also be written as

131

For this problem, Forchheimer [29] suggested

aF <sup>¼</sup> aEc<sup>3</sup>ν<sup>f</sup> ð Þ 1 � c 2

> <sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup>sd<sup>2</sup> ρ<sup>f</sup> ν<sup>f</sup>

for c ¼ 0:5. The parameter bE varies from 1.8 to 3.6 or more [28, 31, 32].

� ∂pf

where κ<sup>p</sup> is the permeability. Combining Eqs. (69) and (73) gives

Comparing Eqs. (69) and (70) and using Eq.(71) give

� ∂pf

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

Combining Eqs. (59) and (61) yields

$$-\frac{\partial p\_f}{\partial \mathbf{z}} = \frac{-c\rho\_s w\_s}{(\mathbf{1} - c)^2 \tau\_p} + \rho\_f \mathbf{g}.\tag{62}$$

Substituting Eqs. (61) and (62) into Eq. (60) leads to

$$\pi\_p = \frac{\rho\_s w\_s}{(1-c)^2 \Big(\rho\_s - \rho\_f\Big) \mathbf{g}},\tag{63}$$

where the solid-phase velocity ws is also called the hindered settling velocity [23]. The hindered velocity is smaller than the terminal velocity of a single particle, w0, due to the influence of volumetric concentration (including many-body hydrodynamic interactions). Richardson and Zaki [24] suggested

$$\frac{w\_s}{w\_0} = \left(\mathbf{1} - c\right)^n,\tag{64}$$

where the coefficient n is related to the particle Reynolds number Res ¼ w0d=ν<sup>f</sup>

$$n = \begin{cases} 4.65, & \text{Re}\_{\circ} < 0.2 \\ 4.4 Re\_{\circ}^{-0.33}, & \text{0.2 \le Re\_{\circ} < 1} \\ 4.4 Re\_{\circ}^{-0.1}, & \text{1 \le Re\_{\circ} < 500} \\ 2.4, & 500 \le Re\_{\circ} \end{cases} \tag{65}$$

The terminal velocity of a single particle w<sup>0</sup> can be computed by

$$w\_0 = \sqrt{\frac{4\text{dg}}{3\text{C}\_d} \frac{\rho\_s - \rho\_f}{\rho\_f}},\tag{66}$$

where Cd is the drag coefficient for steady flows passing a single particle [25, 26]. For spheres, the following formula of White [27] can be used:

$$C\_d = \frac{24}{Re\_p} + \frac{6}{1 + \sqrt{Re\_p}} + 0.4,\tag{67}$$

where Rep ¼ ∣u<sup>f</sup> � us∣d=ν<sup>f</sup> . Combing Eqs. (63)–(67) yields

$$\tau\_p = \frac{\rho\_s}{\rho\_f} \frac{d^2}{\nu\_f} \frac{\left(1 - c\right)^{n-2}}{18 + \left(4.5/\left(1 + \sqrt{Re\_p}\right) + 0.3\right) Re\_p}.\tag{68}$$

It is remarked that Eq. (64) is validated only for c , 0:4 [28]. When the concentration c is so high that contact networks form among particles, ws, becomes zero; when this happens, Eq. (64) is no longer valid any more.

where wf and ws are the vertical velocities of the fluid and solid phases, respec-

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

<sup>∂</sup><sup>z</sup> <sup>¼</sup> �cρsws ð Þ 1 � c 2 τp

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup>sws ð Þ 1 � c 2

where the solid-phase velocity ws is also called the hindered settling velocity [23]. The hindered velocity is smaller than the terminal velocity of a single particle, w0, due to the influence of volumetric concentration (including many-body hydrodynamic interactions). Richardson and Zaki [24] suggested

<sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>n</sup>

:65, Res , 0:2 :4Re�0:<sup>33</sup> <sup>s</sup> , 0:2 ≤Res , 1 :4Re�0:<sup>1</sup> <sup>s</sup> , 1≤ Res , 500 :4, 500 ≤Res

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4dg 3Cd

6 <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi Rep

ð Þ <sup>1</sup> � <sup>c</sup> <sup>n</sup>�<sup>2</sup>

Rep � � <sup>p</sup> <sup>þ</sup> <sup>0</sup>:<sup>3</sup> � �

ρ<sup>s</sup> � ρ<sup>f</sup> ρf

where the coefficient n is related to the particle Reynolds number Res ¼ w0d=ν<sup>f</sup>

ws w<sup>0</sup>

ρ<sup>s</sup> � ρ<sup>f</sup> � �

g

� ∂pf

Substituting Eqs. (61) and (62) into Eq. (60) leads to

n ¼

8 >>><

>>>:

The terminal velocity of a single particle w<sup>0</sup> can be computed by

For spheres, the following formula of White [27] can be used:

Cd <sup>¼</sup> <sup>24</sup> Rep þ

where Rep ¼ ∣u<sup>f</sup> � us∣d=ν<sup>f</sup> . Combing Eqs. (63)–(67) yields

when this happens, Eq. (64) is no longer valid any more.

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup><sup>s</sup> ρf d2 νf

130

w<sup>0</sup> ¼

s

where Cd is the drag coefficient for steady flows passing a single particle [25, 26].

<sup>18</sup> <sup>þ</sup> <sup>4</sup>:5<sup>=</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi

It is remarked that Eq. (64) is validated only for c , 0:4 [28]. When the concentration c is so high that contact networks form among particles, ws, becomes zero;

Because net volume flux through any horizontal plane must be zero, we have

ð Þ 1 � c wf þ cws ¼ 0: (61)

þ ρ<sup>f</sup> g: (62)

, (63)

, (64)

, (66)

<sup>p</sup> <sup>þ</sup> <sup>0</sup>:4, (67)

Rep

: (68)

(65)

:

tively.

Combining Eqs. (59) and (61) yields

#### 3.3.2 A model based on the pressure drop in steady flows through a homogeneous porous media

Another model for particle response time can be derived by examining the pressure drop in the steady flow through a porous media. For a one-dimensional problem of a horizontal, steady flow through porous media, the terms containing the stresses of the fluid phase disappear, and Eq. (38) reduces to

$$-\frac{\partial p\_f}{\partial \mathbf{x}} = \frac{c\rho\_s u\_f}{(\mathbf{1} - c)\tau\_p},\tag{69}$$

where the horizontal coordinate x points in the direction of the flow and u is the velocity component in x-direction.

For this problem, Forchheimer [29] suggested

$$-\frac{\partial p\_f}{\partial \mathbf{x}} = a\_F \rho\_f (\mathbf{1} - c) u\_f + b\_F \rho\_f (\mathbf{1} - c)^2 u\_f^2,\tag{70}$$

where aF and bF are two model parameters. Several formulas for computing aF and bF can be found in previous studies. The following two expressions for aF and bF suggested by Engelund [25] are recommended for the applications presented at the end of this chapter:

$$a\_F = \frac{a\_E c^3 \nu\_f}{(1 - c)^2 d^2}, \quad b\_F = \frac{b\_E c}{\text{g}(1 - c)^3 d},\tag{71}$$

Comparing Eqs. (69) and (70) and using Eq.(71) give

$$
\pi\_p = \frac{\rho\_r d^2}{\rho\_f \nu\_f} \frac{1}{a\_E c^2 + b\_E \text{Re}\_p},
\tag{72}
$$

where aE and bE are two model parameters depending on the composition of the solid phase. The parameter aE is associated with kp as will be shown later. For <sup>d</sup>≈<sup>2</sup> � <sup>10</sup>�<sup>4</sup> m, kp <sup>≈</sup> <sup>10</sup>�<sup>10</sup> � <sup>10</sup>�11m2 [30], which gives aE <sup>≈</sup>1:<sup>6</sup> � <sup>10</sup><sup>3</sup> � <sup>1</sup>:<sup>6</sup> � <sup>10</sup><sup>4</sup> for c ¼ 0:5. The parameter bE varies from 1.8 to 3.6 or more [28, 31, 32].

For flow in a porous media, the particle response time can also be related to its permeability κp. According to Darcy's law for seepage [29], the pressure gradient can also be written as

$$-\frac{\partial p\_f}{\partial \mathbf{x}} = \frac{\rho\_f \nu\_f (\mathbf{1} - \mathbf{c}) u\_f}{\kappa\_p},\tag{73}$$

where κ<sup>p</sup> is the permeability. Combining Eqs. (69) and (73) gives

$$
\pi\_p = \frac{c\rho\_s\kappa\_p}{\left(1-c\right)^2\rho\_f\nu\_f} \tag{74}
$$

When the flow is very slow, Eqs. (70), (71), and (73) suggest that

$$a\_E = \frac{d^2}{k\_p \left(1 - c\right)^2},\tag{75}$$

which means that the particle response time can be related to the permeability.

#### 3.3.3 A hybrid model

Equation (64) is validated only for c , 0:4 [28]. To extend Eq. (64) to high concentration regions, Camenen [33] modified Eq. (64) to

$$\frac{w}{w\_s} = (\mathbf{1} - \mathbf{c})^{n-1} [\max(\mathbf{1} - \mathbf{c}/c\_m, \mathbf{0})]^{\varepsilon\_m},\tag{76}$$

∂uf ∂t

and

forms:

þ ∇ � ufu<sup>f</sup>

∂us ∂t

<sup>u</sup><sup>f</sup> <sup>¼</sup> <sup>A</sup><sup>f</sup> H Af D

<sup>u</sup><sup>s</sup> <sup>¼</sup> <sup>A</sup><sup>s</sup> H As D þ g As D

diagonal matrix, A<sup>β</sup>

for velocity.

133

þ g Af D

4.3 A prediction-correction method

splitting Eq. (83) into a predictor u<sup>∗</sup>

u∗ <sup>s</sup> <sup>¼</sup> <sup>A</sup><sup>s</sup> H As D þ g As D

<sup>u</sup><sup>s</sup> <sup>¼</sup> <sup>u</sup><sup>∗</sup>

obtain the following equation describing the evolution of c:

<sup>s</sup> � ps

∇c ρsA<sup>s</sup> Dc

This predictor-corrector scheme can improve the numerical stability by introducing a numerical diffusion term. To see this, we combine Eqs. (39) and (85) to

which is corrected by the following corrector

� � � <sup>∇</sup> � <sup>u</sup><sup>f</sup>

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

> � <sup>c</sup>ρ<sup>s</sup> ρfð Þ 1 � c

<sup>þ</sup> <sup>∇</sup> � ð Þ� <sup>u</sup>su<sup>s</sup> ð Þ <sup>∇</sup> � <sup>u</sup><sup>s</sup> <sup>u</sup><sup>s</sup> <sup>¼</sup> <sup>g</sup> � <sup>1</sup>

þ 1 ρs

� <sup>∇</sup>pf <sup>ρ</sup>fA<sup>f</sup> D

� <sup>∇</sup>pf ρsA<sup>s</sup> D

� �u<sup>f</sup> <sup>¼</sup> <sup>g</sup> � <sup>1</sup>

ρf

u<sup>f</sup> � u<sup>s</sup> � � τp

<sup>∇</sup> � <sup>τ</sup><sup>s</sup> <sup>þ</sup> <sup>τ</sup><sup>s</sup> � <sup>∇</sup><sup>c</sup> ρsc þ

<sup>þ</sup> <sup>ρ</sup>scu<sup>s</sup> <sup>ρ</sup>fA<sup>f</sup>

� <sup>∇</sup>ps ρsA<sup>s</sup> D � ps ∇c ρsA<sup>s</sup> Dc

<sup>D</sup>, and an off-diagonal matrix, A<sup>w</sup>

(83). OpenFOAM® built-in functions are used to compute A<sup>β</sup>

The solutions of Eqs. (80) and (81) are expressed in the following semidiscretized

<sup>D</sup>ð Þ 1 � c τ<sup>p</sup>

where A<sup>β</sup> (β = s or f ) denotes the systems of linear algebraic equations arising from the discretization of either Eqs. (82) or (83). The matrix A<sup>β</sup> is decomposed into a

b<sup>β</sup> relating to the second to final terms on the right-hand side of either Eqs. (82) or

If Eq. (83) is directly used to calculate u<sup>s</sup> and Eq. (39) to calculate c, then c may increase rapidly toward cc, leading to an infinite ps for large c. This can be avoided by using a prediction-correction method to compute u<sup>f</sup> and us. This is achieved by

> � <sup>∇</sup>pf ρsAs D

> > � <sup>1</sup> As Dcτ<sup>p</sup>

!

<sup>s</sup> and a corrector. The predictor is

<sup>þ</sup> <sup>ρ</sup>su<sup>f</sup> As <sup>D</sup>τ<sup>p</sup>

> νt f σc ∇c

depend on the discretization schemes. For example, Lee et al. [16] and Lee and Huang [35] used a second-order time-implicit scheme and a limited linear interpolation scheme for all variables except for velocity. To interpolate velocities, the total-variation-diminishing (TVD) limited linear interpolation scheme is adopted

∇pf þ

<sup>þ</sup> <sup>ρ</sup><sup>s</sup> ρfð Þ 1 � c τ<sup>p</sup>

> <sup>∇</sup>pf � <sup>1</sup> ρsc ∇cps

u<sup>f</sup> � u<sup>s</sup> � � τp

<sup>þ</sup> <sup>ρ</sup><sup>s</sup> <sup>ρ</sup>fA<sup>f</sup>

<sup>þ</sup> <sup>ρ</sup>su<sup>f</sup> As <sup>D</sup>τ<sup>p</sup>

ρs

1 ρf <sup>∇</sup> � <sup>τ</sup><sup>f</sup> � <sup>τ</sup><sup>f</sup> � <sup>∇</sup><sup>c</sup>

νt f σc ∇c

� 1 cτ<sup>p</sup>

<sup>D</sup>ð Þ 1 � c τ<sup>p</sup>

� <sup>1</sup> As Dcτ<sup>p</sup>

<sup>O</sup>. Also, A<sup>w</sup>

νt f σc ∇c

> νt f σc

νt f σc

<sup>H</sup> <sup>¼</sup> <sup>b</sup><sup>w</sup> � <sup>A</sup><sup>β</sup>

<sup>D</sup> and <sup>A</sup><sup>β</sup>

ρfð Þ 1 � c

(80)

(81)

∇c (82)

∇c (83)

<sup>H</sup>, which

<sup>O</sup>u<sup>β</sup> with

(84)

(85)

where cm is the maximum concentration at which w ¼ 0. In this study, cm ¼ co is adopted because when c≥co, contact networks can form in the granular material.

Combining Eqs. (63), (76), and (66)–(67) gives

$$\tau\_p = \frac{\rho\_s}{\rho\_f} \frac{d^2}{\nu\_f} \frac{(\mathbf{1} - \mathbf{c})^{n-3} [\max(\mathbf{1} - \mathbf{c}/c\_m, \mathbf{0})]^{c\_m}}{\mathbf{1} \mathbf{8} + \left(4.5/\left(\mathbf{1} + \sqrt{Re\_p}\right) + \mathbf{0}.3\right) Re\_p}. \tag{77}$$

We stress that c ¼ cm will lead to τ<sup>p</sup> ¼ 0 and thus an infinite drag force. Physically, when the volumetric concentration is greater than some critical value, say cr, Eq. (63) ceases to be valid, and Eq. (72) should be used. To avoid unnaturally large drag force between the two phases, we propose the following model for particle response time:

$$\tau\_p = \begin{cases} \frac{\rho\_s}{\rho\_f} \frac{d^2}{\nu\_f} \frac{(1 - c)^{n-3} [\max(1 - c/c\_m, 0)]^{c\_m}}{18 + \left(4.5/\left(1 + \sqrt{Re\_p}\right) + 0.3\right) Re\_p}, & \text{for } c < c\_r\\ \frac{\rho\_s d^2}{\rho\_f \nu\_f} \frac{1}{a\_E c^2 + b\_E Re\_p}, & \text{for } c \ge c\_r \end{cases} \tag{78}$$

where cr is the concentration at the intercept point of Eq. (72) and Eq. (77). The transition from Eq. (77) to Eq. (72) is continuous at the intercept point where c ¼ cr. The concentration at the point joining the two models (cr) is problem-dependent and can be found in principle by solving the following equation:

$$\frac{\left(\mathbf{1} - c\_r\right)^{n-3} \left[\max\left(\mathbf{1} - c\_r/c\_m, \mathbf{0}\right)\right]^{c\_m}}{\mathbf{1}\mathbf{8} + \left(4.5/\left(\mathbf{1} + \sqrt{Re\_p}\right) + \mathbf{0}.\mathbf{3}\right) Re\_p} = \frac{\mathbf{1}}{a\_E c\_r^2 + b\_E Re\_p}.\tag{79}$$

For given values of aE and bE, Eq. (79) implicitly defines cr as a function of Rep.

#### 4. Numerical implementation with OpenFOAM

#### 4.1 Introduction to OpenFOAM

This section introduces how to use OpenFOAM® to solve the governing equations with the closure models presented in the previous section. OpenFOAM® is a C++ toolbox developed based on the finite-volume method; it allows CFD code developers to sidestep the discretization of derivative terms on unstructured grids.

#### 4.2 Semidiscretized forms of the governing equations

To avoid numerical noises occurring when c ! 0, Rusche [34] suggests that the momentum equations (Eqs. (38) and (40)) should be converted into the following "phase-intensive" form by dividing ρfð Þ 1 � c and ρsc:

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

$$\begin{aligned} \frac{d\mathbf{u}\_f}{dt} + \nabla \cdot (\mathbf{u}\_f \mathbf{u}\_f) &- (\nabla \cdot \mathbf{u}\_f) \mathbf{u}\_f = \mathbf{g} - \frac{\mathbf{1}}{\rho\_f} \nabla p\_f + \frac{\mathbf{1}}{\rho\_f} \nabla \cdot \mathbf{\tau}\_f - \frac{\mathbf{\tau}\_f \cdot \nabla c}{\rho\_f (\mathbf{1} - c)} \\ &- \frac{c \rho\_s}{\rho\_f (\mathbf{1} - c)} \frac{(\mathbf{u}\_f - \mathbf{u}\_s)}{\tau\_p} + \frac{\rho\_s}{\rho\_f (\mathbf{1} - c) \tau\_p} \frac{\nu\_f^t}{\sigma\_c} \nabla c \end{aligned} \tag{80}$$

and

3.3.3 A hybrid model

response time:

τ<sup>p</sup> ¼

ρs ρf d2 νf

8 >>>>><

>>>>>:

4.1 Introduction to OpenFOAM

132

ρsd<sup>2</sup> ρ<sup>f</sup> ν<sup>f</sup>

Equation (64) is validated only for c , 0:4 [28]. To extend Eq. (64) to high

where cm is the maximum concentration at which w ¼ 0. In this study, cm ¼ co is adopted because when c≥co, contact networks can form in the granular material.

<sup>18</sup> <sup>þ</sup> <sup>4</sup>:5<sup>=</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi

We stress that c ¼ cm will lead to τ<sup>p</sup> ¼ 0 and thus an infinite drag force. Physically, when the volumetric concentration is greater than some critical value, say cr, Eq. (63) ceases to be valid, and Eq. (72) should be used. To avoid unnaturally large drag force between the two phases, we propose the following model for particle

½ � max 1ð Þ � <sup>c</sup>=cm; <sup>0</sup> cm

Rep � � <sup>p</sup> <sup>þ</sup> <sup>0</sup>:<sup>3</sup> � �

where cr is the concentration at the intercept point of Eq. (72) and Eq. (77). The transition from Eq. (77) to Eq. (72) is continuous at the intercept point where c ¼ cr. The concentration at the point joining the two models (cr) is problem-dependent

½ � max 1ð Þ � cr=cm; <sup>0</sup> cm

Rep � � <sup>p</sup> <sup>þ</sup> <sup>0</sup>:<sup>3</sup> � �

ð Þ <sup>1</sup> � <sup>c</sup> <sup>n</sup>�<sup>3</sup>

½ � max 1ð Þ � <sup>c</sup>=cm; <sup>0</sup> cm , (76)

Rep

, for c , cr

: (77)

(78)

: (79)

½ � max 1ð Þ � <sup>c</sup>=cm; <sup>0</sup> cm

Rep

, for c≥cr

Rep

For given values of aE and bE, Eq. (79) implicitly defines cr as a function of Rep.

This section introduces how to use OpenFOAM® to solve the governing equations with the closure models presented in the previous section. OpenFOAM® is a C++ toolbox developed based on the finite-volume method; it allows CFD code developers to sidestep the discretization of derivative terms on unstructured grids.

To avoid numerical noises occurring when c ! 0, Rusche [34] suggests that the momentum equations (Eqs. (38) and (40)) should be converted into the following

<sup>¼</sup> <sup>1</sup> aEc<sup>2</sup>

<sup>r</sup> þ bERep

Rep � � <sup>p</sup> <sup>þ</sup> <sup>0</sup>:<sup>3</sup> � �

concentration regions, Camenen [33] modified Eq. (64) to

<sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>n</sup>�<sup>1</sup>

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

w ws

Combining Eqs. (63), (76), and (66)–(67) gives

ð Þ <sup>1</sup> � <sup>c</sup> <sup>n</sup>�<sup>3</sup>

1 aEc<sup>2</sup> þ bERep

ð Þ <sup>1</sup> � cr <sup>n</sup>�<sup>3</sup>

<sup>18</sup> <sup>þ</sup> <sup>4</sup>:5<sup>=</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi

4. Numerical implementation with OpenFOAM

4.2 Semidiscretized forms of the governing equations

"phase-intensive" form by dividing ρfð Þ 1 � c and ρsc:

<sup>18</sup> <sup>þ</sup> <sup>4</sup>:5<sup>=</sup> <sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi

and can be found in principle by solving the following equation:

<sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>ρ</sup><sup>s</sup> ρf d2 νf

$$\begin{split} \frac{\partial \mathbf{u}\_{s}}{\partial t} + \nabla \cdot (\mathbf{u}\_{s} \mathbf{u}\_{s}) \quad - (\nabla \cdot \mathbf{u}\_{s}) \mathbf{u}\_{s} &= \mathbf{g} - \frac{1}{\rho\_{s}} \nabla p\_{f} - \frac{1}{\rho\_{s} c} \nabla c p\_{s} \\ &+ \frac{1}{\rho\_{s}} \nabla \cdot \boldsymbol{\tau}\_{s} + \frac{\boldsymbol{\tau}\_{s} \cdot \nabla c}{\rho\_{s} c} + \frac{(\mathbf{u}\_{f} - \mathbf{u}\_{s})}{\tau\_{p}} - \frac{\mathbf{1}}{c \boldsymbol{\tau}\_{p}} \frac{\nu\_{f}^{t}}{\sigma\_{c}} \nabla c \end{split} \tag{81}$$

The solutions of Eqs. (80) and (81) are expressed in the following semidiscretized forms:

$$\mathbf{u}\_{f} = \frac{\mathbf{A}\_{H}^{f}}{\mathbf{A}\_{D}^{f}} + \frac{\mathbf{g}}{\mathbf{A}\_{D}^{f}} - \frac{\nabla p\_{f}}{\rho\_{f}\mathbf{A}\_{D}^{f}} + \frac{\rho\_{s}c\mathbf{u}^{\star}}{\rho\_{f}\mathbf{A}\_{D}^{f}(1-c)\tau\_{p}} + \frac{\rho\_{s}}{\rho\_{f}\mathbf{A}\_{D}^{f}(1-c)\tau\_{p}}\frac{\nu\_{f}^{\sharp}}{\sigma\_{c}}\nabla c \tag{82}$$

$$\mathbf{u}\_{s} = \frac{\mathbf{A}\_{H}^{s}}{\mathbf{A}\_{D}^{s}} + \frac{\mathbf{g}}{\mathbf{A}\_{D}^{s}} - \frac{\nabla p\_{f}}{\rho\_{s}\mathbf{A}\_{D}^{s}} - \frac{\nabla p\_{s}}{\rho\_{s}\mathbf{A}\_{D}^{s}} - \frac{p\_{s}\nabla c}{\rho\_{s}\mathbf{A}\_{D}^{s}c} + \frac{\rho\_{s}\mathbf{u}\_{f}}{\mathbf{A}\_{D}^{s}\tau\_{p}} - \frac{\mathbf{1}}{\mathbf{A}\_{D}^{s}c\tau\_{p}}\frac{\nu\_{f}^{t}}{\sigma\_{c}}\nabla c\tag{83}$$

where A<sup>β</sup> (β = s or f ) denotes the systems of linear algebraic equations arising from the discretization of either Eqs. (82) or (83). The matrix A<sup>β</sup> is decomposed into a diagonal matrix, A<sup>β</sup> <sup>D</sup>, and an off-diagonal matrix, A<sup>w</sup> <sup>O</sup>. Also, A<sup>w</sup> <sup>H</sup> <sup>¼</sup> <sup>b</sup><sup>w</sup> � <sup>A</sup><sup>β</sup> <sup>O</sup>u<sup>β</sup> with b<sup>β</sup> relating to the second to final terms on the right-hand side of either Eqs. (82) or (83). OpenFOAM® built-in functions are used to compute A<sup>β</sup> <sup>D</sup> and <sup>A</sup><sup>β</sup> <sup>H</sup>, which depend on the discretization schemes. For example, Lee et al. [16] and Lee and Huang [35] used a second-order time-implicit scheme and a limited linear interpolation scheme for all variables except for velocity. To interpolate velocities, the total-variation-diminishing (TVD) limited linear interpolation scheme is adopted for velocity.

#### 4.3 A prediction-correction method

If Eq. (83) is directly used to calculate u<sup>s</sup> and Eq. (39) to calculate c, then c may increase rapidly toward cc, leading to an infinite ps for large c. This can be avoided by using a prediction-correction method to compute u<sup>f</sup> and us. This is achieved by splitting Eq. (83) into a predictor u<sup>∗</sup> <sup>s</sup> and a corrector. The predictor is

$$\mathbf{u}\_s^\* = \frac{\mathbf{A}\_H^s}{\mathbf{A}\_D^s} + \frac{\mathbf{g}}{\mathbf{A}\_D^s} - \frac{\nabla p\_f}{\rho\_s A\_D^s} + \frac{\rho\_s \mathbf{u}\_f}{\mathbf{A}\_D^s \tau\_p} \tag{84}$$

which is corrected by the following corrector

$$\mathbf{u}\_{s} = \mathbf{u}\_{s}^{\*} - \left(\frac{p\_{s}\nabla c}{\rho\_{s}\mathbf{A}\_{D}^{s}c} - \frac{\mathbf{1}}{\mathbf{A}\_{D}^{s}c\tau\_{p}}\frac{\nu\_{f}^{t}}{\sigma\_{c}}\nabla c\right) \tag{85}$$

This predictor-corrector scheme can improve the numerical stability by introducing a numerical diffusion term. To see this, we combine Eqs. (39) and (85) to obtain the following equation describing the evolution of c:

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

$$\frac{\partial \mathbf{c}}{\partial t} + \nabla \cdot (c \mathbf{u}\_s') = \nabla \cdot \left(\frac{p\_s}{\rho\_s \mathbf{A}\_D^s} + \frac{\mathbf{1}}{\mathbf{A}\_D^s \boldsymbol{\pi}\_p} \frac{\nu\_f^t}{\sigma\_c}\right) \nabla \mathbf{c} \tag{86}$$

we can set u<sup>s</sup> ¼ u<sup>f</sup> , which means the solid particles completely follow the water particles; this does not affect the computations of other variables because the momentum of the solid phase cu<sup>s</sup> is very small when c≤10�6. Because the maximum value of c is always smaller than 1, there is no singularity issue with Eq. (82). An iteration procedure is needed to solve the governing equations at each time step for the values of c,u<sup>f</sup> , u^s, and pf obtained at the previous time step, and it is

outlined below:

2. Compute u<sup>∗</sup>

6. Compute u<sup>∗</sup>

1. Solve Eqs. (80) and (81).

4. Compute u<sup>s</sup> from Eq. (85).

5. Compute u^<sup>s</sup> from Eq. (90).

9. Compute u<sup>f</sup> from Eq. (88).

3. Solve Eq. (86) for c.

7. Solve Eq. (91) for pf .

u<sup>s</sup> ¼ u<sup>f</sup> is enforced in step 10.

defined as

135

<sup>s</sup> from Eq. (84).

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

<sup>f</sup> from Eq. (87).

8. Repeat Eqs. (5)–(7) for n times (say n = 1).

10.Set <sup>u</sup><sup>s</sup> <sup>¼</sup> <sup>u</sup><sup>f</sup> for very dilute region, specifically <sup>c</sup>≤10�6.

Figure 1 is a flowchart showing these 12 solution steps.

11. Repeat Eqs. (1)–(10) with the updated c, u<sup>f</sup> , u^s, and pf until the residuals of

12. Solve Eqs. (45) and (46) for k and ϵ, and compute the related coefficients.

In the absence of the solid phase, the numerical scheme outlined here reduces to the "PIMPLE" scheme, which is a combination of the "pressure implicit with splitting of operator" (PISO) scheme and the "semi-implicit method for pressure-linked

).

Δt, (92)

Eqs. (80), (86), and (91) are smaller than the tolerance (say 10�<sup>5</sup>

equations" (SIMPLE) scheme. Iterations need to be done separately to solve Eq. (80) for u<sup>f</sup> , Eq. (81) for us, Eq. (86) for c, Eq. (91) for pf , Eq. (45) for k, and (46) for ϵ; the convergence criteria are set at the residuals not exceeding 10�8. Because Eqs. (80), (81), (86), and (87) are coupled, additional residual checks need to be performed at step 11; however, the residual for Eq. (81) is not checked because

To ensure the stability of the overall numerical scheme, the Courant-Friedrichs-Lewy (CFL) condition must be satisfied for each cell. The local Courant number for each cell, which is related to the ratio between the distance of a particle moving within Δt and the size of the cell where such particle is located, is

CFL <sup>¼</sup> <sup>∑</sup> abs <sup>u</sup> <sup>j</sup> � <sup>S</sup> <sup>j</sup>

2V

The right-hand side of Eq. (86) now has a diffusive term introduced by the numerical scheme. High sediment concentration and large ps increase the numerical diffusion (the right-hand side of Eq. (86)) and thus can avoid a rapid increase of c and the numerical instability due to high sediment concentration.

For the velocity-pressure coupling, Eq. (82) is similarly solved using a predictor u∗ <sup>f</sup> and a corrector. The predictor is

$$\mathbf{u}\_f^\* = \frac{\mathbf{A}\_H^f}{\mathbf{A}\_D^f} + \frac{\mathbf{g}}{\mathbf{A}\_D^f} + \frac{\rho\_s c \mathbf{u}\_s}{\rho\_f \mathbf{A}\_D^f (1 - c) \tau\_p} + \frac{\rho\_s}{\rho\_f \mathbf{A}\_D^f (1 - c) \tau\_p} \frac{\nu\_f^t}{\sigma\_c} \nabla c \tag{87}$$

which is corrected by the following corrector

$$\mathbf{u}\_{\circ} = \mathbf{u}\_{\circ}^\* - \frac{\nabla p\_f}{\rho\_f \mathbf{A}\_D^{\zeta}} \tag{88}$$

Substituting Eq. (88) into Eq. (37) gives a pressure equation. However, when using this pressure equation to simulate air-water flows, numerical experiments have shown that the lighter material is poorly conserved [36]. The poor conservation of lighter material can be avoided by combining Eqs. (37) and (39) into the following Eq. (37):

$$\nabla \cdot \left[ (\mathbf{1} - \mathbf{c}) \mathbf{u}\_f + \mathbf{c} \mathbf{u}\_t \right] = \mathbf{0} \tag{89}$$

and using Eq. (89) to correct pf . The method proposed [37] can help avoid the numerical instability. To show this, we follow Carver [37] and define

$$\hat{\mathbf{u}}\_{\varepsilon} = \frac{\mathbf{A}\_{H}^{\varepsilon}}{\mathbf{A}\_{D}^{\varepsilon}} + \frac{\mathbf{g}}{\mathbf{A}\_{D}^{\varepsilon}} - \frac{\nabla p\_{\varepsilon}}{\rho\_{\varepsilon}\mathbf{A}\_{D}^{\varepsilon}} - \frac{p\_{\varepsilon}\nabla c}{\rho\_{\varepsilon}\mathbf{A}\_{D}^{\varepsilon}c} + \frac{\rho\_{\varepsilon}\mathbf{u}\_{f}}{\mathbf{A}\_{D}^{\varepsilon}\tau\_{p}} - \frac{\mathbf{1}}{\mathbf{A}\_{D}^{\varepsilon}c\tau\_{p}}\frac{\nu\_{f}^{t}}{\sigma\_{c}}\nabla c \tag{90}$$

and combine Eqs. (83) and (88)–(90) to obtain the following equation

$$\nabla \cdot \left[ (\mathbf{1} - \boldsymbol{c}) \hat{\mathbf{u}}\_f + \boldsymbol{c} \hat{\mathbf{u}}\_s \right] = \nabla \cdot \left[ \frac{\mathbf{1} - \boldsymbol{c}}{\rho\_f \mathbf{A}\_D^f} + \frac{\boldsymbol{c}}{\rho\_s \mathbf{A}\_D^s} \right] \nabla p\_f \tag{91}$$

The numerical diffusion term on the right-hand side of Eq. (91) can help improve the numerical stability.

The prediction-correction method presented here deals with velocity-pressure coupling and avoids the numerical instability caused by high concentration. The turbulence closure k � ϵ model is also solved in "phase-intensive" forms. For other details relating to the numerical treatments, the reader is referred to "twoPhaseEulerFoam," a two-phase solver provided by OpenFOAM®.

#### 4.3.1 Outline of the solution procedure

When c ! 0, Eq. (83) becomes singular. To avoid this, 1=c is replaced by <sup>1</sup>=ð Þ <sup>c</sup> <sup>þ</sup> <sup>δ</sup><sup>c</sup> in numerical computations, where <sup>δ</sup><sup>c</sup> is a very small number, say 10�6. When c≤ δc, only a very small amount of solid particles are moving with the fluid; replacing 1=<sup>c</sup> by 1=ð Þ <sup>c</sup> <sup>þ</sup> <sup>δ</sup><sup>c</sup> may introduce error in computing <sup>u</sup><sup>s</sup> ; to avoid this error, we can set u<sup>s</sup> ¼ u<sup>f</sup> , which means the solid particles completely follow the water particles; this does not affect the computations of other variables because the momentum of the solid phase cu<sup>s</sup> is very small when c≤10�6. Because the maximum value of c is always smaller than 1, there is no singularity issue with Eq. (82).

An iteration procedure is needed to solve the governing equations at each time step for the values of c,u<sup>f</sup> , u^s, and pf obtained at the previous time step, and it is outlined below:


∂c ∂t

<sup>f</sup> and a corrector. The predictor is

u∗ <sup>f</sup> <sup>¼</sup> <sup>A</sup><sup>f</sup> H Af D

u∗

Eq. (37):

134

þ ∇ � cu<sup>0</sup> s

the numerical instability due to high sediment concentration.

þ g Af D

which is corrected by the following corrector

<sup>u</sup>^<sup>s</sup> <sup>¼</sup> <sup>A</sup><sup>s</sup> H As D þ g As D

improve the numerical stability.

4.3.1 Outline of the solution procedure

� � <sup>¼</sup> <sup>∇</sup> � ps

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

<sup>þ</sup> <sup>ρ</sup>scu<sup>s</sup> <sup>ρ</sup>fA<sup>f</sup>

<sup>u</sup><sup>f</sup> <sup>¼</sup> <sup>u</sup><sup>∗</sup>

∇ � ð Þ 1 � c u<sup>f</sup> þ cu<sup>s</sup>

and using Eq. (89) to correct pf . The method proposed [37] can help avoid the

numerical instability. To show this, we follow Carver [37] and define

� <sup>∇</sup>ps ρsA<sup>s</sup> D � ps ∇c ρsA<sup>s</sup> Dc

∇ � ð Þ 1 � c u^<sup>f</sup> þ cu^<sup>s</sup>

and combine Eqs. (83) and (88)–(90) to obtain the following equation

details relating to the numerical treatments, the reader is referred to "twoPhaseEulerFoam," a two-phase solver provided by OpenFOAM®.

replacing 1=<sup>c</sup> by 1=ð Þ <sup>c</sup> <sup>þ</sup> <sup>δ</sup><sup>c</sup> may introduce error in computing <sup>u</sup><sup>s</sup>

� � <sup>¼</sup> <sup>∇</sup> � <sup>1</sup> � <sup>c</sup>

The numerical diffusion term on the right-hand side of Eq. (91) can help

When c ! 0, Eq. (83) becomes singular. To avoid this, 1=c is replaced by <sup>1</sup>=ð Þ <sup>c</sup> <sup>þ</sup> <sup>δ</sup><sup>c</sup> in numerical computations, where <sup>δ</sup><sup>c</sup> is a very small number, say 10�6. When c≤ δc, only a very small amount of solid particles are moving with the fluid;

The prediction-correction method presented here deals with velocity-pressure coupling and avoids the numerical instability caused by high concentration. The turbulence closure k � ϵ model is also solved in "phase-intensive" forms. For other

ρsA<sup>s</sup> D þ 1 As <sup>D</sup>τ<sup>p</sup>

The right-hand side of Eq. (86) now has a diffusive term introduced by the numerical scheme. High sediment concentration and large ps increase the numerical diffusion (the right-hand side of Eq. (86)) and thus can avoid a rapid increase of c and

For the velocity-pressure coupling, Eq. (82) is similarly solved using a predictor

<sup>D</sup>ð Þ 1 � c τ<sup>p</sup>

Substituting Eq. (88) into Eq. (37) gives a pressure equation. However, when using this pressure equation to simulate air-water flows, numerical experiments have shown that the lighter material is poorly conserved [36]. The poor conservation of lighter material can be avoided by combining Eqs. (37) and (39) into the following

<sup>f</sup> � <sup>∇</sup>pf <sup>ρ</sup>fA<sup>f</sup> D

!

<sup>þ</sup> <sup>ρ</sup><sup>s</sup> <sup>ρ</sup>fA<sup>f</sup>

<sup>D</sup>ð Þ 1 � c τ<sup>p</sup>

� � <sup>¼</sup> <sup>0</sup> (89)

� <sup>1</sup> As Dcτ<sup>p</sup> νt f σc

∇c (90)

∇pf (91)

; to avoid this error,

<sup>þ</sup> <sup>ρ</sup>su<sup>f</sup> As <sup>D</sup>τ<sup>p</sup>

<sup>ρ</sup>fA<sup>f</sup> D þ c ρsA<sup>s</sup> D

" #

νt f σc

νt f σc

∇c (86)

∇c (87)

(88)


12. Solve Eqs. (45) and (46) for k and ϵ, and compute the related coefficients.

Figure 1 is a flowchart showing these 12 solution steps.

In the absence of the solid phase, the numerical scheme outlined here reduces to the "PIMPLE" scheme, which is a combination of the "pressure implicit with splitting of operator" (PISO) scheme and the "semi-implicit method for pressure-linked equations" (SIMPLE) scheme. Iterations need to be done separately to solve Eq. (80) for u<sup>f</sup> , Eq. (81) for us, Eq. (86) for c, Eq. (91) for pf , Eq. (45) for k, and (46) for ϵ; the convergence criteria are set at the residuals not exceeding 10�8. Because Eqs. (80), (81), (86), and (87) are coupled, additional residual checks need to be performed at step 11; however, the residual for Eq. (81) is not checked because u<sup>s</sup> ¼ u<sup>f</sup> is enforced in step 10.

To ensure the stability of the overall numerical scheme, the Courant-Friedrichs-Lewy (CFL) condition must be satisfied for each cell. The local Courant number for each cell, which is related to the ratio between the distance of a particle moving within Δt and the size of the cell where such particle is located, is defined as

$$\text{CFL} = \sum \frac{\text{abs}\left(\mathbf{u}^j \cdot \mathbf{S}^j\right)}{2V} \Delta t,\tag{92}$$

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

scour downstream of a sluice gate is caused by the horizontal submerged water jet issuing from the sluice gate. It is of practical importance to understand the maximum scour depth for the safety of a sluice gate structure. Many experimental studies have been done to investigate the maximum scour depth and the evolution of scour profile (e.g., Chatterjee et al. [39]). For numerical simulations, this problem includes water (fluid phase) and sediment (solid phase) and is best modeled by a liquid-solid two-phase flow approach. In the following, the numerical setup and main conclusions used in Lee et al. [38] are briefly described. The experimental setup of Chatterjee et al. [39] is shown in Figure 2. To numerically simulate the experiment of [8], we use the same sand and dimensions to set up the numerical simulations: quartz sand with <sup>ρ</sup><sup>s</sup> <sup>¼</sup> 2650 kg/m<sup>3</sup> and <sup>d</sup> <sup>¼</sup> <sup>0</sup>:76 mm is placed in the sediment reservoir, with its top surface being on the same level as the top surface of the apron; the sluice gate opening is 2 cm; the length of apron is 0:66 m; the sediment reservoir length is 2:1 m; the overflow weir on the right end has a height of 0:239 m; the upstream inflow discharge rate at the sluice opening is 0:204 m<sup>2</sup>

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

which translates into an average horizontal flow velocity V ¼ 1:02 m/s under the sluice gate. As an example, the computed development of scour depth ds is shown in

The problem involves also an air-water surface, which can be tracked using a modified volume-of-fluid method introduced in [38]. A nonuniform mesh is used in the two-phase flow simulation because of the air-water interface, the interfacial momentum transfer at the bed, and the large velocity variation due to the water jet. The finest mesh with a vertical mesh resolution of 2d is used in the vicinity of the sediment-fluid interface; this fine mesh covers the dynamic sediment-fluid

Figure 3 together with the measurement of Chatterjee et al. [39].

A sketch of the experimental setup for scour induced by a submerged water jet.

Comparison of the computed scour depth with measurements of Chatterjee et al. [39].

Figure 2.

Figure 3.

137

/s,

#### Figure 1.

A flow chart showing the solution procedure using OpenFOAM®.

where in <sup>u</sup> <sup>j</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>u</sup><sup>j</sup> <sup>f</sup> <sup>þ</sup> <sup>c</sup>u<sup>j</sup> <sup>s</sup>, the subscript "j" represents the j th face of the cell, S <sup>j</sup> is a unit normal vector, V is the volume of the cell, and Δt is the time step. The Courant number must be less than 1 to avoid numerical instability. Generally, max (CFL) <0.1 is suggested. The values of CFL for high concentration regions should be much smaller than those for low concentration regions so that rapid changes of c can be avoided. Therefore, it is recommended that max CFLj c>co <sup>&</sup>lt; 0.005. The time step is recommended to be in the range of 10�<sup>5</sup> and 10�<sup>4</sup> s.

#### 5. Applications

This section briefly describes two examples that have been studied using the two-phase flow models described. The problem descriptions and numerical setups for these two problems are included here; for other relevant information, the reader is referred to Lee and Huang [35] and Lee et al. [38].

#### 5.1 Scour downstream of a sluice gate

A sluice gate is a hydraulic structure used to control the flow in a water channel. Sluice gate structures usually have a rigid floor followed by an erodible bed. The

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

scour downstream of a sluice gate is caused by the horizontal submerged water jet issuing from the sluice gate. It is of practical importance to understand the maximum scour depth for the safety of a sluice gate structure. Many experimental studies have been done to investigate the maximum scour depth and the evolution of scour profile (e.g., Chatterjee et al. [39]). For numerical simulations, this problem includes water (fluid phase) and sediment (solid phase) and is best modeled by a liquid-solid two-phase flow approach. In the following, the numerical setup and main conclusions used in Lee et al. [38] are briefly described. The experimental setup of Chatterjee et al. [39] is shown in Figure 2. To numerically simulate the experiment of [8], we use the same sand and dimensions to set up the numerical simulations: quartz sand with <sup>ρ</sup><sup>s</sup> <sup>¼</sup> 2650 kg/m<sup>3</sup> and <sup>d</sup> <sup>¼</sup> <sup>0</sup>:76 mm is placed in the sediment reservoir, with its top surface being on the same level as the top surface of the apron; the sluice gate opening is 2 cm; the length of apron is 0:66 m; the sediment reservoir length is 2:1 m; the overflow weir on the right end has a height of 0:239 m; the upstream inflow discharge rate at the sluice opening is 0:204 m<sup>2</sup> /s, which translates into an average horizontal flow velocity V ¼ 1:02 m/s under the sluice gate. As an example, the computed development of scour depth ds is shown in Figure 3 together with the measurement of Chatterjee et al. [39].

The problem involves also an air-water surface, which can be tracked using a modified volume-of-fluid method introduced in [38]. A nonuniform mesh is used in the two-phase flow simulation because of the air-water interface, the interfacial momentum transfer at the bed, and the large velocity variation due to the water jet. The finest mesh with a vertical mesh resolution of 2d is used in the vicinity of the sediment-fluid interface; this fine mesh covers the dynamic sediment-fluid

Figure 2. A sketch of the experimental setup for scour induced by a submerged water jet.

Figure 3. Comparison of the computed scour depth with measurements of Chatterjee et al. [39].

where in <sup>u</sup> <sup>j</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup> <sup>u</sup><sup>j</sup>

c>co 

that max CFLj

Figure 1.

10�<sup>5</sup> and 10�<sup>4</sup> s.

5. Applications

136

<sup>f</sup> <sup>þ</sup> <sup>c</sup>u<sup>j</sup>

A flow chart showing the solution procedure using OpenFOAM®.

is referred to Lee and Huang [35] and Lee et al. [38].

5.1 Scour downstream of a sluice gate

<sup>s</sup>, the subscript "j" represents the j

< 0.005. The time step is recommended to be in the range of

is a unit normal vector, V is the volume of the cell, and Δt is the time step. The Courant number must be less than 1 to avoid numerical instability. Generally, max (CFL) <0.1 is suggested. The values of CFL for high

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

concentration regions should be much smaller than those for low concentration regions so that rapid changes of c can be avoided. Therefore, it is recommended

This section briefly describes two examples that have been studied using the two-phase flow models described. The problem descriptions and numerical setups for these two problems are included here; for other relevant information, the reader

A sluice gate is a hydraulic structure used to control the flow in a water channel. Sluice gate structures usually have a rigid floor followed by an erodible bed. The

th face of the cell, S <sup>j</sup>

interface during the entire simulation. In regions away from the sediment-fluid interface or regions where the scouring is predicted to be negligible (e.g., further downstream the scour hole), the mesh sizes with a vertical resolution ranging from 3 to 5 mm are used. The aspect ratio of the mesh outside the wall jet region is less than 3.0. Since in the wall jet, horizontal velocity is significantly larger than the vertical velocity, the aspect ratio of the local mesh in the wall jet region is less than 5.0.

The scour process is sensitive to the model for particle response time used in the simulation. Because Eq. (72) can provide a better prediction of sediment transport rate for small values of Shields parameter, it is recommended for this problem. The two-phase flow model can reproduce well the measured scour depth and the location of sand dune downstream of the scour hole.

#### 5.2 Collapse of a deeply submerged granular column

Another application of the fluid-solid two-phase flow simulation is the simulation of the collapse of a deeply submerged granular column. The problem is best described as a granular flow problem, which involves sediment (a solid phase) and water (fluid phase). Many experimental studies have been reported in the literature on this topic. This section describes a numerical simulation using the fluid-solid two-phase flow model described in this chapter.

Figure 4 shows the experimental setup of Rondon et al. [40]. A 1:1 scale twophase flow simulation was performed by Lee and Huang [35] using the fluid-solid two-phase flow model presented in this chapter. The diameter and the density of the sand grain are 0.225 mm and 2500 kg/m<sup>3</sup> , respectively. The density and the dynamic viscosity of the liquid are 1010 kg/m<sup>3</sup> and 12 mPa s, respectively. Note that the viscosity of the liquid in the experiment is ten times larger than that for water at room temperature. For this problem, using a mesh of 1.0 1.0 mm and the particle response model given by Eq. (78), the fluid-solid two-phase flow model presented in this chapter can reproduce well the collapse process reported in Rondon et al. [40]. Figure 5 shows the simulated collapsing processes compared with the measurement for two initial packing conditions: initially loosely packed condition and initially densely packed condition.

6. Summary

Figure 5.

phase flow models presented in this chapter.

of Rondon et al. [40]. The figure is adapted from Lee and Huang [35].

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

Acknowledgements

Foundation.

139

This chapter presented a brief introduction to the equations and closure models suitable for fluid-solid two-phase flow problems such as sediment transport, submarine landslides, and scour at hydraulic structures. Two averaging operations were performed to derive the governing equations so that the turbulent dispersion, important for geophysical flow problems, can be considered. A new model for the rheological characteristics of sediment phase was used when computing the stresses of the solid phase. The k ϵ model was used to determine the Reynolds stresses. A hybrid model to compute the particle response time was introduced, and the numerical implementation in the framework of OpenFOAM® was discussed. A numerical scheme was introduced to avoid numerical instability when the concentration is high. Two applications were describe to show the capacity of the two-

The simulated collapsing processes for the initially loose condition (a)–(d) and the initially dense condition (e)–(h). The lines represent contours of the computed concentrations, and the symbols were experimental data

This material presented here is partially based upon work supported by the National Science Foundation under Grant No. 1706938 and the Ministry of Science and Technology, Taiwan [MOST 107-2221-E-032-018-MY3]. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science

The two-phase model and closure models presented in this chapter are able to deal with both initially loose packing and initially dense packing conditions and reveal the roles played by the contractancy inside the granular column with a loose packing and dilatancy inside a granular column with a dense packing. One of the conclusions of Lee and Huang [35] is that the collapse process of a densely packed granular column is more sensitive to the model used for particle response time than that of a loosely packed granular column. The particle response model given by Eq. (78) performs better than other models; this is possibly because the liquid used in Rondon et al. [40] is much viscous than water.

Figure 4. A sketch of the experimental setup for the collapse of a deeply submerged granular column.

Figure 5.

interface during the entire simulation. In regions away from the sediment-fluid interface or regions where the scouring is predicted to be negligible (e.g., further downstream the scour hole), the mesh sizes with a vertical resolution ranging from 3 to 5 mm are used. The aspect ratio of the mesh outside the wall jet region is less than 3.0. Since in the wall jet, horizontal velocity is significantly larger than the vertical velocity, the aspect ratio of the local mesh in the wall jet region is less than 5.0.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

tion of sand dune downstream of the scour hole.

two-phase flow model described in this chapter.

the sand grain are 0.225 mm and 2500 kg/m<sup>3</sup>

in Rondon et al. [40] is much viscous than water.

A sketch of the experimental setup for the collapse of a deeply submerged granular column.

initially densely packed condition.

Figure 4.

138

5.2 Collapse of a deeply submerged granular column

The scour process is sensitive to the model for particle response time used in the simulation. Because Eq. (72) can provide a better prediction of sediment transport rate for small values of Shields parameter, it is recommended for this problem. The two-phase flow model can reproduce well the measured scour depth and the loca-

Another application of the fluid-solid two-phase flow simulation is the simulation of the collapse of a deeply submerged granular column. The problem is best described as a granular flow problem, which involves sediment (a solid phase) and water (fluid phase). Many experimental studies have been reported in the literature on this topic. This section describes a numerical simulation using the fluid-solid

Figure 4 shows the experimental setup of Rondon et al. [40]. A 1:1 scale twophase flow simulation was performed by Lee and Huang [35] using the fluid-solid two-phase flow model presented in this chapter. The diameter and the density of

dynamic viscosity of the liquid are 1010 kg/m<sup>3</sup> and 12 mPa s, respectively. Note that the viscosity of the liquid in the experiment is ten times larger than that for water at room temperature. For this problem, using a mesh of 1.0 1.0 mm and the particle response model given by Eq. (78), the fluid-solid two-phase flow model presented in this chapter can reproduce well the collapse process reported in Rondon et al. [40]. Figure 5 shows the simulated collapsing processes compared with the measurement for two initial packing conditions: initially loosely packed condition and

The two-phase model and closure models presented in this chapter are able to deal with both initially loose packing and initially dense packing conditions and reveal the roles played by the contractancy inside the granular column with a loose packing and dilatancy inside a granular column with a dense packing. One of the conclusions of Lee and Huang [35] is that the collapse process of a densely packed granular column is more sensitive to the model used for particle response time than that of a loosely packed granular column. The particle response model given by Eq. (78) performs better than other models; this is possibly because the liquid used

, respectively. The density and the

The simulated collapsing processes for the initially loose condition (a)–(d) and the initially dense condition (e)–(h). The lines represent contours of the computed concentrations, and the symbols were experimental data of Rondon et al. [40]. The figure is adapted from Lee and Huang [35].

#### 6. Summary

This chapter presented a brief introduction to the equations and closure models suitable for fluid-solid two-phase flow problems such as sediment transport, submarine landslides, and scour at hydraulic structures. Two averaging operations were performed to derive the governing equations so that the turbulent dispersion, important for geophysical flow problems, can be considered. A new model for the rheological characteristics of sediment phase was used when computing the stresses of the solid phase. The k ϵ model was used to determine the Reynolds stresses. A hybrid model to compute the particle response time was introduced, and the numerical implementation in the framework of OpenFOAM® was discussed. A numerical scheme was introduced to avoid numerical instability when the concentration is high. Two applications were describe to show the capacity of the twophase flow models presented in this chapter.

#### Acknowledgements

This material presented here is partially based upon work supported by the National Science Foundation under Grant No. 1706938 and the Ministry of Science and Technology, Taiwan [MOST 107-2221-E-032-018-MY3]. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

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103-136

115:F03015

261-291

108:C33057

141

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Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

> [10] Chandrasekharaiah DS, Debnath L. Continuum Mechanics. California:

[11] Hwang GJ, Shen HH. Modeling the phase interaction in the momentum equations of a fluid solid mixture. International Journal of Multiphase

[12] Drew DA. Turbulent sediment transport over a flat bottom using momentum balance. Journal of Applied

[13] Revil-Baudard T, Chauchat J. A two phase model for sheet flow regime based on dense granular flow rheology. Journal of Geophysical Research. 2013;

[14] Chiodi F, Claudin P, Andreotti B. A two-phase flow model of sediment transport: Transition from bedload to suspended load. Journal of Fluid Mechanics. 2014;755:561-581

[15] Launder BE, Sharma BI. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and

[16] Lee CH, Low YM, Chiew YM. Multidimensional rheology-based two-phase model for sediment transport and applications to sheet flow and pipeline scour. Physics of Fluids. 2016;28:053305

Mass Transfer. 1974;1(2):131-137

[17] Song C, Wang P, Makse HA. A phase diagram for jammed matter.

turbulence modulation in fluid–particle

[19] Hsu TJ, Jenkins JT, Liu PLF. On two-phase sediment transport: Sheet

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[18] Crowe CT. On models for

flows. International Journal of Multiphase Flow. 2000;26(5):719-727

Mechanics. 1975;42(1):38-44

Academic Press; 1994

Flow. 1991;17(1):45-57

118:1-16

[2] Picano P, Breugem WP, Brandt L. Turbulent channel flow of dense suspensions of neutrally buoyant

[4] Hsu TJ, Liu PLF. Toward modeling turbulent suspension of sand in the nearshore. Journal of Geophysical Research. 2004;109:C06018

[5] Jha SK, Bombardelli FA. Toward two-phase flow modeling of nondilute sediment transport in open channels. Journal of Geophysical Research. 2010;

[6] Jha SK, Bombardelli FA. Theoretical/ numerical model for the transport of non-uniform suspended sediment in open channels. Advances in Water

[7] Lee CH, Huang ZH, Chiew YM. A multi-scale turbulent dispersion model for dilute flows with suspended sediment. Advances in Water Resources. 2015;79:18-34

[8] Drew DA. Mathematical modeling of two-phase flow. Annual Review of Fluid Mechanics. 1983;15(1):

[9] Hsu TJ, Jenkins JT, Liu PLF. On twophase sediment transport: Dilute flow. Journal of Geophysical Research. 2003;

Resources. 2011;34:577-591

spheres. Journal of Fluid Mechanics. 2015;764:463-487

[3] Vreman AW. Turbulence attenuation in particle-laden flow in smooth and rough channels. Journal of Fluid Mechanics. 2015;73:

### Author details

Zhenhua Huang<sup>1</sup> \* and Cheng-Hsien Lee<sup>2</sup>

1 Department of Ocean and Resources Engineering, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu, HI, USA

2 Department of Water Resources and Environmental Engineering, Tamkang University, New Taipei City, Taiwan

\*Address all correspondence to: zhenhua@hawaii.edu

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

#### References

[1] Balachandar S, Eaton JK. Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics. 2010;42: 111-133

[2] Picano P, Breugem WP, Brandt L. Turbulent channel flow of dense suspensions of neutrally buoyant spheres. Journal of Fluid Mechanics. 2015;764:463-487

[3] Vreman AW. Turbulence attenuation in particle-laden flow in smooth and rough channels. Journal of Fluid Mechanics. 2015;73: 103-136

[4] Hsu TJ, Liu PLF. Toward modeling turbulent suspension of sand in the nearshore. Journal of Geophysical Research. 2004;109:C06018

[5] Jha SK, Bombardelli FA. Toward two-phase flow modeling of nondilute sediment transport in open channels. Journal of Geophysical Research. 2010; 115:F03015

[6] Jha SK, Bombardelli FA. Theoretical/ numerical model for the transport of non-uniform suspended sediment in open channels. Advances in Water Resources. 2011;34:577-591

[7] Lee CH, Huang ZH, Chiew YM. A multi-scale turbulent dispersion model for dilute flows with suspended sediment. Advances in Water Resources. 2015;79:18-34

[8] Drew DA. Mathematical modeling of two-phase flow. Annual Review of Fluid Mechanics. 1983;15(1): 261-291

[9] Hsu TJ, Jenkins JT, Liu PLF. On twophase sediment transport: Dilute flow. Journal of Geophysical Research. 2003; 108:C33057

[10] Chandrasekharaiah DS, Debnath L. Continuum Mechanics. California: Academic Press; 1994

[11] Hwang GJ, Shen HH. Modeling the phase interaction in the momentum equations of a fluid solid mixture. International Journal of Multiphase Flow. 1991;17(1):45-57

[12] Drew DA. Turbulent sediment transport over a flat bottom using momentum balance. Journal of Applied Mechanics. 1975;42(1):38-44

[13] Revil-Baudard T, Chauchat J. A two phase model for sheet flow regime based on dense granular flow rheology. Journal of Geophysical Research. 2013; 118:1-16

[14] Chiodi F, Claudin P, Andreotti B. A two-phase flow model of sediment transport: Transition from bedload to suspended load. Journal of Fluid Mechanics. 2014;755:561-581

[15] Launder BE, Sharma BI. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Letters in Heat and Mass Transfer. 1974;1(2):131-137

[16] Lee CH, Low YM, Chiew YM. Multidimensional rheology-based two-phase model for sediment transport and applications to sheet flow and pipeline scour. Physics of Fluids. 2016;28:053305

[17] Song C, Wang P, Makse HA. A phase diagram for jammed matter. Nature. 2008;453:629-632

[18] Crowe CT. On models for turbulence modulation in fluid–particle flows. International Journal of Multiphase Flow. 2000;26(5):719-727

[19] Hsu TJ, Jenkins JT, Liu PLF. On two-phase sediment transport: Sheet

Author details

Zhenhua Huang<sup>1</sup>

140

\* and Cheng-Hsien Lee<sup>2</sup>

Tamkang University, New Taipei City, Taiwan

provided the original work is properly cited.

\*Address all correspondence to: zhenhua@hawaii.edu

1 Department of Ocean and Resources Engineering, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu, HI, USA

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 Department of Water Resources and Environmental Engineering,

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

flow of massive particles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science. 2004; 460(2048):2223-2250

[20] Hinze J. Turbulence. New York: McGraw Hill; 1959

[21] Trulsson M, Andreotti B, Claudin P. Transition from the viscous to inertial regime in dense suspensions. Physical Review Letters. 2012;109(11):118305

[22] Boyer F, Guazzelli É, Pouliquen O. Unifying suspension and granular rheology. Physical Review Letters. 2011; 107(18):188301

[23] Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences. 2005;363(1832): 1573-1601

[24] Richardson JF, Zaki WN. Sedimentation and fluidisation: Part I. Chemical Engineering Research and Design. 1954;32:S82-S100

[25] Engelund F. On the Laminar and Turbulent Flows of Ground Water Through Homogeneous Sand. Copenhagen: Danish Academy of Technical Sciences; 1953

[26] Chien N, Wan Z. Mechanics of Sediment Transport. Reston: American Society of Civil Engineers; 1999

[27] White FM. Viscous Fluid Flow. Singapore: McGraw-Hill; 2000

[28] Yin X, Koch DL. Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers. Physics of Fluids. 2007;19:093302

[29] Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier; 1972

[30] Das BM. Principles of Geotechnical Engineering. Stamford: Cengage Learning; 2013

[40] Rondon L, Pouliquen O, Aussillous P. Granular collapse in a fluid: Role of the initial volume fraction. Physics of

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

Fluids. 2004;23:73301

143

[31] Burcharth HF, Andersen OH. On the one-dimensional steady and unsteady porous flow equations. Coastal Engineering. 1995;24:233-257

[32] Higuera P, Lara JL, Losada IJ. Threedimensional interaction of waves and porous coastal structures using OpenFOAM (R). Part I: Formulation and validation. Coastal Engineering. 2014;83:243-258

[33] Camenen B. Settling velocity of sediments at high concentrations. Coastal Engineering. 2005;51(1):91-100

[34] Rusche H. Computational fluid dynamics of dispersed two-phase flows at high phase fractions [PhD thesis]. London: University of London; 2003

[35] Lee CH, Huang ZH. A two-phase flow model for submarine granular flows: With an application to collapse of deeply-submerged granular columns. Advances in Water Resources. 2018;115: 286-300

[36] Hancox WT, Banerjee S. Numerical standards for flow-boiling analysis. Nuclear Science and Engineering. 1977; 64(1):106-123

[37] Carver MB. Numerical computation of phase separation in two fluid flow. Journal of Fluids Engineering. 1984; 106(2):147-153

[38] Lee CH, Xu CH, Huang ZH. A three-phase flow simulation of local scour caused by a submerged wall-jet with a water-air interface. Advances in Water Resources. 2018. DOI: 10.1016/j. advwaters.2017.07.017. In Press

[39] Chatterjee SS, Ghosh SN, Chatterjee M. Local scour due to submerged horizontal jet. Journal of Hydraulic Engineering. 1994;120(8):973-992

Modeling of Fluid-Solid Two-Phase Geophysical Flows DOI: http://dx.doi.org/10.5772/intechopen.81449

[40] Rondon L, Pouliquen O, Aussillous P. Granular collapse in a fluid: Role of the initial volume fraction. Physics of Fluids. 2004;23:73301

flow of massive particles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science. 2004;

Advanced Computational Fluid Dynamics for Emerging Engineering Processes…

[30] Das BM. Principles of Geotechnical Engineering. Stamford: Cengage

[31] Burcharth HF, Andersen OH. On the one-dimensional steady and

Engineering. 1995;24:233-257

unsteady porous flow equations. Coastal

[32] Higuera P, Lara JL, Losada IJ. Threedimensional interaction of waves and porous coastal structures using OpenFOAM (R). Part I: Formulation and validation. Coastal Engineering.

[33] Camenen B. Settling velocity of sediments at high concentrations. Coastal Engineering. 2005;51(1):91-100

[34] Rusche H. Computational fluid dynamics of dispersed two-phase flows at high phase fractions [PhD thesis]. London: University of London; 2003

[35] Lee CH, Huang ZH. A two-phase flow model for submarine granular flows: With an application to collapse of deeply-submerged granular columns. Advances in Water Resources. 2018;115:

[36] Hancox WT, Banerjee S. Numerical standards for flow-boiling analysis. Nuclear Science and Engineering. 1977;

[37] Carver MB. Numerical computation of phase separation in two fluid flow. Journal of Fluids Engineering. 1984;

[38] Lee CH, Xu CH, Huang ZH. A three-phase flow simulation of local scour caused by a submerged wall-jet with a water-air interface. Advances in Water Resources. 2018. DOI: 10.1016/j.

advwaters.2017.07.017. In Press

[39] Chatterjee SS, Ghosh SN, Chatterjee M. Local scour due to submerged horizontal jet. Journal of Hydraulic Engineering. 1994;120(8):973-992

Learning; 2013

2014;83:243-258

286-300

64(1):106-123

106(2):147-153

[20] Hinze J. Turbulence. New York:

[21] Trulsson M, Andreotti B, Claudin P. Transition from the viscous to inertial regime in dense suspensions. Physical Review Letters. 2012;109(11):118305

[22] Boyer F, Guazzelli É, Pouliquen O. Unifying suspension and granular rheology. Physical Review Letters. 2011;

[23] Pitman EB, Le L. A two-fluid model

Engineering Sciences. 2005;363(1832):

Sedimentation and fluidisation: Part I. Chemical Engineering Research and

[25] Engelund F. On the Laminar and Turbulent Flows of Ground Water Through Homogeneous Sand. Copenhagen: Danish Academy of

[26] Chien N, Wan Z. Mechanics of Sediment Transport. Reston: American Society of Civil Engineers; 1999

[27] White FM. Viscous Fluid Flow. Singapore: McGraw-Hill; 2000

[28] Yin X, Koch DL. Hindered settling

[29] Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier;

velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers. Physics of

Fluids. 2007;19:093302

1972

142

for avalanche and debris flows. Philosophical Transactions. Series A,

Mathematical, Physical, and

[24] Richardson JF, Zaki WN.

Design. 1954;32:S82-S100

Technical Sciences; 1953

460(2048):2223-2250

McGraw Hill; 1959

107(18):188301

1573-1601

**145**

**Chapter 7**

Pumps

**Abstract**

**1. Introduction**

in virtual space

flowing medium)

CFD Simulation of Flow

Centrifugal Pumps, Industrial

Fans and Positive Displacement

*Wieslaw Fiebig, Paulina Szwemin and Maciej Zawislak*

studies for various operating parameters of pumps have been presented.

**Keywords:** CFD simulation, pumps, cavitation, industrial fans, flow analysis

Industrial machines and devices with rotating operating parts are difficult to model due to their complex geometry, the transition of elements of the discrete model between the rotating and non-rotating parts, the importance of the quality of elements of the discrete model, and the fact that in most cases, it is necessary to take into account the time step (elements rotate in relation to the casing). It is also troublesome that very often the calculations are stabilised only after a few rotations of the operating element. However, the use of computational fluid dynamics methods to model this group of machines and equipment is justified, as it enables:

• Determining the internal and external characteristics of machines and devices

• Imaging and observing the flow phenomena in the machine itself (especially when for various reasons it is impossible to measure physical quantities of the

The chapter presents simulation models for the analysis of centrifugal pumps, fans and positive displacement pumps. In centrifugal pumps based on the "sliding mesh" method, a CFD model was created to calculate the flow characteristics, and the pump operating parameters were determined at which an unfavourable phenomenon of cavitation occurs. In the case of a radial fan, the CFD model was used to determine the influence of inlet channel geometry on the efficiency of an industrial installation. The main purpose of the CFD simulation was to obtain the pressure distributions and determine the areas in which cavitation may occur. To investigate the flow phenomena that occur in external gear pumps and double-acting vane pumps, the "immersed solid" method was used. The results of 2D and 3D simulation

Phenomena in Selected

### **Chapter 7**

## CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans and Positive Displacement Pumps

*Wieslaw Fiebig, Paulina Szwemin and Maciej Zawislak*

### **Abstract**

The chapter presents simulation models for the analysis of centrifugal pumps, fans and positive displacement pumps. In centrifugal pumps based on the "sliding mesh" method, a CFD model was created to calculate the flow characteristics, and the pump operating parameters were determined at which an unfavourable phenomenon of cavitation occurs. In the case of a radial fan, the CFD model was used to determine the influence of inlet channel geometry on the efficiency of an industrial installation. The main purpose of the CFD simulation was to obtain the pressure distributions and determine the areas in which cavitation may occur. To investigate the flow phenomena that occur in external gear pumps and double-acting vane pumps, the "immersed solid" method was used. The results of 2D and 3D simulation studies for various operating parameters of pumps have been presented.

**Keywords:** CFD simulation, pumps, cavitation, industrial fans, flow analysis

#### **1. Introduction**

Industrial machines and devices with rotating operating parts are difficult to model due to their complex geometry, the transition of elements of the discrete model between the rotating and non-rotating parts, the importance of the quality of elements of the discrete model, and the fact that in most cases, it is necessary to take into account the time step (elements rotate in relation to the casing). It is also troublesome that very often the calculations are stabilised only after a few rotations of the operating element. However, the use of computational fluid dynamics methods to model this group of machines and equipment is justified, as it enables:


In this chapter, selected examples of numerical calculations will be described, showing the possibility of using CFD methods to solve machine and equipment problems with a rotating operating element, often found in industrial practice.

#### **2. Vane pump: flow analysis**

The innovative vane pump described in study [1] was subjected to the analysis of flow phenomena. In this solution, the pump is integrated into the BLDC permanent magnet electric motor. Due to its design, which differs from the standard solutions, it was necessary to check whether cavitation could occur in the suction channel of the pump. The main objective of the CFD simulation was to determine the areas where cavitation is likely to occur [2] and its intensity depending on the rotational velocity. The subject of the study is a positive displacement pump with integrated electric drive, consisting of an impeller embedded in a casing. Unlike conventional gear and vane pumps [3–10], the pump impeller and motor stator are immovable components, while the pump casing rotates with the rotor of the electric motor. **Figure 1** shows the 3D model of the analysed pump.

An important problem is to examine the flow in the suction channel of the pump, as it is exposed to the adverse effects of cavitation, which can develop as a result of a too high value of negative pressure occurring in the suction area.

On the basis of the three-dimensional model of the pump, a geometric model of the volume of operating fluid filling its interior was prepared (**Figure 1b**). As expected, the result is a very complex structure in terms of geometry. Due to the particular interest in the phenomena occurring in the suction channel of the pump, the calculations used a fragment of the geometric model of the operating fluid volume filling the interior of the pump, which is the volume of oil filling the pump from the inlet to the suction kidneys supplying the fluid to the inter-vane spaces (**Figure 2a**). The separated volume is contained in the immovable elements of the structure, which further simplifies the formulation of the flow problem and the choice of calculation parameters.

#### **Figure 1.**

*(a) 3D model of the vane pump with integrated mechatronic electric drive and (b) 3D model of the operating fluid volume filling the pump.*

**147**

**Figure 3.**

**Figure 2.**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

Based on the three-dimensional pump model, the simplified geometric model of the operating fluid volume filling the suction channel was discretized using a tetrahedral grid. The result is a geometric model divided into 144,390 tetrahedral

The next step in formulating the flow problem is to select the type and define the boundary conditions for relevant fragments of the geometry. In the analysed case, the conditions concerning the fluid inflow and outflow were set as shown in **Figure 2a**.

*Generic geometry—suction channel pressure distribution for different rotational velocities: (a) 500 rpm, (b)* 

*1000 rpm, (c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

*(a) Suction channel geometry and (b) discrete model.*

elements with 29,711 nodes, as shown in **Figure 2b**.

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

Based on the three-dimensional pump model, the simplified geometric model of the operating fluid volume filling the suction channel was discretized using a tetrahedral grid. The result is a geometric model divided into 144,390 tetrahedral elements with 29,711 nodes, as shown in **Figure 2b**.

The next step in formulating the flow problem is to select the type and define the boundary conditions for relevant fragments of the geometry. In the analysed case, the conditions concerning the fluid inflow and outflow were set as shown in **Figure 2a**.

#### **Figure 2.**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

• Improving the efficiency of machinery and equipment

**Figure 1** shows the 3D model of the analysed pump.

tials, mixers)

**2. Vane pump: flow analysis**

choice of calculation parameters.

trial practice.

• Designing equipment for which there are no design guidelines (e.g. differen-

The innovative vane pump described in study [1] was subjected to the analysis of flow phenomena. In this solution, the pump is integrated into the BLDC permanent magnet electric motor. Due to its design, which differs from the standard solutions, it was necessary to check whether cavitation could occur in the suction channel of the pump. The main objective of the CFD simulation was to determine the areas where cavitation is likely to occur [2] and its intensity depending on the rotational velocity. The subject of the study is a positive displacement pump with integrated electric drive, consisting of an impeller embedded in a casing. Unlike conventional gear and vane pumps [3–10], the pump impeller and motor stator are immovable components, while the pump casing rotates with the rotor of the electric motor.

An important problem is to examine the flow in the suction channel of the pump, as it is exposed to the adverse effects of cavitation, which can develop as a result of a too high value of negative pressure occurring in the suction area.

On the basis of the three-dimensional model of the pump, a geometric model of the volume of operating fluid filling its interior was prepared (**Figure 1b**). As expected, the result is a very complex structure in terms of geometry. Due to the particular interest in the phenomena occurring in the suction channel of the pump, the calculations used a fragment of the geometric model of the operating fluid volume filling the interior of the pump, which is the volume of oil filling the pump from the inlet to the suction kidneys supplying the fluid to the inter-vane spaces (**Figure 2a**). The separated volume is contained in the immovable elements of the structure, which further simplifies the formulation of the flow problem and the

*(a) 3D model of the vane pump with integrated mechatronic electric drive and (b) 3D model of the operating* 

In this chapter, selected examples of numerical calculations will be described, showing the possibility of using CFD methods to solve machine and equipment problems with a rotating operating element, often found in indus-

**146**

**Figure 1.**

*fluid volume filling the pump.*

*(a) Suction channel geometry and (b) discrete model.*

#### **Figure 3.**

*Generic geometry—suction channel pressure distribution for different rotational velocities: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

*Generic geometry—suction channel velocity distribution for different velocities: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

In order to obtain the most accurate results of the simulation, the "pressure inlet" condition at the inlet and the "mass flow rate" at the outlet were assumed. The mass flow rate was determined using the formula:

$$q\_1 = 2xb\left[\frac{\pi}{2}\left(R\_2^2 - R\_1^2\right) - w\left(R\_2 - R\_1\right)\right] \tag{1}$$

**149**

**Figure 5.**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

The results of the calculations, apart from pressure distributions, were presented in the form of velocity distributions in the considered area, which are presented in **Figure 4**. From the obtained velocity distributions, it appears that the rotational velocity of the pump significantly influences the velocity of fluid flow in one of the supply channels for both the generic and the modified geometry. It is worth noting that the area where the highest velocities were identified corresponds to the area of the lowest pressures observed in the suction channel. The velocity of the fluid decreases with the lowering of

*Generic geometry—streamlines in the investigated suction channel area for different velocities: (a) 500 rpm,* 

*(b) 1000 rpm, (c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

**Figure 5** shows the fluid flow in the form of streamlines, for which the inflow plane to the domain is assigned as the beginning. The results obtained confirm the previous assumptions that the fluid flows evenly and without major turbulences through both inlet channels. Uneven velocity distribution and different pressure values due to asymmetrical layout of channels did not affect the fluid flow. The results obtained on the basis of numerical calculations are the basis for evaluation

Another object under consideration with rotating operating elements was a radial fan. The aim of the numerical simulation was to improve its efficiency. The flow of real gas through a fan with a finite amount of blades is carried out by the cost of loss of energy, called hydraulic losses. Those losses are a consequence of the friction of air molecules occurring on the blade walls and fan housing, vortexes developed in the gas stream, etc. The influence of hydraulic losses on the working characteristic of the radial fan is described by a hydraulic efficiency coefficient, which is defined as the ratio of the useful power to the power delivered by the impeller. This coefficient also defines the real delivery height to the theoretical delivery height—obtained for the finite amount of impeller blades. The impeller geometry considered in possible options, i.e. with eight (factory option) and nine

the rotational velocity, but in the case of simplified geometry, it is slightly lower.

of the structure of channels supplying fluid to the inter-vane volumes.

**3. Radial fan: characteristics and performance improvement**

(suggested option) vanes, are shown in **Figures 6** and **7**.

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

where *q* is the specific mass flow rate; z is the number of vanes; b is the width of a vane; w is the thickness of a vane; *R1* is the small race radius; and R2 is the large race radius.

On this basis, the numerical values entered into the simulation for each impeller velocity were obtained. Within the framework of the study, the analysis of the operating medium flow through the suction channel of the vane pump was performed for various rotational velocity values—changed within the range of 500–3000 rpm.

**Figure 3** shows the pressure distributions in the suction channel of the tested pump for the generic geometry. For each of the cases considered, the lowest pressure occurs in one of the channels supplying fluid to the suction kidneys directly at the inlet to the channel. It was found that the negative pressures for the whole range of rotational velocities are higher than the pressure of oil evaporation, which prevents the occurrence of cavitation phenomena.

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

#### **Figure 5.**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

*Generic geometry—suction channel velocity distribution for different velocities: (a) 500 rpm, (b) 1000 rpm,* 

In order to obtain the most accurate results of the simulation, the "pressure inlet" condition at the inlet and the "mass flow rate" at the outlet were assumed.

where *q* is the specific mass flow rate; z is the number of vanes; b is the width of a vane; w is the thickness of a vane; *R1* is the small race radius; and R2 is the large

On this basis, the numerical values entered into the simulation for each impeller velocity were obtained. Within the framework of the study, the analysis of the operating medium flow through the suction channel of the vane pump was performed for various rotational velocity values—changed within the range of 500–3000 rpm. **Figure 3** shows the pressure distributions in the suction channel of the tested pump for the generic geometry. For each of the cases considered, the lowest pressure occurs in one of the channels supplying fluid to the suction kidneys directly at the inlet to the channel. It was found that the negative pressures for the whole range of rotational velocities are higher than the pressure of oil evaporation, which

) − *w*(*R*<sup>2</sup> − *R*1)] (1)

\_\_ *π <sup>z</sup>* (*R*<sup>2</sup> <sup>2</sup> − *R*<sup>1</sup> 2

*(c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

*q* = 2*zb*[

The mass flow rate was determined using the formula:

prevents the occurrence of cavitation phenomena.

**148**

**Figure 4.**

race radius.

*Generic geometry—streamlines in the investigated suction channel area for different velocities: (a) 500 rpm, (b) 1000 rpm, (c) 1500 rpm, (d) 2000 rpm, (e) 2500 rpm and (f) 3000 rpm.*

The results of the calculations, apart from pressure distributions, were presented in the form of velocity distributions in the considered area, which are presented in **Figure 4**. From the obtained velocity distributions, it appears that the rotational velocity of the pump significantly influences the velocity of fluid flow in one of the supply channels for both the generic and the modified geometry. It is worth noting that the area where the highest velocities were identified corresponds to the area of the lowest pressures observed in the suction channel. The velocity of the fluid decreases with the lowering of the rotational velocity, but in the case of simplified geometry, it is slightly lower.

**Figure 5** shows the fluid flow in the form of streamlines, for which the inflow plane to the domain is assigned as the beginning. The results obtained confirm the previous assumptions that the fluid flows evenly and without major turbulences through both inlet channels. Uneven velocity distribution and different pressure values due to asymmetrical layout of channels did not affect the fluid flow. The results obtained on the basis of numerical calculations are the basis for evaluation of the structure of channels supplying fluid to the inter-vane volumes.

#### **3. Radial fan: characteristics and performance improvement**

Another object under consideration with rotating operating elements was a radial fan. The aim of the numerical simulation was to improve its efficiency. The flow of real gas through a fan with a finite amount of blades is carried out by the cost of loss of energy, called hydraulic losses. Those losses are a consequence of the friction of air molecules occurring on the blade walls and fan housing, vortexes developed in the gas stream, etc. The influence of hydraulic losses on the working characteristic of the radial fan is described by a hydraulic efficiency coefficient, which is defined as the ratio of the useful power to the power delivered by the impeller. This coefficient also defines the real delivery height to the theoretical delivery height—obtained for the finite amount of impeller blades. The impeller geometry considered in possible options, i.e. with eight (factory option) and nine (suggested option) vanes, are shown in **Figures 6** and **7**.

#### *Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

**Figure 6.** *Impeller shape: eight vanes.*

#### **Figure 8.**

*Discrete model with division into tetrahedral elements on the impeller and vanes.*

#### **Figure 9.**

*Comparison of the calculation results and the results of the technical documentation for the impeller with the eight and nine vanes.*

**151**

**Figure 10.**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

For the calculations, the model of impeller according to the enclosed documentation was used as the output model. Calculations have been made for both impeller variants. For both of the cases, the discrete model was based on tetrahedral elements (as exemplary shown in **Figure 8**). Elements near walls were compacted. The flow was modelled as turbulent, using the RANS method and the two-equation turbulence model k-ε. In the first stage of the study, the analysis of the impeller with eight (**Figure 6**)

and nine (**Figure 7**) vanes was carried out. For the eight vanes, the results of the simulation were also compared with the available results in the technical

*Total velocity [m/s] distribution for calculated impeller operating points with eight and nine vanes.*

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

**Figure 10.** *Total velocity [m/s] distribution for calculated impeller operating points with eight and nine vanes.*

For the calculations, the model of impeller according to the enclosed documentation was used as the output model. Calculations have been made for both impeller variants. For both of the cases, the discrete model was based on tetrahedral elements (as exemplary shown in **Figure 8**). Elements near walls were compacted. The flow was modelled as turbulent, using the RANS method and the two-equation turbulence model k-ε.

In the first stage of the study, the analysis of the impeller with eight (**Figure 6**) and nine (**Figure 7**) vanes was carried out. For the eight vanes, the results of the simulation were also compared with the available results in the technical

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

*Discrete model with division into tetrahedral elements on the impeller and vanes.*

*Comparison of the calculation results and the results of the technical documentation for the impeller with the* 

**150**

**Figure 9.**

*eight and nine vanes.*

**Figure 8.**

**Figure 7.**

**Figure 6.**

*Impeller shape: nine vanes.*

*Impeller shape: eight vanes.*

documentation and found to be similar (**Figure 9**). Furthermore, the overall performance of the two types of impellers found with aid of CFD calculation maintains in similar level.

In order to verify the correctness of the calculation of the main dimensions of the impeller, a theoretical design process was carried out. On the basis of known designs, the influence of impeller parameters on its performance, compression and efficiency was simulated. It was necessary to maintain the existing parameters of the impeller, improving only its efficiency. The modifications were limited by the external dimensions of the impeller in order to be able to work with the existing collecting volute.

After a number of variant combinations, the outlet angle of the vane was changed to 23° and the vane profile modified to improve efficiency. The results show that by changing the outlet angle, the average efficiency for the eight-vane impeller was increased by 2.3% and for the nine-vane impeller by 2.9% in relation to the basic eight-vane impeller.

**Figure 10** shows a comparison of the flow images for the impellers with eight and nine vanes with a 23° outlet angle.

The best results were obtained for the nine-vane impeller and the changed outlet angle. An average efficiency increase of 2.9% was achieved in relation to the impeller from the technical documentation. The flow images are correct. There are no particularly dangerous phenomena, such as interruption of flow or turbulence.

#### **4. Centrifugal pump with collecting channel: undetermined flow with cavitation**

Another object of the study was a single-stage centrifugal pump with a spiral volute cooperating with two similar types of impellers, commonly used in such a device. Those impellers are denoted as W13 and W17. The W17 impeller differs from the W13 impeller only by the shape of a vane. Both impellers had eight vanes each. The analysis of the impellers with the two-dimensional peculiarity method for non-viscous medium suggested higher cavitation resistance of the W13 impeller.

In the first stage, calculations were made of the undetermined flow through the pump without cavitation in order to determine the most favourable boundary conditions to be applied when analysing the flow through the pump and determining the calculation characteristics of the pump and the impeller.

The calculations reflect the full three-dimensional geometry of the pump (**Figure 11**) consisting of a straight section of the pipeline before the inlet to the impeller, a centrifugal impeller, a spiral collecting volute, a diffuser, and a short section of pipeline after the pump.

Separate discreet models have been built in the inlet and outlet impeller areas. On the cylindrical surface between the impeller and the volute, these models were not connected by common nodes and remained unfit. Thus, during the calculation it was possible to use the "sliding mesh" technique, which is used to model the rotation of the impeller in relation to the stationary casing. The discrete model is built with approximately 1.3 million tetrahedral elements in total. The elements were also compacted near the vane surface and in the area between the impeller and the collecting channel (**Figure 12**).

The mathematical model of the flow is described by the Reynolds-averaged Navier-Stokes equations (RANS). For the description of the turbulence, a twoequation k-ε model was used. The following control surfaces were used, where static pressure was monitored during the calculation:

**153**

ANSYS Fluent.

**Figure 11.**

**Figure 12.**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

• The inlet section at the beginning of the suction channel (A-A)

*Discrete model by type of tetrahedral element of the impeller surface, on the hub and rear disc side.*

*The calculation area under consideration and its characteristic cross sections (W13 impeller pump).*

• The cross section at the end of the diffuser (C-C)

Calculations were made according to the scheme:

• The cylindrical surface at the outlet from the impeller inter-vane channel (B-B)

• The outlet section at the end of the cylindrical section of the pipeline (D-D)

In the inlet section (A-A), a homogeneous velocity field was set with the value resulting from the flow rate and the channel section area c = Q/A and the direction corresponding to the connector axis ("velocity inlet" boundary condition). In cross section (D-D), a high static pressure of 1000 kPa was set so that the pressure in the impeller would not drop below the saturation vapour pressure ("pressure outlet" boundary condition). A two-phase flow "mixture" model was selected for the calculations. During the calculations, equations describing the formation of the gaseous phase (cavitation) were excluded. This approach is suggested by

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

#### **Figure 12.**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

in similar level.

collecting volute.

**cavitation**

impeller.

the basic eight-vane impeller.

and nine vanes with a 23° outlet angle.

section of pipeline after the pump.

collecting channel (**Figure 12**).

static pressure was monitored during the calculation:

documentation and found to be similar (**Figure 9**). Furthermore, the overall performance of the two types of impellers found with aid of CFD calculation maintains

After a number of variant combinations, the outlet angle of the vane was changed to 23° and the vane profile modified to improve efficiency. The results show that by changing the outlet angle, the average efficiency for the eight-vane impeller was increased by 2.3% and for the nine-vane impeller by 2.9% in relation to

**Figure 10** shows a comparison of the flow images for the impellers with eight

**4. Centrifugal pump with collecting channel: undetermined flow with** 

Another object of the study was a single-stage centrifugal pump with a spiral volute cooperating with two similar types of impellers, commonly used in such a device. Those impellers are denoted as W13 and W17. The W17 impeller differs from the W13 impeller only by the shape of a vane. Both impellers had eight vanes each. The analysis of the impellers with the two-dimensional peculiarity method for non-viscous medium suggested higher cavitation resistance of the W13

In the first stage, calculations were made of the undetermined flow through the pump without cavitation in order to determine the most favourable boundary conditions to be applied when analysing the flow through the pump and determin-

The calculations reflect the full three-dimensional geometry of the pump (**Figure 11**) consisting of a straight section of the pipeline before the inlet to the impeller, a centrifugal impeller, a spiral collecting volute, a diffuser, and a short

Separate discreet models have been built in the inlet and outlet impeller areas. On the cylindrical surface between the impeller and the volute, these models were not connected by common nodes and remained unfit. Thus, during the calculation it was possible to use the "sliding mesh" technique, which is used to model the rotation of the impeller in relation to the stationary casing. The discrete model is built with approximately 1.3 million tetrahedral elements in total. The elements were also compacted near the vane surface and in the area between the impeller and the

The mathematical model of the flow is described by the Reynolds-averaged Navier-Stokes equations (RANS). For the description of the turbulence, a twoequation k-ε model was used. The following control surfaces were used, where

ing the calculation characteristics of the pump and the impeller.

The best results were obtained for the nine-vane impeller and the changed outlet angle. An average efficiency increase of 2.9% was achieved in relation to the impeller from the technical documentation. The flow images are correct. There are no particularly dangerous phenomena, such as interruption of flow or turbulence.

In order to verify the correctness of the calculation of the main dimensions of the impeller, a theoretical design process was carried out. On the basis of known designs, the influence of impeller parameters on its performance, compression and efficiency was simulated. It was necessary to maintain the existing parameters of the impeller, improving only its efficiency. The modifications were limited by the external dimensions of the impeller in order to be able to work with the existing

**152**

*Discrete model by type of tetrahedral element of the impeller surface, on the hub and rear disc side.*


Calculations were made according to the scheme:

In the inlet section (A-A), a homogeneous velocity field was set with the value resulting from the flow rate and the channel section area c = Q/A and the direction corresponding to the connector axis ("velocity inlet" boundary condition). In cross section (D-D), a high static pressure of 1000 kPa was set so that the pressure in the impeller would not drop below the saturation vapour pressure ("pressure outlet" boundary condition). A two-phase flow "mixture" model was selected for the calculations. During the calculations, equations describing the formation of the gaseous phase (cavitation) were excluded. This approach is suggested by ANSYS Fluent.

**Figure 13.** *Example pressure pulsation diagram as pressure difference between vane outlet (interface\_2) and inlet (inlet), depending on iteration (time).*

On the internal walls of the flow channel, the condition of zero velocity of the fluid in relation to the wall was set. The increase of static pressure (increase of hydrostatic height) between inlet and outlet cross sections of the pump was the expected value and allowed to reproduce flow characteristics. During the calculations, the average static pressure was monitored on the four control surfaces mentioned above. The calculations were interrupted after repeated oscillations of the static pressure on these surfaces were obtained, which took place after 6–8 rotations of the impeller. An example of a pressure pulsation diagram is shown in **Figure 13**. A fixed time step of Δt = 5,75E–5 s, corresponding to an impeller rotation by 1°, was used for the calculations.

The calculated flow characteristics of the entire pump and the W13 impeller are presented in **Figure 14**. The course of the relevant experimental characteristics is also presented.

The pump characteristics indicate a pressure increase between the cross sections A-A and C-C, characteristics of the impeller—between sections A-A and B-B. The pressure drop in the suction channel is insignificant compared to the pressure drop in the impeller.

#### **Figure 14.**

*Pump and impeller flow characteristics W13 determined by calculation of the transient flow (spiral collecting channel model): comparison with experimental data.*

**155**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

Cavitation in the pump is associated with a pressure drop in the suction area of the first degree [11]. This causes the fluid-vapour biphasic flow to occur and the continuity of the flow through the pump to be interrupted. In centrifugal pumps, cavitation shall be characterised by a clearly visible disturbance in the following characteristics: flow H = f(Q ), power consumption P = f(Q ) and efficiency η = f(Q ). If the suction height increases at a given velocity and flow rate (or the intake height decreases), then the boundary value of the suction height at which the pump enters the cavitation state is obtained. In this way, taking into account a certain safety margin, it is possible to obtain a curve of the required excess of the energy of a fluid at the pump inlet section over the energy of evaporation of this fluid in the form of NPSH = f(Q ) (net positive suction head). The NPSH parameter

> *ps* − *pv* <sup>γ</sup> <sup>+</sup> *cs* 2 \_\_

where *ps* is the absolute pressure at the inlet cross section of the pump and cs is the fluid velocity at the pump inlet cross section (average). Typically, this surplus is

Due to the symmetry of geometry, the flow through the impeller is determined. The elimination of pressure pulsations has significantly accelerated the iterative calculation process. The flow field in the impeller still remained a periodic-symmetric field, but it was the same in all the vane channels. This allowed the calculation area to be limited to one inter-vane channel of the impeller. As a result, the calculation

A discrete model consisting of about 300,000 hexahedral cells was used. Since the discrete model remains stationary during the calculation, a *moving reference* 

For the calculations, the "velocity inlet" and "pressure outlet" boundary condi-

Cavitation test in the impeller was performed for several selected values of the flow rate. Calculations were carried out in which equations describing cavitation and two-phase flow were included. The static pressure at the outlet was gradually

It was found that the lowest pressure in the impeller was initially higher than the saturated vapour pressure pmin > pv; then it was already limited by the pv value. For each set outlet pressure, the static inlet pressure was recorded. In the W13 impeller, cavitation occurs on the impeller vanes, close to the incidence edge on the concave side of the vane. In the W17 impeller, cavitation appears on the convex side of

When the outlet pressure is further reduced, it reaches a constant boundary value, depending on the flow rate—fully developed cavitation. Further lowering of the outlet pressure leads to a loss of convergence and interruption of the calculation.

/h). Selected images of the development of cavitation are

tions were used on the outer surface of the annular collecting channel and the two-phase flow "mixture" model. During the calculation, the average static pressure value at the cross sections A-A (inlet) and B-B (outlet from the impeller inter-vane

Determination of the cavitation state in the impeller for a given flow rate requires many calculations of the pressure distribution in the inter-vane space at the decreasing inlet pressure. The simulation assumes that a simplified geometric model of a collective channel can be used to determine the flow characteristics of the impeller itself. Instead of a spiral, an axial-symmetrical guide was used as a

related to a state where the first-stage total head drops by 3% (NPSH3).

time corresponding to one characteristic point has been reduced.

*frame* was used which rotates at the impeller velocity.

<sup>2</sup>*<sup>g</sup>* (2)

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

expresses the "suction power" of the pump:

drainage element for the medium.

channel) was monitored.

the vane (for Q = 70 m3

presented in **Figures 15** and **16**.

reduced from 800 to 580 kPa.

*NPSH* = \_\_\_\_\_

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

Cavitation in the pump is associated with a pressure drop in the suction area of the first degree [11]. This causes the fluid-vapour biphasic flow to occur and the continuity of the flow through the pump to be interrupted. In centrifugal pumps, cavitation shall be characterised by a clearly visible disturbance in the following characteristics: flow H = f(Q ), power consumption P = f(Q ) and efficiency η = f(Q ). If the suction height increases at a given velocity and flow rate (or the intake height decreases), then the boundary value of the suction height at which the pump enters the cavitation state is obtained. In this way, taking into account a certain safety margin, it is possible to obtain a curve of the required excess of the energy of a fluid at the pump inlet section over the energy of evaporation of this fluid in the form of NPSH = f(Q ) (net positive suction head). The NPSH parameter expresses the "suction power" of the pump:

$$\text{NPSH} \quad = \frac{p\_s - p\_v}{\text{\textdegree\uparrow}} + \frac{c\_s^2}{\text{\textdegree\downarrow}} \tag{2}$$

where *ps* is the absolute pressure at the inlet cross section of the pump and cs is the fluid velocity at the pump inlet cross section (average). Typically, this surplus is related to a state where the first-stage total head drops by 3% (NPSH3).

Determination of the cavitation state in the impeller for a given flow rate requires many calculations of the pressure distribution in the inter-vane space at the decreasing inlet pressure. The simulation assumes that a simplified geometric model of a collective channel can be used to determine the flow characteristics of the impeller itself. Instead of a spiral, an axial-symmetrical guide was used as a drainage element for the medium.

Due to the symmetry of geometry, the flow through the impeller is determined. The elimination of pressure pulsations has significantly accelerated the iterative calculation process. The flow field in the impeller still remained a periodic-symmetric field, but it was the same in all the vane channels. This allowed the calculation area to be limited to one inter-vane channel of the impeller. As a result, the calculation time corresponding to one characteristic point has been reduced.

A discrete model consisting of about 300,000 hexahedral cells was used. Since the discrete model remains stationary during the calculation, a *moving reference frame* was used which rotates at the impeller velocity.

For the calculations, the "velocity inlet" and "pressure outlet" boundary conditions were used on the outer surface of the annular collecting channel and the two-phase flow "mixture" model. During the calculation, the average static pressure value at the cross sections A-A (inlet) and B-B (outlet from the impeller inter-vane channel) was monitored.

Cavitation test in the impeller was performed for several selected values of the flow rate. Calculations were carried out in which equations describing cavitation and two-phase flow were included. The static pressure at the outlet was gradually reduced from 800 to 580 kPa.

It was found that the lowest pressure in the impeller was initially higher than the saturated vapour pressure pmin > pv; then it was already limited by the pv value. For each set outlet pressure, the static inlet pressure was recorded. In the W13 impeller, cavitation occurs on the impeller vanes, close to the incidence edge on the concave side of the vane. In the W17 impeller, cavitation appears on the convex side of the vane (for Q = 70 m3 /h). Selected images of the development of cavitation are presented in **Figures 15** and **16**.

When the outlet pressure is further reduced, it reaches a constant boundary value, depending on the flow rate—fully developed cavitation. Further lowering of the outlet pressure leads to a loss of convergence and interruption of the calculation.

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

*Example pressure pulsation diagram as pressure difference between vane outlet (interface\_2) and inlet (inlet),* 

On the internal walls of the flow channel, the condition of zero velocity of the fluid in relation to the wall was set. The increase of static pressure (increase of hydrostatic height) between inlet and outlet cross sections of the pump was the expected value and allowed to reproduce flow characteristics. During the calculations, the average static pressure was monitored on the four control surfaces mentioned above. The calculations were interrupted after repeated oscillations of the static pressure on these surfaces were obtained, which took place after 6–8 rotations of the impeller. An example of a pressure pulsation diagram is shown in **Figure 13**. A fixed time step of Δt = 5,75E–5 s,

The calculated flow characteristics of the entire pump and the W13 impeller are presented in **Figure 14**. The course of the relevant experimental characteristics is

The pump characteristics indicate a pressure increase between the cross sections A-A and C-C, characteristics of the impeller—between sections A-A and B-B. The pressure drop in the suction channel is insignificant compared to the pressure drop in the impeller.

corresponding to an impeller rotation by 1°, was used for the calculations.

*Pump and impeller flow characteristics W13 determined by calculation of the transient flow (spiral collecting* 

**154**

**Figure 14.**

*channel model): comparison with experimental data.*

**Figure 13.**

*depending on iteration (time).*

also presented.

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

#### **Figure 15.**

*Cavitation development image on the impeller W13 disc surface at Q = 70 m3 /h and decreasing static pressure at the inlet (percentage of gas phase is given).*

Cavitation image – model with spiral collection channel and immovable impeller (*Moving Reference Frame*) – **Figure 17**. Cavitation image – model with spiral collecting channel and rotating impeller (*Moving Mesh*) – **Figure 18**.

The cavitation fields for the axial-symmetric model are correctly symmetrical. However, the behaviour of the tested impellers is different:


The calculations converge quickly. However, the cavitation fields in the *moving reference frame* model are non-physical, and the cavitation area expands very

#### **Figure 16.**

*Cavitation development image on the impeller W17 disc surface at Q = 70 m3 /h and decreasing static inlet pressure (percentage of gas phase is given).*

**157**

duration.

**Figure 18.**

**Figure 17.**

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

*Cavitation area for parameters: (a) inlet = 189 kPa and outlet = 800 kPa, (b) inlet = 95 kPa and outlet = 700 kPa, (c) inlet = 58 kPa and outlet = 650 kPa, (d) inlet = 27 kPa and outlet = 622 kPa and (e)* 

*inlet = 27 kPa and outlet = 600 kPa. The percentage of gas phase is given.*

quickly. Cavitation starts in the direction of the smallest radius of the collecting spiral. The *moving mesh* model produces the best results (mainly physical). However, the problem is the slow convergence of calculations and their long

*Cavitation area for parameters: (a) inlet = 28.3 kPa and outlet = 675 kPa, (b) inlet = 27.8 kPa and outlet = 650 kPa and (c) inlet = 27.8 kPa and outlet = 622 kPa. The percentage of gas phase is given.*

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

#### **Figure 17.**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

Cavitation image – model with spiral collection channel and immovable impeller (*Moving Reference Frame*) – **Figure 17**. Cavitation image – model with spiral collect-

The cavitation fields for the axial-symmetric model are correctly symmetrical.

The calculations converge quickly. However, the cavitation fields in the *moving reference frame* model are non-physical, and the cavitation area expands very

• Impeller W13: cavitation is formed on the concave side of the vane.

• Impeller W17: cavitation is formed on the convex side of the vane.

ing channel and rotating impeller (*Moving Mesh*) – **Figure 18**.

However, the behaviour of the tested impellers is different:

*Cavitation development image on the impeller W13 disc surface at Q = 70 m3*

**156**

**Figure 16.**

**Figure 15.**

*the inlet (percentage of gas phase is given).*

*pressure (percentage of gas phase is given).*

*Cavitation development image on the impeller W17 disc surface at Q = 70 m3*

*/h and decreasing static inlet* 

*/h and decreasing static pressure at* 

*Cavitation area for parameters: (a) inlet = 189 kPa and outlet = 800 kPa, (b) inlet = 95 kPa and outlet = 700 kPa, (c) inlet = 58 kPa and outlet = 650 kPa, (d) inlet = 27 kPa and outlet = 622 kPa and (e) inlet = 27 kPa and outlet = 600 kPa. The percentage of gas phase is given.*

#### **Figure 18.**

*Cavitation area for parameters: (a) inlet = 28.3 kPa and outlet = 675 kPa, (b) inlet = 27.8 kPa and outlet = 650 kPa and (c) inlet = 27.8 kPa and outlet = 622 kPa. The percentage of gas phase is given.*

quickly. Cavitation starts in the direction of the smallest radius of the collecting spiral. The *moving mesh* model produces the best results (mainly physical). However, the problem is the slow convergence of calculations and their long duration.

#### **5. Cavitation resistance of the pump**

The different cavitation properties of the two impellers can be explained by the significantly different inlet angle of the β1 vane—30°40′ (W13) and 21° (W17)—as with the same other geometric data, resulted in a very different position of the ideal inflow point. This is confirmed by the experimental characteristics of the pumps H = f(Q ) and η = f(Q ) from operation.

The analyses indicated the possibility of obtaining information on cavitation resistance of the designed structure through the rational use of CFD programs. The alternative solution of designing a prototype pump and carrying out a series of experiments may be challenging.

#### **6. Conclusion**

The CFD analysis made it possible to identify areas where cavitation is more likely to occur and to assess its intensity in relation to the rotational velocity. The results showed that one of the inlet channels has both negative pressure and increased fluid flow velocity. Calculations made for different pump rotational velocities and different suction channel geometries have shown that the intensity of these phenomena increases with the rotational velocity. However, these phenomena are not strong enough to contribute to the development of the phenomenon of cavitation. A series of simulations for different suction channel geometries have confirmed that no modification of the suction channel geometry is required. Considering the designs presented here the cavitation occurred either on the convex or concave side of the vane. The main difference between the vanes was the angle of its inclination. Hence there is a specific angle between 30 and 21° at which the transition occurs. The volumetric flow rate was unchanged in both of the impeller designs, although the inlet pressures were found to be different. For blades inclined at 30°, the inlet pressures were almost twice lower than in the case of 21°. Hence lower inclination of blades is more immune to cavitation development.

The CFD calculations were made to check the selection of the main dimensions of the radial fan. After performing many variant calculations, it was found that by changing the number of blades and the outlet angle of the blades, it is possible to increase the efficiency of the fan. It appeared that the efficiency is greater for impellers with greater amount of vanes. Furthermore the efficiency increased when the vanes were inclined to 23°, and as stated above, at such angle the cavitation occurs at higher inlet pressures and represent higher immunity to cavitation. Therefore the increase of efficiency may be partially a consequence of lack of cavitation.

It was found that the characteristics of the centrifugal pump from CFD calculations are consistent with the characteristics obtained experimentally. Based on the CFD analysis, cavitation resistance of the designed centrifugal pump was determined.

**159**

**Author details**

Wroclaw, Poland

provided the original work is properly cited.

Wieslaw Fiebig\*, Paulina Szwemin and Maciej Zawislak

\*Address all correspondence to: wieslaw.fiebig@pwr.edu.pl

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Faculty of Mechanical Engineering, Wroclaw University of Science and Technology,

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans…*

*DOI: http://dx.doi.org/10.5772/intechopen.82266*

*CFD Simulation of Flow Phenomena in Selected Centrifugal Pumps, Industrial Fans… DOI: http://dx.doi.org/10.5772/intechopen.82266*

### **Author details**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

The different cavitation properties of the two impellers can be explained by the significantly different inlet angle of the β1 vane—30°40′ (W13) and 21° (W17)—as with the same other geometric data, resulted in a very different position of the ideal inflow point. This is confirmed by the experimental characteristics of the pumps

The analyses indicated the possibility of obtaining information on cavitation resistance of the designed structure through the rational use of CFD programs. The alternative solution of designing a prototype pump and carrying out a series of

The CFD analysis made it possible to identify areas where cavitation is more likely to occur and to assess its intensity in relation to the rotational velocity. The results showed that one of the inlet channels has both negative pressure and increased fluid flow velocity. Calculations made for different pump rotational velocities and different suction channel geometries have shown that the intensity of these phenomena increases with the rotational velocity. However, these phenomena are not strong enough to contribute to the development of the phenomenon of cavitation. A series of simulations for different suction channel geometries have confirmed that no modification of the suction channel geometry is required. Considering the designs presented here the cavitation occurred either on the convex or concave side of the vane. The main difference between the vanes was the angle of its inclination. Hence there is a specific angle between 30 and 21° at which the transition occurs. The volumetric flow rate was unchanged in both of the impeller designs, although the inlet pressures were found to be different. For blades inclined at 30°, the inlet pressures were almost twice lower than in the case of 21°. Hence lower inclination of blades is more immune to cavitation

The CFD calculations were made to check the selection of the main dimensions of the radial fan. After performing many variant calculations, it was found that by changing the number of blades and the outlet angle of the blades, it is possible to increase the efficiency of the fan. It appeared that the efficiency is greater for impellers with greater amount of vanes. Furthermore the efficiency increased when the vanes were inclined to 23°, and as stated above, at such angle the cavitation occurs at higher inlet pressures and represent higher immunity to cavitation. Therefore the increase of efficiency may be partially a consequence of lack of

It was found that the characteristics of the centrifugal pump from CFD calculations are consistent with the characteristics obtained experimentally. Based on the CFD analysis, cavitation resistance of the designed centrifugal pump was

**5. Cavitation resistance of the pump**

H = f(Q ) and η = f(Q ) from operation.

experiments may be challenging.

**6. Conclusion**

development.

cavitation.

determined.

**158**

Wieslaw Fiebig\*, Paulina Szwemin and Maciej Zawislak Faculty of Mechanical Engineering, Wroclaw University of Science and Technology, Wroclaw, Poland

\*Address all correspondence to: wieslaw.fiebig@pwr.edu.pl

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Fiebig W, Cependa P, Jedraszczyk P, Kuczwara H. Innovative solution of an integrated motor pump assembly. In: ASME/BATH 2017 Symposium on Fluid Power and Motion Control. Sarasota, USA: ASME; 2017

[2] Frosina E. A three dimensional cfd modeling methodology applied to improve hydraulic components performance. Energy Procedia. 2015;**82**:950-956

[3] Mancò S, Nervegna N, Rundo M, Armenio G. Modelling and Simulation of Variable Displacement Vane Pumps for IC Engine Lubrication, SAE Technical Paper 2004-01-16012004

[4] Inaguma Y. Theoretical analysis of mechanical efficiency in vane pump. JTEKT Engineering Journal, English Edition No. 1007E Technical paper. 2010:28-35

[5] Frosina E, Stelson KA, et al. Vane pump power split transmission: Three dimensional computational fluid dynamics modeling. In: ASME/BATH 2015 Symposium on Fluid Power & Motion Control. Chicago, USA; 2015

[6] Houzeaux G, Codina R. A finite element method for the solution of rotary pumps. Computers and Fluids. 2007;**36**:667-679

[7] Hyun K, Hazel M, Suresh P. Two-Dimensional CFD Analysis of a Hydraulic Gear Pump. Washington: American Society for Engineeguide Education; 2007

[8] Mochala M. Intermittent CFD simulation of interlocked hydraulic pumps industrial use, basic conditions and prospect. In: FLUIDON Konferenz "Simulation in mechanischen Umfeld", Aachen; 2009

[9] Stryczek J et al. Visualisation research of the flow processes in the outlet chamber—outlet bridge—inlet chamber zone of the gear pumps. Archives of Civil and Mechanical Engineering. 2014;**15**(1):95-108. DOI: 10.1016/j.ACME 2014.02.010

[10] Jedraszczyk P, Fiebig W. CFD Model of an External Gear Pump Proceedings of 13th International Conference Computer Aided. Springer International Publishing; 2017. Available from: https://www.spguideerprofessional. de/en/cfd-model-of-an-external-gearpump/12181546

[11] Tabaczek T, Zawiślak M, Zieliński AK. Flow and cavitation analysis of a centrifugal pump. Systems: Journal of Transdisciplinary Systems Science. 2012;**16**(2):385-394

**160**

*Advanced Computational Fluid Dynamics for Emerging Engineering Processes...*

[9] Stryczek J et al. Visualisation research of the flow processes in the outlet chamber—outlet bridge—inlet chamber zone of the gear pumps. Archives of Civil and Mechanical Engineering. 2014;**15**(1):95-108. DOI:

10.1016/j.ACME 2014.02.010

[10] Jedraszczyk P, Fiebig W. CFD Model of an External Gear Pump Proceedings of 13th International Conference

Computer Aided. Springer International

[11] Tabaczek T, Zawiślak M, Zieliński AK. Flow and cavitation analysis of a centrifugal pump. Systems: Journal of Transdisciplinary Systems Science.

Publishing; 2017. Available from: https://www.spguideerprofessional. de/en/cfd-model-of-an-external-gear-

pump/12181546

2012;**16**(2):385-394

**References**

USA: ASME; 2017

2015;**82**:950-956

2010:28-35

2007;**36**:667-679

Education; 2007

Aachen; 2009

[1] Fiebig W, Cependa P, Jedraszczyk P, Kuczwara H. Innovative solution of an integrated motor pump assembly. In: ASME/BATH 2017 Symposium on Fluid Power and Motion Control. Sarasota,

[2] Frosina E. A three dimensional cfd modeling methodology applied to improve hydraulic components performance. Energy Procedia.

[3] Mancò S, Nervegna N, Rundo M, Armenio G. Modelling and Simulation of Variable Displacement Vane Pumps for IC Engine Lubrication, SAE Technical Paper 2004-01-16012004

[4] Inaguma Y. Theoretical analysis of mechanical efficiency in vane pump. JTEKT Engineering Journal, English Edition No. 1007E Technical paper.

[5] Frosina E, Stelson KA, et al. Vane pump power split transmission: Three dimensional computational fluid dynamics modeling. In: ASME/BATH 2015 Symposium on Fluid Power & Motion Control. Chicago, USA; 2015

[6] Houzeaux G, Codina R. A finite element method for the solution of rotary pumps. Computers and Fluids.

[7] Hyun K, Hazel M, Suresh P. Two-Dimensional CFD Analysis of a Hydraulic Gear Pump. Washington: American Society for Engineeguide

[8] Mochala M. Intermittent CFD simulation of interlocked hydraulic pumps industrial use, basic conditions and prospect. In: FLUIDON Konferenz "Simulation in mechanischen Umfeld",

### *Edited by Albert S. Kim*

As researchers deal with processes and phenomena that are geometrically complex and phenomenologically coupled the demand for high-performance computational fluid dynamics (CFD) increases continuously. The intrinsic nature of coupled irreversibility requires computational tools that can provide physically meaningful results within a reasonable time. This book collects the state-of-the-art CFD research activities and future R&D directions of advanced fluid dynamics. Topics covered include in-depth fundamentals of the Navier–Stokes equation, advanced multi-phase fluid flow, and coupling algorithms of computational fluid and particle dynamics. In the near future, true multi-physics and multi-scale simulation tools must be developed by combining micro-hydrodynamics, fluid dynamics, and chemical reactions within an umbrella of irreversible statistical physics.

Published in London, UK © 2019 IntechOpen © Vanessa Ives / unsplash

Advanced Computational Fluid Dynamics for Emerging Engineering Processes -

Eulerian vs. Lagrangian

Advanced Computational

Fluid Dynamics for Emerging

Engineering Processes

Eulerian vs. Lagrangian

*Edited by Albert S. Kim*