**2. Governing equations and numerical method**

The present 3D RSTM model is based on the two-way coupling *k*-*L* model [8] and applies the 3D RANS equations and the RSTM closure momentum equations.

The sketch of the computational flow domain is shown in Figure 1 for the case of the downward grid-generated turbulent particulate flow in the channel of square cross-section. Here *u* and *us* are the longitudinal components of velocities of gas and particles, respectively.

number. Then, these models have been expanded for the free-surface flows. As opposed to the *k*-*ε* models, [7, 8] considered both the turbulence augmentation caused by the velocity slip between gas and particles and the turbulence attenuation due to the change of the turbulence macroscale occurred in the particulate flow as compared to the unladen flow. The given

Currently, the probability dense function (PDF) approach is widely applied for the numerical modeling of the particulate flows. The PDF models, for example, [9-13] contain more complete differential transport equations, which are written for various velocity correlations and

As opposed to the pipe flows, the rectangular and square channel flows, even in case of unladen flows, are considerably anisotropic with respect to the components of the turbulence energy, that is vividly expressed near the channel walls and corners being notable as for the secondary flows. In addition, the presence of particles aggravates such anisotropy. Such flows are studied by the Reynolds stress turbulence models (RSTM), which are based on the transport equations for all components of the Reynolds stress tensor and the turbulence dissipation rate. RSTM approach allows to completely analyze the influence of particles on longitudinal, radial and azimuthal components of the turbulence kinetic energy, including also possible modifications

A few studies based on the RSTM approach showed its good performance and capability for simulation of the complicated flows, e.g., [14], as well for the turbulent particulate flows, for example, see [15]. Recently, the nonlinear algebraic Reynolds stress model based on the PDF approach has been proposed in [16] for the gas flow laden with small heavy particles. The original equations written for each component of Reynolds stress were reduced to their general form in terms of the turbulence energy and its dissipation rate with additional effect of the particulate phase. Eventually, the model [16] operated with the *k*-*ε* solution and did not allow

The 3D RSTM model, being presented in this chapter, is intended to apply for a simulation of the downward turbulent particulate flow in channel of the square cross-section (the aspect

In order to approve and validate the developed model, the separate investigations have been carried out. The first study was the simulation of the downward unladen gas flow in channel of the rectangular cross-section with the smooth and rough walls. The second study relates to the downward grid-generated turbulent particulate flow in the same channel with the smooth

The further stage of this study will be the development of the present model for implementa‐ tion to the particulate channel flow with the rough walls and the initial level of turbulence.

The present 3D RSTM model is based on the two-way coupling *k*-*L* model [8] and applies the

to analyze the particles effect on each component of the Reynolds stress.

**2. Governing equations and numerical method**

3D RANS equations and the RSTM closure momentum equations.

approach has been successfully tested for various pipe and channel particulate flows.

consider both the turbulence augmentation and attenuation due to the particles.

of the cross-correlation velocity moments.

42 Computational and Numerical Simulations

ratio of 1:6) with rough walls.

walls.

**Figure 1.** Downward channel grid-generated turbulent particulate flow.

### **2.1. Governing equations for the Reynolds stress turbulence model**

The numerical simulation of the stationary incompressible 3D turbulent particulate flow in the square cross-section channel was performed by the 3D RANS model with applying of the 3D Reynolds stress turbulence model for the closure of the governing equations of gas, while the particulate phase was modeled in a frame of the 3D Euler approach with the equations closed by the two-way coupling model [8] and the eddy-viscosity concept.

The particles were brought into the developed isotropic turbulent flow set-up in channel domain, which has been preliminary computed to obtain the flow velocity field. The system of the momentum and closure equations of the gas phase are identical for the unladen while the particle-laden flows under impact of the viscous drag force. Therefore, here is only presented the system of equations of the gas phase written for the case of the particle-laden flow in the Cartesian coordinates.

3D governing equations for the stationary gas phase of the laden flow are written together with the closure equations as follows:

continuity equation:

$$
\frac{
\partial \mu
}{
\partial \chi
} + \frac{
\partial \upsilon
}{
\partial y
} + \frac{
\partial \upsilon
}{
\partial z
} = 0,
\tag{1}
$$

the transport equation of the *x*-normal component of the Reynolds stress:

2 2 2

*<sup>u</sup> u u C Tv CT uv vw y y x z*

é æöù ¶ ¶ ¶¶ ¢ ¢¢ <sup>+</sup> <sup>ê</sup> ¢ + + ç ÷ ¢¢ ¢ ¢ <sup>+</sup> <sup>ú</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû

*<sup>u</sup> u u C Tw CT uw vw z z x y*

<sup>é</sup> <sup>æ</sup> öù ¶ ¶ ¢ ¶ ¶ ¢ ¢ <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup> ÷ú <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû

2 2 2

the transport equation of the *y*-normal component of the Reynolds stress:

2 2 2

*<sup>v</sup> v v C Tv CT uv vw y y x z*

é æöù ¶ ¶ ¶¶ ¢ ¢¢ <sup>+</sup> <sup>ê</sup> ¢ + + ç ÷ ¢¢ ¢ ¢ <sup>+</sup> <sup>ú</sup> <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû

*<sup>v</sup> v v C Tw CT uw vw z z x y*

e

<sup>é</sup> <sup>æ</sup> öù ¶ ¶ ¢ ¶ ¶ ¢ ¢ <sup>+</sup> <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup> ÷ú ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ <sup>÷</sup> <sup>ë</sup> <sup>è</sup> øû

2 2 2

the transport equation of the *z*-normal component of the Reynolds stress:

2 2 2

2 2 2

( )

*x y zx x y z*

¶¶¶ ¢¢ ¢ é ù æ ö ¶ ¶ ¶¶ ¢ ¢ + + = ++ + ê ú ¢ ç ÷ ¢¢ ¢ ¢ ¶ ¶ ¶¶ ¶ ¶ ¶ ê ú ë û è ø

2

*s s*

n

( )

¶¶¶ ¢¢ ¢ é ù æ ö ¶ ¶ ¶¶ ¢ ¢¢ + + = ++ + ê ú ¢ ç ÷ ¢¢ ¢ ¢ ¶ ¶ ¶¶ ¶ ¶ ¶ ê ú ë û è ø

*uu vu wu <sup>u</sup> u u C Tu CT uv uw x y zx x y z*

2

( )

*x y zx x y z*

¶¶¶ ¢¢ ¢ é ù æ ö ¶ ¶ ¶¶ ¢ ¢¢ + + = ++ + ê ú ¢ ç ÷ ¢¢ ¢ ¢ ¶ ¶ ¶¶ ¶ ¶ ¶ ê ú ë û è ø

2

*s s*

n

*s s*

n

2 2 2

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

( ) 2 *s*

t

*u u*

*p*

e

http://dx.doi.org/10.5772/57047

(5)

45

(6)

(7)

*uu uu D h*

2 2 2

2 2 2

*PR C*

( ) 2 *s*

t

*w w*

*p*

e

*ww ww D h*


a

*<sup>w</sup> w w C Tu CT uv uw*


*PR C*a

*<sup>v</sup> v v C Tu CT uv uw*

ú

( ) ( ) ( )

2 2 2

*s s*

n

*s s*

( ) ( ) ( )

*uv vv wv*

2 2 2

*s s*

n

*s s*

( ) 2 *s*

t

*v v*

*p*

*<sup>w</sup> w w C Tv CT uv vw y y x z*

é æöù ¶ ¶ ¶¶ ¢ ¢¢ <sup>+</sup> <sup>ê</sup> ¢ + + ç ÷ ¢¢ ¢ ¢ <sup>+</sup> <sup>ú</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû

*<sup>w</sup> w w C Tw CT uw vw z z x y*

<sup>é</sup> <sup>æ</sup> öù ¶ ¶ ¢ ¶ ¶ ¢ ¢ <sup>+</sup> <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup> ÷ú <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû

n

*vv vv D h*


( )

2

( )

( ) ( ) ( )

*uw vw ww*

2 2 2

*s s*

n

*s s*

n

( )

2

( )

2

*PR C*a

2

n

( )

2

( )

2

where *u*, *v* and *w* are the axial, transverse and spanwise time-averaged velocity components of the gas phase, respectively.

*x*-component of the momentum equation:

$$\begin{split} \frac{\partial \boldsymbol{u}^{2}}{\partial \mathbf{x}} + \frac{\partial \boldsymbol{u} \boldsymbol{v}}{\partial \boldsymbol{y}} + \frac{\partial \boldsymbol{u} \boldsymbol{w}}{\partial \boldsymbol{z}} &= \frac{\partial}{\partial \mathbf{x}} \left( 2\nu \frac{\partial \boldsymbol{u}}{\partial \mathbf{x}} - \overline{\boldsymbol{u}'^{2}} \right) + \frac{\partial}{\partial \boldsymbol{y}} \bigg[ \nu \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{y}} + \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{x}} \right) - \overline{\boldsymbol{u}' \boldsymbol{v}'} \bigg] + \\ + \frac{\partial}{\partial \boldsymbol{z}} \bigg[ \nu \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{z}} + \frac{\partial \boldsymbol{w}}{\partial \boldsymbol{x}} \right) - \overline{\boldsymbol{u}' \boldsymbol{w}'} \bigg] - \frac{\partial p}{\rho \hat{\boldsymbol{\alpha}} \boldsymbol{x}} - \alpha \mathcal{C}\_{D}' \frac{\left( \boldsymbol{u} - \boldsymbol{u}\_{s} \right)}{\boldsymbol{\tau}\_{p}} \end{split} \tag{2}$$

*y*-component of the momentum equation:

$$\begin{split} \frac{\partial \boldsymbol{u} \boldsymbol{v}}{\partial \boldsymbol{x}} &+ \frac{\partial \boldsymbol{v}^{2}}{\partial \boldsymbol{y}} + \frac{\partial \boldsymbol{v} \boldsymbol{w} \boldsymbol{v}}{\partial \boldsymbol{z}} = \frac{\partial}{\partial \boldsymbol{x}} \bigg[ \nu \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{y}} + \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{x}} \right) - \overline{\boldsymbol{u}' \boldsymbol{v}'} \bigg] + \frac{\partial}{\partial \boldsymbol{y}} \bigg( 2\nu \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{y}} - \overline{\boldsymbol{v}'^{2}} \bigg) + \\ &+ \frac{\partial}{\partial \boldsymbol{z}} \bigg[ \nu \left( \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{z}} + \frac{\partial \boldsymbol{w}}{\partial \boldsymbol{y}} \right) - \overline{\boldsymbol{v}' \boldsymbol{w}'} \bigg] - \frac{\partial p}{\rho \partial \boldsymbol{y}} - \alpha \boldsymbol{C}\_{D}' \frac{\left( \boldsymbol{v} - \boldsymbol{v}\_{s} \right)}{\boldsymbol{\tau}\_{p}} \end{split} \tag{3}$$

*z*-component of the momentum equation:

$$\begin{aligned} \frac{\partial \overline{u}w}{\partial x} + \frac{\partial \overline{v}w}{\partial y} + \frac{\partial \overline{w}^2}{\partial z} &= \frac{\partial}{\partial x} \left[ \nu \left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right) - \overline{u'w'} \right] + \frac{\partial}{\partial y} \left[ \nu \left( \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \right) - \overline{v'w'} \right] \\ + \frac{\partial}{\partial z} \left( 2\nu \frac{\partial w}{\partial z} - \overline{w'^2} \right) - \frac{\partial p}{\rho \overline{\partial z}} - \alpha C\_D' \frac{\left( w - w\_s \right)}{\tau\_p} \end{aligned} \tag{4}$$

the transport equation of the *x*-normal component of the Reynolds stress:

the particle-laden flows under impact of the viscous drag force. Therefore, here is only presented the system of equations of the gas phase written for the case of the particle-laden

3D governing equations for the stationary gas phase of the laden flow are written together

0, *uvw xy z* ¶¶¶ ++ =

<sup>2</sup> 2

¶ ¶ ¶ ¶ ¶ ¶ ¶¶ æ ö é æ ö ù ++ = -+ +- + ç ÷ ¢ <sup>ê</sup> ç ÷ ¢ ¢ú ¶ ¶ ¶ ¶ ¶ ¶ ¶¶ è ø êë è ø ûú

> r

*uv v vw u v v*

¶ ¶ ¶ ¶ ¶¶ é æ ö ù ¶ ¶ æ ö ++ = +- + -+ <sup>ê</sup> ¢ ¢ú ¢ ç ÷ ç ÷ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ êë è ø úû è ø

n

¶ ¶¶ <sup>é</sup> æ ö <sup>ù</sup> ¶ - + + - -- <sup>ê</sup> ¢ ¢ú ¢ ç ÷ ¶ ¶¶ <sup>ê</sup> <sup>ú</sup> ¶ <sup>ë</sup> è ø <sup>û</sup>

*x y z x yx y y v w p v v vw C*

> r

*u uv uw u u v*

n

*x y z x x y yx u w p u u uw C*

*z zx x*

*z zy y*

<sup>2</sup> 2 *<sup>s</sup>*

r

*w p w w w C*

n

¶ ¶¶ <sup>é</sup> æ ö <sup>ù</sup> ¶ - + + - -- ¢ ¢ ¢ <sup>ê</sup> ç ÷ <sup>ú</sup> ¶ ¶¶ è ø ¶ <sup>ë</sup> <sup>û</sup>

where *u*, *v* and *w* are the axial, transverse and spanwise time-averaged velocity components

( )

n

*p*

( )

*p*

t

*D*

a

( )

¶ ¶ ¶ ¶ ¶¶ é æ ö ù ¶ ¶¶ é æ ö ù ++= +- + +- <sup>ê</sup> ç ÷ ¢ ¢ú <sup>ê</sup> ç ÷ ¢ ¢ú ¶ ¶ ¶ ¶ ¶¶ <sup>ë</sup> è ø <sup>û</sup> ¶ ¶¶ êë è ø úû

*p*

t

*D*

a

*uw vw w u w v w*

*x y z x zx y zy*

, *s*

*u v v*

n

*u w v w*

n

t

*D*

a

, *s*

<sup>2</sup> 2

*u u v*

¶¶ ¶ (1)

(2)

(3)

(4)

flow in the Cartesian coordinates.

44 Computational and Numerical Simulations

continuity equation:

of the gas phase, respectively.

2

n

*y*-component of the momentum equation:

2

n

*z*-component of the momentum equation:

2

¶ ¶ <sup>æ</sup> <sup>ö</sup> ¶ - + - -- ¢ ¢ <sup>ç</sup> <sup>÷</sup> ¶ ¶ <sup>è</sup> <sup>ø</sup> ¶

*z z z*

n

*x*-component of the momentum equation:

with the closure equations as follows:

( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 *s s s s s s uu vu wu <sup>u</sup> u u C Tu CT uv uw x y zx x y z <sup>u</sup> u u C Tv CT uv vw y y x z <sup>u</sup> u u C Tw CT uw vw z z x y* n n n ¶¶¶ ¢¢ ¢ é ù æ ö ¶ ¶ ¶¶ ¢ ¢¢ + + = ++ + ê ú ¢ ç ÷ ¢¢ ¢ ¢ ¶ ¶ ¶¶ ¶ ¶ ¶ ê ú ë û è ø é æöù ¶ ¶ ¶¶ ¢ ¢¢ <sup>+</sup> <sup>ê</sup> ¢ + + ç ÷ ¢¢ ¢ ¢ <sup>+</sup> <sup>ú</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû <sup>é</sup> <sup>æ</sup> öù ¶ ¶ ¢ ¶ ¶ ¢ ¢ <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup> ÷ú <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶ ÷ú <sup>ë</sup> <sup>è</sup> øû ( ) 2 *s uu uu D h p u u PR C*a e t - ++ - ¢ (5)

the transport equation of the *y*-normal component of the Reynolds stress:

$$\begin{aligned} &\frac{\partial\left(\overline{u\,v'^2}\right)}{\partial\overline{x}} + \frac{\partial\left(\overline{v\,v'^2}\right)}{\partial y} + \frac{\partial\left(\overline{w\,v'^2}\right)}{\partial z} = \frac{\partial}{\partial x}\bigg[\left(\mathbf{C}\_s T \overline{u'^2} + \nu\right) \frac{\partial\overline{v\,v'^2}}{\partial x} + \mathbf{C}\_s T \bigg(\overline{u'v'\,\overline{\partialv'^2}} + \overline{u'w'\,\overline{\partialv'^2}}\bigg)\bigg] \\ &+ \frac{\partial}{\partial y}\bigg[\left(\mathbf{C}\_s T \overline{v'^2} + \nu\right) \frac{\partial\overline{v\,v'^2}}{\partial y} + \mathbf{C}\_s T \bigg(\overline{u'v'\,\overline{\partialv'^2}} + \overline{v\,w'}\frac{\partial\overline{w'^2}}{\partial z}\bigg)\bigg] + \\ &+ \frac{\partial}{\partial z}\bigg[\left(\mathbf{C}\_s T \overline{w\,v'^2} + \nu\right) \frac{\partial\overline{w\,v'^2}}{\partial z} + \mathbf{C}\_s T \bigg(\overline{u'w'\,\overline{\partialw'}} + \overline{v\,w'w'\,\overline{\partialw'^2}}\bigg)\bigg] \\ &+ P\_{vv} + R\_{vv} + \alpha \mathbf{C}\_D' \frac{\left(\mathbf{v} - \overline{v}\_s\right)^2}{\tau\_p} - \varepsilon\_h\end{aligned} \tag{6}$$

the transport equation of the *z*-normal component of the Reynolds stress:

$$\begin{aligned} &\frac{\partial\left(\overline{u\,w'^2}\right)}{\partial x} + \frac{\partial\left(\overline{v\,w'^2}\right)}{\partial y} + \frac{\partial\left(\overline{w\,w'^2}\right)}{\partial z} = \frac{\partial}{\partial x}\bigg[\left(C\_s T \overline{u'^2} + \nu\right) \frac{\partial\overline{w'^2}}{\partial x} + C\_s T \left(\overline{u'\nu}' \frac{\partial\overline{w'^2}}{\partial y} + \overline{u'w'} \frac{\partial\overline{w'^2}}{\partial z}\right)\bigg] \\ &+ \frac{\partial}{\partial y}\bigg[\left(C\_s T \overline{v'^2} + \nu\right) \frac{\partial\overline{w'^2}}{\partial y} + C\_s T \left(\overline{u'\nu'} \frac{\partial\overline{w'^2}}{\partial x} + \overline{v'w'} \frac{\partial\overline{w'^2}}{\partial z}\right)\bigg] \\ &+ \frac{\partial}{\partial z}\bigg[\left(C\_s T \overline{w'^2} + \nu\right) \frac{\partial\overline{w'^2}}{\partial z} + C\_s T \left(\overline{u'w'} \frac{\partial\overline{w'^2}}{\partial x} + \overline{v'w'} \frac{\partial\overline{w'^2}}{\partial y}\right)\bigg] + P\_{ww} + R\_{ww} + \alpha C\_D' \frac{\left(w - w\_s\right)^2}{\tau\_p} - \varepsilon\_h \end{aligned} \tag{7}$$

the transport equation of the *xy* shear stress component of the Reynolds stress:

$$\begin{split} &\frac{\partial\left(\overline{u'u'}\right)}{\partial x} + \frac{\partial\left(\overline{vu'v'}\right)}{\partial y} + \frac{\partial\left(\overline{u'w'}\right)}{\partial z} = \\ &= \frac{\partial}{\partial x}\bigg[\left(C\_s T \overline{u'^2} + \nu\right) \frac{\partial\overline{u'v'}}{\partial x} + C\_s T \bigg(\overline{u'v'} \frac{\partial\overline{u'v'}}{\partial y} + \overline{u'w'} \frac{\partial\overline{u'v'}}{\partial z}\bigg)\bigg] + \frac{\partial}{\partial y}\bigg[\left(C\_s T \overline{v'^2} + \nu\right) \frac{\partial\overline{u'v'}}{\partial y}\bigg] \\ &+ C\_s T \bigg(\overline{u'v'} \frac{\partial\overline{u'w'}}{\partial x} + \overline{v'w'} \frac{\partial\overline{u'w'}}{\partial z}\bigg)\bigg] + \frac{\partial}{\partial z}\bigg[\left(C\_s T \overline{w'^2} + \nu\right) \frac{\partial\overline{u'v'}}{\partial z} + C\_s T \bigg(\overline{u'w'} \frac{\partial\overline{u'w'}}{\partial x} + \overline{v'w'} \frac{\partial\overline{u'v'}}{\partial y}\bigg)\bigg] + \\ &+ P\_{uv} + R\_{uv} \end{split} \tag{8}$$

( )

*uvw CTu CT uv uw x y zx x y z*

e

e

ù

úû

 e

0 0

 e

> e

n

æ ¶ ¶ ö

¶ ¶ è ø

*x y*

and *<sup>T</sup>* <sup>=</sup> *<sup>k</sup>*

0 00 00

*k*0 *ε*0

distance *λ*. Here *λ* =*δ*( *πρ<sup>p</sup>* / 6*ρα* <sup>3</sup> −1), *L* <sup>0</sup> =

averaged velocity flow field (*u*, *v*, *w*).

particle-laden flows, respectively. *<sup>k</sup>* =0.5(*<sup>u</sup>* ′ ¯2

e

*CT uw vw*

+ ç ¢¢ ¢¢ +

e

ee

e

*Cε*2=1.92. Here *T*<sup>0</sup> =

( ) ( )

0 0 0 00 0 0 0 0

*<sup>P</sup> C C*

e

*CTv CT uv vw CTw y y x zz z*

> e

unladen flows, respectively; *τp* is the Stokesian particle response time, *τ<sup>p</sup>* =

viscosity; (*u* −*us*), (*v* −*vs*) and (*w* −*ws*) are the components of the slip velocity.

÷ú + -

e

¶ é ¶ æ ¶ ¶ ö é ù ¶ ¶ + ê ¢ + + ç ¢¢ ¢ ¢ + ÷ú + ê ¢ + ¶ ê ¶ ¶ ¶¶ ú ¶ ë è ø ë û

2 0 0 0 2 0

*k k*

The given system of the transport equations (Eqs. 1 – 11) is based on the model [17] with applying of the numerical constants taken from [18]: *CR*=1.8, *C*2=0.6, *Cs*=0.22, *Cε*=0.18, *Cε*1=1.44,

+ *v* ′ ¯2

turbulence kinetic energy of gas in the particle-laden and in the unladen flows, respectively; *ε* and *ε*0 are the dissipation rates of the turbulence kinetic energy in the particle-laden and

The additional terms of Eqs. (2 – 7) pertain to presence of particles in the flow and contain the particle mass concentration *α*. The influence of particles on gas is considered by the aerody‐ namic drag force in the momentum equations (the last term of the right-hand sides of Eqs. 2 – 4), and by the turbulence generation and attenuation effects contained in the transport equations of components of the Reynolds stress (the penultimate and last terms of the righthand sides of Eqs. (5 – 7), respectively). The given model applies the two-way coupling approach [8], where the turbulence generation terms are proportional to the squared slip velocity, and the turbulence attenuation terms are expressed via the hybrid length scale *L <sup>h</sup>* and the hybrid dissipation rate *εh* of the particle-laden flow, where *L <sup>h</sup>* is calculated as the harmonic average of the integral length scale of the unladen flow *L* 0 and the interparticle

> *k*0 3/2 *ε*0

<sup>2</sup> 2 , *uu uu u P u uv uw*

*xy z* æ ö ¶¶ ¶

The production terms *P* are determined according to [18] as follows:

on the shear Reynolds stress components is considered in Eqs. (8 – 10) indirectly via the

, *<sup>L</sup> <sup>h</sup>* <sup>=</sup> <sup>2</sup>*<sup>L</sup>* <sup>0</sup>*<sup>λ</sup>*

*<sup>L</sup>* <sup>0</sup> <sup>+</sup> *<sup>λ</sup>* , *ε<sup>h</sup>* <sup>=</sup> *<sup>k</sup>* 3/2

=- + + ¢ ¢¢ ¢ ¢ ç ÷ ¶¶ ¶ è ø (12)

*L <sup>h</sup>*

+ *w* ′ ¯2

e

e

00 00 00 2 0 0 0

¶¶¶ ¶ é ¶ æ ¶ ¶ öù ++ = <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú ¶ ¶ ¶¶ êë ¶ <sup>è</sup> ¶ ¶ øúû

n

0 0 0 00 0 0

e

*<sup>ε</sup>* are the turbulence integral time scales for the unladen and

) and *k*<sup>0</sup> =0.5(*<sup>u</sup>* ′

e

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

 e

n

0 ¯2 + *v* ′ 0 ¯2 + *w* ′ 0 ¯2

*<sup>ρ</sup>p<sup>δ</sup>* <sup>2</sup>

e

http://dx.doi.org/10.5772/57047

(11)

47

) are the

<sup>18</sup>*ρν* ; *ν* is the gas

. The particles influence

e

 e

2 0 0 1 2 0 0

 e

the transport equation of the *xz* shear stress component of the Reynolds stress:

$$\begin{split} &\frac{\partial\left(\overline{u'u'}\overline{w'}\right)}{\partial x} + \frac{\partial\left(\overline{vu'w'}\right)}{\partial y} + \frac{\partial\left(\overline{wu'w'}\right)}{\partial z} = \frac{\partial}{\partial x}\left[\left(\mathbf{C}\_{s}\overline{T}\overline{u'^2} + \nu\right)\frac{\partial\overline{u'w'}}{\partial x} + \mathbf{C}\_{s}T\left(\overline{u'v'}\frac{\partial\overline{u'w'}}{\partial y} + \overline{u'w'}\frac{\partial\overline{u'w'}}{\partial z}\right)\right] \\ &+ \frac{\partial}{\partial y}\left[\left(\mathbf{C}\_{s}\overline{T}\overline{v'^2} + \nu\right)\frac{\partial\overline{u'w'}}{\partial y} + \mathbf{C}\_{s}T\left(\overline{u'v'}\frac{\partial\overline{u'w'}}{\partial x} + \overline{v'w'}\frac{\partial\overline{u'w'}}{\partial z}\right)\right] \\ &+ \frac{\partial}{\partial z}\left[\left(\mathbf{C}\_{s}\overline{T}\overline{w'^2} + \nu\right)\frac{\partial\overline{u'w'}}{\partial z} + \mathbf{C}\_{s}T\left(\overline{u'w'}\frac{\partial\overline{u'w'}}{\partial x} + \overline{v'w'}\frac{\partial\overline{u'w'}}{\partial y}\right)\right] + P\_{uw} + R\_{uw} \end{split} \tag{9}$$

the transport equation of the *yz* shear stress component of the Reynolds stress:

$$\begin{split} &\frac{\partial\left(\overline{u'w'w'}\right)}{\partial x} + \frac{\partial\left(\overline{v'w'}\right)}{\partial y} + \frac{\partial\left(\overline{w'w'}\right)}{\partial z} \\ &= \frac{\partial}{\partial x}\bigg[\left(C\_s T \overline{u'^2} + \nu\right) \frac{\partial\overline{v'w'}}{\partial x} + C\_s T \bigg(\overline{u'v'} \frac{\partial\overline{v'w'}}{\partial y} + \overline{u'w'} \frac{\partial\overline{v'w'}}{\partial z}\bigg)\bigg] + \frac{\partial}{\partial y}\bigg[\left(C\_s T \overline{v'^2} + \nu\right) \frac{\partial\overline{v'w'}}{\partial y} \\ &+ C\_s T \left(\overline{u'w'} \frac{\partial\overline{v'w'}}{\partial x} + \overline{v'w'} \frac{\partial\overline{v'w'}}{\partial z}\right)\bigg] + \frac{\partial}{\partial z}\bigg[\left(C\_s T \overline{w'^2} + \nu\right) \frac{\partial\overline{v'w'}}{\partial z} + C\_s T \bigg(\overline{u'w'} \frac{\partial\overline{v'w'}}{\partial x} + \overline{v'w'} \frac{\partial\overline{v'w'}}{\partial y}\bigg)\bigg] \\ &+ P\_{vw} + R\_{vw} \end{split} \tag{10}$$

the transport equation of the dissipation rate of the turbulence kinetic energy:

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall http://dx.doi.org/10.5772/57047 47

( ) ( ) ( ) 00 00 00 2 0 0 0 0 0 0 00 0 0 2 0 0 0 2 0 0 0 0 00 0 0 0 0 0 0 0 00 00 *uvw CTu CT uv uw x y zx x y z CTv CT uv vw CTw y y x zz z CT uw vw x y* e e e e e e ee e e e e n e e e e n n e e ¶¶¶ ¶ é ¶ æ ¶ ¶ öù ++ = <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú ¶ ¶ ¶¶ ëê ¶ <sup>è</sup> ¶ ¶ øúû ¶ é ¶ æ ¶ ¶ ö é ù ¶ ¶ + ê ¢ + + ç ¢¢ ¢ ¢ + ÷ú + ê ¢ + ¶ ê ¶ ¶ ¶¶ ú ¶ ë è ø ë û æ ¶ ¶ ö + ç ¢¢ ¢¢ + ¶ ¶ è ø 2 0 0 1 2 0 0 *<sup>P</sup> C C k k* e e ù e e ÷ú + úû (11)

the transport equation of the *xy* shear stress component of the Reynolds stress:

( ) ( )

<sup>é</sup> <sup>æ</sup> ö é <sup>ù</sup> ¶ ¶ ¢ ¢ ¶ ¶¶ ¢¢ ¢¢ ¶ ¢ ¢ <sup>=</sup> <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú <sup>+</sup> <sup>ê</sup> ¢ <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶¶ ÷ú <sup>ê</sup> ¶ <sup>ë</sup> <sup>è</sup> ø ë <sup>û</sup>

*s s s*

*u w uw uw C Tv CT uv vw y y x z*

<sup>é</sup> <sup>æ</sup> öù ¶ ¶ ¢ ¢ ¶ ¶ ¢¢ ¢¢ <sup>+</sup> <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú ¶ ¶ <sup>ç</sup> ¶ ¶ <sup>÷</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>è</sup> øû

¶ ¶ ¢ ¢ <sup>æ</sup> ¶ ¶ ¢¢ ¢¢ <sup>+</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup>

*u w uw uw C Tw CT uw vw z z x y*

¶ ¶ <sup>ç</sup> ¶ ¶ <sup>è</sup> *uw uw P R* <sup>é</sup> öù <sup>ê</sup> ÷ú + + <sup>÷</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> øû

the transport equation of the *yz* shear stress component of the Reynolds stress:

( ) ( )

the transport equation of the dissipation rate of the turbulence kinetic energy:

¶ é ù ¶ ¢ ¢ <sup>æ</sup> ¶ ¶¶ ¢¢ ¢¢ö é ¶ ¢ ¢ <sup>=</sup> <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú <sup>+</sup> <sup>ê</sup> ¢ <sup>+</sup> ¶ <sup>ê</sup> ¶ <sup>ç</sup> ¶ ¶¶ ÷ú <sup>ê</sup> ¶ <sup>ë</sup> <sup>è</sup> ø ë <sup>û</sup>

*s s s*

*s s s*

2 2

*v w vw vw v w C Tu CT uv uw C Tv x x y zy y*

( )

*vw vw v w vw vw CT uv vw C Tw CT uw vw*

<sup>æ</sup> ¶ ¶¶ ¢¢ ¢¢öù ¶ ¢ ¢ <sup>æ</sup> ¶ ¶ ¢¢ ¢¢ <sup>+</sup> <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú <sup>+</sup> ¢ + + <sup>ç</sup> ¢¢ ¢¢ <sup>+</sup> <sup>ç</sup> ¶ ¶¶ ÷ú ¶ <sup>ç</sup> ¶ ¶ <sup>è</sup> øû <sup>è</sup>

2

*x zz z x y*

n

2 2

*u v uv uv u v C Tu CT uv uw C Tv x x y zy y*

( )

*uv uv u v uv uv CT uv vw C Tw CT uw vw*

<sup>æ</sup> ¶ ¶¶ ¢¢ ¢¢öù ¶ ¢ ¢ <sup>æ</sup> ¶ ¶ ¢¢ ¢¢ <sup>+</sup> <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú <sup>+</sup> ¢ + + ¢¢ ¢¢ <sup>+</sup> <sup>ç</sup> ¶ ¶¶ ÷ú ¶ ¶ ¶ <sup>è</sup> øû <sup>è</sup>

the transport equation of the *xz* shear stress component of the Reynolds stress:

2

( )

*uu w vu w wu w u w uw uw C Tu CT uv uw x y zx x y z*

¶¶¶ ¢¢ ¢¢ ¢¢ ¶ ¶ ¶¶ é ù ¢ ¢ æ ö ¢¢ ¢¢ ++ = <sup>ê</sup> ¢ + + <sup>ç</sup> ¢¢ ¢ ¢ <sup>+</sup> ÷ú ¶ ¶ ¶¶ ¶ <sup>ç</sup> ¶ ¶ <sup>÷</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>è</sup> øû

2

*s s*

n

*x zz z x y*

n

n

n

<sup>é</sup> öù <sup>ê</sup> ÷ú <sup>÷</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> øû (8)

(9)

(10)

<sup>é</sup> öù <sup>ê</sup> <sup>ç</sup> ÷ú <sup>+</sup> <sup>ç</sup> <sup>÷</sup> <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> øû

, *<sup>s</sup> <sup>s</sup> <sup>s</sup>*

( ) ( ) ( )

++ =

n

( ) ( ) ( )

*s s*

n

*s s*

( ) ( ) ( )

*uv w vv w wv w xy z*

¶¶¶ ¢¢ ¢¢ ¢¢ + + ¶¶ ¶

> n

*vw vw P R*

+ +

n

( )

2

( )

2

*uu v vu v wu v xy z*

¶¶¶ ¢¢ ¢¢ ¢¢

46 Computational and Numerical Simulations

¶¶ ¶

*uv uv P R*

+ +

The given system of the transport equations (Eqs. 1 – 11) is based on the model [17] with applying of the numerical constants taken from [18]: *CR*=1.8, *C*2=0.6, *Cs*=0.22, *Cε*=0.18, *Cε*1=1.44,

*Cε*2=1.92. Here *T*<sup>0</sup> = *k*0 *ε*0 and *<sup>T</sup>* <sup>=</sup> *<sup>k</sup> <sup>ε</sup>* are the turbulence integral time scales for the unladen and particle-laden flows, respectively. *<sup>k</sup>* =0.5(*<sup>u</sup>* ′ ¯2 + *v* ′ ¯2 + *w* ′ ¯2 ) and *k*<sup>0</sup> =0.5(*<sup>u</sup>* ′ 0 ¯2 + *v* ′ 0 ¯2 + *w* ′ 0 ¯2 ) are the turbulence kinetic energy of gas in the particle-laden and in the unladen flows, respectively; *ε* and *ε*0 are the dissipation rates of the turbulence kinetic energy in the particle-laden and

unladen flows, respectively; *τp* is the Stokesian particle response time, *τ<sup>p</sup>* = *<sup>ρ</sup>p<sup>δ</sup>* <sup>2</sup> <sup>18</sup>*ρν* ; *ν* is the gas viscosity; (*u* −*us*), (*v* −*vs*) and (*w* −*ws*) are the components of the slip velocity.

The additional terms of Eqs. (2 – 7) pertain to presence of particles in the flow and contain the particle mass concentration *α*. The influence of particles on gas is considered by the aerody‐ namic drag force in the momentum equations (the last term of the right-hand sides of Eqs. 2 – 4), and by the turbulence generation and attenuation effects contained in the transport equations of components of the Reynolds stress (the penultimate and last terms of the righthand sides of Eqs. (5 – 7), respectively). The given model applies the two-way coupling approach [8], where the turbulence generation terms are proportional to the squared slip velocity, and the turbulence attenuation terms are expressed via the hybrid length scale *L <sup>h</sup>* and the hybrid dissipation rate *εh* of the particle-laden flow, where *L <sup>h</sup>* is calculated as the harmonic average of the integral length scale of the unladen flow *L* 0 and the interparticle

distance *λ*. Here *λ* =*δ*( *πρ<sup>p</sup>* / 6*ρα* <sup>3</sup> −1), *L* <sup>0</sup> = *k*0 3/2 *ε*0 , *<sup>L</sup> <sup>h</sup>* <sup>=</sup> <sup>2</sup>*<sup>L</sup>* <sup>0</sup>*<sup>λ</sup> <sup>L</sup>* <sup>0</sup> <sup>+</sup> *<sup>λ</sup>* , *ε<sup>h</sup>* <sup>=</sup> *<sup>k</sup>* 3/2 *L <sup>h</sup>* . The particles influence on the shear Reynolds stress components is considered in Eqs. (8 – 10) indirectly via the averaged velocity flow field (*u*, *v*, *w*).

The production terms *P* are determined according to [18] as follows:

$$P\_{uu} = -2\left(\overline{u'^2}\frac{\partial u}{\partial x} + \overline{u'v'}\frac{\partial u}{\partial y} + \overline{u'w'}\frac{\partial u}{\partial z}\right)\_{\prime} \tag{12}$$

$$P\_{vv} = -2\left(\overline{\mu'\upsilon'}\frac{\partial\upsilon}{\partial x} + \overline{\upsilon'^2}\frac{\partial\upsilon}{\partial y} + \overline{\upsilon'w'}\frac{\partial\upsilon}{\partial z}\right)\tag{13}$$

( )

( )

( )

3D governing equations for the particulate phase are written as follows:

The relative friction coefficient *C* ′

Re*<sup>s</sup>* =*δ* (*u* −*us*)<sup>2</sup> + (*v* −*vs*)<sup>2</sup> + (*w* −*ws*)<sup>2</sup> / *ν*.

the particle mass conservation equation:

*uvw*

*s s s*

*s s s*

a

*v w v v C*

 a

a

é ù - æ ö ¶ æ ö ¶ ¶ <sup>+</sup> + + ¢ - - ç ÷ <sup>ê</sup> ç ÷ú ¶ ¶¶ ç ÷ êë è øúû è ø

*v w v v*

 a

¶¶¶ aaa

*x*-component of the momentum equation:

( ) ( ) ( )

*y*-component of the momentum equation:

( ) ( ) ( )

a

2

*s D*

¶ <sup>é</sup> æ ö ¶ ¶ <sup>ù</sup> - <sup>+</sup> <sup>ê</sup> + +<sup>ú</sup> ¢ ç ÷ ¶ ¶¶ êë è øúû

*z*-component of the momentum equation:

a

*z zy*

*z zy*

an

an

a

a

2

*s D*

*uv uv C uv R C P T*

*uw uw C uw R C P T*

*vw vw C vw R C P T*

*<sup>D</sup>* is expressed as *C* ′

( ) ( ) ( ) , *ss s*

++ = + +

*s s s s s s s s*

*u v <sup>v</sup> v w u v <sup>v</sup> <sup>k</sup> x y z x yx y y*

an

¶ ¶ ¶ ¶ ¶ é ù æ öæ ö ¶ ¶ ¶ ++ = <sup>ê</sup> ç ÷ç + +<sup>ú</sup> - <sup>÷</sup> ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ëê è øè ûú <sup>ø</sup>

( )

t

( )

*p*

t

,

*s s s s s s s s*

*u v <sup>v</sup> v w u v <sup>v</sup> <sup>k</sup> x y z x yx y y*

an

¶ ¶ ¶ ¶ ¶ é ù æ öæ ö ¶¶ ¶ ++ = <sup>ê</sup> ç ÷ç + +<sup>ú</sup> - <sup>÷</sup> ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ êë è øè ûú <sup>ø</sup>

*C g*

*x y z x xy yz z*

streamlining of particle. The particle Reynolds number Re*s* is calculated according to [19] as

2 <sup>1</sup> , *<sup>R</sup>*

2

2

<sup>1</sup> , *<sup>R</sup>*

<sup>1</sup> . *<sup>R</sup>*


RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall



0.687 for the non-Stokesian

http://dx.doi.org/10.5772/57047

49

*<sup>D</sup>* =1 + 0.15Re*<sup>s</sup>*

*sss*

aaa

*s s s*

*s s s*

e n

a n

<sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup>

3

3

(26)

(27)

*DDD*

¶ ¶ ¶ ¶ ¶¶ ¶¶ ¶ (25)

¶¶ ¶¶ ¶¶

1

r

 r

*p p*

a

$$P\_{uvw} = -2\left(\overline{u'w'}\frac{\partial w}{\partial x} + \overline{v'w'}\frac{\partial w}{\partial y} + \overline{w'^2}\frac{\partial w}{\partial z}\right),\tag{14}$$

$$P\_{uv} = -\left(\overline{u'^2}\frac{\partial v}{\partial \mathbf{x}} + \overline{u'v'}\frac{\partial v}{\partial y} + \overline{u'w'}\frac{\partial v}{\partial \mathbf{z}} + \overline{u'v'}\frac{\partial u}{\partial \mathbf{x}} + \overline{v'^2}\frac{\partial u}{\partial y} + \overline{v'w'}\frac{\partial u}{\partial \mathbf{z}}\right),\tag{15}$$

$$P\_{uv} = -\left(\overline{u'^2}\,\frac{\partial w}{\partial \mathbf{x}} + \overline{u'v'}\,\frac{\partial w}{\partial y} + \overline{u'w'}\,\frac{\partial w}{\partial z} + \overline{u'w'}\,\frac{\partial u}{\partial \mathbf{x}} + \overline{v'w'}\,\frac{\partial u}{\partial y} + \overline{w'^2}\,\frac{\partial u}{\partial z}\right),\tag{16}$$

$$P\_{vw} = -\left(\overline{u'v'}\frac{\partial w}{\partial \mathbf{x}} + \overline{v'^2}\frac{\partial w}{\partial y} + \overline{v'w'}\frac{\partial w}{\partial z} + \overline{u'w'}\frac{\partial v}{\partial \mathbf{x}} + \overline{v'w'}\frac{\partial v}{\partial y} + \overline{w'^2}\frac{\partial v}{\partial z}\right),\tag{17}$$

$$P = \frac{1}{2} \left( P\_{uu} + P\_{vv} + P\_{uw} \right) = \overline{u'^2} \frac{\partial u}{\partial x} + \overline{u'v'} \frac{\partial u}{\partial y} + \overline{u'w'} \frac{\partial u}{\partial z} + \overline{u'v'} \frac{\partial v}{\partial x} + \overline{v'^2} \frac{\partial v}{\partial y} + \overline{v'w'} \frac{\partial v}{\partial z} \tag{18}$$
 
$$+ \overline{u'w'} \frac{\partial w}{\partial x} + \overline{v'w'} \frac{\partial w}{\partial y} + \overline{w'^2} \frac{\partial w}{\partial z}$$

The diffusive or second order partial differentiation over Cartesian coordinates, i.e. the first three terms in Eqs. (5 – 11) are given, e.g. in [18]. The anisotropy terms *R* of the normal and shear components of the Reynolds stress *<sup>u</sup>* ′ ¯2 , *v* ′ ¯2 , *w* ′ ¯2 , *u* ′ *v* ¯′ , *u* ′ *w*¯′ , *v* ′ *w*¯′ , are defined by various pressure-rate-of-strain models of the isotropic turbulence written in terms of variation of constants *CR* and *C*2 [18] as follows:

$$R\_{\rm uu} = -\frac{\left(C\_R - 1\right)}{T} \left(\overline{\mu'^2} - \frac{2}{3}k\right) - C\_2 \left(P\_{\rm uu} - \frac{2}{3}P\right) \tag{19}$$

$$R\_{vv} = -\frac{\left(\mathcal{C}\_R - 1\right)}{T} \left(\overline{v'^2} - \frac{2}{3}k\right) - \mathcal{C}\_2 \left(P\_{vv} - \frac{2}{3}P\right) \tag{20}$$

$$R\_{uvw} = -\frac{\left(C\_R - 1\right)}{T} \left(\overline{w'^2} - \frac{2}{3}k\right) - C\_2 \left(P\_{uvw} - \frac{2}{3}P\right) \tag{21}$$

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall http://dx.doi.org/10.5772/57047 49

$$R\_{uv} = -\frac{\left(\mathbb{C}\_R - 1\right)\overline{u'v'}}{T} - \mathbb{C}\_2 P\_{uv'} \tag{22}$$

$$R\_{\mu\nu} = -\frac{(\mathcal{C}\_R - 1)\overline{\mu' w'}}{T} - \mathcal{C}\_2 P\_{\mu w'} \tag{23}$$

$$\mathcal{R}\_{vw} = -\frac{\left(\mathcal{C}\_R - 1\right)\overline{v'w'}}{T} - \mathcal{C}\_2 P\_{vw}.\tag{24}$$

The relative friction coefficient *C* ′ *<sup>D</sup>* is expressed as *C* ′ *<sup>D</sup>* =1 + 0.15Re*<sup>s</sup>* 0.687 for the non-Stokesian streamlining of particle. The particle Reynolds number Re*s* is calculated according to [19] as Re*<sup>s</sup>* =*δ* (*u* −*us*)<sup>2</sup> + (*v* −*vs*)<sup>2</sup> + (*w* −*ws*)<sup>2</sup> / *ν*.

3D governing equations for the particulate phase are written as follows:

the particle mass conservation equation:

<sup>2</sup> 2 , *vv vv v P uv v vw*

<sup>2</sup> 2 , *ww w ww P uw vw w*

<sup>2</sup> <sup>2</sup> , *uv v v v uu u P u uv uw uv v vw*

<sup>2</sup> <sup>2</sup> , *uw ww w u uu P u uv uw uw vw w*

<sup>2</sup> <sup>2</sup> , *vw ww w v v v P uv v vw uw vw w*

*u u u vv v P P P P u uv uw uv v vw*

= ++ = + + + + + ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶¶ ++ + ¢¢ ¢¢ ¢ ¶ ¶¶

*w ww*

*x yz*

The diffusive or second order partial differentiation over Cartesian coordinates, i.e. the first three terms in Eqs. (5 – 11) are given, e.g. in [18]. The anisotropy terms *R* of the normal and

, *v* ′ ¯2 , *w* ′ ¯2 , *u* ′ *v* ¯′

pressure-rate-of-strain models of the isotropic turbulence written in terms of variation of

( ) <sup>2</sup> <sup>2</sup>

*uw vw w*

( ) <sup>2</sup>

( ) <sup>2</sup>

( ) <sup>2</sup>

*uu uu*

*vv vv*

*ww ww*

*R w k CP P*

*R v k CP P*


*R u k CP P*


*R*

*R*

*R*

*T*

*C*

*T*

*C*

*T*

*C*

1

48 Computational and Numerical Simulations

2 *uu vv ww*

shear components of the Reynolds stress *<sup>u</sup>* ′ ¯2

constants *CR* and *C*2 [18] as follows:

*xy z* æ ¶¶ ¶ ö

*x yz* æ ö ¶ ¶¶

*x y z xy z* æ ¶ ¶ ¶ ¶¶ ¶ ö

*x y z x yz* æ ¶ ¶ ¶ ¶ ¶¶ ö

*xy z x yz* æ ¶¶ ¶ ¶ ¶¶ ö

=- + + + + + ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢ <sup>ç</sup> <sup>÷</sup> ¶ ¶ ¶ ¶¶ ¶ <sup>è</sup> <sup>ø</sup> (15)

=- + + + + + ¢ ¢¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ <sup>ç</sup> <sup>÷</sup> ¶ ¶ ¶ ¶ ¶¶ <sup>è</sup> <sup>ø</sup> (16)

=- + + + + + ¢¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ <sup>ç</sup> <sup>÷</sup> ¶¶ ¶ ¶ ¶¶ <sup>è</sup> <sup>ø</sup> (17)

*x y z xy z*

, *u* ′ *w*¯′

, *v* ′ *w*¯′

è øè ø (19)

è øè ø (20)

(18)

, are defined by various

¶ ¶ ¶ ¶¶ ¶

2

2 <sup>1</sup> 2 2 , 3 3

2 <sup>1</sup> 2 2 , 3 3

2 <sup>1</sup> <sup>2</sup> <sup>2</sup> , <sup>3</sup> <sup>3</sup>


=- + + ¢¢ ¢ ¢ ¢ <sup>ç</sup> <sup>÷</sup> ¶¶ ¶ <sup>è</sup> <sup>ø</sup> (13)

=- + + ¢¢ ¢¢ ¢ ç ÷ ¶ ¶¶ è ø (14)

$$\frac{\partial \left(a u\_s\right)}{\partial \mathbf{x}} + \frac{\partial \left(a v v\_s\right)}{\partial y} + \frac{\partial \left(a w\_s\right)}{\partial \mathbf{z}} = \frac{\partial}{\partial \mathbf{x}} D\_s \frac{\partial a}{\partial \mathbf{x}} + \frac{\partial}{\partial y} D\_s \frac{\partial a}{\partial y} + \frac{\partial}{\partial \mathbf{z}} D\_s \frac{\partial a}{\partial \mathbf{z}},\tag{25}$$

*x*-component of the momentum equation:

$$\begin{split} \frac{\partial \left( a u\_s v\_s \right)}{\partial x} + \frac{\partial \left( a v\_s^2 \right)}{\partial y} + \frac{\partial \left( a v\_s w\_s \right)}{\partial z} &= \frac{\partial}{\partial x} \left[ a \nu\_s \left( \frac{\partial u\_s}{\partial y} + \frac{\partial v\_s}{\partial x} \right) \right] + \frac{\partial}{\partial y} a \left( 2 \nu\_s \frac{\partial v\_s}{\partial y} - \frac{2}{3} k\_s \right) \\ + \frac{\partial}{\partial z} \left[ a \nu\_s \left( \frac{\partial v\_s}{\partial z} + \frac{\partial w\_s}{\partial y} \right) \right] + a C\_D' \frac{\left( v - v\_s \right)}{\tau\_p} - a g \left( 1 - \frac{\rho}{\rho\_p} \right) \end{split} \tag{26}$$

*y*-component of the momentum equation:

$$\begin{split} \frac{\partial \left( a u\_s v\_s \right)}{\partial x} + \frac{\partial \left( a v\_s^2 \right)}{\partial y} + \frac{\partial \left( a v\_s w\_s \right)}{\partial z} &= \frac{\partial}{\partial x} \bigg[ a \nu\_s \left( \frac{\partial u\_s}{\partial y} + \frac{\partial v\_s}{\partial x} \right) \bigg] + \frac{\partial}{\partial y} \varepsilon \left( 2 \nu\_s \frac{\partial v\_s}{\partial y} - \frac{2}{3} k\_s \right) \\ + \frac{\partial}{\partial z} \bigg[ a \nu\_s \left( \frac{\partial v\_s}{\partial z} + \frac{\partial w\_s}{\partial y} \right) \bigg] + a \he{\mathsf{C}}\_D' \frac{\left( v - v\_s \right)}{\tau\_p} \,, \end{split} \tag{27}$$

*z*-component of the momentum equation:

$$\begin{split} &\frac{\partial\left(au\_{s}w\_{s}\right)}{\partial\mathbf{x}} + \frac{\partial\left(av\_{s}w\_{s}\right)}{\partial\mathbf{y}} + \frac{\partial\left(av\_{s}^{2}\right)}{\partial\mathbf{z}} = \frac{\partial}{\partial\mathbf{x}}\bigg[av\_{s}\left(\frac{\partial u\_{s}}{\partial\mathbf{z}} + \frac{\partial w\_{s}}{\partial\mathbf{x}}\right)\bigg] + \frac{\partial}{\partial\mathbf{y}}\bigg[av\_{s}\left(\frac{\partial v\_{s}}{\partial\mathbf{z}} + \frac{\partial w\_{s}}{\partial\mathbf{y}}\right)\bigg] \\ &+ \frac{\partial}{\partial\mathbf{z}}\bigg(a\left(2\,\nu\_{s}\,\frac{\partial w\_{s}}{\partial\mathbf{z}} - \frac{2}{3}k\_{s}\right) + aC\_{D}'\frac{\left(w - w\_{s}\right)}{\tau\_{p}}.\end{split} \tag{28}$$

The wall conditions are written as follows:

\* 1 , <sup>1</sup> ln

<sup>ï</sup> <sup>+</sup> <sup>î</sup>

+

+

2

[21] as *v*\*=(*cμ*/2)0.25 *k*, where *cμ* is the numerical constant of the *k*-*ε* model, *cμ*=0.09; *B*1=5.2 for the smooth wall and *B*2=8.5 for the rough wall; *Δy* is the grid step of the control volume.

For the normal and shear stresses and dissipation rate of the unladen flow calculated at the wall, the boundary conditions are set based on the "wall-function" according to [18] with the

> , 0, *uu vv ww uv uw vw <sup>u</sup> P uv P P P P P*

> > 0.75 1.5 <sup>2</sup> , *c k æ y* m

, 0, *uu vv ww uv uw vw <sup>u</sup> P uw P P P P P*

> 0.75 1.5 <sup>2</sup> . *c k æ z* m

= = + == (33)

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

is the friction velocity of gas; *æ* is von Karman's constant, *æ*

¶ = - ¢ ¢ = == = = ¶ (34)

¶ = - ¢ ¢ = == = = ¶ (36)

<sup>=</sup> <sup>D</sup> (35)

correspond to the transverse and spanwise directions,

is determined according to

http://dx.doi.org/10.5772/57047

(32)

51

(37)

*v y B æ*

*<sup>y</sup> <sup>u</sup>*

<sup>1</sup> <sup>4</sup> ln ; 0 *<sup>u</sup> <sup>y</sup> <sup>u</sup> Bvw*

ì <sup>ï</sup> = = <sup>í</sup>

*u*

+

\*

and *z* <sup>+</sup>

respectively; *s* is a roughness height. The friction velocity of gas *v*\*

following relationships for the production and dissipation terms:

*y*

*z*

e<sup>=</sup> <sup>D</sup>

e

*væ s* <sup>+</sup> D

for *y* =0.5*h* (smooth wall):

for *y* = −0.5*h* (rough wall):

where *h* is the channel width; *v*\*

=0.41; the wall coordinates *y* <sup>+</sup>

for *y* =0.5*h* :

for *z* =0.5*h* :

The closure model for the transport equations of the particulate phase was applied to the PDF model [20], where the turbulent kinetic energy of dispersed phase, the coefficients of the turbulent viscosity and turbulent diffusion of the particulate phase are determined as follows, respectively:

$$\begin{split} \boldsymbol{k}\_{s} &= \left[ 1 - \exp\left( -\frac{T\_{0}}{\tau\_{p}^{\prime}} \right) \right] \boldsymbol{k}\_{\prime} \quad \boldsymbol{\nu}\_{s} = \left( \boldsymbol{\nu}\_{t} + \frac{\tau\_{p}^{\prime} k}{3} \right) \left[ 1 - \exp\left( -\frac{T\_{0}}{\tau\_{p}^{\prime}} \right) \right] \quad \boldsymbol{D}\_{s} = \frac{2k}{3} \left( 1 + \frac{T\_{0}}{\tau} \right) \left[ 1 - \exp\left( -\frac{T\_{0}}{\tau} \right) \right] \\ \boldsymbol{\nu}\_{s} &= \left( \boldsymbol{\nu}\_{t} + \frac{\tau\_{p}^{\prime} k}{3} \right) \left[ 1 - \exp\left( -\frac{T\_{0}}{\tau\_{p}^{\prime}} \right) \right] , \end{split} \tag{29}$$

where *νt* is the turbulent viscosity, *ν<sup>t</sup>* =0.09 *k*0 2 *ε*0 and *τ* ′ *<sup>p</sup>* <sup>=</sup>*τ<sup>p</sup>* / *<sup>C</sup>* ′ *<sup>D</sup>* is the particle response time with respect of correction of the particles motion to the non-Stokesian regime.

### **2.2. Boundary conditions for the Reynolds stress turbulence model**

The grid-generated turbulent flow is vertical, and it is symmetrical with respect to the vertical axis for both *y*-and *z*-directions. Therefore, the symmetry conditions are set at the flow axis, and the wall conditions are set at the wall. In case of the rough and smooth walls the flow was asymmetrical over the *y*-direction and symmetrical over the *z*-direction.

The axisymmetric conditions are written as follows:

for *z*=0:

$$\frac{\partial u}{\partial z} = \frac{\overline{\partial u'^2}}{\partial z} = \frac{\overline{\partial v'^2}}{\partial z} = \frac{\overline{\partial w'^2}}{\partial z} = \frac{\partial \varepsilon}{\partial z} = \frac{\partial u\_s}{\partial z} = \frac{\partial \alpha}{\partial z} = \upsilon = \overline{w'\upsilon'} = \overline{u'w'} = \overline{\upsilon'w'} = \upsilon\_s = w\_s = 0; \tag{30}$$

for*z* =0.5*h* :

$$
\mu^{+} = \frac{\mu}{v\_{\*}} = \begin{cases} z^{+} \\ \frac{1}{\mathfrak{a}\mathfrak{b}} \ln z^{+} + B\_{1} \end{cases} \tag{31}
$$

The wall conditions are written as follows:

for *y* =0.5*h* (smooth wall):

( ) ( ) ( )

 a

3

¶ <sup>æ</sup> ¶ <sup>ö</sup> - <sup>+</sup> - + ¢ <sup>ç</sup> <sup>÷</sup> ¶ ¶ <sup>è</sup> <sup>ø</sup>

*z z*

50 Computational and Numerical Simulations

a n

respectively:

*s t*

for *z*=0:

for*z* =0.5*h* :

n n *p*

t

a

*s sD*

<sup>2</sup> 2 .

0

n n

è ø ¢ <sup>ú</sup> ç ÷

*p*

The axisymmetric conditions are written as follows:

22 2

*uu v w u*

*z z z z zzz* ¶¶ ¶ ¶ ¶ ¶ ¢¢ ¢

e

*u*

+

t

*s s t s p p*

1 exp , <sup>3</sup>

where *νt* is the turbulent viscosity, *ν<sup>t</sup>* =0.09

*k T*

æ ö ¢ é æ öù =+ - - ç ÷ê ç ÷ú ç ÷ê ç ÷ ¢ <sup>ú</sup> è øë è øû

t *w w w k C*

*s s*

a

a

( )

*p*

t

*s ss ss s s s s*

The closure model for the transport equations of the particulate phase was applied to the PDF model [20], where the turbulent kinetic energy of dispersed phase, the coefficients of the turbulent viscosity and turbulent diffusion of the particulate phase are determined as follows,

0 0 0 0

and *τ* ′

*<sup>p</sup>* <sup>=</sup>*τ<sup>p</sup>* / *<sup>C</sup>* ′

0; *<sup>s</sup>*

*v w uv uw vw v w*

t

<sup>2</sup> 1 exp , 1 exp , 1 1 exp , <sup>3</sup> <sup>3</sup>

è ø ¢ <sup>ú</sup> è ø èø êë úû <sup>ë</sup> <sup>û</sup> <sup>ë</sup> <sup>û</sup>

*k*0 2 *ε*0

The grid-generated turbulent flow is vertical, and it is symmetrical with respect to the vertical axis for both *y*-and *z*-directions. Therefore, the symmetry conditions are set at the flow axis, and the wall conditions are set at the wall. In case of the rough and smooth walls the flow was

<sup>é</sup> æ öù æ ö ¢ <sup>é</sup> æ öù æ ö æö <sup>é</sup> <sup>ù</sup> =- - <sup>ê</sup> ç ÷ú =+ - - = + - - ç ÷ê ç ÷ú ç ÷ ç÷ <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> ç ÷

*<sup>T</sup> <sup>k</sup> <sup>T</sup> <sup>k</sup> <sup>T</sup> <sup>T</sup> <sup>k</sup> <sup>k</sup> <sup>D</sup>*

è ø<sup>ê</sup> ç ÷

with respect of correction of the particles motion to the non-Stokesian regime.

**2.2. Boundary conditions for the Reynolds stress turbulence model**

asymmetrical over the *y*-direction and symmetrical over the *z*-direction.

 a

ì <sup>ï</sup> = = <sup>í</sup>

*u*

¶ = = = = = = == = = = = = = ¢¢ ¢ ¢ ¢ ¢ ¶ ¶ ¶ ¶ ¶¶¶ (30)

+

+

\* 1 . <sup>1</sup> ln *z*

<sup>ï</sup> <sup>+</sup> <sup>î</sup>

*v z B æ*

*uw vw w uw vw x y z x zx y zy*

¶ ¶ ¶ ¶ é ù æ ö ¶ ¶ ¶ é ù æ ö ¶ ¶ + += <sup>ê</sup> ç ÷ + +<sup>ú</sup> <sup>ê</sup> ç ÷ <sup>+</sup> <sup>ú</sup> ¶ ¶ ¶ ¶ ¶¶ ¶ ¶¶ êë è øûú ëê è øúû

an

*s s*

an

t t

*<sup>D</sup>* is the particle response time

*s s*

(28)

(29)

(31)

2

*p*

t

$$\mu^{+} = \frac{\mu}{v\_{\*}} = \left| \frac{y^{+}}{\frac{1}{\mathfrak{a}\mathfrak{r}} \ln y^{+} + B\_{1}^{'}} \right. \tag{32}$$

for *y* = −0.5*h* (rough wall):

$$
\mu^{+} = \frac{\mu}{\upsilon\_{\*}} = \frac{1}{\mathfrak{a}\mathfrak{r}} \ln \frac{4\Delta\mathfrak{y}}{\mathfrak{s}} + B\_{2}; \; \upsilon = \mathfrak{z}\upsilon = 0 \tag{33}
$$

where *h* is the channel width; *v*\* is the friction velocity of gas; *æ* is von Karman's constant, *æ* =0.41; the wall coordinates *y* <sup>+</sup> and *z* <sup>+</sup> correspond to the transverse and spanwise directions, respectively; *s* is a roughness height. The friction velocity of gas *v*\* is determined according to [21] as *v*\*=(*cμ*/2)0.25 *k*, where *cμ* is the numerical constant of the *k*-*ε* model, *cμ*=0.09; *B*1=5.2 for the smooth wall and *B*2=8.5 for the rough wall; *Δy* is the grid step of the control volume.

For the normal and shear stresses and dissipation rate of the unladen flow calculated at the wall, the boundary conditions are set based on the "wall-function" according to [18] with the following relationships for the production and dissipation terms:

for *y* =0.5*h* :

$$P\_{uu} = -\overline{\mathfrak{u}'v'} \frac{\partial \mathfrak{u}}{\partial y}, \ P\_{vv} = P\_{uvw} = P\_{uv} = P\_{uw} = P\_{vw} = \mathcal{O}\_{\prime} \tag{34}$$

$$
\omega = \frac{2c\_{\mu}^{0.75}k^{1.5}}{\text{ze}\Delta y},
\tag{35}
$$

for *z* =0.5*h* :

, 0, *uu vv ww uv uw vw <sup>u</sup> P uw P P P P P z* ¶ = - ¢ ¢ = == = = ¶ (36)

$$
\varepsilon = \frac{2c\_{\mu}^{0.75}k^{1.5}}{\mathfrak{av}\Delta\mathfrak{z}}.\tag{37}
$$

The boundary conditions for the particulate phase are set at the wall as follows:

for *y* =0.5*h* :

$$\frac{\partial u\_s}{\partial y} = -\lambda u\_{s'} \quad \frac{\partial w\_s}{\partial y} = -\lambda w\_{s'} \quad \frac{\partial \alpha}{\partial y} = D\_s \alpha\_{\prime} \quad v\_s = 0,\tag{38}$$

**3. Numerical results**

and the roughness height of 3.18 mm.

gas over the channel cross-section.

The validation of the present model took place in two stages.

In case of the unladen flow, the model was validated by comparison of the kinetic (normal) components of stresses with the experimental data [22] obtained for the specially constructed horizontal turbulent gas flow in the channel of rectangular cross-section (the aspect ratio of 1:6) of 54 mm width with the smooth and rough walls for the flow Reynolds number *Re*=56000

RSTM Numerical Simulation of Channel Particulate Flow with Rough Wall

http://dx.doi.org/10.5772/57047

53

**Figure 2.** Numerical and experimental [22] distributions of the longitudinal component of the averaged velocity of

Figure 2 shows the distributions of the longitudinal component of the averaged velocity of gas *u*0 over the channel cross-section for two cases: i) the smooth walls and ii) the left wall is rough and the right wall is smooth for the mean flow velocity 15.5 m/s. Figure 3 shows the distribu‐ tions of the normalized Reynolds normal stress tensor components obtained for the same conditions as Figure 2. The radial distance *y*/*h*=0 corresponds to the rough wall and *y*/*h*=1 corresponds to the smooth wall. The subscript "0" denotes the unladen flow conditions.

One can see that in case of the smooth channel walls, the mean flow velocity and the compo‐ nents of the turbulence kinetic energy demonstrate the representative symmetrical turbulent distributions over the cross-section of the rectangular channel. The transfer to the rough walls results in transformation of the given distributions. The maximum of the distribution of the time-averaged flow velocity moves towards the smooth wall. The similar change relates to the distributions of each component of the turbulence kinetic energy. These numerical results

demonstrates the satisfactory agreement with the experimental data [22].

for *z* =0.5*h* :

$$\frac{\partial u\_s}{\partial \mathbf{z}} = -\lambda u\_s; \quad \frac{\partial v\_s}{\partial \mathbf{z}} = -\lambda v\_s, \quad \frac{\partial a}{\partial \mathbf{z}} = D\_s \alpha \quad \text{or} \quad w\_s = 0; \tag{39}$$

At the exit of the channel the following boundary conditions are set:

$$\begin{split} \frac{\partial u}{\partial \mathbf{x}} &= \frac{\partial v}{\partial \mathbf{x}} = \frac{\partial w}{\partial \mathbf{x}} = \frac{\partial \overline{u'^2}}{\partial \mathbf{x}} = \frac{\partial \overline{v'^2}}{\partial \mathbf{x}} = \frac{\partial \overline{w'^2}}{\partial \mathbf{x}} = \frac{\partial \overline{u'v'}}{\partial \mathbf{x}} = \\ \frac{\partial \overline{u'w'}}{\partial \mathbf{x}} &= \frac{\partial \overline{v'w'}}{\partial \mathbf{x}} = \frac{\partial \overline{x}}{\partial \mathbf{x}} = \frac{\partial u\_s}{\partial \mathbf{x}} = \frac{\partial v\_s}{\partial \mathbf{x}} = \frac{\partial w\_s}{\partial \mathbf{x}} = \frac{\partial w\_s}{\partial \mathbf{x}} = \frac{\partial \alpha}{\partial \mathbf{x}} = 0. \end{split} \tag{40}$$

Additionally, the initial boundary conditions are set for three specific cases:


### **2.3. Numerical method**

The control volume method was applied to solve the 3D partial differential equations written for the unladen flow (Eqs. 1 – 11) and the particulate phase (Eqs. 26 – 29), respectively, with taking into account the boundary conditions (Eqs. 30 – 40). The governing equations were solved using the implicit lower and upper (ILU) matrix decomposition method with the fluxblending differed-correction and upwind-differencing schemes [21]. This method is utilized for the calculations of the particulate turbulent flows in channels of the rectangular and square cross-sections. The calculations were performed in the dimensional form for all the flow conditions. The number of the control volumes was 1120000.
