**3. Governing equations**

The fundamental equations of the Fluid Mechanics applied to a three-dimensional flow of an incompressible and viscous fluid, with sediment in suspension, are written:

Turbulent Boundary Layer Models: Theory and Applications 209

attempts have been made to add turbulence conservation relations to the time-averaged

According to the Boussinesq hypothesis, the turbulent shear stresses *' ' u ui <sup>j</sup>* are modelled in

*j i i <sup>u</sup> <sup>u</sup> <sup>ρ</sup> u u ν δ K u ρ γ i, j = , , x x x*

the turbulent viscosity, and *<sup>t</sup> γ* is the turbulent diffusivity. In contrast to the molecular

state of the turbulence and may vary considerably over the flow field. A turbulence model

relating the turbulence correlations to the averaged dependent variables. As a first order turbulence closure, the turbulent viscosity *<sup>t</sup> ν* is obtained through the *mixing-length* theory of Prandtl (1925), who, by analogy with kinetic theory, proposed that each turbulent

> *<sup>u</sup> u v l l z zz*

A derivation of the turbulent shear stresses, where *i j* , involves subtracting the above time-averaged equation (2-b) from its instantaneous value (1-b), for both the *<sup>i</sup> x* and *<sup>j</sup> x* directions. The *i*th result is then multiplied by *' uj* and added to the *j*th result multiplied by *'ui* . This relation is then time-averaged to yield the following *Reynolds stress equation ' ' u ui <sup>j</sup>* :

*uu u uu uu uu gu <sup>ρ</sup> g u <sup>ρ</sup> t x xx <sup>ρ</sup>*

*k kk*

 *uuu u u ν u u ν*

*<sup>j</sup> ' ' '' '' '' <sup>i</sup> '' '' ij k ij ik jk ij ji*

*u u*

 

 

*x ρ x x x x x*

 

0

In equation (5), the three terms of different nature *'''*

1

 

<sup>123</sup> 2 2 *'' ' ' ' K u u u u u i j* is the turbulent kinetic energy, per mass unit; *<sup>t</sup> ν* is

; ; 1 2 3 <sup>3</sup>

2 2

is the boundary layer thickness.

0 2 2

*p p u u*

 *i j k*

*uuu*

,

2

*k*

*x* 

are to be either neglected or related to other variables. Let us

 

 *<sup>j</sup> '' ' ' ' ' ' i ijk i j l ij l k j i k k k*

1

(3)

(4)

, where

*''*

> 

 

(5)

and

 

 

 *' ' i j j i p p u u ρ x x* 

0 1

*<sup>t</sup>* over the flow field, by

*<sup>t</sup>* is not a fluid property, but depends strongly on the

2

**3.2 Boussinesq hypothesis (first order turbulence closure model)** 

*j ' ' i ' ' ij t ij i t*

fluctuation could be related to a length *lm* scale and a velocity gradient,

2 2 *<sup>i</sup> tm m*

For the *lm* scale different relations have been proposed. We suggest *l kz z z <sup>m</sup>* 1

 

thus usually has the task of determining the distribution of

 

terms of the gradients of the mean flow velocities through (3),

where <sup>222</sup>

*<sup>l</sup>* , the turbulent viscosity

*k* 0.4 is the von Kármán constant and *z*

**3.3 Second order turbulence closure model** 

 

*' 'j i*

*k k u u*

*x x*

 

equations above.

viscosity

 <sup>2</sup> 2

*' ' l i j k ν u u x* 

2

*l*

consider these terms in some detail.

 

*ν*

$$\begin{aligned} \text{a) } \frac{\partial \boldsymbol{u}\_{i}}{\partial \mathbf{x}\_{i}} &= 0 \\ \text{b) } \frac{\partial \boldsymbol{u}\_{i}}{\partial t} + \boldsymbol{u}\_{j} \frac{\partial \boldsymbol{u}\_{i}}{\partial \mathbf{x}\_{j}} &= -\frac{1}{\rho\_{0}} \frac{\partial p}{\partial \mathbf{x}\_{i}} + \nu\_{l} \frac{\partial^{2} \boldsymbol{u}\_{i}}{\partial \mathbf{x}\_{j}^{2}} + g\_{i} \frac{\rho - \rho\_{0}}{\rho\_{0}} \\ \text{c) } \frac{\partial \mathbf{C}}{\partial t} + \left(\boldsymbol{u}\_{i} + \boldsymbol{u}\_{s\_{i}}\right) \frac{\partial \mathbf{C}}{\partial \mathbf{x}\_{i}} &= \boldsymbol{\chi}\_{m} \frac{\partial^{2} \mathbf{C}}{\partial \mathbf{x}\_{i}^{2}} \\ \text{d) } \rho = \rho\_{s} \mathbf{C} + \left(1 - \mathbf{C}\right) \rho\_{0} \end{aligned} \tag{1}$$

where *u* is the instantaneous velocity of the flow; *C* is the volumetric concentration of the sediment; *p* is the pressure; *<sup>l</sup>* is the kinematic viscosity; *gi* is the acceleration due to gravity; is the density; 0 is the density of the fluid; *<sup>s</sup>* is the density of the sediment; *ws* is the sediment settling velocity, and *<sup>m</sup>* is the molecular diffusivity.

#### **3.1 Turbulence closure model with sediment in suspension**

Following the classical Osborne Reynolds procedure, and assuming that the fluid is in a randomly unsteady turbulent state and applying time averaging to the basic equations of motion, the fundamental equations of incompressible turbulent motion are obtained. These are known as the Reynolds equations, and involve both mean and fluctuating quantities – the turbulent inertia tensor components. We consider only incompressible turbulent flow with constant transport properties but with possible significant fluctuations in velocity, pressure, and concentration, i.e.:

*<sup>i</sup> ' uuu i i* ; *<sup>i</sup> ' p p p i i* ; *' CCC iii*

Substituting these functions into the basic equations (1), and taking the time average of each entire equation, we obtain (2) (Rodi, 1984):

$$\begin{aligned} \text{a)} \quad & \frac{\partial \,\overline{u}\_{i}}{\partial \mathbf{x}\_{i}} = 0 \\ \text{b)} \quad & \frac{\partial \,\overline{u}\_{i}}{\partial t} + \overline{u}\_{j} \frac{\partial \,\overline{u}\_{i}}{\partial \mathbf{x}\_{j}} = -\frac{1}{\rho\_{0}} \frac{\partial \overline{p}}{\partial \mathbf{x}\_{i}} + \frac{\partial}{\partial \mathbf{x}\_{j}} \bigg[ \nu\_{l} \frac{\partial \,\overline{u}\_{i}}{\partial \mathbf{x}\_{j}} - \overline{u\_{i} \, u\_{j}^{\circ}} \bigg] + \text{g}\_{i} \frac{\overline{\rho} - \rho\_{0}}{\rho\_{0}} \\ \text{d)} \quad & \frac{\partial \overline{\mathbf{C}}}{\partial t} + \left( \overline{u}\_{i} + w\_{s\_{i}} \right) \frac{\partial \overline{\mathbf{C}}}{\partial \mathbf{x}\_{i}} = \frac{\partial}{\partial \mathbf{x}\_{i}} \bigg[ \nu\_{m} \frac{\partial \overline{\mathbf{C}}}{\partial \mathbf{x}\_{i}} - \overline{u\_{i} \, \rho} \bigg] \\ \text{d)} \quad & \overline{\rho} = \rho\_{s} \overline{\mathbf{C}} + \left( 1 - \overline{\mathbf{C}} \right) \rho\_{0} \end{aligned} \tag{2}$$

where - *' ' u ui <sup>j</sup>* are the tensor components of the Reynolds stresses, and - *'* ' *ui ρ* are the tensor components of density-velocity correlations. Thus the mean momentum equation and the equation for the concentration are complicated by new terms involving the turbulent inertia tensor *' ' u ui <sup>j</sup>* and density fluctuations *'* ' *ui ρ* . The new terms are never negligible in any turbulent flow with sediment in suspension, and can be defined only through knowledge of the detailed turbulent structure, which is, in its turn, unavailable. These turbulent quantities are related not only to the fluid physical properties but also to local flow conditions. As no physical laws are available, most attempts have been made to resolve this dilemma. Many 208 Advanced Fluid Dynamics

2

2 0 0 2 2

*<sup>l</sup>* is the kinematic viscosity; *gi* is the acceleration due to gravity;

0

*<sup>s</sup>* is the density of the sediment; *ws* is the

(1)

(2)

*ρ ρ C C ρ*

0 is the density of the fluid;

<sup>1</sup> b)

d) 1

*s*

**3.1 Turbulence closure model with sediment in suspension** 

*ρ ρ C C ρ*

<sup>1</sup> b)

 

*i*

0

*C CC u w <sup>γ</sup> <sup>u</sup> <sup>ρ</sup> t xx x*

a) 0

*u x*

*i i*

c)

sediment; *p* is the pressure;

sediment settling velocity, and

pressure, and concentration, i.e.:

entire equation, we obtain (2) (Rodi, 1984):

c)

a) 0

*u x*

*i i*

d) 1

*s*

is the density;

*i*

 

*is m*

*C CC uw <sup>γ</sup> t x <sup>x</sup>*

0

where *u* is the instantaneous velocity of the flow; *C* is the volumetric concentration of the

*<sup>m</sup>* is the molecular diffusivity.

Following the classical Osborne Reynolds procedure, and assuming that the fluid is in a randomly unsteady turbulent state and applying time averaging to the basic equations of motion, the fundamental equations of incompressible turbulent motion are obtained. These are known as the Reynolds equations, and involve both mean and fluctuating quantities – the turbulent inertia tensor components. We consider only incompressible turbulent flow with constant transport properties but with possible significant fluctuations in velocity,

*<sup>i</sup> ' uuu i i* ; *<sup>i</sup> ' p p p i i* ; *' CCC iii*

Substituting these functions into the basic equations (1), and taking the time average of each

0

*j l ij i j ij j*

*u u <sup>p</sup> <sup>u</sup> <sup>ρ</sup> <sup>ρ</sup> <sup>u</sup> <sup>ν</sup> uu g t x <sup>ρ</sup> xx x <sup>ρ</sup>*

*i i i ' '*

 

*i s m i ii i*

where - *' ' u ui <sup>j</sup>* are the tensor components of the Reynolds stresses, and - *'* ' *ui ρ* are the tensor components of density-velocity correlations. Thus the mean momentum equation and the equation for the concentration are complicated by new terms involving the turbulent inertia tensor *' ' u ui <sup>j</sup>* and density fluctuations *'* ' *ui ρ* . The new terms are never negligible in any turbulent flow with sediment in suspension, and can be defined only through knowledge of the detailed turbulent structure, which is, in its turn, unavailable. These turbulent quantities are related not only to the fluid physical properties but also to local flow conditions. As no physical laws are available, most attempts have been made to resolve this dilemma. Many

0 0

*' '*

*i i*

*i i i j l i j i j*

*uu u <sup>p</sup> <sup>ρ</sup> <sup>ρ</sup><sup>u</sup> <sup>ν</sup> <sup>g</sup> t x <sup>ρ</sup> <sup>x</sup> <sup>x</sup> <sup>ρ</sup>*

 

attempts have been made to add turbulence conservation relations to the time-averaged equations above.

### **3.2 Boussinesq hypothesis (first order turbulence closure model)**

According to the Boussinesq hypothesis, the turbulent shear stresses *' ' u ui <sup>j</sup>* are modelled in terms of the gradients of the mean flow velocities through (3),

$$\overrightarrow{-u\_i \cdot \overrightarrow{u\_j}} = \nu\_t \left[ \frac{\partial \,\overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \,\overline{u}\_j}{\partial \mathbf{x}\_i} \right] - \frac{2}{3} \delta\_{\overrightarrow{\eta}} \mathcal{K} \; ; \; \overline{-u\_i \cdot \overrightarrow{\rho} \; ;} = \chi\_t \frac{\partial \,\overline{\rho}}{\partial \mathbf{x}\_i} \; ; \; i, j = 1, 2, 3 \tag{3}$$

where <sup>222</sup> <sup>123</sup> 2 2 *'' ' ' ' K u u u u u i j* is the turbulent kinetic energy, per mass unit; *<sup>t</sup> ν* is the turbulent viscosity, and *<sup>t</sup> γ* is the turbulent diffusivity. In contrast to the molecular viscosity *<sup>l</sup>* , the turbulent viscosity *<sup>t</sup>* is not a fluid property, but depends strongly on the state of the turbulence and may vary considerably over the flow field. A turbulence model thus usually has the task of determining the distribution of *<sup>t</sup>* over the flow field, by relating the turbulence correlations to the averaged dependent variables. As a first order turbulence closure, the turbulent viscosity *<sup>t</sup> ν* is obtained through the *mixing-length* theory of Prandtl (1925), who, by analogy with kinetic theory, proposed that each turbulent fluctuation could be related to a length *lm* scale and a velocity gradient,

$$\left| \nu\_t = l\_m^2 \right| \frac{\left| \hat{\boldsymbol{\mathcal{O}}} \, \boldsymbol{u}\_i \right|}{\left| \hat{\boldsymbol{\mathcal{O}}} \boldsymbol{z} \right|} = l\_m^2 \sqrt{\left( \frac{\hat{\boldsymbol{\mathcal{O}}} \boldsymbol{u} \right)^2 + \left( \frac{\hat{\boldsymbol{\mathcal{O}}} \boldsymbol{v} \right)^2}{\left( \hat{\boldsymbol{\mathcal{O}}} \boldsymbol{z} \right)^2}}} \tag{4}$$

For the *lm* scale different relations have been proposed. We suggest *l kz z z <sup>m</sup>* 1 , where *k* 0.4 is the von Kármán constant and *z*is the boundary layer thickness.

#### **3.3 Second order turbulence closure model**

A derivation of the turbulent shear stresses, where *i j* , involves subtracting the above time-averaged equation (2-b) from its instantaneous value (1-b), for both the *<sup>i</sup> x* and *<sup>j</sup> x* directions. The *i*th result is then multiplied by *' uj* and added to the *j*th result multiplied by *'ui* . This relation is then time-averaged to yield the following *Reynolds stress equation ' ' u ui <sup>j</sup>* :

 0 2 2 0 1 12 *<sup>j</sup> ' ' '' '' '' <sup>i</sup> '' '' ij k ij ik jk ij ji k kk ' ' <sup>j</sup> '' ' ' ' ' ' i ijk i j l ij l k j i k k k u u uu u uu uu uu gu <sup>ρ</sup> g u <sup>ρ</sup> t x xx <sup>ρ</sup> p p u u uuu u u ν u u ν x ρ x x x x x* (5)

In equation (5), the three terms of different nature *''' i j k k uuu x* , 0 1 *' ' i j j i p p u u ρ x x* and

 <sup>2</sup> 2 *' ' l i j k ν u u x* 2 *' ' j i l k k u u ν x x* are to be either neglected or related to other variables. Let us

consider these terms in some detail.

Turbulent Boundary Layer Models: Theory and Applications 211

1

*L L qL <sup>u</sup> <sup>ρ</sup> C qL C C g*

where 0.75 *Cq* , 0.1125 *Cr* , 0.35 *Cl* , 0.075 *Cs* and 0.80 *Cz* . As can be easily seen, an

 

 

*<sup>i</sup> <sup>t</sup> <sup>q</sup> z i*

*x x qx ρ q*

*<sup>i</sup>* PRODUCTION DISSIPATION DIFFUSION BUOYANCY

(10)

2

  2

*l ij*

*<sup>L</sup> <sup>u</sup> C uu q x*

<sup>2</sup> <sup>0</sup>

*j*

, as suggested by Lewellen (1977), and

being the wave length;

*' '*

(9)

and *C qs* , respectively,

An equation for the turbulent length scale (or macroscale of the eddies), *L* , is written:

2

 

*L LL <sup>u</sup> u C uu Cq tx x <sup>q</sup>*

*i*

in suspension, and considering: 1. A sinusoidal wave <sup>ˆ</sup>*U , T, L w w* .

thickness ( *kN* z

( <sup>ˆ</sup>*Uw gh* ).

of the velocity correlations.

*L L u t x*

the diffusion terms by

 

Modelling the production and dissipation terms by 2 ' ' *<sup>i</sup>*

**3.4 Simplified turbulent boundary layer models** 

2. The following boundary layer approximations: - Small boundary layer thickness *z 2*

> ).

3. Small wave amplitude and Stokes hypothesis, which assumes that:

without stratification, the following approximations (11) result:

<sup>1</sup> ' ' *u P*

 

 

0

 

*t xz*

2 *t q*

 

*ii i <sup>L</sup> qL C qL C x x qx*

adding the buoyancy term, the approximation (9) above is newly obtained.

*' ' i i l ij s i j*

2

*ii i*

equation for the turbulent length scale *L* is, like all other approximations, of the form:

1

 

In summary, equations (2), along with equations (6), (7), (8) and (9) for turbulence closure, constitute a complete 3D turbulent boundary layer model with sediment in suspension.

Proceeding with a non-dimensional analysis of the mean flow equations, without sediment



 , 

4. Local equilibrium turbulence, along with the turbulent kinetic energy is equivalent to the viscous dissipation. Assuming local equilibrium there is no time evolution or spatial diffusion of the correlations, and the Reynolds stress equation *' ' u ui <sup>j</sup>* can be reduced. In summary, assuming these hypotheses we can: *i*) consider a horizontal flow (*u*, *v*, *w* = 0); *ii*) neglect the convective and horizontal diffusion transport, and *iii*) simplify the turbulent transport equations, cancelling the remaining time variation terms and the diffusion terms

Considering the above hypotheses in the pure hydrodynamic Reynolds equations (2-b),

; <sup>0</sup>

 <sup>1</sup> ' ' *v P*

 

 

*t yz*

 

*v w*

(11)

*u w*


$$\begin{split} \frac{\mathcal{\overline{\mathcal{C}}}}{\mathcal{\overline{\mathcal{C}}}} \overline{\left(\overline{\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{u}\_{j}^{\cdot}}\right)} + \overline{\boldsymbol{u}\_{k}} \frac{\mathcal{\overline{\mathcal{C}}}}{\mathcal{\overline{\mathcal{C}}}\mathbf{x}\_{k}} \overline{\left(\overline{\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{u}\_{j}^{\cdot}}\right)} &= -\overline{\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{u}\_{k}} \frac{\mathcal{\mathcal{C}}}{\mathcal{\overline{\mathcal{C}}}\mathbf{x}\_{j}} - \overline{\boldsymbol{u}\_{j}^{\cdot}\boldsymbol{u}\_{k}} \frac{\mathcal{\mathcal{C}}}{\mathcal{\overline{\mathcal{C}}}\mathbf{x}\_{i}} + \frac{1}{\rho\_{\boldsymbol{o}}} \bigg(\overline{\boldsymbol{g}\_{i}\boldsymbol{u}\_{j}^{\cdot}\boldsymbol{\rho}} + \overline{\boldsymbol{g}\_{j}\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{\rho}}\Big) \\ &+ \mathcal{C}\_{t} \frac{\mathcal{\overline{\mathcal{C}}}{\mathcal{\overline{\mathcal{C}}}\mathbf{x}\_{k}} \bigg[\overline{\boldsymbol{\eta}}\mathbf{L} \frac{\mathcal{\overline{\mathcal{C}}}{\mathcal{\overline{\mathcal{C}}}\mathbf{x}\_{k}} \bigg(\overline{\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{u}\_{j}^{\cdot}}\Big)\bigg] - \mathcal{C}\_{p} \frac{\sqrt{\overline{\boldsymbol{q}}^{2}}}{L} \bigg(\overline{\boldsymbol{u}\_{i}^{\cdot}\boldsymbol{u}\_{j}^{\cdot}} - \frac{1}{3}\delta\_{i,j}\overline{\boldsymbol{q}^{\cdot}}\Big) - \mathcal{C}\_{v} \frac{1}{L} \delta\_{i,j} \bigg(\overline{\boldsymbol{q}^{\cdot}}\Big)^{3/2} \end{split} \tag{6}$$

where 2 2 *' ' <sup>i</sup> <sup>j</sup> q = K uu* , and the constants have the following values: 0.30 *Ct* , 1.0 *Cp* and 1 12 *Cv* . By analogy to equation (6), a density-velocity correlations tensor *' ' ui ρ* is obtained:

$$\begin{split} \frac{\partial}{\partial t} \overline{\left(\overline{\dot{u}\_{i} \rho} \right)} + \overline{\overline{u}\_{j}} \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\left(\overline{\dot{u}\_{i} \rho} \right)} &= -\overline{\dot{u}\_{i} \dot{\overline{u}\_{j}}} \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\frac{\partial}{\partial \mathbf{x}\_{j}}} - \overline{\overline{\dot{u}\_{j} \rho}} \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\mathbf{c}} \frac{\overline{\mathbf{u}\_{i}}}{\partial \mathbf{x}\_{j}} \\ &+ \mathbf{g}\_{i} \frac{\overline{\rho^{2}}}{\rho\_{0}} + \mathbf{C}\_{t} \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \overline{q} L \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\left(\overline{\dot{u}\_{i} \rho} \right)} \right] - \mathbf{C}\_{q} \frac{\overline{q}}{L} \overline{\overline{\dot{u}\_{i} \rho}} \overline{\overline{\dot{\rho}}}. \end{split} \tag{7}$$

with the quadratic term *'*<sup>2</sup> *ρ* calculated through (8),

$$\begin{split} \frac{\partial}{\partial t} \overline{\left(\rho^{\square}\right)} + \overline{u}\_{j} \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\left(\rho^{\square}\right)} &= -2 \overline{u\_{j} \rho} \frac{\partial}{\partial \mathbf{x}\_{j}} \overline{\rho} \frac{\partial}{\partial \mathbf{x}\_{j}} \\ &+ \mathbf{C}\_{t} \frac{\partial}{\partial \mathbf{x}\_{j}} \bigg[ \overline{q} L \frac{\partial}{\partial \mathbf{x}\_{j}} \bigg(\overline{\rho^{\square}}\bigg) \bigg] - \mathbf{C}\_{r} \frac{\overline{q}}{L} \bigg(\overline{\rho^{\square}}\bigg) \end{split} \tag{8}$$

210 Advanced Fluid Dynamics

*k*

*t i j k k C qL u u x x*

 

 

 

 

> 

 

, where *i,j δ* is the Kronecker symbol and *Cp* is a constant;

*' 'j i*

*k k u u*

3

0

1

2 3 2

1 1

*' ' ' '*

*ρ u*

*' '*

 

2 2

2 2

(6)

(7)

(8)

*x x*

*x* 

Reynolds stresses are conservatively transmitted from one region of the flow to another;

the root-mean-square value of the total velocity fluctuation, *L* is the macroscale of the

 *' ' i j j i <sup>p</sup> <sup>p</sup> u u ρ x x* 

flow the turbulent energy produced, are commonly approximated by

*l*

*ν*

of the mean flow energy by viscous effects, are jointly modelled through

3 2 <sup>1</sup> <sup>2</sup> *Cv i, j <sup>δ</sup> <sup>q</sup> <sup>L</sup>* , where *Cv* is a constant. Inserting these approximations in (5), the

*u u*

*<sup>i</sup> <sup>j</sup> q = K uu* , and the constants have the following values: 0.30 *Ct* , 1.0 *Cp* and

*j jj*

*i t i qi j j*

 

*ρ xx L*

 

2

*j j*

 

*t r j j*

 

 

0 1

and 2

*uu u uu u u u u gu <sup>ρ</sup> g u <sup>ρ</sup> t x xx <sup>ρ</sup>*

*q C qL u u C u u <sup>δ</sup> q C <sup>δ</sup> <sup>q</sup> xx L <sup>L</sup>*

*j ' ' '' ' ' ' ' i ' ' '' ij k ij ik jk ij ji k kk*

> *' ' ' ' t i j p i j i, j v i, j*

1 12 *Cv* . By analogy to equation (6), a density-velocity correlations tensor *' ' ui ρ* is

*' ' '' '' '' i i j i ij j*

*<sup>u</sup> <sup>ρ</sup> u u <sup>ρ</sup> uu u <sup>ρ</sup> t x xx*

*' '' ' j j*

*<sup>ρ</sup> <sup>ρ</sup> <sup>u</sup> <sup>ρ</sup> <sup>u</sup> <sup>ρ</sup> tx x*

<sup>2</sup>

*<sup>ρ</sup> q g C qL u <sup>ρ</sup> C u <sup>ρ</sup>*

*q C qL <sup>ρ</sup> <sup>C</sup> <sup>ρ</sup> xx L*

 *i j k*

express the process as the

, which represent the destruction

(Lewellen, 1977), where *q* is

, which redistribute to the mean

*uuu*


eddies, and *Ct* is a constant;


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


*' ' p i j i,j <sup>q</sup> C uu <sup>δ</sup> <sup>q</sup> <sup>L</sup>* 

where 2 2 *' '*

obtained:

they are usually obtained through  *' '*

2

following equation (6) for the *' ' u ui <sup>j</sup>* correlations is obtained:

 

0

2 2

*'*

 

 

with the quadratic term *'*<sup>2</sup> *ρ* calculated through (8),

*k k*

 

 

*' ' l i j k ν u u x* 

An equation for the turbulent length scale (or macroscale of the eddies), *L* , is written:

$$\begin{split} \frac{\partial \mathcal{L}}{\partial \boldsymbol{t}} + \overline{\boldsymbol{u}}\_{i} \frac{\partial \mathcal{L}}{\partial \mathbf{x}\_{i}} &= \mathcal{C}\_{l} \frac{\mathcal{L}}{q^{2}} \overline{\boldsymbol{u}\_{i}^{\top} \boldsymbol{u}\_{j}^{\top}} \frac{\partial \overline{\boldsymbol{u}\_{i}}}{\partial \mathbf{x}\_{j}} + \mathcal{C}\_{s} \overline{\boldsymbol{q}} \\ &+ \mathcal{C}\_{t} \frac{\mathcal{C}}{\partial \mathbf{x}\_{i}} \bigg[ \overline{\boldsymbol{q}} \, \mathrm{L} \frac{\mathcal{C} \mathrm{L}}{\partial \mathbf{x}\_{i}} \bigg] - \mathcal{C}\_{q} \frac{\mathbf{1}}{2 \overline{\boldsymbol{q}}} \bigg( \frac{\mathcal{C} \, \overline{\boldsymbol{q}} \mathrm{L}}{\partial \mathbf{x}\_{i}} \bigg)^{2} + \mathcal{C}\_{z} \frac{\mathcal{L}}{\overline{\boldsymbol{q}}^{2}} \mathrm{g}\_{i} \overline{\frac{\boldsymbol{u}\_{i}^{\top} \boldsymbol{p}}{\rho\_{0}}} \end{split} \tag{9}$$

where 0.75 *Cq* , 0.1125 *Cr* , 0.35 *Cl* , 0.075 *Cs* and 0.80 *Cz* . As can be easily seen, an equation for the turbulent length scale *L* is, like all other approximations, of the form:

$$\frac{\partial \mathbf{L}}{\partial \mathbf{t}} + \overline{\mathbf{u}}\_{i} \frac{\partial \mathbf{L}}{\partial \mathbf{x}\_{i}} = \text{PRODUCTION} \; + \text{ DISIPANION} + \text{DIFFUSION} \; + \text{ BUOYANCY} \; \text{(10)}$$

Modelling the production and dissipation terms by 2 ' ' *<sup>i</sup> l ij j <sup>L</sup> <sup>u</sup> C uu q x* and *C qs* , respectively,

the diffusion terms by 2 1 2 *t q ii i <sup>L</sup> qL C qL C x x qx* , as suggested by Lewellen (1977), and

adding the buoyancy term, the approximation (9) above is newly obtained.

In summary, equations (2), along with equations (6), (7), (8) and (9) for turbulence closure, constitute a complete 3D turbulent boundary layer model with sediment in suspension.

#### **3.4 Simplified turbulent boundary layer models**

Proceeding with a non-dimensional analysis of the mean flow equations, without sediment in suspension, and considering:

	- Small boundary layer thickness *z 2* , being the wave length;
	- Nikuradse equivalent bottom rugosity much inferior to the boundary layer thickness ( *kN* z ).
	- The maximum wave velocity amplitude is much inferior to the celerity ( <sup>ˆ</sup>*Uw gh* ).

In summary, assuming these hypotheses we can: *i*) consider a horizontal flow (*u*, *v*, *w* = 0); *ii*) neglect the convective and horizontal diffusion transport, and *iii*) simplify the turbulent transport equations, cancelling the remaining time variation terms and the diffusion terms of the velocity correlations.

Considering the above hypotheses in the pure hydrodynamic Reynolds equations (2-b), without stratification, the following approximations (11) result:

$$\frac{\partial \mathcal{D}u}{\partial t} = -\frac{1}{\rho\_\flat} \frac{\partial P}{\partial \mathbf{x}} - \frac{\mathcal{D}}{\partial z} (\overline{u'w'}) \\ \vdots \\ \frac{\partial \, v}{\partial t} = -\frac{1}{\rho\_\flat} \frac{\partial P}{\partial y} - \frac{\mathcal{C}}{\partial z} (\overline{v'w'}) \tag{11}$$

Turbulent Boundary Layer Models: Theory and Applications 213

<sup>t</sup> <sup>2</sup> 3 = 3 2 4 *v v*

*uv uv q K CL <sup>L</sup>*

Considering now a horizontal flow along *x*-direction (*u*, *v* = 0, *w* = 0) with sediment in suspension, so with the buoyancy terms, and local equilibrium turbulence, the system (12) is

> 0 2 2 3

*<sup>w</sup> g qqq w Cw C t L L*

' ' 0 ' '' ''

*u w <sup>u</sup> g q <sup>w</sup> u C uw tz L*

<sup>2</sup> 0 '' '

0

 

*u u <sup>q</sup> w uw C u t zz L*

 <sup>2</sup> 0 2 2

' ' 0 '' '' ''

' 0 2'' '

*t z L*

;

. As shown before and from (3) '' *<sup>t</sup>*

*. Ω . Ω*

1 19.778 3 1 16 444 3 *t t qL*

1 16 444 1 16 444 2 1 19 778 1 4 1 19 778 1 4

1 4 1

*. Ω qL . Ω KL*

(20)

 

*u w*

*u*

and '' *<sup>t</sup>*

*v w*

*v*

*z* 

(17)

(18)

*t <sup>ρ</sup> <sup>w</sup> ρ γ*

*z* 

;

it is

*z* 

*K L*

*C qL C K L* (16)

22 22

2

*p v*

3

*q*

*q*

*r*

 

1 19.778 3 z

*u*

*z* 

and ' *'*

(19)

*zz zz*

*p*

' ' '

we obtain (19):

<sup>1</sup> ' '

*u w*

 

*. Ω*

(21)

*qL <sup>w</sup>*

*z L*

*<sup>q</sup> w C*

*g q C w*

Comparing the last two equations of (15) with '' *<sup>t</sup>*

In addition, it can be seen from the first three equations of (15) that:

 

2 2 <sup>2</sup> 2 24 2 *<sup>v</sup>*

 

*. Ω qL u*

 

**3.5 1DV turbulent boundary layer models** 

'

*w*

*t*

2

2

Solving this equation system for *u w*' ' and *w*' '

1 16 444 ' '

2 2

0

*g L z q* 

*t*

4 3

' ' 0 '

*w*

 

*u w . <sup>Ω</sup> <sup>z</sup>*

1 19 778 1 4

comparing with the expressions (19) above we can write (20) and (21):

 

written in the following form (18):

 

 

 

where

clear that:

On the other hand, under the same assumptions, the *Reynolds stress equation* ' *' u ui <sup>j</sup>* (6) can be written explicitly:

 2 2 2 2 2 3 2 2 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2' ' ' <sup>3</sup> ' 2' ' *t p t p t pv t uw u u w <sup>q</sup> w C qL C u w t zz z L vw v v w <sup>q</sup> w C qL C v w t zz z L uu u qq q u w C qL C u C t zzz L L v v v w C qL t zz* 2 2 3 2 2 2 2 3 2 ' ' 3 ' ' ' <sup>3</sup> *p v t pv <sup>v</sup> qq q Cv C zL L w w qqq C qL C w C <sup>t</sup> zz L L* (12)

Adding the last three equations we get (13) for <sup>2</sup> *q* :

$$\frac{\partial^2 \boldsymbol{q}^2}{\partial t \boldsymbol{t}} = -2\overline{\boldsymbol{u}^\prime \boldsymbol{w}^\prime} \frac{\partial^2 \boldsymbol{u}}{\partial \boldsymbol{z}} - 2\overline{\boldsymbol{v}^\prime \boldsymbol{w}^\prime} \frac{\partial}{\partial \boldsymbol{z}} \boldsymbol{v} + \mathbf{C}\_t \frac{\partial}{\partial \boldsymbol{z}} \bigg( \overline{\boldsymbol{q}} \boldsymbol{L} \frac{\partial \boldsymbol{q}^2}{\partial \boldsymbol{z}} \bigg) - \mathfrak{K}\_v \frac{\boldsymbol{q}^3}{\boldsymbol{L}} \tag{13}$$

Taking now into account local equilibrium turbulence (Sheng, 1984), which can be assumed when the scale *L/q* is much smaller than the time scale of the mean flow and when the turbulent quantities have a small variation on the macroscale of the eddies *L*. In addition, neglecting both variations in time and diffusive transport terms, from equations system (12) the following equations (14) are obtained:

$$\begin{aligned} \overline{w^2} \frac{\partial u}{\partial z} + \frac{q}{L} \overline{u^\* w^\*} &= 0 \end{aligned} \begin{aligned} \overline{w^2} \frac{\partial \overline{v}}{\partial \overline{z}} + \frac{q}{L} \overline{v^\* w^\*} &= 0 \end{aligned}$$

$$2 \overline{u^\* w^\*} \frac{\partial \overline{u}}{\partial \overline{z}} + \frac{q}{L} \left( \overline{u^{\*2}} - \frac{q^2}{3} \right) + C\_v \frac{q^3}{L} = 0$$

$$2 \overline{v^\* w^\*} \frac{\partial \overline{v}}{\partial \overline{z}} + \frac{q}{L} \left( \overline{v^{\*2}} - \frac{q^2}{3} \right) + C\_v \frac{q^3}{L} &= 0 \end{aligned} \tag{14}$$

$$\frac{q}{L} \left( \overline{w^{\*2}} - \frac{q^2}{3} \right) + C\_v \frac{q^3}{L} \tag{15}$$

This system of equations allows us to obtain (15):

$$\begin{split} \overline{u^{\prime 2}} &= 6 \, \mathcal{C}\_{v} \mathcal{L}^{2} \left( \frac{\mathcal{C}u}{\mathcal{C}z} \right)^{2} + 3 \, \mathcal{C}\_{v} \, q^{2} \quad ; \; \overline{v^{\prime 2}} = 6 \, \mathcal{C}\_{v} \, \mathcal{L}^{2} \left( \frac{\mathcal{C}v}{\mathcal{C}z} \right)^{2} + 3 \, \mathcal{C}\_{v} \, q^{2} ; \\ \overline{\mathcal{w}^{\prime 2}} &= 3 \, \mathcal{C}\_{v} \, q^{2} \quad ; \; -\overline{u^{\prime}w^{\prime}} = 3 \, \mathcal{C}\_{v} \, q \, \mathcal{L} \frac{\mathcal{C}u}{\mathcal{C}z} \quad ; \; -\overline{v^{\prime}w^{\prime}} = 3 \, \mathcal{C}\_{v} \, q \, \mathcal{L} \frac{\mathcal{C}v}{\mathcal{C}z} \end{split} \tag{15}$$

212 Advanced Fluid Dynamics

On the other hand, under the same assumptions, the *Reynolds stress equation* ' *' u ui <sup>j</sup>* (6) can be

*t p*

 

 

' ' ' ' ' ' '

*uw u u w <sup>q</sup> w C qL C u w t zz z L*

 

 

' ' ' ' ' ' '

*vw v v w <sup>q</sup> w C qL C v w t zz z L*

 

 

2 2

2

' 3 ; ' ' 3 ; ' ' 3

2

 

*u v q q w uw w vw z L z L <sup>u</sup> qq q uw u C z L L*

2' ' ' 0

2

*<sup>v</sup> qq q vw v C z L L*

 ' 0 3

*qqq w C L L*

 2' ' ' 0 3

' ' 2' ' ' <sup>3</sup>

' ' ' <sup>3</sup>

Taking now into account local equilibrium turbulence (Sheng, 1984), which can be assumed when the scale *L/q* is much smaller than the time scale of the mean flow and when the turbulent quantities have a small variation on the macroscale of the eddies *L*. In addition, neglecting both variations in time and diffusive transport terms, from equations system (12)

' ' ' 0 ; ' ' ' 0

3

*t*

 

This system of equations allows us to obtain (15):

2 2

*v v v w C qL t zz*

 

> 

 

> 

 

 

 

2 2 2 3 2

*uu u qq q u w C qL C u C t zzz L L*

2 2 2 3

*w w qqq C qL C w C <sup>t</sup> zz L L*

 

2 2 3 2' ' 2' ' 3 *t v q q u v <sup>q</sup> u w v w C qL C t z zzz L*

2 3

 

 

*v*

*v*

*v*

*z z*

 

 

 

 

2 3

2 3

2 2 2 2 22 2 2

*v vv v*

' 6 3 ; ' 6 3 ;

*u v u CL <sup>C</sup> <sup>q</sup> v CL C <sup>q</sup> z z u v w C q uw C qL vw C qL*

*vv v*

 

'

*t pv*

'

 

*t pv*

2 2 3 2

 

*<sup>v</sup> qq q Cv C zL L*

2

3

(12)

(13)

(14)

(15)

*p v*

*t p*

 

written explicitly:

2

2

2' '

Adding the last three equations we get (13) for <sup>2</sup> *q* :

the following equations (14) are obtained:

2

'

Comparing the last two equations of (15) with '' *<sup>t</sup> u u w z* and '' *<sup>t</sup> v v w z* it is clear that:

$$\text{Cov}\_v = \text{3 C}\_v \text{ } qL = \text{3 C}\_v \sqrt{2K} \text{ } L = \frac{\sqrt{2K} \text{ } L}{4} \tag{16}$$

In addition, it can be seen from the first three equations of (15) that:

$$q^2 = 2K = 24 \text{ C}\_v \, L^2 \left[ \left( \frac{\mathcal{O} \, u}{\mathcal{O} \, z} \right)^2 + \left( \frac{\mathcal{O} \, v}{\mathcal{O} \, z} \right)^2 \right] = 2L^2 \left[ \left( \frac{\mathcal{O} \, u}{\mathcal{O} \, z} \right)^2 + \left( \frac{\mathcal{O} \, v}{\mathcal{O} \, z} \right)^2 \right] \tag{17}$$

### **3.5 1DV turbulent boundary layer models**

Considering now a horizontal flow along *x*-direction (*u*, *v* = 0, *w* = 0) with sediment in suspension, so with the buoyancy terms, and local equilibrium turbulence, the system (12) is written in the following form (18):

$$\begin{aligned} \frac{\partial \,\,\,u^{\prime}\,w^{\prime} = 0 = -\overline{w^{\prime}}^{2} \frac{\partial \,\,\,u}{\partial z} & -\frac{\mathcal{g}}{\rho\_{0}} \overline{u^{\prime} \,\,\rho^{\prime}} - C\_{p} \frac{q}{L} \overline{u^{\prime} \,\,w^{\prime}},\\ \frac{\partial \,\,\overline{w^{\prime}}^{2}}{\partial t} = 0 & -\frac{2\underline{g}}{\rho\_{0}} \overline{w^{\prime} \,\rho^{\prime}} - C\_{p} \frac{q}{L} \overline{\left(\overline{w^{\prime}}^{2} - \frac{q^{2}}{3}\right)} - C\_{v} \frac{q^{3}}{L} \\ \frac{\partial \,\,\overline{u^{\prime} \,\rho^{\prime}}}{\partial t} = 0 & -\overline{w^{\prime}} \overline{\rho^{\prime}} \frac{\partial}{\partial z} - \overline{u^{\prime} \,w^{\prime}} \frac{\partial}{\partial z} \rho & -C\_{q} \frac{q}{L} \overline{u^{\prime} \,\rho^{\prime}},\\ \frac{\partial \,\,\overline{w^{\prime}}^{\prime} \overline{\rho^{\prime}}^{\prime}}{\partial t} = 0 & -\overline{w^{\prime}} \frac{\partial}{\partial z} \overline{\rho} & -\frac{\mathcal{g}}{\rho\_{0}} \overline{\rho^{\prime 2}} & -C\_{q} \frac{q}{L} \overline{w^{\prime}} \overline{\rho^{\prime}} \\ \frac{\partial \,\,\overline{\rho^{\prime}}^{\prime 2}}{\partial t} = 0 & -2\overline{w^{\prime}} \overline{\rho^{\prime}} \frac{\partial}{\partial z} \rho & -C\_{r} \frac{q}{L} \overline{\rho^{\prime 2}} \end{aligned} \tag{18}$$

Solving this equation system for *u w*' ' and *w*' ' we obtain (19):

$$\overline{u'w'} = -\frac{\begin{pmatrix} 1 - 16.444 \ \Omega \end{pmatrix}}{\begin{pmatrix} 1 - 19.778 \ \Omega \end{pmatrix} \begin{pmatrix} 1 - \Omega \end{pmatrix}} \frac{qL}{4} \frac{\partial \, u}{\partial z}; \; \overline{w' \rho'} = -\frac{1}{\begin{pmatrix} 1 - 19.778 \ \Omega \end{pmatrix}} \frac{qL}{3} \frac{\partial \, \rho}{\partial z} \tag{19}$$

where 2 2 0 4 3 *g L z q* . As shown before and from (3) '' *<sup>t</sup> u u w z* and ' *' t <sup>ρ</sup> <sup>w</sup> ρ γ z* ; comparing with the expressions (19) above we can write (20) and (21):

$$\mathbf{v}\_t = \frac{\begin{pmatrix} 1 - 16.444 \,\Omega \end{pmatrix}}{\begin{pmatrix} 1 - 19.778 \,\Omega \end{pmatrix} \begin{pmatrix} 1 - \Omega \end{pmatrix}} \frac{qL}{4} = \frac{\begin{pmatrix} 1 - 16.444 \,\Omega \end{pmatrix}}{\begin{pmatrix} 1 - 19.778 \,\Omega \end{pmatrix} \begin{pmatrix} 1 - \Omega \end{pmatrix}} \frac{\sqrt{2K}L}{4} \tag{20}$$

$$\gamma\_t = \frac{1}{\left(1 - 19.778 \,\Omega\right)} \frac{qL}{3} = \frac{\left(1 - \Omega\right)}{\left(1 - 16.444 \,\Omega\right)} \frac{4}{3} \nu\_t \tag{21}$$

Turbulent Boundary Layer Models: Theory and Applications 215

A number of empirical equations for the length scale *L* could be found in the literature; some

, where 26 *A* , *wz* is the distance to the wall and *z zu w wT l*

is the Richardson number and *Lo* is the length scale *L*

 

 

(26)

 (27)

, , ; , , *u u Uv v V d d zt zt t zt zt t* (28)

 

   

(29)

 

 

 

, where 0.4 *k* and *z*

 

 

 

 

*m m* ;

1 , with 10, 14 and 0.5, 1.5 *<sup>n</sup>*

,

(25)

is the boundary layer

being the boundary layer thickness.

The influence of a stable stratification on *L* can be taken into account through (25),

*<sup>L</sup> R n*

 

*u P uu v P vv l l t x z zz t y z zz*

Stable stratification effects on *lm* could be taken into account through the relation

*m mo <sup>i</sup> ll R* , where *<sup>m</sup>*<sup>0</sup> *<sup>l</sup>* is the mixing length *lm* value without stratification, and

1 1 and *PU PV x t y t*

where *U* and *V* are the velocity components outside of the boundary layer. Defining the

 2 2 ; *d d dd d <sup>d</sup> m m u uu v vv l l t z zz t z zz*

These equations are non-linear and no analytical solutions are available, so they have to be

 

> 

 

 

thickness, equations (26) for the *u* and *v* variables in the boundary layer are obtained:

1 1 2 2

0 0

 

0 0

*Ri* is the Richardson number, as defined above. We now assume in (26):

 

and substituting in (26) the following equations (29) are obtained,

   

> 

 

 

examples are (with *k* = 0.4): <sup>4</sup> *L k cz <sup>l</sup>* , where 0.08 *<sup>l</sup> c* .

*z*

 , *z*

*uT* being the friction velocity.

*o*

*<sup>g</sup> u v <sup>R</sup>*

value without stratification.

0.5 1 10 <sup>2</sup>

*zz z*

**3.5.3 Zero-equation boundary layer model**  Defining the mixing length as *l kz z z <sup>m</sup>* 1

> 

 

deficit velocity components , *d d u v* as (28),

solved numerically, as will be shown later.

 

 

*L*

2

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

*oo o o <sup>z</sup> L k K K dz z K K K z t* .

*i*

2 2

*L kz z z* 1

<sup>1</sup> *wz A L kz e <sup>w</sup>*

where

*i*

### **3.5.1 Two-equation K-L 1DV boundary layer model**

Taking into account the assumptions stated before, a complete set of governing equations (22) for the two-equation *K L* model is written (Tran-Thu & Temperville, 1994):

 0 1 *t uP u t xz z* ; 0 1 *t vP v t yz z* 0 2 2 rate of dissipation diffusion buoyancy change production 2 rate of change <sup>2</sup> 0.30 2 4 0.35 2 *t t t K uv K <sup>K</sup> <sup>g</sup> <sup>K</sup> KL t zz L z z z L uv t Kz* 0 2 dissipation production 2 diffusion buoyancy 0.075 2 0.375 0.30 2 2 0.80 2 2 *<sup>t</sup> s t L K z L L <sup>g</sup> KL KL z z z Kz K C C w C t z zz* (22)

where *u* and *v* are horizontal components of flow velocity in the boundary layer; *C* is the volumetric concentration; *ws* is the sediment settling velocity; *K* is the turbulent kinetic energy, and *L* is the length scale of the large vortices.

The turbulent viscosity *<sup>t</sup>* and the turbulent diffusivity *<sup>t</sup>* are given by equations (20) and (21), respectively. The hydrodynamic equations and the concentration equation are coupled through the equation (23) for the density:

$$
\rho = \rho\_o + (\rho\_\* - \rho\_o)\mathbf{C} \tag{23}
$$

where 0 *ρ* and *<sup>s</sup> ρ* are the densities of the fluid and sediment, respectively.

#### **3.5.2 One-equation K-L 1DV boundary layer model**

With *L f kzK* (,, ) , a complete one-equation *K L* turbulence closure model is simply written:

$$\begin{aligned} \frac{\partial^{\square}u}{\partial t} &= -\frac{1}{\rho\_{\text{o}}} \frac{\partial P}{\partial \mathbf{x}} + \frac{\partial}{\partial z} \left(\nu\_{t} \frac{\partial \mathbf{u}}{\partial z}\right) \quad ; \quad \frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}u}{\partial \mathbf{t}} = -\frac{1}{\rho\_{\text{o}}} \frac{\partial P}{\partial \mathbf{y}} + \frac{\partial}{\partial z} \left(\nu\_{t} \frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}{\partial z}}{\partial z}\right) \\ \frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}K}{\partial t} &= \nu\_{t} \left[ \left(\frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}u}{\partial z}\right)^{2} + \left(\frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}v}{\partial z}\right)^{2} \right] - \frac{\sqrt{2K}}{4L}K + 0.30 \frac{\partial}{\partial z} \left(\sqrt{2KL} \frac{\partial K}{\partial z}\right) + \frac{\mathcal{g}}{\rho\_{\text{o}}} \nu\_{t} \frac{\partial \overline{\rho}}{\partial z} \\ L &= f(k, z, K) \quad ; \quad \frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}u}{\partial t} = \frac{\partial \left(w\_{s} \mathbb{C}\right)}{\partial z} + \frac{\partial}{\partial z} \left(\nu\_{t} \frac{\partial \operatorname{\boldsymbol{\mathcal{E}}}{\partial z}}{\partial z}\right) \end{aligned} \tag{24}$$

where (20), (21) and (23) apply.

A number of empirical equations for the length scale *L* could be found in the literature; some examples are (with *k* = 0.4):

<sup>4</sup> *L k cz <sup>l</sup>* , where 0.08 *<sup>l</sup> c* .

214 Advanced Fluid Dynamics

Taking into account the assumptions stated before, a complete set of governing equations

;

rate of dissipation diffusion buoyancy change production

2

*z*

4

*K uv K <sup>K</sup> <sup>g</sup> <sup>K</sup> KL t zz L z z z*

*t t*

0 1

 

*L L <sup>g</sup> KL KL z z z Kz K*

*t*

 

diffusion buoyancy

*ρ ρ ρρ* 0 0 *<sup>s</sup> C* (23)

2

2 2 *<sup>t</sup>*

dissipation

0.075 2

*L K*

<sup>2</sup> 0.30 2

*vP v t yz z*

 

*t*

 

0

(22)

0

are given by equations (20) and

0

 

   

(24)

(22) for the two-equation *K L* model is written (Tran-Thu & Temperville, 1994):

*t*

 

*C C w C t z zz*

where *u* and *v* are horizontal components of flow velocity in the boundary layer; *C* is the volumetric concentration; *ws* is the sediment settling velocity; *K* is the turbulent kinetic

(21), respectively. The hydrodynamic equations and the concentration equation are coupled

With *L f kzK* (,, ) , a complete one-equation *K L* turbulence closure model is simply

<sup>2</sup> 0.30 2

 

   

> 

> >

 

*t t*

*t*

*t t*

*s*

*<sup>t</sup>* and the turbulent diffusivity *<sup>t</sup>*

where 0 *ρ* and *<sup>s</sup> ρ* are the densities of the fluid and sediment, respectively.

*s*

4

*K uv K <sup>K</sup> <sup>g</sup> <sup>K</sup> KL t zz L z z z*

*u P uv P v t xz z t yz z*

 

 

> 

 

0 0

 

 

1 1 ;

2 2

 

> 

 

*C C w C L f kzK t z zz*

0.375 0.30 2 2 0.80

**3.5.1 Two-equation K-L 1DV boundary layer model** 

0 1

*t*

*L uv*

 

0.35 2

*t Kz*

energy, and *L* is the length scale of the large vortices.

**3.5.2 One-equation K-L 1DV boundary layer model** 

( , , ) ;

 

through the equation (23) for the density:

*uP u t xz z*

 

2 2

2

production

rate of change

The turbulent viscosity

where (20), (21) and (23) apply.

written:

  *L kz z z* 1 , *z*being the boundary layer thickness.

$$\bullet \qquad L = k \sqrt{K} \left\{ \int\_{z\_o}^{z} K^{-1/2} dz + z\_o K\_o^{-1/2} \right\} \; \; K\_o = K \left( z\_o, t \right) \; .$$

	- *uT* being the friction velocity.

The influence of a stable stratification on *L* can be taken into account through (25),

$$\left(\frac{L}{L\_o}\right)^2 = \left(1 + \beta R\_i\right)^n,\text{ with }\beta \approx \begin{pmatrix} 10, \ 14 \end{pmatrix}\text{ and }n \approx \begin{pmatrix} -0.5, \ -1.5 \end{pmatrix}\tag{25}$$

where 2 2 *i <sup>g</sup> u v <sup>R</sup> zz z* is the Richardson number and *Lo* is the length scale *L*

value without stratification.

#### **3.5.3 Zero-equation boundary layer model**

Defining the mixing length as *l kz z z <sup>m</sup>* 1 , where 0.4 *k* and *z* is the boundary layer thickness, equations (26) for the *u* and *v* variables in the boundary layer are obtained:

$$\frac{\partial \mathcal{D} \mathbf{u}}{\partial \mathbf{f}} = -\frac{1}{\rho\_o} \frac{\partial P}{\partial \mathbf{x}} + \frac{\partial}{\partial z} \left( l\_m^2 \left| \frac{\partial \mathbf{u}}{\partial z} \right| \frac{\partial \mathbf{u}}{\partial z} \right) \; ; \; \frac{\partial \mathbf{v}}{\partial \mathbf{f}} = -\frac{1}{\rho\_o} \frac{\partial P}{\partial \mathbf{y}} + \frac{\partial}{\partial z} \left( l\_m^2 \left| \frac{\partial \mathbf{v}}{\partial z} \right| \frac{\partial \mathbf{v}}{\partial z} \right) \tag{26}$$

Stable stratification effects on *lm* could be taken into account through the relation 0.5 1 10 <sup>2</sup> *m mo <sup>i</sup> ll R* , where *<sup>m</sup>*<sup>0</sup> *<sup>l</sup>* is the mixing length *<sup>l</sup> <sup>m</sup>* value without stratification, and *Ri* is the Richardson number, as defined above. We now assume in (26):

$$-\frac{1}{\rho\_{\text{o}}} \frac{\partial P}{\partial \mathbf{x}} = \frac{\partial \mathcal{U}}{\partial t} \quad \text{and} \quad -\frac{1}{\rho\_{\text{o}}} \frac{\partial P}{\partial \mathbf{y}} = \frac{\partial V}{\partial \mathbf{t}} \tag{27}$$

where *U* and *V* are the velocity components outside of the boundary layer. Defining the deficit velocity components , *d d u v* as (28),

$$u\_d(z,t) = u(z,t) - \mathcal{U}(t) \ \ \vdots \ v\_d(z,t) = v(z,t) - V(t) \tag{28}$$

and substituting in (26) the following equations (29) are obtained,

$$\frac{\partial \mathcal{D} \mathbf{u}\_d}{\partial t} = \frac{\partial}{\partial z} \left( l\_m^2 \left| \frac{\partial \operatorname{\boldsymbol{u}}\_d}{\partial z} \right| \frac{\partial \operatorname{\boldsymbol{u}}\_d}{\partial z} \right) \; ; \; \frac{\partial \operatorname{\boldsymbol{v}}\_d}{\partial \mathbf{t}} = \frac{\partial}{\partial z} \left( l\_m^2 \left| \frac{\partial \operatorname{\boldsymbol{v}}\_d}{\partial z} \right| \frac{\partial \operatorname{\boldsymbol{v}}\_d}{\partial z} \right) \tag{29}$$

These equations are non-linear and no analytical solutions are available, so they have to be solved numerically, as will be shown later.

Turbulent Boundary Layer Models: Theory and Applications 217

Over movable beds, the interaction of flow and sediment transport creates a variety of bed forms such as ripples, dunes, antidunes or other irregular shapes and obstacles. Their presence, in general, causes flow separation and recirculation, which can alter the overall flow resistance and, consequently, can affect sediment transport within the water mass and bottom erosion. For dunes, in particular, the flow is characterized by an attached flow on their windward side, separation at their crest and formation of a recirculation eddy in their leeside (Fourniotis *et al*., 2006). A detailed description of the flow over a dune is then of fundamental interest because the pressure and friction (shear-stress) distributions on the bed determine the total resistance on the bottom and the rate of sediment transport. Over bed forms a 1DV version of the turbulent boundary layer is no able to describe the main processes that occur above and close to the bed surface. Consequently, a 2DV turbulent

Considering a two-dimensional mean non-stratified flow in the vertical plane *uv w* , 0, , only non-zero *y*-derivatives are present. The physical problem is outlined in figure 1 below, under the action of a wave. Knowing that the wavelength is always greater than the length of the ripples, i.e. *L L w r* , we can restrict the domain of calculation, instead of investigate

Fig. 1. Scheme of the physical system (Huynh-Thanh & Temperville, 1991)

 

> 

The basic equations of the model are derived from the previous ones (2). In order to simplify the numerical resolution of the equations we make use of the stream function ( ) and

domain into a rectangular one. Considering that only two-independent spatial derivatives are involved in the flow, in the *xz*-plane, i.e., a flow with only velocity components *u xzt* , , and *w xzt* ,, , the equations of motion are restricted to the continuity equation and the two components of the Reynolds equations. Under these assumptions, from (2) the two components (32) and (33) of the pure hydrodynamic momentum equation are written:

> <sup>1</sup> <sup>2</sup>' '' *uu u <sup>p</sup> u w u uw tx z ρ xx z*

 

) variables, instead of the velocities *u* and *v*, and a transformation of the physical

(32)

**3.7 2DV turbulent boundary layer model** 

boundary layer model is developed herein.

all the domain over of the whole wavelength.

vorticity (
