**3.6 Boundary conditions for 1DV turbulent boundary layer models**

	- *uz vz* 0 0 0 ; *K z* 0 ; *Lz z* 0 0 , with 0.67 (empirical constant).
	- At the hydraulic rough regime, the level 0*z* is taken to be *kN* 30 , with *k d <sup>N</sup>* 2.5 the Nikuradse equivalent roughness of a bed of sand with diameter *d*. In the transitory regime, *kN* and 0*z* are calculated following Sleath (1984) (Tran Thu and Temperville, 1994).
	- For the reference concentration at the bottom, *Cb* , the following relations may be used: 0.63 *C C <sup>b</sup>* , or *C C ψ* , where *ψ τ <sup>b</sup> t ρs gd* 1 .
	- *uz Ut* , *U t* may contain a component of the mean current *Uc* as well as oscillatory components of the wave;
	- *K z* 0 (pure wave), or 0 *<sup>z</sup> K z* (combined wave and current);
	- *L z* 0 (pure wave), or *Lz z* (combined wave and current);
	- Depending on the problem, the condition 0 *<sup>z</sup> L z* may be also adequate;


 Estimation of the boundary layer thickness, *z*

Considering a pure current ( <sup>ˆ</sup> <sup>0</sup> *Uw* ) in a channel with a water column *h*, the boundary layer thickness is *z h* .

Assuming now a pure wave (*Uc* = *Vc* = 0) propagating in a channel, the boundary layer thickness reaches its minimum value and can be approximated by 0.81 0.246 ˆ *z k ak N N* (Huynh-Thanh, 1990), where the orbital amplitude is given by <sup>ˆ</sup> <sup>ˆ</sup> <sup>2</sup> *weq ch aU T* for an equivalent sinusoidal wave with ˆ*Uweq* , and during a characteristic signal period *Tch* (Antunes do Carmo *et al*., 1996). The relation proposed for *z k <sup>N</sup>* corresponds to the thickness beyond which *K* is zero.

A general rough estimation for *z*can be obtained by (30):

$$z\_{\delta} = \frac{\left| \vec{\mathcal{U}}\_{c} \right| h + \left| \hat{\mathcal{U}}\_{w} \right| z\_{\delta w}}{\left| \vec{\mathcal{U}}\_{c} \right| + \left| \hat{\mathcal{U}}\_{w} \right|} \implies z\_{\delta} = \frac{\left| \vec{\mathcal{U}}\_{c} \right| h + 0.246 \, k\_{N} \left( \hat{a} / k\_{N} \right)^{0.81} \left| \hat{\mathcal{U}}\_{w} \right|}{\left| \vec{\mathcal{U}}\_{c} \right| + \left| \hat{\mathcal{U}}\_{w} \right|} \tag{30}$$

#### **3.6.2 Zero-equation model**

The following conditions (31) are imposed at the lower limit 0 *z z* and at the upper limit *z z* of the boundary layer:

$$\mu\_d(z\_o, t) = -\mathcal{U} \; ; \; u\_d(z\_\delta, t) = 0 \; ; \; v\_d(z\_o, t) = -V \; ; \; v\_d(z\_\delta, t) = 0 \tag{31}$$

### **3.7 2DV turbulent boundary layer model**

216 Advanced Fluid Dynamics



, *U t* may contain a component of the mean current *Uc* as well as

*b*

Initial values for *u*, *v*, *K* and *L* are the solution for the initial field current velocities (*Uc*,

Considering a pure current ( <sup>ˆ</sup> <sup>0</sup> *Uw* ) in a channel with a water column *h*, the boundary

Assuming now a pure wave (*Uc* = *Vc* = 0) propagating in a channel, the boundary layer thickness reaches its minimum value and can be approximated by

*N N* (Huynh-Thanh, 1990), where the orbital amplitude is given by

characteristic signal period *Tch* (Antunes do Carmo *et al*., 1996). The relation proposed

=> 0.81 <sup>ˆ</sup> 0.246 ˆ

can be obtained by (30):

The following conditions (31) are imposed at the lower limit 0 *z z* and at the upper limit

*u z ,t U u z ,t v z ,t V v z ,t d o* ; *<sup>d</sup>*

*<sup>N</sup>* corresponds to the thickness beyond which *K* is zero.

*z*

ˆ *c ww c w*

ˆ

 

*UhU z*

*U U*

(combined wave and current);

for an equivalent sinusoidal wave with ˆ*Uweq* , and during a

ˆ *c NN w c w*

 0 ; *d o* ;0 *<sup>d</sup>* (31)

(30)

*U h k ak U*

*U U*

(combined wave and current);

(combined wave and current).

may be also adequate;

used: 0.63 *C C <sup>b</sup>* , or *C C ψ* , where *ψ τ <sup>b</sup> t ρs gd* 1 .

Assuming that the instantaneous velocity *U t* is given at a level *z z*

*q t <sup>z</sup> wC C t* 

 

 , with 0.67 

(empirical constant).

outside the

**3.6 Boundary conditions for 1DV turbulent boundary layer models 3.6.1 One- and two-equation boundary layer models of the K-L type** 

 At the lower limit of the boundary layer, 0 *z z* - *uz vz* 0 0 0 ; *K z* 0 ; *Lz z* 0 0

At the upper limit of the boundary layer, *z z*

boundary layer, the boundary conditions are:

oscillatory components of the wave;

Estimation of the boundary layer thickness, *z*

.

(pure wave), or 0 *<sup>z</sup> K z*

(pure wave), or *Lz z*


(pure wave), or 0

Temperville, 1994).





layer thickness is *z h*

0.81 0.246 ˆ *z k ak*

A general rough estimation for *z*

*z*

**3.6.2 Zero-equation model** 

of the boundary layer:

<sup>ˆ</sup> <sup>ˆ</sup> <sup>2</sup> *weq ch aU T*

*Vc*).

*z z* 

for *z k* 

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 boundary layer model is developed herein.

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 all the domain over of the whole wavelength.

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

$$\frac{\partial^2 u}{\partial t} + u \frac{\partial^2 u}{\partial \mathbf{x}} + w \frac{\partial^2 u}{\partial z} = -\frac{1}{\rho} \frac{\partial^2 p}{\partial \mathbf{x}} + \frac{\partial}{\partial \mathbf{x}} \left( -\overline{u'^2} \right) + \frac{\partial}{\partial z} \left( -\overline{u'w'} \right) \tag{32}$$

Turbulent Boundary Layer Models: Theory and Applications 219

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

*u uu qq q u uw C u C t xzL L*

' ' 0 ' ' ''

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

0 2 ' '' '

0 '

2 2 3

<sup>2</sup> 2 3 2 2 0 2 '' ' ' <sup>3</sup> *p v*

*t x zL L*

 

The third equation of this system allows us to obtain 2 2 *vq K* ' 42 , with 1.0 *Cp* and

Assuming identical production along both *x*- and *z*-directions, from the second and fourth equations we find that 2 2 *u w* ' ' . This hypothesis is supported by laboratory experiments over a bottom with ripples conducted by Sato *et al*. (1984), among others. Therefore, as 2 22 2 *q Ku v w* 2 '' ' , the above results show that 22 2 *uw q K* ' ' 3 8 3 4 . On the other

> *L w uL w u w u <sup>q</sup> uw u w KL q x zq x z x z*

> > <sup>3</sup> <sup>2</sup> 8

The equation for the turbulent kinetic energy, *K*, is obtained through the earlier already

2 2

dissipation diffu

2 2 2 22

2 2

*tt t*

*K KK L xx zz*

*K K <sup>K</sup> <sup>K</sup> KL KL L xx zz*

*t x z x zx z*

*K K K u uw w*

2 0.30 2 0.30 2

 

 

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

2

*p*

*w w qqq uw w C w C*

8 8

2 22

(42)

3

(41)

*p v*

*p v*

3

 

 

*<sup>t</sup> KL* (43)

 

 

sion 

 

(44)

(45)

2 2

hand, from the first equation of the system (41) we find that:

presented in two-dimensions in the vertical plane:

*u w*

 

*K K*

4

, 4 ,

*t*

Inserting the stream function ( ) in equation (44), we find (45):

 

> 

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

 

> 

> > *t*

advection production

*t xz xz z x*

3

2

2 0.80 0.80

 

'

1 12 *Cv* .

Therefore,

'

'

*w*

$$\frac{\partial^\gamma w}{\partial t} + \mu \frac{\partial^\gamma w}{\partial \mathbf{x}} + w \frac{\partial^\gamma w}{\partial z} = -\frac{1}{\rho} \frac{\partial^\gamma p}{\partial z} + \frac{\partial}{\partial \mathbf{x}} \left( -\overline{u'w'} \right) + \frac{\partial}{\partial z} \left( -\overline{w'^2} \right) \tag{33}$$

Substituting in (32) and (33) the approximations (34),

$$-\overline{u'^2} = 2\ \nu\_t \frac{\partial \overline{u}}{\partial \mathbf{x}} ; -\overline{u'w'} = \nu\_t \left(\frac{\partial \overline{u}}{\partial \mathbf{z}} + \frac{\partial \overline{w}}{\partial \mathbf{x}}\right) \text{ and } -\overline{w'^2} = 2\ \nu\_t \frac{\partial \overline{w}}{\partial \mathbf{z}}\tag{34}$$

the governing equations (35) and (36) result:

$$\frac{\partial^2 \mathbf{u}}{\partial t} + \mathbf{u} \frac{\partial^2 \mathbf{u}}{\partial \mathbf{x}} + w \frac{\partial \mathbf{u}}{\partial \mathbf{z}} = -\frac{1}{\rho} \frac{\partial}{\partial \mathbf{x}} \frac{p}{\partial \mathbf{x}} + 2 \frac{\partial}{\partial \mathbf{x}} \left( \nu\_t \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{z}} \left| \nu\_t \left( \frac{\partial \mathbf{u}}{\partial \mathbf{z}} + \frac{\partial \mathbf{w}}{\partial \mathbf{x}} \right) \right| \tag{35}$$

$$\frac{\partial^{\gamma}w}{\partial t} + u\frac{\partial^{\gamma}w}{\partial x} + w\frac{\partial^{\gamma}w}{\partial z} = -\frac{1}{\rho}\frac{\partial^{\gamma}p}{\partial z} + \frac{\partial}{\partial x}\left[\nu\_t\left(\frac{\partial \, \boldsymbol{u}}{\partial z} + \frac{\partial \, \boldsymbol{w}}{\partial x}\right)\right] + 2\frac{\partial}{\partial z}\left(\nu\_t\frac{\partial \, \boldsymbol{w}}{\partial z}\right) \tag{36}$$

The unknown pressure gradient due to the bed forms can now be eliminated from equations (35) and (36) by cross-differentiation, i.e., taking the curl of the two-dimensional vector momentum equations. The result reads:

$$\begin{split} \frac{\partial}{\partial t} \left( \frac{\partial \, u}{\partial z} - \frac{\partial \, w}{\partial \mathbf{x}} \right) + \mu \frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial \, u}{\partial z} - \frac{\partial \, w}{\partial \mathbf{x}} \right) + w \frac{\partial}{\partial z} \left( \frac{\partial \, u}{\partial z} - \frac{\partial \, w}{\partial \mathbf{x}} \right) = \\ - \frac{\partial^2}{\partial \mathbf{x}^2} \left[ \nu\_t \left( \frac{\partial \, u}{\partial \mathbf{z}} + \frac{\partial \, w}{\partial \mathbf{x}} \right) \right] + 2 \frac{\partial^2}{\partial \mathbf{x} \partial \mathbf{z}} \left[ \nu\_t \left( \frac{\partial \, u}{\partial \mathbf{x}} - \frac{\partial \, w}{\partial \mathbf{z}} \right) \right] + \frac{\partial^2}{\partial \mathbf{z}^2} \left[ \nu\_t \left( \frac{\partial \, u}{\partial \mathbf{z}} + \frac{\partial \, w}{\partial \mathbf{x}} \right) \right] \end{split} \tag{37}$$

By definition, the following relations (38) account:

$$\mu = \frac{\partial \Psi}{\partial \mathbf{z}} ; \; w = -\frac{\partial \Psi}{\partial \mathbf{x}} ; \; \xi = \frac{\partial \ln}{\partial \mathbf{z}} - \frac{\partial \ln}{\partial \mathbf{x}} \tag{38}$$

Inserting the stream function ( ) and vorticity ( ) variables in equation (37) the following result (39) for the vorticity is obtained (Huynh-Thanh, 1990; Tran-Thu, 1995):

$$\frac{\partial \tilde{\xi}}{\partial t} - \frac{\partial \left( \Psi, \tilde{\xi} \right)}{\partial \left( \mathbf{x}, z \right)} = \nabla^2 \left( \nu\_t \tilde{\xi} \right) - 2 \left( \frac{\partial^2 \nu\_t}{\partial \mathbf{x}^2} \frac{\partial^2 \Psi}{\partial z^2} - 2 \frac{\partial^2 \nu\_t}{\partial \mathbf{x} \partial z} \frac{\partial^2 \Psi}{\partial \mathbf{x} \partial z} + \frac{\partial^2 \nu\_t}{\partial z^2} \frac{\partial^2 \Psi}{\partial \mathbf{x}^2} \right) \tag{39}$$

where , *xz x z z x* , and 2 2 2 2 2 *x z* .

An equation for the stream function is obtained through the definitions (38), substituting *u* and *v* in :

$$
\nabla^2 \Psi = \xi \tag{40}
$$

which is known as the Poisson equation. The turbulent viscosity *<sup>t</sup>* is obtained assuming local equilibrium turbulence. Once more in the vertical plane *uv w* , 0, , the following equations system (41) can be written:

$$\begin{aligned} \frac{\partial}{\partial t} \frac{u'w'}{t} &= 0 = -\left(\overline{u'^2} \frac{\partial}{\partial x} + \overline{w'^2} \frac{\partial}{\partial z}\right) & -C\_p \frac{q}{L} \overline{u'w'} \\ \frac{\partial}{\partial t} \frac{\overline{u'^2}}{t} &= 0 = -2\left(\overline{u'^2} \frac{\partial}{\partial x} + \overline{u'w'} \frac{\partial}{\partial z}\right) & -C\_p \frac{q}{L} \left(\overline{u'^2} - \frac{q^2}{3}\right) - C\_v \frac{q^3}{L} \\ \frac{\partial}{\partial t} \frac{\overline{v'^2}}{t} &= 0 & -C\_p \frac{q}{L} \left(\overline{v'^2} - \frac{q^2}{3}\right) - C\_v \frac{q^3}{L} \\ \frac{\partial}{\partial t} \frac{\overline{w'^2}}{t} &= 0 = -2\left(\overline{u'w'} \frac{\partial}{\partial x} + \overline{w'^2} \frac{\partial}{\partial z}\right) - C\_p \frac{q}{L} \left(\overline{w'^2} - \frac{q^2}{3}\right) - C\_v \frac{q^3}{L} \end{aligned} \tag{41}$$

The third equation of this system allows us to obtain 2 2 *vq K* ' 42 , with 1.0 *Cp* and 1 12 *Cv* .

Assuming identical production along both *x*- and *z*-directions, from the second and fourth equations we find that 2 2 *u w* ' ' . This hypothesis is supported by laboratory experiments over a bottom with ripples conducted by Sato *et al*. (1984), among others. Therefore, as 2 22 2 *q Ku v w* 2 '' ' , the above results show that 22 2 *uw q K* ' ' 3 8 3 4 . On the other hand, from the first equation of the system (41) we find that:

$$-\overline{u'w'} = \frac{L}{q} \left( \overline{u'^2} \frac{\partial^\* w}{\partial \mathbf{x}} + \overline{w'^2} \frac{\partial^\* u}{\partial \mathbf{z}} \right) = \frac{L}{q} \frac{3q^2}{8} \left( \frac{\partial^\* w}{\partial \mathbf{x}} + \frac{\partial^\* u}{\partial \mathbf{z}} \right) = \frac{3}{8} \sqrt{2KL} \left( \frac{\partial^\* w}{\partial \mathbf{x}} + \frac{\partial^\* u}{\partial \mathbf{z}} \right) \tag{42}$$

Therefore,

218 Advanced Fluid Dynamics

 <sup>1</sup> <sup>2</sup> '' ' *ww w <sup>p</sup> u w uw w tx z ρ zx z*

*u w*

*z x*

2 *t t*

(36)

and <sup>2</sup>

*w*

 

2 *t t*

(33)

*w*

(34)

(35)

(37)

*z* 

' 2 *<sup>t</sup>*

 

(38)

) variables in equation (37) the following

(40)

*<sup>t</sup>* is obtained assuming

 

;

*u w*

*u w*

2 ' 2 *<sup>t</sup>*

the governing equations (35) and (36) result:

*u*

momentum equations. The result reads:

By definition, the following relations (38) account:

Inserting the stream function ( ) and vorticity (

*xz x z z x* ,

equations system (41) can be written:

 

where

and *v* in

 

> :

 ,

*u*

2

 

and

which is known as the Poisson equation. The turbulent viscosity

*z* ; *<sup>w</sup>*

result (39) for the vorticity is obtained (Huynh-Thanh, 1990; Tran-Thu, 1995):

, 2 2 ,

2

Substituting in (32) and (33) the approximations (34),

*u*

*x* 

> 

> >

 

> 

 

> 

*u w*

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

1

 

 

 2 22 2 2 2 *t tt*

*uw uw uw u w tz x xz x zz x*

1

*uu u p u uw*

*tx z ρ x x x z zx*

*ww w p uw w*

*tx z ρ zx z x z z*

The unknown pressure gradient due to the bed forms can now be eliminated from equations (35) and (36) by cross-differentiation, i.e., taking the curl of the two-dimensional vector

*u w uw uw*

; *u w*

*z x*

2 22 2 22

(39)

2 2 2 2

*x z z x xz x z z x*

*x*

*t tt <sup>t</sup> t xz x z xz xz z x*

2 2

2 2 *x z* .

An equation for the stream function is obtained through the definitions (38), substituting *u*

<sup>2</sup> 

local equilibrium turbulence. Once more in the vertical plane *uv w* , 0, , the following

 

$$
\omega\_t = \frac{3}{8} \sqrt{2KL} \tag{43}
$$

The equation for the turbulent kinetic energy, *K*, is obtained through the earlier already presented in two-dimensions in the vertical plane:

$$\begin{aligned} \frac{\partial \mathcal{K}}{\partial t} + u \frac{\partial \mathcal{K}}{\partial \mathbf{x}} + w \frac{\partial \mathcal{K}}{\partial z} &= \nu\_t \left[ 2 \left( \frac{\partial \mathcal{u}}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial \mathcal{u}}{\partial z} + \frac{\partial \mathcal{w}}{\partial \mathbf{x}} \right)^2 + 2 \left( \frac{\partial \mathcal{w}}{\partial z} \right)^2 \right] \\ &- \frac{K \sqrt{2 \mathcal{K}}}{4L} + 0.30 \frac{\mathcal{C}}{\partial \mathbf{x}} \left( \sqrt{2 \mathcal{K}} L \frac{\partial \mathcal{K}}{\partial \mathbf{x}} \right) + 0.30 \frac{\mathcal{C}}{\partial z} \left( \sqrt{2 \mathcal{K}} L \frac{\partial \mathcal{K}}{\partial z} \right) \end{aligned} \tag{44}$$

Inserting the stream function ( ) in equation (44), we find (45):

$$\begin{aligned} \frac{\partial \mathcal{R}}{\partial t} - \underbrace{\frac{\partial \left(\Psi, K\right)}{\partial \left(\mathbf{x}, z\right)}}\_{\text{advection}} &= \underbrace{\nu\_t \left[4 \left(\frac{\partial^2 \Psi}{\partial \mathbf{x} \partial \mathbf{z}}\right)^2 + \left(\frac{\partial^2 \Psi}{\partial z^2} - \frac{\partial^2 \Psi}{\partial \mathbf{x}^2}\right)^2\right]}\_{\text{production}} \\ &- \underbrace{\frac{2}{\mathcal{R}} \nu\_t \frac{K}{L^2}}\_{\text{dissipation}} + \underbrace{0.80 \frac{\partial}{\partial \mathbf{x}} \left(\nu\_t \frac{\partial K}{\partial \mathbf{x}}\right)}\_{\text{diffusion}} + 0.80 \frac{\partial}{\partial z} \left(\nu\_t \frac{\partial K}{\partial z}\right)}\_{\text{diffusion}} \end{aligned} \tag{45}$$

Turbulent Boundary Layer Models: Theory and Applications 221

*t tt <sup>t</sup> t XZ z x xz xz x z*

<sup>2</sup> *J XZ*

<sup>3</sup> <sup>2</sup> 8

 <sup>2</sup> , <sup>2</sup> 0.80 , 3 *t tt*

*P* represents the production of *K*, and for C:

*C xC xC w C w C t XZ Z X ZX X Z* 

**3.8 Boundary conditions for a 2DV turbulent boundary layer model** 


time precedent through 2 22 

**3.9 Other simplified two-equation turbulence closure models** 

is the turbulent dissipation rate defined by


, if the flow is known at the level *z z*

At the lateral boundaries (*X* = 0 and *X* = *L*), a spatially periodic condition for ,

A relation for the turbulent viscosity, equivalent to (16), can be written as <sup>2</sup>

*J JZ XZ* .



 

*K K K KK t XZ X X ZZ L* 

2 22 2 22

2 *J ZZ* , where 1 is the stream function

.

0 (it is assumed non-rotational flow outside of the

*l*

*ν*

*<sup>t</sup>* with (16), a relation between *L* and

*' 'j i*

. Comparing this

*k k u u*

*x x*

can be also obtained from the one obtained at the

*<sup>J</sup> J P* (53)

ψ ψ<sup>0</sup> *s t s t*

*J J* (54)

*J J* (50)

2 2 2 2

*<sup>t</sup> KL* (52)

(51)

. For *K* we get (53),

 

> , or

and *K* is

*<sup>t</sup> ν CK ε* ,

is found

where

where an algebraic equation for *L* is used, *L Z Zz* 0.67 1

2 2 2 22

At the lower limit of the boundary layer, 0 *zz k <sup>N</sup>* 30

 

2 2 <sup>4</sup>*<sup>t</sup> x z z x*

value at height *Z*1.. Value for 0

At the upper limit of the boundary layer, *z z*

(combined wave and current).


boundary layer).

definition of the eddy viscosity

assumed.

where

 *z t Qt* 

2

 

, 2 2 ,

The length scale *L* is directly imposed by the analytical solution (46):

$$L = 0.67 \ z \sqrt{1 - z/z\_{\delta}} \tag{46}$$

In order to describe the space-time distribution of the sediments concentration over a bottom with ripples, an equation for *C* is included, considering in it the advection and diffusion terms in both x-horizontal and z-vertical directions:

$$\frac{\partial \mathbf{C}}{\partial t} + \frac{\partial}{\partial \mathbf{x}} (\mu \mathbf{C}) + \frac{\partial}{\partial z} [(w - w\_s) \mathbf{C}] = \frac{\partial}{\partial \mathbf{x}} \left( \chi\_t \frac{\partial \mathbf{C}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial z} \left( \chi\_t \frac{\partial \mathbf{C}}{\partial z} \right) \tag{47}$$

In order to simplify the numerical resolution of the equations, as well as the description of the boundary conditions at the ripples surface, the physical domain in coordinates (*x*, *z*) is transformed into a rectangular one (the computation domain) utilizing orthogonal curvilinear coordinates (*X*, *Z*) (Figure 2), using the following transformations (48) (Sato *et al*., 1984; Huynh-Thanh, 1990; Tran-Thu, 1995; Silva, 2001):

$$\begin{aligned} X &= \mathbf{x} + \sum\_{n=1}^{N} a\_n \exp\left(-n\frac{2\pi}{L\_r} Z\right) \sin\left(n\frac{2\pi}{L\_r} X - \theta\_n\right) \\ Z &= \mathbf{z} - \sum\_{n=1}^{N} a\_n \exp\left(-n\frac{2\pi}{L\_r} Z\right) \cos\left(n\frac{2\pi}{L\_r} X - \theta\_n\right) \end{aligned} \tag{48}$$

where *N*, *na* and *<sup>n</sup>* are coefficients to be determined in such a way that the curve *Z =* 0 represents the real ripple.

Fig. 2. Physical and computational domains. Transformation of coordinates *xz XZ* , ,

The Jacobian of the transformation is defined by (49):

$$\mathbf{J} = \frac{\partial \left( X, Z \right)}{\partial \left( \mathbf{x}, z \right)} = \frac{\partial X}{\partial \mathbf{x}} \frac{\partial Z}{\partial z} - \frac{\partial X}{\partial z} \frac{\partial Z}{\partial \mathbf{x}} = \left( \frac{\partial X}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial X}{\partial z} \right)^2 \tag{49}$$

which is calculated from the inverse transformation of the Jacobian <sup>1</sup> 0 0 *J JJ* . After carried out the transformation of coordinates *xz XZ* , , , the above equations (39), (40), (43), (45) and (47) are written and solved iteratively as will be shown later (see Huynh-Thanh, 1990, Tran-Thu, 1995, and Silva, 2001, for details):

220 Advanced Fluid Dynamics

*L z zz* 0.67 1

In order to describe the space-time distribution of the sediments concentration over a bottom with ripples, an equation for *C* is included, considering in it the advection and

> *st t C C <sup>C</sup> uC w w C tx z xxzz*

> > 2 2 exp sin

exp cos

*n r r*

Fig. 2. Physical and computational domains. Transformation of coordinates *xz XZ* , ,

which is calculated from the inverse transformation of the Jacobian <sup>1</sup>

2 2 ,

*<sup>J</sup>* (49)

*X Z XZ XZ X X xz x z z x x z*

carried out the transformation of coordinates *xz XZ* , , , the above equations (39), (40), (43), (45) and (47) are written and solved iteratively as will be shown later (see Huynh-

*n r r*

*Xx a n Z n X*

*Zz a n Z n X*

2 2

*L L*

*L L*

*<sup>n</sup>* are coefficients to be determined in such a way that the curve *Z =* 0

 

 

*n n*

*n n*

In order to simplify the numerical resolution of the equations, as well as the description of the boundary conditions at the ripples surface, the physical domain in coordinates (*x*, *z*) is transformed into a rectangular one (the computation domain) utilizing orthogonal curvilinear coordinates (*X*, *Z*) (Figure 2), using the following transformations (48) (Sato *et al*.,

(47)

(46)

 

(48)

0 0

*J JJ* . After

The length scale *L* is directly imposed by the analytical solution (46):

diffusion terms in both x-horizontal and z-vertical directions:

1984; Huynh-Thanh, 1990; Tran-Thu, 1995; Silva, 2001):

The Jacobian of the transformation is defined by (49):

(,)

Thanh, 1990, Tran-Thu, 1995, and Silva, 2001, for details):

where *N*, *na* and

represents the real ripple.

1

*N*

*N*

1

$$\frac{\partial \mathcal{\tilde{\xi}}}{\partial t} - \mathbf{J} \frac{\partial \left( \Psi, \mathcal{\tilde{\xi}} \right)}{\partial \left( X, Z \right)} = \mathbf{J} \,\nabla^2 \left( \nu\_t \mathcal{\tilde{\xi}} \right) - 2 \left( \frac{\partial^2 \nu\_t}{\partial z^2} \frac{\partial^2 \Psi}{\partial x^2} - 2 \frac{\partial^2 \nu\_t}{\partial x \partial z} \frac{\partial^2 \Psi}{\partial x \partial z} + \frac{\partial^2 \nu\_t}{\partial x^2} \frac{\partial^2 \Psi}{\partial z^2} \right) \tag{50}$$

$$\mathbf{J}\,\nabla\_{\times\mathbb{Z}}^{2}\Psi = \mathfrak{J}\tag{51}$$

$$
\omega\_t = \frac{3}{8} \sqrt{2KL} \tag{52}
$$

where an algebraic equation for *L* is used, *L Z Zz* 0.67 1 . For *K* we get (53),

$$\frac{\partial \mathcal{K}}{\partial t} - \mathbf{J} \frac{\partial \left( \Psi, K \right)}{\partial \left( X, Z \right)} = 0.80 \left[ \frac{\partial}{\partial X} \left( \nu\_t \frac{\partial \mathcal{K}}{\partial X} \right) + \frac{\partial}{\partial Z} \left( \nu\_t \frac{\partial \mathcal{K}}{\partial Z} \right) \right] - \frac{2}{3} \nu\_t \frac{\mathcal{K}}{L^2} + \mathbf{P} \tag{53}$$

where 2 2 2 22 2 2 <sup>4</sup>*<sup>t</sup> x z z x P* represents the production of *K*, and for C:

$$
\frac{\partial \mathbf{C}}{\partial t} + \mathbf{J} \frac{\partial}{\partial \mathbf{X}} \left[ \left( \frac{\partial \boldsymbol{\upmu}}{\partial \mathbf{Z}} + \boldsymbol{w}\_s \frac{\partial \mathbf{x}}{\partial \mathbf{Z}} \right) \mathbf{C} - \boldsymbol{\upmu}\_t \frac{\partial \mathbf{C}}{\partial \mathbf{X}} \right] - \mathbf{J} \frac{\partial}{\partial \mathbf{Z}} \left[ \left( \frac{\partial \boldsymbol{\upmu}}{\partial \mathbf{X}} + \boldsymbol{w}\_s \frac{\partial \mathbf{x}}{\partial \mathbf{X}} \right) \mathbf{C} + \boldsymbol{\upmu}\_t \frac{\partial \mathbf{C}}{\partial \mathbf{Z}} \right] = \mathbf{0} \tag{54}
$$

#### **3.8 Boundary conditions for a 2DV turbulent boundary layer model**

	- conditions for the stream current: *X Z* 0 ; 0 .
	- condition for the turbulent kinetic energy: *K Z* 0 .
	- condition for the vorticity: 2 0 110 2 *J ZZ* , where 1 is the stream function value at height *Z*1.. Value for 0 can be also obtained from the one obtained at the time precedent through 2 22 *J JZ XZ* .
	- condition for the stream current: *Z Ut* , where *Ut U U t c w* sin , or *z t Qt* , if the flow is known at the level *z z* .
	- condition for the turbulent kinetic energy. *K* = 0 (pure current), or *K Z* 0 (combined wave and current).
	- condition for the vorticity: 0 (it is assumed non-rotational flow outside of the boundary layer).

At the lateral boundaries (*X* = 0 and *X* = *L*), a spatially periodic condition for , and *K* is assumed.

#### **3.9 Other simplified two-equation turbulence closure models**

A relation for the turbulent viscosity, equivalent to (16), can be written as <sup>2</sup> *<sup>t</sup> ν CK ε* , where is the turbulent dissipation rate defined by *' ' j i l k k u u ν x x* . Comparing this definition of the eddy viscosity *<sup>t</sup>* with (16), a relation between *L* and is found

Turbulent Boundary Layer Models: Theory and Applications 223

*z*

1 1 ; 2 1 *n nn Au Bu Cu D j J j dj j dj j dj j*

1 2

;

1 *B AC j jj* ; n n 12 12

*z*

*t t*

*u u*

*t t*

*u u*

2 *m j j d j j j <sup>l</sup> <sup>t</sup> C u z z*

1 1 2 2 *<sup>n</sup> D u Au Cu j dj j dj j dj*

if 0

if 0

2 12 n

1 2

 

1 2

;

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

Considering the case 0 *ud* , a discretized form of this equation reads:

2 12 n

Fig. 3. Flowchart for the 1DV two-equation *K L* boundary layer model

Applications of 1DV boundary layer models are presented later, in this chapter.

1 2

 

with n nn *u uu <sup>d</sup>* 12 1 *<sup>j</sup> <sup>d</sup> <sup>j</sup> <sup>d</sup> <sup>j</sup>* and n nn *u uu <sup>d</sup> <sup>j</sup>* 12 1 *<sup>d</sup> <sup>j</sup> <sup>d</sup> <sup>j</sup>* .

*u u*

=>

1 11

*u*

where the coefficients *Aj* , *Bj* , *Cj* and *Dj* are:

2 *m j j d j j j <sup>l</sup> <sup>t</sup> A u z z*

*t z*

3 2 *<sup>K</sup> <sup>ε</sup> CK L* . Any other combination of the form *m n K L* can be utilized, for example the specific dissipation rate 1 2 *<sup>ω</sup> C K L CK <sup>ω</sup><sup>L</sup>* . This suggests the use of different variables, other than the macroscale of the eddies *L*, with all approximations of the form (10). One of these turbulence closure schemes, possibly the best known, is the two-equation *K* model; its governing equations are written:

$$\frac{\partial \mathcal{E}}{\partial t} + \overline{u}\_j \frac{\partial \mathcal{K}}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left( \frac{\nu\_t}{\sigma\_K} \frac{\partial \mathcal{K}}{\partial \mathbf{x}\_j} \right) + \nu\_t \left( \frac{\partial \,\overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \,\overline{u}\_j}{\partial \mathbf{x}\_i} \right) \frac{\partial \,\overline{u}\_i}{\partial \mathbf{x}\_j} - \boldsymbol{\varepsilon} \; \; ; \; i, j = 1, 2, 3 \tag{55}$$

$$\frac{\partial \overline{\mathbf{x}}}{\partial t} + \overline{\mathbf{u}}\_{j} \frac{\partial \overline{\mathbf{x}}}{\partial \mathbf{x}\_{j}} = \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \frac{\nu\_{t}}{\sigma\_{e}} \frac{\partial \overline{\boldsymbol{\varepsilon}}}{\partial \mathbf{x}\_{j}} \right) + \mathbf{C}\_{1x} \frac{\overline{\mathbf{z}}}{K} \nu\_{t} \left( \frac{\partial \overline{\mathbf{u}}\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial \overline{\mathbf{u}}\_{j}}{\partial \mathbf{x}\_{i}} \right) \frac{\partial}{\partial \mathbf{x}\_{j}} - \mathbf{C}\_{2x} \frac{\mathbf{z}^{2}}{K}; \text{ i.} \ j = 1, 2, 3 \tag{56}$$

where, 0.08 0.09 *C C <sup>K</sup>* , 1.0 *C<sup>L</sup>* , 1.0 *<sup>K</sup>* , 1.30 , 1 *C* 1.44 and 2 *C* 1.92 . The turbulent viscosity is calculated by <sup>2</sup> *<sup>t</sup> ν CK ε* , where 0.09 *C*.

The Wilcox (1993) model is a two-equation *K* turbulence closure scheme. The *K* and equations are determined through (57) and (58):

$$\frac{\partial \mathcal{R}}{\partial \mathbf{\hat{t}}} + \overline{\boldsymbol{u}}\_{j} \frac{\partial \mathcal{K}}{\partial \mathbf{x}\_{j}} = \frac{\partial}{\partial \mathbf{x}\_{j}} \bigg[ (\mathbf{v}\_{l} + \mathbf{C}\_{1K} \boldsymbol{\nu}\_{t}) \frac{\partial \mathcal{K}}{\partial \mathbf{x}\_{j}} \bigg] + \boldsymbol{\nu}\_{t} \bigg[ \frac{\partial}{\partial \mathbf{x}\_{j}} + \frac{\partial}{\partial \mathbf{x}\_{i}} \bigg] \frac{\partial \overline{\mathbf{u}}\_{i}}{\partial \mathbf{x}\_{j}} - \mathbf{C}\_{2K} \boldsymbol{K} \ \boldsymbol{\nu} \ \mathbf{ \ddots \ \mathbf{i} \ \mathbf{y} = 1,2,3} \tag{57}$$

$$\frac{\partial \boldsymbol{\hat{\alpha}} \boldsymbol{\alpha}}{\partial t} + \overline{\boldsymbol{\mu}}\_{j} \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}\_{j}} = \frac{\partial}{\partial \mathbf{x}\_{j}} \bigg[ (\boldsymbol{\nu}\_{l} + \mathbf{C}\_{1o} \boldsymbol{\nu}\_{t}) \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}\_{j}} \bigg] + \mathbf{C}\_{2o} \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \boldsymbol{\nu}\_{t} \bigg[ \frac{\partial \, \overline{\boldsymbol{u}}\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial}{\partial \mathbf{x}\_{i}} \bigg] \frac{\partial \, \overline{\boldsymbol{u}}\_{i}}{\partial \mathbf{x}\_{j}} - \mathbf{C}\_{3o} \boldsymbol{\alpha}^{2}; \ \mathbf{i}, \boldsymbol{j} = 1, 2, 3 \tag{58}$$

where 1 0.50 *C <sup>K</sup>* , 2 0.09 *C <sup>K</sup>* , 1 *C* 0.50 , 2 *C* 5 9 and 3 *C* 3 40 . The turbulent viscosity is calculated by *<sup>t</sup> ν K* .
