**4. Numerical approaches**

#### **4.1 1DV boundary layer models**

#### **4.1.1 One- and two-equation models of the K-L type**

Equations system (22) can be easily solved applying an implicit finite-difference approach in the raw unknowns (*u*, *v*, *K*, *L*, *C*, *<sup>t</sup>* , and *<sup>t</sup>* ) of five differential equations, both in space and time, and two algebraic ones.

Final solution for the vertical profiles of the horizontal components of the velocity (*u*, *v*), turbulent kinetic energy (*K*), macroscale of the eddies (*L*), concentration (*C*) and turbulent viscosity ( *<sup>t</sup>* ), is obtained iteratively during the time-period *T* of the signal introduced at the upper limit of the boundary layer. A flowchart representing the numerical solution implemented is presented in figure 3.

#### **4.1.2 Zero-equation model**

The model equations (29) are to be solved in this section. Considering the *ud* , we note that the non-linear term should be linearized in time using Taylor series. With <sup>2</sup> *u lu z t md* , the following form of the *ud* -equation show how the solution could be obtained:

222 Advanced Fluid Dynamics

*<sup>K</sup> <sup>ε</sup> CK L* . Any other combination of the form *m n K L* can be utilized, for example the

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

*j jKj j i j*

*<sup>u</sup> <sup>C</sup> C ij t xx x K x K* 

> *<sup>L</sup>* , 1.0

*jj j j ij*

 

 

*<sup>u</sup> i j t xx x x*

 

*j t i i*

*<sup>t</sup> ν CK ε* 

1 2 j

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

 

*KK K u u u*

*j t ii*

. This suggests the use of different

(55)

and 2 *C* 1.92

; , 1,2,3

; , 1,2,3

. The turbulent

.

(56)

(57)

(58)

; , 1,2,3

2 1 2 ; , 1,2,3

, where 0.09 *C*

x x *j i i*

> x x *j i i*

and 3 *C* 3 40

) of five differential equations, both in space and

*u u u*

 

> 

 , 1 *C* 1.44 

> .

turbulence closure scheme. The *K* and

x x

x x

 *<sup>K</sup>* , 1.30 

*u u u*

3 2

where, 0.08 0.09 *C C <sup>K</sup>* 

> 

 

 

 

viscosity is calculated by *<sup>t</sup> ν K*

**4. Numerical approaches 4.1 1DV boundary layer models** 

the raw unknowns (*u*, *v*, *K*, *L*, *C*,

implemented is presented in figure 3.

time, and two algebraic ones.

**4.1.2 Zero-equation model** 

viscosity (

*K* 

specific dissipation rate 1 2 *<sup>ω</sup> C K L CK <sup>ω</sup><sup>L</sup>*

 

 

model; its governing equations are written:

*j t*

 

*K K K u u u*

*j lt t*

**4.1.1 One- and two-equation models of the K-L type** 

.

> *<sup>t</sup>* , and *<sup>t</sup>*

*j l Kt t K j j j i j*

*j j j i j*

 

*<sup>u</sup> <sup>C</sup> C K <sup>ω</sup> i j t xx x x*

*<sup>u</sup> C C C ij t xx x K x*

 

   

 , 2 *C* 5 9 

Equations system (22) can be easily solved applying an implicit finite-difference approach in

Final solution for the vertical profiles of the horizontal components of the velocity (*u*, *v*), turbulent kinetic energy (*K*), macroscale of the eddies (*L*), concentration (*C*) and turbulent

upper limit of the boundary layer. A flowchart representing the numerical solution

The model equations (29) are to be solved in this section. Considering the *ud* , we note that the non-linear term should be linearized in time using Taylor series. With <sup>2</sup> *u lu z t md* , the

following form of the *ud* -equation show how the solution could be obtained:

*<sup>t</sup>* ), is obtained iteratively during the time-period *T* of the signal introduced at the

 

*j t*

 

, 1.0 *C*

The turbulent viscosity is calculated by <sup>2</sup>

The Wilcox (1993) model is a two-equation *K*

equations are determined through (57) and (58):

where 1 0.50 *C <sup>K</sup>* , 2 0.09 *C <sup>K</sup>* , 1 *C* 0.50

$$\frac{\partial^\* u\_t}{\partial t} = \frac{\partial}{\partial z} (|u\_t| u\_t) \implies \begin{cases} \frac{\partial}{\partial z} (u\_t) & \text{if } u\_t > 0 \\ -\frac{\partial}{\partial z} (u\_t) & \text{if } u\_t < 0 \end{cases}$$

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

$$A\_j \mu\_{d \cdot j-1}^{n+1} + B\_j \mu\_{d \cdot j}^{n+1} + C\_j \mu\_{d \cdot j+1}^{n+1} = D\_j \; ; \; 2 \le j \le J-1$$

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

$$A\_{j} = -2\frac{\Delta t}{\Delta z\_{j}} \left(\frac{l\_{m,j-1/2}}{\Delta z\_{j-1/2}}\right)^{2} \left|\Delta \,\boldsymbol{u}\_{d}^{\mathrm{n}}\_{d,j-1/2}\right|;\ \mathcal{C}\_{j} = -2\frac{\Delta t}{\Delta z\_{j}} \left(\frac{l\_{m,j+1/2}}{\Delta z\_{j+1/2}}\right)^{2} \left|\Delta \,\boldsymbol{u}\_{d}^{\mathrm{n}}\_{d,j+1/2}\right|;$$

$$B\_{j} = 1 - A\_{j} - \mathcal{C}\_{j}:\,\,\boldsymbol{D}\_{j} = \boldsymbol{u}\_{d,j}^{\mathrm{n}} - \frac{1}{2}A\_{j}\Delta \,\boldsymbol{u}\_{d}^{\mathrm{n}}\_{d,j-1/2} - \frac{1}{2}\mathcal{C}\_{j}\Delta \,\boldsymbol{u}\_{d}^{\mathrm{n}}\_{d,j+1/2}$$

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>* .

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.

Turbulent Boundary Layer Models: Theory and Applications 225

() 1 2 exp

Formula *c1 c2 n1*

*HTfwr* 0.00278 4.6500 -0.2200

*CTfwr* 0.00140 4.5840 -0.1340

Table 1. Fitting coefficients 1*c* , 2*c* and 1*n* , for model of Huynh Thanh (1990) (= *HTfwr*) and

In the case of a current alone, Huynh Thanh found that the friction coefficient *c r f* coincides

Considering rough turbulent flows, for values of the wave friction coefficient, *w r f* , Antunes do Carmo *et al*. (2003) proposed formula (59) with *CTfwr* coefficients (table 1); Tanaka & Thu (1994) suggested formula (61), Swart (1974) formula (62) and Soulsby *et al.* (1994) formula (63):

( ) 0.00251 exp 5.21 *f A w r KN*

According to Sleath (1991), bottom shear stress may be split into two components:

 

A comparison between formulae (59), with *HT*fwr and *CT*fwr coefficients, (61), (62) and (63) is shown in Antunes do Carmo *et al*. (2003). The same figure also shows experimental measurements of Sleath (1987), Kamphuis (1975), Jensen *et al.* (1989), Sumer *et al.* (1987) and

> ˆ ˆˆ *wp w p*

0

0.10

0.52

0.19

2

( ) 0 exp 7.53 8.07 *w r f A <sup>z</sup>* (61)

(62)

( ) <sup>0</sup> 1.39 *w r f Az* (63)

(64)

*<sup>A</sup> fc c <sup>K</sup>* 

where *A* is the wave excursion amplitude, and with the empirical coefficients 1*c* , 2 *c* and 1*n* determined by Huynh Thanh, and presented in table 1 (formula *HT*fwr). Using the same boundary layer model (22), considering the best overall fit with a large number of the model results, in the interval 1 3 6.4 10 *A kN* 3.4 10 , Antunes do Carmo *et al*. (2003) proposed formula (59) with the empirical coefficients determined in that study, listed in table 1 as

*w r*

formula *CT*fwr.

Coeff.

proposed by Antunes do Carmo *et al*. (2003) (= *CTfwr*)

with the value obtained by the theoretical formula (60):

**5.1 Sinusoidal wave alone** 

Jonsson & Carlsen (1976).

( )

2 1 *c r <sup>k</sup> <sup>f</sup> Ln h z*

1

(59)

(60)

*n*

*N*
