**5. Parametric formulations**

Following we show how different parametric approaches are derived, and tested with experimental data, using the two-equation *K L* boundary layer model (22). Using this model, Huynh Thanh (1990) proposed formula (59) below for the wave friction coefficient, *w r f* , in the rough turbulent flow case:

224 Advanced Fluid Dynamics

Equations (50) to (54) are easily solved applying an implicit finite-difference approach centred in space and forward in time. The alternating direction implicit (ADI) method is

cyclic reduction method (Roache, 1976), which allows a huge saving in calculation time compared to the Gauss-Seidel iteration method (Huynh-Thanh & Temperville, 1991). Final solution 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

and *K*. The Poisson equation for is solved by the bloc-

Fig. 4. Flowchart for the 2DV one-equation *K L* boundary layer model

Comparisons of laboratory experiments with numerical results of the 2DV boundary layer

Following we show how different parametric approaches are derived, and tested with experimental data, using the two-equation *K L* boundary layer model (22). Using this model, Huynh Thanh (1990) proposed formula (59) below for the wave friction coefficient,

**4.2 2DV boundary layer model** 

used to solve the equations for

implemented is presented in figure 4.

model are presented later, in section 6.

*w r f* , in the rough turbulent flow case:

**5. Parametric formulations** 

$$f\_{w(r)} = c\_1 \exp\left[c\_2 \left(\frac{A}{K\_N}\right)^{n\_1}\right] \tag{59}$$

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 formula *CT*fwr.


Table 1. Fitting coefficients 1*c* , 2*c* and 1*n* , for model of Huynh Thanh (1990) (= *HTfwr*) and proposed by Antunes do Carmo *et al*. (2003) (= *CTfwr*)

In the case of a current alone, Huynh Thanh found that the friction coefficient *c r f* coincides with the value obtained by the theoretical formula (60):

$$f\_{c(r)} = 2\left[\frac{k}{Ln\left(h/z\_0\right) - 1}\right]^2\tag{60}$$

#### **5.1 Sinusoidal wave alone**

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):

$$f\_{w(r)} = \exp\left(-7.53 + 8.07 \left(A/z\_0\right)^{-0.10}\right) \tag{61}$$

$$f\_{w(r)} = 0.00251 \exp\left(5.21 \left(A/K\_N\right)^{-0.19}\right) \tag{62}$$

$$f\_{w(r)} = 1.39 \left( A / z\_0 \right)^{-0.52} \tag{63}$$

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 Jonsson & Carlsen (1976).

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

$$
\hat{\boldsymbol{\tau}}\_{wp} = \hat{\boldsymbol{\tau}}\_{w} + \hat{\boldsymbol{\tau}}\_{p} \tag{64}
$$

Turbulent Boundary Layer Models: Theory and Applications 227

propagation of a sinusoidal wave, making use of a numerical Boussinesq-type model

The values of the friction coefficient for a sinusoidal wave are shown in figure 6. Close agreement is evident between results 1 and 2. The instantaneous bottom shear stresses

have been calculated using model (59) with *CTfwr* coefficients. In figure 7, results given by the *K L* model (22) (result 2) are compared both with those of model (59) (result 1) and with those obtained by a constant friction coefficient without the phase shift (result 3). Computed shear stresses for the sinusoidal wave case are presented in figure 7-a). Results of the model (59) with *CT*fwr coefficients (result 1) are in close agreement with those of the

Fig. 5. Instantaneous velocity records: a – Sinusoidal wave (orbital velocity amplitude = 0.225 m/s, period = 3.6 sec); b – Cnoidal wave (total velocity amplitude = 1.107 m/s, period = 9.0 sec); c – Irregular wave (resulting from the non-linear propagation of a sinusoidal wave with a period = 3.0 sec in a channel 0.30 m depth) (Antunes do Carmo *et al*., 2003)

*t*

(Antunes do Carmo *et al.*, 1993), with a 3.0 sec period in a channel 0.30 m depth.

*K L* model (22).

The shear stress in the fluid, ˆ *w* , is taken into account by the model, but the value of ˆ *p* , due to the mean pressure gradient acting on the bed roughness, is not. Using Sleath's experiments, it can be seen that a global friction coefficient may be split into the following two components:

$$f\_{wp} = f\_w + f\_p \tag{65}$$

where *wf* represents the friction coefficient obtained by the *K-L* model, and *pf* represents the pressure gradient contribution. Assuming 50 *K d <sup>N</sup>* 2.5 , Sleath (1991) presented the formula (66):

$$f\_p = 0.48 \left( A / K\_N \right)^{-1} \tag{66}$$

The pressure gradient was not taken into account in experiments conducted by Sleath, Sumer, Jensen and Jonsson. Therefore, results of their experimental data are compared with model (59) considering CTfwr coefficients. Excluding a small part of the Sleath's experiments, all other cases show a close agreement model (Antunes do Carmo *et al*., 2003). Discrepancies are explained as a consequence of some of Sleath's experiments being in the smooth-laminar transition regime. The pressure gradient is taken into account in Kamphuis' experiments, so this data should be compared with values for the following expression (67):

$$f\_{wp} = f\_w + f\_p = 0.0014 \cdot \exp\left[ 4.584 \left( \frac{A}{K\_N} \right)^{-0.134} \right] + 0.48 \left( \frac{A}{K\_N} \right)^{-1} \tag{67}$$

For values of *A KN* 100 , the *pf* term is negligible and expression (59) with *CTfwr* coefficients (table 1) is in close agreement with results (Antunes dio Carmo *et al*., 2003).

#### **5.2 Time-dependent shear stress**

For the purpose of calculating time-dependent shear stress *t* in the case of an irregular wave whose instantaneous velocity is given by *U t* , Soulsby *et al.* (1994) propose calculating the value of the friction coefficient *wf* for the equivalent sinusoidal wave with orbital velocity amplitude equal to 2 *Urms* and period *Tp*. It can therefore be deduced (Antunes do Carmo *et al*., 2003):

$$f\_w = 1.39 \left(\frac{A}{z\_0}\right)^{-0.52} A = \frac{\sqrt{2} \text{ } \mathcal{U}\_{rms}}{2\,\pi} \tag{68}$$

where *Urms* = root-mean-square of orbital velocities. For a sinusoidal wave, this formulation correctly represents, in parametric form, the bottom shear stress obtained using *K L* model (22), but does not take into account the phase shift between *<sup>t</sup>* and *U <sup>t</sup>* . For an asymmetric wave, or an irregular wave, more important differences appear between this parametric formulation and the results calculated directly by the *K L* model.

To illustrate these phenomena, we consider the instantaneous velocity records presented in figure 5 for three cases (Antunes do Carmo *et al*., 2003): a) a sinusoidal wave, with orbital velocity amplitude 0.225 m/s and period 3.6 sec; b) a cnoidal wave, with a total velocity amplitude 1.107 m/s and period 9 sec, and c) an irregular wave obtained by the non-linear 226 Advanced Fluid Dynamics

to the mean pressure gradient acting on the bed roughness, is not. Using Sleath's experiments, it can be seen that a global friction coefficient may be split into the following

where *wf* represents the friction coefficient obtained by the *K-L* model, and *pf* represents the pressure gradient contribution. Assuming 50 *K d <sup>N</sup>* 2.5 , Sleath (1991) presented the

<sup>1</sup> 0.48 *f AK p N*

The pressure gradient was not taken into account in experiments conducted by Sleath, Sumer, Jensen and Jonsson. Therefore, results of their experimental data are compared with model (59) considering CTfwr coefficients. Excluding a small part of the Sleath's experiments, all other cases show a close agreement model (Antunes do Carmo *et al*., 2003). Discrepancies are explained as a consequence of some of Sleath's experiments being in the smooth-laminar transition regime. The pressure gradient is taken into account in Kamphuis' experiments, so

this data should be compared with values for the following expression (67):

0.0014 exp 4.584 0.48 *wp w p*

*A A f ff K K*

For values of *A KN* 100 , the *pf* term is negligible and expression (59) with *CTfwr* coefficients (table 1) is in close agreement with results (Antunes dio Carmo *et al*., 2003).

wave whose instantaneous velocity is given by *U t* , Soulsby *et al.* (1994) propose calculating the value of the friction coefficient *wf* for the equivalent sinusoidal wave with orbital velocity amplitude equal to 2 *Urms* and period *Tp*. It can therefore be deduced

0.52

where *Urms* = root-mean-square of orbital velocities. For a sinusoidal wave, this formulation correctly represents, in parametric form, the bottom shear stress obtained using *K L*

asymmetric wave, or an irregular wave, more important differences appear between this

To illustrate these phenomena, we consider the instantaneous velocity records presented in figure 5 for three cases (Antunes do Carmo *et al*., 2003): a) a sinusoidal wave, with orbital velocity amplitude 0.225 m/s and period 3.6 sec; b) a cnoidal wave, with a total velocity amplitude 1.107 m/s and period 9 sec, and c) an irregular wave obtained by the non-linear

0

*z* 

1.39 *<sup>w</sup> <sup>A</sup> <sup>f</sup>*

model (22), but does not take into account the phase shift between

parametric formulation and the results calculated directly by the *K L* model.

, is taken into account by the model, but the value of ˆ

*wp <sup>w</sup> <sup>p</sup> f ff* (65)

(66)

0.134 1

*N N*

2 2 *Urms <sup>A</sup>* 

*p* , due

(67)

*t* in the case of an irregular

*<sup>t</sup>* and *U <sup>t</sup>* . For an

(68)

The shear stress in the fluid, ˆ

**5.2 Time-dependent shear stress** 

(Antunes do Carmo *et al*., 2003):

For the purpose of calculating time-dependent shear stress

two components:

formula (66):

*w*

propagation of a sinusoidal wave, making use of a numerical Boussinesq-type model (Antunes do Carmo *et al.*, 1993), with a 3.0 sec period in a channel 0.30 m depth.

The values of the friction coefficient for a sinusoidal wave are shown in figure 6. Close agreement is evident between results 1 and 2. The instantaneous bottom shear stresses *t* have been calculated using model (59) with *CTfwr* coefficients. In figure 7, results given by the *K L* model (22) (result 2) are compared both with those of model (59) (result 1) and with those obtained by a constant friction coefficient without the phase shift (result 3). Computed shear stresses for the sinusoidal wave case are presented in figure 7-a). Results of the model (59) with *CT*fwr coefficients (result 1) are in close agreement with those of the *K L* model (22).

Fig. 5. Instantaneous velocity records: a – Sinusoidal wave (orbital velocity amplitude = 0.225 m/s, period = 3.6 sec); b – Cnoidal wave (total velocity amplitude = 1.107 m/s, period = 9.0 sec); c – Irregular wave (resulting from the non-linear propagation of a sinusoidal wave with a period = 3.0 sec in a channel 0.30 m depth) (Antunes do Carmo *et al*., 2003)

Turbulent Boundary Layer Models: Theory and Applications 229

result 1 is closer to result 2 than it is to result 3, for both phase and negative values. However, asymmetries are not reproduced and a discrepancy can be seen for the maximum value. Several observations can be made concerning these results (Antunes do Carmo *et al*.,

It may be assumed that a "*turbulence memory*" created for this main peak influences what happens afterwards; *ii*) If the maximum velocity value is considered to be *U*1 and the

2

0.08 *<sup>U</sup>*

2 2 1 1

 

replacing the maximum velocity with the instantaneous velocity *U t*

(59), with *CT*fwr coefficients (table 1), where *A* is given by (70):

*U*

Therefore, as in the case of a sinusoidal wave, the friction coefficient does not remain constant when velocity changes, assuming increasing values with decreasing velocity. Antunes do Carmo *et al*. (2003) propose calculating a time-dependent friction coefficient by

account the phase shift. The coefficient *f t* will accordingly be calculated using expression

<sup>2</sup>

 represents the phase lag between *U t* and the bottom shear stress *τ t* at the upper limit of the boundary layer. Computed shear stresses for the more complex velocity case (irregular wave obtained by the non-linear propagation of an input sinusoidal wave) is presented in figure 7-c). A comparison of results 1 and 3 with result 2 shows that result 1 is still closer to that of the *K L* model (22) than to result 3. Also, a slight discrepancy can be seen for the maximum value. Despite the "*turbulence memory effects*", the model (59) with *CTfwr* coefficients fits closely with the boundary layer model results for the three cases analysed. Comparisons were made, however, assuming that results given by the *K L* model correctly represent the real conditions. Moreover, some discrepancies occur,

Following closely Antunes do Carmo *et al*. (1996), an application of the *K L* turbulence model is presented, which corresponds to a sinusoidal mass oscillation where the velocity at the top of the bottom boundary layer is a pure sinusoidal wave with amplitude *<sup>u</sup>* <sup>170</sup> cm/s and period 7.2 sec. The following values were considered: 2.6 *ws* cm/s, 50 *<sup>d</sup>* 0.021

<sup>0</sup>*<sup>z</sup>* 0.175 10 cm, 50 2 0.042 *<sup>a</sup> z d* cm and *<sup>z</sup>* 16.2

2 2 *U T rms p U t <sup>θ</sup> <sup>A</sup> π U*

*f t*

 max

*t* does not present the symmetry of velocities *U t* .

0.24 is greater than the value calculated by (69).

(70)

*τ t Ut θ U t θ* (71)

(69)

cm. Figure 8-a) to d) show the

, which takes into

*t* are more important after the main positive peak than before it.

2003): *i*) The representative curve

minimum velocity value *U*<sup>2</sup> , it follows that:

Figure 7-b) shows that the relation 2 1

*t* is defined by (71):

especially for the maximum values.

**6.1 K-L 1DV boundary layer model** 

**6. Applications** 

cm, <sup>2</sup>

The negative values of

and 

Fig. 6. Comparisons between the parameterized friction coefficient and the *K L* model result for a sinusoidal wave. Model (59) with *CTfwr* coefficients (result 1: ----; result 3: …..…) and that obtained by *K L* model (result 2: \_\_\_\_\_) (Antunes do Carmo *et al*., 2003)

Fig. 7. Comparisons between the parameterized shear stress and the *K L* model result: a) Sinusoidal wave; b) Cnoidal wave; c) Irregular wave. Model (59) with *CTfwr* coefficients (result 1: ----; result 3: .…....) and that obtained with *K L* model (result 2: \_\_\_\_\_) (Antunes do Carmo *et al*., 2003)

A phase error between result 3 and result 2 ( *K L* model) is evident. In the cnoidal wave case, the bottom shear stress calculated by the numerical boundary layer model is represented in figure 7-b) by the continuous line. As can be seen, for this case (figure 7-b), 228 Advanced Fluid Dynamics

Fig. 6. Comparisons between the parameterized friction coefficient and the *K L* model result for a sinusoidal wave. Model (59) with *CTfwr* coefficients (result 1: ----; result 3: …..…) and that obtained by *K L* model (result 2: \_\_\_\_\_) (Antunes do Carmo *et al*., 2003)

Fig. 7. Comparisons between the parameterized shear stress and the *K L* model result: a) Sinusoidal wave; b) Cnoidal wave; c) Irregular wave. Model (59) with *CTfwr* coefficients (result 1: ----; result 3: .…....) and that obtained with *K L* model (result 2: \_\_\_\_\_)

A phase error between result 3 and result 2 ( *K L* model) is evident. In the cnoidal wave case, the bottom shear stress calculated by the numerical boundary layer model is represented in figure 7-b) by the continuous line. As can be seen, for this case (figure 7-b),

(Antunes do Carmo *et al*., 2003)

result 1 is closer to result 2 than it is to result 3, for both phase and negative values. However, asymmetries are not reproduced and a discrepancy can be seen for the maximum value. Several observations can be made concerning these results (Antunes do Carmo *et al*., 2003): *i*) The representative curve *t* does not present the symmetry of velocities *U t* . The negative values of *t* are more important after the main positive peak than before it. It may be assumed that a "*turbulence memory*" created for this main peak influences what happens afterwards; *ii*) If the maximum velocity value is considered to be *U*1 and the minimum velocity value *U*<sup>2</sup> , it follows that:

$$\frac{\tau\_2}{\tau\_1} = \left(\frac{\mathcal{U}\_2}{\mathcal{U}\_1}\right)^2 = 0.08\tag{69}$$

Figure 7-b) shows that the relation 2 1 0.24 is greater than the value calculated by (69). Therefore, as in the case of a sinusoidal wave, the friction coefficient does not remain constant when velocity changes, assuming increasing values with decreasing velocity. Antunes do Carmo *et al*. (2003) propose calculating a time-dependent friction coefficient by replacing the maximum velocity with the instantaneous velocity *U t* , which takes into account the phase shift. The coefficient *f t* will accordingly be calculated using expression (59), with *CT*fwr coefficients (table 1), where *A* is given by (70):

$$A = \frac{\sqrt{2}\,\mathrm{U}\_{rms}\,\mathrm{T}\_p\,\mathrm{U}\,(t+\theta)}{2\pi} \frac{\mathrm{U}\,(t+\theta)}{\mathrm{U}\_{\mathrm{max}}} \tag{70}$$

and *t* is defined by (71):

$$
\tau(t) = \frac{f\left(t\right)}{2} \mathcal{U}\left(t+\theta\right) \left| \mathcal{U}\left(t+\theta\right) \right| \tag{71}
$$

 represents the phase lag between *U t* and the bottom shear stress *τ t* at the upper limit of the boundary layer. Computed shear stresses for the more complex velocity case (irregular wave obtained by the non-linear propagation of an input sinusoidal wave) is presented in figure 7-c). A comparison of results 1 and 3 with result 2 shows that result 1 is still closer to that of the *K L* model (22) than to result 3. Also, a slight discrepancy can be seen for the maximum value. Despite the "*turbulence memory effects*", the model (59) with *CTfwr* coefficients fits closely with the boundary layer model results for the three cases analysed. Comparisons were made, however, assuming that results given by the *K L* model correctly represent the real conditions. Moreover, some discrepancies occur, especially for the maximum values.

### **6. Applications**

#### **6.1 K-L 1DV boundary layer model**

Following closely Antunes do Carmo *et al*. (1996), an application of the *K L* turbulence model is presented, which corresponds to a sinusoidal mass oscillation where the velocity at the top of the bottom boundary layer is a pure sinusoidal wave with amplitude *<sup>u</sup>* <sup>170</sup> cm/s and period 7.2 sec. The following values were considered: 2.6 *ws* cm/s, 50 *<sup>d</sup>* 0.021 cm, <sup>2</sup> <sup>0</sup>*<sup>z</sup>* 0.175 10 cm, 50 2 0.042 *<sup>a</sup> z d* cm and *<sup>z</sup>* 16.2 cm. Figure 8-a) to d) show the

Turbulent Boundary Layer Models: Theory and Applications 231

ii. the pick concentration in the time series occurs with larger and larger phase the further

iii. at the upper levels, a time phase shift between the computed values of concentration

iv. in the vicinity of the bottom (figure 8-a)) the time series of concentration shows the

The flow in the bottom boundary layer established over a rippled bed was investigated through experiments and numerical calculations with a 2DV model. Experiments were conducted in an oscillatory flow tunnel illustrated in figure 9. This device was built from an existing wave flume at the Department of Civil Engineering of the University of Coimbra, Portugal. The wave tunnel has a rectangular cross section with 0.30 m width and 0.20 m

0.6 m

7.5 m

At the left end (A) the vertical motion of a wave paddle produces an oscillatory flow within the tunnel. Five artificial symmetrical ripples have been placed on the tunnel's bed: each of the ripples has a length (*Lr*) of 7 cm and height (*Hr*) equal to 1.2 cm. The ripples were made

4 4 0 2 *r r r r*

(72)

*H H z x x H ; x L /*

Sediment with a median grain diameter of 0.27 mm was glue to the surface of the ripples in order to simulate the skin roughness. Velocities were measured with an acoustic Doppler system (ADV) under sinusoidal oscillations at the wave paddle, over one ripple crest and one trough. Table 2 presents the experimental conditions considered in one of the tests made, being *z*1 the height above the crest where the measurements were done. With the configuration of the ADV used, the measurements could only be done for heights above 4 cm from the bed. Figure 10 represents the mean values of the measured values of *u* and *w* at different levels during the wave cycle: *u* and *w* represent, respectively, the horizontal

0.2 m

A B

2 2

*L L*

velocity in wave's tunnel direction and the vertical velocity.

*r r*

v. the maximum values of sediment concentration agree well with data at all levels.

i. the vertical distribution of sediment agrees well with experimental data;

The analyses of results show that:

intermittence phenomena;

**6.2 2DV boundary layer model** 

Fig. 9. Wave tunnel

away the level is located from the bed;

and the experimental ones is observed;

high. The total length of the tunnel is 7.5 m.

in aluminium with the following profile (72):

time series of sediment concentration computed at different levels above *<sup>a</sup> z* (z = 0.10, 1.62, 2.08 and 4.54 cm). In figure 8-e) the vertical profiles of sediment concentration with phase shift of 60º are plotted (full lines), as well as the mean values over a wave period (dash lines). In each case the numerical solutions are compared to experimental data obtained by Ribberink & Al-Salem (Tran-Thu, 1995). Finally, in figure 8-f) the eddy diffusivity vertical profile averaged over a wave period is plotted.

Fig. 8. Sinusoidal mass oscillation *(*Antunes do Carmo *et al*., 1996)

The analyses of results show that:

230 Advanced Fluid Dynamics

time series of sediment concentration computed at different levels above *<sup>a</sup> z* (z = 0.10, 1.62, 2.08 and 4.54 cm). In figure 8-e) the vertical profiles of sediment concentration with phase shift of 60º are plotted (full lines), as well as the mean values over a wave period (dash lines). In each case the numerical solutions are compared to experimental data obtained by Ribberink & Al-Salem (Tran-Thu, 1995). Finally, in figure 8-f) the eddy diffusivity vertical

profile averaged over a wave period is plotted.

Fig. 8. Sinusoidal mass oscillation *(*Antunes do Carmo *et al*., 1996)


#### **6.2 2DV boundary layer model**

The flow in the bottom boundary layer established over a rippled bed was investigated through experiments and numerical calculations with a 2DV model. Experiments were conducted in an oscillatory flow tunnel illustrated in figure 9. This device was built from an existing wave flume at the Department of Civil Engineering of the University of Coimbra, Portugal. The wave tunnel has a rectangular cross section with 0.30 m width and 0.20 m high. The total length of the tunnel is 7.5 m.

Fig. 9. Wave tunnel

At the left end (A) the vertical motion of a wave paddle produces an oscillatory flow within the tunnel. Five artificial symmetrical ripples have been placed on the tunnel's bed: each of the ripples has a length (*Lr*) of 7 cm and height (*Hr*) equal to 1.2 cm. The ripples were made in aluminium with the following profile (72):

$$\mathbf{x} = \frac{4H\_r}{L\_r}\mathbf{x}^2 - \frac{4H\_r}{L\_r}\mathbf{x} + H\_r \; ; \; \; 0 \le \mathbf{x} \le L\_r \; / \; \mathbf{2} \tag{72}$$

Sediment with a median grain diameter of 0.27 mm was glue to the surface of the ripples in order to simulate the skin roughness. Velocities were measured with an acoustic Doppler system (ADV) under sinusoidal oscillations at the wave paddle, over one ripple crest and one trough. Table 2 presents the experimental conditions considered in one of the tests made, being *z*1 the height above the crest where the measurements were done. With the configuration of the ADV used, the measurements could only be done for heights above 4 cm from the bed. Figure 10 represents the mean values of the measured values of *u* and *w* at different levels during the wave cycle: *u* and *w* represent, respectively, the horizontal velocity in wave's tunnel direction and the vertical velocity.

Turbulent Boundary Layer Models: Theory and Applications 233

These values were divided by the amplitude of the horizontal velocity outside the boundary layer, *Uw*, and were obtained by averaging *equi*-phase data over approximately 20 wave periods. The analysis of figure 10 shows that: (1) The oscillatory flow in the wave tunnel does not correspond to a sinusoidal oscillation as we can observe from the velocities measured at the highest levels (a), and (2) At the lower levels (c, d) the measured values of *w* show oscillations with a time scale inferior to the ones observed in the highest levels: this suggest that the flow at those levels is perturbed by the lee vortex developed during the

A numerical simulation of the flow in the bottom boundary layer was done with the 2DV model (50) to (54). In figure 11 the numerical results are compared with the experimental

It is seen that there is a good agreement between the computed and measured horizontal velocity. The computed vertical velocity shows small oscillations after flow reversal, between 0º-90º and 210º-300º. The shape of these oscillations is similar to the observed one, although there is a phase shift between them. The amplitude of these oscillations is also lower than the amplitude of the measured values of *w*: this means that the model dissipates the kinetic energy of the ejected vortex at a rate that is superior to what it is observed. This feature has also been noted in other comparisons. To analyse with more detail the flow in the bottom boundary layer, namely the vortex paths during the wave cycle, we have plotted in figure 12 the vorticity field at different wave phases. It is seen that the lowest level of measurements over the ripple crest is above the track of the vortex that is carried by the flow after it is ejected. This justifies the poor agreement between the numerical and experimental

wave cycle and that are ejected from the bottom after flow reversal (0º and 180º).

data. The results are only plotted for the lower level of measurements.

Fig. 11. Numerical results *vs* experimental data (Silva, 2001)

results in the figure 11 (iv).


Table 2. Experimental conditions

Fig. 10. Measured time series of *U* and *W* for different levels above the crest (i, ii) - (z1 (a) = 8.9 cm, z1 (b) = 7.9 cm, z1 (c) = 4.0 cm, z1 (d) = 3.9 cm) and above the trough (iii, iv) – (z1 (a) = 9.9 cm, z1 (b) = 8.1 cm, z1 (c) = 2.8 cm, z1 (d) = 2.6 cm) (Silva, 2001)

232 Advanced Fluid Dynamics

**Serie Nr Crest/Trough T (s) Zi (cm)**  S1 1 Cr 3.60 3.9 - 2 Cr 3.60 4.0 - 3 Cr 3.60 4.8 - 4 Cr 3.60 5.7 - 5 Cr 3.60 6.8 - 6 Cr 3.60 7.9 - 7 Cr 3.60 8.9 - 8 T 3.60 2.6 - 9 T 3.60 2.8 - 10 T 3.60 3.1 - 11 T 3.60 3.6 - 12 T 3.60 4.1 - 13 T 3.60 4.6 - 14 T 3.60 5.1 - 15 T 3.60 6.1 - 16 T 3.60 8.1 - 17 T 3.60 9.9

Fig. 10. Measured time series of *U* and *W* for different levels above the crest (i, ii) - (z1 (a) = 8.9 cm, z1 (b) = 7.9 cm, z1 (c) = 4.0 cm, z1 (d) = 3.9 cm) and above the trough (iii, iv) – (z1 (a) =

9.9 cm, z1 (b) = 8.1 cm, z1 (c) = 2.8 cm, z1 (d) = 2.6 cm) (Silva, 2001)

Table 2. Experimental conditions

These values were divided by the amplitude of the horizontal velocity outside the boundary layer, *Uw*, and were obtained by averaging *equi*-phase data over approximately 20 wave periods. The analysis of figure 10 shows that: (1) The oscillatory flow in the wave tunnel does not correspond to a sinusoidal oscillation as we can observe from the velocities measured at the highest levels (a), and (2) At the lower levels (c, d) the measured values of *w* show oscillations with a time scale inferior to the ones observed in the highest levels: this suggest that the flow at those levels is perturbed by the lee vortex developed during the wave cycle and that are ejected from the bottom after flow reversal (0º and 180º).

A numerical simulation of the flow in the bottom boundary layer was done with the 2DV model (50) to (54). In figure 11 the numerical results are compared with the experimental data. The results are only plotted for the lower level of measurements.

Fig. 11. Numerical results *vs* experimental data (Silva, 2001)

It is seen that there is a good agreement between the computed and measured horizontal velocity. The computed vertical velocity shows small oscillations after flow reversal, between 0º-90º and 210º-300º. The shape of these oscillations is similar to the observed one, although there is a phase shift between them. The amplitude of these oscillations is also lower than the amplitude of the measured values of *w*: this means that the model dissipates the kinetic energy of the ejected vortex at a rate that is superior to what it is observed. This feature has also been noted in other comparisons. To analyse with more detail the flow in the bottom boundary layer, namely the vortex paths during the wave cycle, we have plotted in figure 12 the vorticity field at different wave phases. It is seen that the lowest level of measurements over the ripple crest is above the track of the vortex that is carried by the flow after it is ejected. This justifies the poor agreement between the numerical and experimental results in the figure 11 (iv).

Turbulent Boundary Layer Models: Theory and Applications 235

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. A three-dimensional form of conservation equations for a single Reynolds stress and for the turbulent kinetic energy is derived. However, as the full three-dimensional form of equations is very complex and not easy to solve, with many unknown correlations to model, other much simpler one- and two-dimensional boundary layer forms of these relations are derived. A brief discussion about numerical models based on control volumes and finite difference approximations is presented to solve 1DV versions of the one- and two-equation rough turbulent bottom boundary layer model of the K-L type, and of the 2DV boundary layer model. These numerical models are then used to calibrate general parametric formulations for the instantaneous bottom shear stress due to both a wave and a wave-current interaction cases. They are still used to discuss some important aspects, like the *phase shift* and the *turbulence memory effects*. Mathematical formulations and parametric approaches are extended to include the effect of suspended non cohesive sediments. Comparisons with experimental results show that both 1DV and 2DV boundary layer models are able to predict quite well the complex flow properties. However, these models are strictly valid for permanent flows in the fully developed turbulent regime at high Reynolds numbers. When the flow is oscillatory, the condition of local equilibrium of the turbulence is no longer completely satisfied, particularly at the time when the velocity of the potential flow is small. Therefore, improvements are necessary to obtain more precise

Aldama, A.A., 1990. Filtering techniques for turbulent flows simulation, *Lecture Notes in* 

Antunes do Carmo, J.S., Seabra-Santos, F.J. & Barthélemy, E., 1993. Surface waves

Antunes do Carmo J.S., Silva P. & Seabra-Santos F.J., 1996. The K-L turbulence model *vs*

Antunes do Carmo, J.S., Temperville, A. & Seabra-Santos F.J., 2003. Bottom friction and

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parametric formulations, *Hydraulic Engineering Software VI*, 323–334, Ed. W.R. Blain,

time-dependent shear stress for wave-current interaction, *Journal of Hydraulic* 

**7. Conclusion** 

results for moderate Reynolds numbers.

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**8. References** 

52137-2.

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Fig. 12. Computed vorticity field (*s*-1) at different phases of the flow. The lower levels of the measurements over the ripple crest and trough are marked
