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Figure from:Wave propagation modeling in coastal engineering

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The governing equations for k and € are modeled as (Rodi 1980; Lin and Liu 1998a,b),

Figure 10 The governing equations for k and € are modeled as (Rodi 1980; Lin and Liu 1998a,b),

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It has been demonstrated that with optimal choice of, z,=-0.531h the modified Boussinesq equations are able to simulate wave propagation from intermediate water depth (water depth to wave- length ratio is about 0.5) to shallow water including the wave- current interaction (Chen et al. 1998). It should be also pointed out that the convective acceleration in the momentum equation (13) and (15) has been written in the conservative form. One could replace them by the non-conservative form, i.e. 1: Vi and u,,, Vu, , respectively, without changing the order of magnitude of accuracy of the model equations. plications to shorter waves (or deeper water depth) many modi- fied forms of Boussinesq-type equations have been introduced (e.g., Madsen er al. 1991, Nwogu 1993, Chen and Liu, 1995). Although t dispersion he methods of derivation are different, the resulting relations of the linear components of these modified Boussinesq equations are similar, and may be viewed as a slight modification of the (2,2) Pade approximation of the full disper- sion relation for linear water wave (Witting 1984). Following Nwogu’s a pproach (1993), the depth-integrated continuity equa- tion and momentum equations can be expressed in terms of the free surface displacement 1 and w,, the horizontal velocity vector at the water depth z=z,, can be expressed as: It has been demonstrated that with optimal choice of, z,=-0.531h
It has been demonstrated that with optimal choice of, z,=-0.531h the modified Boussinesq equations are able to simulate wave propagation from intermediate water depth (water depth to wave- length ratio is about 0.5) to shallow water including the wave- current interaction (Chen et al. 1998). It should be also pointed out that the convective acceleration in the momentum equation (13) and (15) has been written in the conservative form. One could replace them by the non-conservative form, i.e. 1: Vi and u,,, Vu, , respectively, without changing the order of magnitude of accuracy of the model equations. plications to shorter waves (or deeper water depth) many modi- fied forms of Boussinesq-type equations have been introduced (e.g., Madsen er al. 1991, Nwogu 1993, Chen and Liu, 1995). Although t dispersion he methods of derivation are different, the resulting relations of the linear components of these modified Boussinesq equations are similar, and may be viewed as a slight modification of the (2,2) Pade approximation of the full disper- sion relation for linear water wave (Witting 1984). Following Nwogu’s a pproach (1993), the depth-integrated continuity equa- tion and momentum equations can be expressed in terms of the free surface displacement 1 and w,, the horizontal velocity vector at the water depth z=z,, can be expressed as: It has been demonstrated that with optimal choice of, z,=-0.531h
These equations are the statements of conservation of mass and momentum respectively. They are derived without making any approximation on the nonlinearity. Therefore, if one were to re- place the conservative form of the inertia term, via, I’, u,-Vu,, inthe momentum equation, (17), additional higher oe terms must be added to maintain the order of magnitude in accu- racy. It is straightforward to show that the conventional Boussinesq equations, (12) and (13), and the modified Boussinesq equations, (14) and (15), are the subsets of the unified model equations shown in (16) and (17). The hichlv nonlinear and dispersive models have been extended
These equations are the statements of conservation of mass and momentum respectively. They are derived without making any approximation on the nonlinearity. Therefore, if one were to re- place the conservative form of the inertia term, via, I’, u,-Vu,, inthe momentum equation, (17), additional higher oe terms must be added to maintain the order of magnitude in accu- racy. It is straightforward to show that the conventional Boussinesq equations, (12) and (13), and the modified Boussinesq equations, (14) and (15), are the subsets of the unified model equations shown in (16) and (17). The hichlv nonlinear and dispersive models have been extended
Table 1. Parameters for spectral tests (Chawla et al. 1998) riod, 6, is the mean wave direction, 6,, determines the width of the directional spreading and I is the width of the frequency spec- trum.
Table 1. Parameters for spectral tests (Chawla et al. 1998) riod, 6, is the mean wave direction, 6,, determines the width of the directional spreading and I is the width of the frequency spec- trum.
‘ig. 3. Numerical simulation of the free surface elevation for Case 3 (narrow-directional spread) Figure 5 presents the comparisons for Case 6 along other Assuming a Rayleigh wave height distribution, the significant wave heights for the data and the model can be obtained. Figure 4 gives the comparison of the normalized wave heights between the calculated values and the experimental data along different transects, for Case 6, which corresponds to a broader directional spread than Case 3. Wave heights have been normalized using the input significant wave height, Hy,. The comparison is quite good except near the top of the shoal where the numerical model slightly over-predicts the experimental data. Te ene i caawaneawnitica the weaeemanreenms £Fiaxn FUanan ZC. AlLAeaew wrth
‘ig. 3. Numerical simulation of the free surface elevation for Case 3 (narrow-directional spread) Figure 5 presents the comparisons for Case 6 along other Assuming a Rayleigh wave height distribution, the significant wave heights for the data and the model can be obtained. Figure 4 gives the comparison of the normalized wave heights between the calculated values and the experimental data along different transects, for Case 6, which corresponds to a broader directional spread than Case 3. Wave heights have been normalized using the input significant wave height, Hy,. The comparison is quite good except near the top of the shoal where the numerical model slightly over-predicts the experimental data. Te ene i caawaneawnitica the weaeemanreenms £Fiaxn FUanan ZC. AlLAeaew wrth
transects. The numerical results (solid lines) show for all transects a large significant wave height at the side-walls due to the reflect- ing wall boundary condition. Clearly, the worst comparisons are obtained on the top of the shoal (F-F) where the focusing and the wave breaking take place. However, in general the comparisons are good in all transects.
transects. The numerical results (solid lines) show for all transects a large significant wave height at the side-walls due to the reflect- ing wall boundary condition. Clearly, the worst comparisons are obtained on the top of the shoal (F-F) where the focusing and the wave breaking take place. However, in general the comparisons are good in all transects.
model considered. Similarly, in the numerical models based on Boussinesq-type equations, adding a new term to the depth-integrated momentum equation parameterizes the wave breaking process. While Zelt (1991), Karambas and Koutitas (1992) and Kennedy et al. (2000) used the eddy viscosity model, Brocchini et al. (1992) and Schaffer et al. (1993) employed a more complicated roller mode based on the surface roller concept for spilling breal roller model the instantaneous roller thickness at each point and the orientation of the roller must be prescribed. Furthermore, in both approximations incipient breaking has to be determined making certain assumptions. By adjusting parameters associated with the breaking models, results of these models all s kers. In the howed very reasonable agreement with the respective laboratory data for free surface profiles. However, these models are unlikely to produce accurate solutions for the velocity field or to determine spatial distributions of the turbulent kinetic energy and therefore, more specific models on breaking waves are needed.
model considered. Similarly, in the numerical models based on Boussinesq-type equations, adding a new term to the depth-integrated momentum equation parameterizes the wave breaking process. While Zelt (1991), Karambas and Koutitas (1992) and Kennedy et al. (2000) used the eddy viscosity model, Brocchini et al. (1992) and Schaffer et al. (1993) employed a more complicated roller mode based on the surface roller concept for spilling breal roller model the instantaneous roller thickness at each point and the orientation of the roller must be prescribed. Furthermore, in both approximations incipient breaking has to be determined making certain assumptions. By adjusting parameters associated with the breaking models, results of these models all s kers. In the howed very reasonable agreement with the respective laboratory data for free surface profiles. However, these models are unlikely to produce accurate solutions for the velocity field or to determine spatial distributions of the turbulent kinetic energy and therefore, more specific models on breaking waves are needed.
flow. In the momentum equation (20), the influence of the turbu- lent fluctuations on the mean flow field is represented by the Reynolds stress tensor —p (uu, . Many second-order turbulence closure models have been developed for different applications, which have been summarized in a recent review article (Jaw and Chen 1998a,b). In the present model, the Reynolds stress, -p (uiu’,) , is expressed by a nonlinear algebraic stress model (Shih et al. 1996; Lin and Liu 1998a,b):
flow. In the momentum equation (20), the influence of the turbu- lent fluctuations on the mean flow field is represented by the Reynolds stress tensor —p (uu, . Many second-order turbulence closure models have been developed for different applications, which have been summarized in a recent review article (Jaw and Chen 1998a,b). In the present model, the Reynolds stress, -p (uiu’,) , is expressed by a nonlinear algebraic stress model (Shih et al. 1996; Lin and Liu 1998a,b):
in which ©;, 6,, Cig, and Cy¢ are empirical coefficients. In the transport equation for the turbulent kinetic energy (22), the left- hand side term denotes the convection, while the first term on the right-hand side represents the diffusion. The second and the third term on the right-hand side of (22) are the production and the dis- sipation of turbulent kinetic energy, respectively. Thea erneffic-iante in annatian (91) tra (92) have heen deatarminedn hy
in which ©;, 6,, Cig, and Cy¢ are empirical coefficients. In the transport equation for the turbulent kinetic energy (22), the left- hand side term denotes the convection, while the first term on the right-hand side represents the diffusion. The second and the third term on the right-hand side of (22) are the production and the dis- sipation of turbulent kinetic energy, respectively. Thea erneffic-iante in annatian (91) tra (92) have heen deatarminedn hy
ig. 7 Free surface evolution of a spilling breaker on an impermeable plane beach. Experimental data (-) and numerical results (---). Wave breaking starts at gauge 16. Even though the computing power is rapidly increasing and that we are getting closer to in a relatively large domain, wave propagation modeling is stil dependent on several a order to simulate comp engineering problems. have been focused on the development of unified mathematica models describing wave As aresult, today several depth-integrated 2-D models for highly nonlinear and dispersive waves are available. Many aspects of be able to solve Navier-Stokes equations ternative mathematical formulations in ex wave conditions in practical coasta n the last ten years tremendous efforts propagation from deep to shallow waters.
ig. 7 Free surface evolution of a spilling breaker on an impermeable plane beach. Experimental data (-) and numerical results (---). Wave breaking starts at gauge 16. Even though the computing power is rapidly increasing and that we are getting closer to in a relatively large domain, wave propagation modeling is stil dependent on several a order to simulate comp engineering problems. have been focused on the development of unified mathematica models describing wave As aresult, today several depth-integrated 2-D models for highly nonlinear and dispersive waves are available. Many aspects of be able to solve Navier-Stokes equations ternative mathematical formulations in ex wave conditions in practical coasta n the last ten years tremendous efforts propagation from deep to shallow waters.
Fig. 8 Comparisons for horizontal and vertical velocities for previous case. Lab (solid line) and numerical model (---). SWL at z=0. problems. these models have been tested satisfactorily comparing numerical results with experimental field and laboratory data. At the present moment, if the model equations are truncated at O(u”), the model equations are adequate up to kh=3. In principle, the higher fre- quency dispersive effects can always be included in the model equations by adding the higher terms in W, resulting in higher spatial and temporal derivative terms. These higher derivatives terms create numerical difficulties as well as uncertainties in boundary conditions. Alternative approach must be taken if the unified models are to be applied to practical coastal engineering Our experience and ability of modeling wave breaking and wave and structure interaction have been improved considerably with the development of numerical models based on the Reynolds Av- eraged Navier Stokes equations. Today it is possible to analyze several dynamic processes in wave and structure interaction, such as wave dissipation and wave overtopping, only possible by means of physical models in the past. However, the development of these models is still at an early stage. The development of 3-D models, the search for adequate turbulence models, the optimiza-
Fig. 8 Comparisons for horizontal and vertical velocities for previous case. Lab (solid line) and numerical model (---). SWL at z=0. problems. these models have been tested satisfactorily comparing numerical results with experimental field and laboratory data. At the present moment, if the model equations are truncated at O(u”), the model equations are adequate up to kh=3. In principle, the higher fre- quency dispersive effects can always be included in the model equations by adding the higher terms in W, resulting in higher spatial and temporal derivative terms. These higher derivatives terms create numerical difficulties as well as uncertainties in boundary conditions. Alternative approach must be taken if the unified models are to be applied to practical coastal engineering Our experience and ability of modeling wave breaking and wave and structure interaction have been improved considerably with the development of numerical models based on the Reynolds Av- eraged Navier Stokes equations. Today it is possible to analyze several dynamic processes in wave and structure interaction, such as wave dissipation and wave overtopping, only possible by means of physical models in the past. However, the development of these models is still at an early stage. The development of 3-D models, the search for adequate turbulence models, the optimiza-
boundary conditions. Alternative approach must be taken if the
boundary conditions. Alternative approach must be taken if the