Internal generation of waves for extended Boussinesq equations

More documents

Recommendations

Info

156 ( ) C. Lee et al.rCoastal Engineering 42 2001 155–162 superiority of the Boussinesq equations to other wave equations has been strengthened. Two types of the extended Boussinesq equations have been developed. The first type was developed by adding correction terms to the momentum equations in order to get the linear dispersion relation close to the exact one Ž Madsen et al., 1991; Madsen and Sorensen, 1992 .. The second type was developed by using the horizontal velocities at a certain level instead of the depth-averaged velocities in order to get the linear dispersion relation close to the exact one ŽNwogu, 1993 .. The model of Nwogu, which is accurate to OŽŽ kh . 2 , arh ., was developed in a different form with the velocity potential instead of the particle velocity by Chen and Liu Ž 1995 .. It was extended to OŽŽ kh . 2 , Ž arh. 3 Ž kh. 2 . by Wei et al. Ž 1995 ., and further to OŽŽ kh . 4 , Ž arh. 5 Ž kh. 4 . by Madsen and Schaffer ¨ Ž 1998 .. It was also extended to consider the case of waves on a current with OŽŽ kh . 2 , arh. by Chen et al. Ž 1998 .. The hyperbolic-type wave equations may be used to solve a lot of engineering problems such as design wave height or harbor resonance because the behavior of water waves is predicted in real time and the reflection conditions at coastal structures can be specified at will. The solution process of the hyperbolic-type wave equations starts with a calm state of surface elevations and continues until the wave energy, which is made at offshore boundaries, propagates to the region of interest. The wave energy would reflect against the coastal structures and propagate back to the offshore boundaries. The open boundary conditions may be specified perfectly if the wave energy which is reflected back to the offshore boundaries is transmitted without any re-reflection back to the computational domain. This is possible by generating waves internally at a wave generation line which is placed inside the computational domain while a sponge layer is placed at offshore boundaries Ž Larsen and Dancy, 1983 .. Then the wave which is generated at the wavemaker would propagate in both directions from the wave generation line and some wave energy which is reflected against the structure would pass through the wave generation line without any numerical distortion, and further be absorbed in the sponge layer at the offshore boundaries. There are two different viewpoints related to the internal wave generation. One is the viewpoint of mass transport that the water mass which is generated at the wave generation line transports in phase velocity. The viewpoint is applied in the Boussinesq equations of Peregrine Ž Larsen and Dancy, 1983. and the mild-slope equations of Copeland Ž 1985. ŽMad- sen and Larsen, 1987; Yoon et al., 1996 .. The other is the viewpoint of energy transport that the wave energy which is generated at the wave generation line transports in energy velocity. This viewpoint is applied in the mild-slope equations of Radder and Dingemans Ž 1985. and Copeland ŽSuh et al., 1997; Lee and Suh, 1998 .. Lee and Suh found that application of the mass transport approach to the mild-slope equations of Radder and Dingemans yields improper wave energy. They also found that, in the case of equations of Copeland, two viewpoints of the mass transport and energy transport give the same results because the energy velocity is the phase velocity. Further, they implied that, in the case of equations of Peregrine, the mass transport approach might be applied properly because the phase velocity in shallow water is almost the same as the energy velocity. In other words, it would be problematic to use the viewpoint of mass transport in deeper water. In this study, it will be found whether the mass transport or energy transport approach yields a proper wave energy for internally generating waves in the Boussinesq equations of Nwogu. In Section 2, the approaches of the mass transport and energy transport are described and the phase and energy velocities are derived. In Section 3, in a horizontally one-dimensional domain, a finite difference method is used to discretize the equations of Nwogu. Then, the cnoidal waves and sinusoidal waves are generated internally with two different approaches. Finally, concluding remarks are made in Section 4. 2. Mass transport and energy transport The extended Boussinesq equations of Nwogu Ž 1993. are written as: Eh z 2 a h 2 q=P Ž hqh. u q=P y h= Ž =Pu. Et 2 6 ž / ½ž / h q z q h= =P Ž hu. s0 Ž 1. 2 a 5
( ) C. Lee et al.rCoastal Engineering 42 2001 155–162 157 Eu q Ž uP= . uqg=h Et a½ ž / ž / 5 za Eu Eu qz = =P q= =P h s0. 2 Et Et Ž 2. In Eqs. Ž. 1 and Ž. 2 , h is the water surface elevation, u is the horizontal velocity vector at a certain elevation of zsz a, h is the mean water depth, g is the gravitational acceleration, = is the horizontal gradient operator, and asŽ z . arh 2 r2qzarh. In this study, asy0.4 is used which gives zarhs y0.55. Waves are generated internally by adding the value of h ) to the computed surface elevation h at the wave generation line at each time step. The value of h ) is given by: ° I CDt 2h cosu , mass transport ) ~ D x h s , Ž 3. CeDt I ¢ 2h cosu , energy transport D x where h I is the water surface elevation of incident waves, C and Ce are the phase and energy velocities of incident waves, respectively, u is the angle of wave rays from the x-axis, D x is the grid size in the x-axis, and Dt is the time step. The wave generation line is assumed to be parallel to the y-axis. The phase velocity and energy velocity of waves can be obtained analytically using the geometric optics approach. In the case that water depths are constant and the nonlinear terms are ignored, differentiating Eq. Ž. 1 in space, differentiating equation Eq. Ž. 2 in time, and eliminating h from two resultant equations yield the following equation as, E 2 u ygh= =Pu Ž . Et 2 ž / 1 y aq 3 Ž . 4 3 gh= =P = =Pu ž / E 2 u 2 qa h = =P s0. Ž 4 2 . Et When we consider the wave field along the direction of wave ray, i.e., s-direction, the wave field may be analyzed in one dimension. Then, the particle velocity u may be defined as, usAe ic , Ž 5. where A is the amplitude of the velocity and the phase function c has the following relations with the wave number k and angular frequency v as, Ec Ec ks , vsy . Ž 6. Es Et Substituting Eq. Ž. 5 into Eq. Ž. 4 and rearranging the real part of the resultant equation to the order of OŽ. 1 yield the phase velocity C as: ) ž / 2 1 2 1y aq Ž kh v . 3 Cs s gh . Ž 7. k 1ya Ž kh. Substituting Eq. Ž. 5 into Eq. Ž. 4 , rearranging the imaginary part of the resultant equation to the order of OŽ E ArEt, E ArEs ., and further multiplying by A yield the transport equation for wave energy given as, E A 2 E A 2 qCe s0, Ž 8. Et Es where the energy velocity Ce is given by: 2 Ž kh. Ce sC 1y . 1 2 2 3½1yž aq / Ž kh. 51ya Ž kh. 4 3 Ž 9. It should be noted that the energy velocity Ce is the same as the group velocity Cg sEvrEk obtained by Nwogu Ž 1993 .. 3. Numerical experiment Numerical experiments are conducted in horizontally one-dimensional cases. The governing equations are discretized in time using the predictor-cor-
Page 1: Ž . Coastal Engineering 42 2001 15
Page 5 and 6: ( ) C. Lee et al.rCoastal Engineeri
Page 7 and 8: ( ) C. Lee et al.rCoastal Engineeri

Internal generation of waves for extended Boussinesq equations

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?