Internal generation of waves for extended Boussinesq equations

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158 ( ) C. Lee et al.rCoastal Engineering 42 2001 155–162 rector scheme. Both cnoidal and sinusoidal waves are generated internally from two viewpoints of the mass transport and energy transport. 3.1. Finite difference method Sponge layers are placed at boundaries in order to minimize wave reflections from the boundaries. Eqs. Ž. 1 and Ž. 2 can be written in a horizontally one-dimensional domain as, ½ž / Eh E E z 2 a h 2 E 2 u q Ž hqh. u4 q y h 2 Et Ex Ex 2 6 Ex ž / h E 2 q z q h Ž hu. s0 Ž 10. 2 Ex a 2 5 ½ ž / 5 Eu 1 Eu 2 Eh za E 3 u E 2 Eu q qg qza q h 2 2 Et 2 Ex Ex 2 Ex Et Ex Et qv Dus0, s Ž 11. where, ° d r S ~ e y1 , inside sponge layer Ds s ey1 . Ž 12. ¢ 0, outside sponge layer In Eq. Ž 12 ., Ss2.5 L Ž L is the wavelength. is the thickness of the sponge layer and d is the distance from the starting point of the sponge layer. The variables h and u in Eqs. Ž 10. and Ž 11. iy1r2 i are placed in a staggered grid system where the subscript i denotes the spatial grid point. The firstorder spatial derivative terms in Eqs. Ž 10. and Ž 11. are discretized to OŽŽ D x . 4 .. Both the second-and third-order spatial derivative terms in Eq. Ž 10. are discretized to OŽŽ D x. 2 . and the second-order spatial derivative terms in Eq. Ž 11. are discretized to OŽŽ D x . 4 .. The time derivative terms in Eqs. Ž 10. and Ž 11. are discretized to OŽŽ Dt. 2 . using the predictor-corrector method. Eq. Ž 10. is discretized explicitly between the n-th and Ž nq1r2. -th time steps to get the p surface elevation h at the Ž nq1r2. -th time step. The superscript p in h stands for the predictor. Eq. Ž 10. is discretized again implicitly between the n-th and Ž nq1r2. -th time steps to get the surface elevanq1r2 tion h at the Ž nq1r2. -th time step. At this time, the surface elevation h and the particle velocity u in the spatial derivatives are replaced by h p and Žu n qu nq1 . r2, respectively. And then, Eq. Ž 11. is discretized between the n-th and Ž nq1. -th time steps. At this time, the surface elevation h is replaced by h nq1r2 . The resulting equation gives a hepta-diagonal matrix of u nq1 given as: Au nq1 qBu nq1 qCu nq1 qDu nq1 qEu nq1 i iy3 i iy2 i iy1 i i i iq1 qFiuiq2 nq1 qGu i iq3 nq1 sH i , Ž 13. where A i, ..., Hi are coefficients of h n and u n . Using the method of LU decomposition ŽPress et al., 1992 ., Eq. Ž 13. is solved to get the particle velocity nq1 u . Again, Eq. Ž 10. is discretized between the n-th and Ž nq1. -th time steps. At this time, the particle velocity u is replaced by Žu n qu nq1 . r2 and the surface elevation h in EŽ hqh. u4 i rEx is replaced Ž n nq1 n nq1 by h qh qh qh . iq1r2 iq1r2 iy1r2 iy1r2 r4. The resulting equation gives a penta-diagonal matrix given as, ah nq1 qbh nq1 qgh nq1 qdh nq1 i iy5r2 i iy3r2 i iy1r2 i iq1r2 qeihiq3r2 nq1 ss i , Ž 14. where a i, ..., si are coefficients of h n , u n , and u nq1 . Using the method of LU decomposition, Eq. Ž . nq1 14 is solved to get the surface elevation h . Waves are generated smoothly by multiplying Ž . Ž . ) tanh nDtrT T is the wave period to h in Eq. Ž. 3 . At both the left and right boundaries, a reflective boundary condition is specified as: u nq1 s0, u nq1 s0, h nq1 sh nq1 , 1 Iy1 1r2 3r2 h nq1 sh nq1 , Ž 15. Iy1r2 Iy3r2 where is1 and isIy1 denote the points of the left and right boundaries, respectively. The grid size is chosen as D xsLr20 in order to get sufficient spatial resolution. The time step is chosen so that the Courant number is CrsCe DtrD xs0.2 and a sta- ble solution is guaranteed.
( ) C. Lee et al.rCoastal Engineering 42 2001 155–162 159 3.2. Uni-directional regular waÕes Firstly, cnoidal waves are generated internally with the condition of Larsen and Dancy Ž 1983 .. The computational domain consists of an inner domain of 400 m and a sponge layer at the right boundary. The wave generation point is located at the mid-point of the inner computational domain. The wave period is Ts20 s, the wave height is Hs0.5 m, and the water depth is hs10 m, which gives the wavelength Ls195 m, the relative water depth khs 0.1p , the wave steepness arhs0.025, and the Ursell number U Ž . Ž . r s arh r kh 2 s0.24. Both the mass transport and energy transport approaches are used to generate waves. Fig. 1 shows the surface elevations at time between ts10T and ts11T with an interval of Tr10. Good agreement is observed between this figure and Fig. 2 of Larsen and Dancy Ž 1983 .. This is because, in shallow water with khs0.1p , the equations of Nwogu Ž 1993. are almost the same as those of Peregrine Ž 1967 .. Numerical solutions of the wave amplitude by the mass transport approach are 3% larger than those by the energy transport approach which matches the ratio of CrCe s1.03 for khs0.1p. In Fig. 1, the incident waves generated at the wave generation point propagate both directions while the waves reflected from the left boundary pass through the wave generation point with no serious numerical distortion. Such good work for internally generating waves wouldn’t be seen if the unstaggered grid scheme is used in the Boussinesq equations of Nwogu Ž Wei et al., 1999 .. This is probably because the unstaggered grid scheme may result in distortion of the solution. Lee and Suh Ž 1998. found that the use of the unstaggered grid scheme in the time-dependent mild-slope equations of Radder and Dingemans Ž 1985. is all right for the internal wave generation. This is probably because the model equations are linear and the distortion of the solution is not serious. However, the resolved wave height was found to fluctuate around the exact solution due to the use of the unstaggered grid Fig. 1. Surface elevations of internally generated cnoidal waves; solid linesenergy transport, dashed linesmass transport.
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