An Efficient Two-Layer Non-Hydrostatic Model for Investigating Wave Run-Up Phenomena

In this paper, we study the maximum run-up of solitary waves on a sloping beach and over a reef through a non-hydrostatic model. We do a modification on the non-hydrostatic model derived by Stelling and Zijlema. The model is approximated by resolving the vertical fluid depth into two-layer system. In contrast to the two-layer model proposed by Stelling, here, we have a block of a tridiagonal matrix for the hydrodynamic pressure. The equations are then solved by applying a staggered finite volume method with predictor-corrector step. For validation, several test cases are presented. The first test is simulating the propagation of solitary waves over a flat bottom. Good results in amplitude and shape preservation are obtained. Furthermore, run-up simulations are conducted for solitary waves climbing up a sloping beach, following the experimental set-up by Synolakis. In this case, two simulations are performed with solitary waves of small and large amplitude. Again, good agreements are obtained, especially for the prediction of run-up height. Moreover, we validate our numerical scheme for wave run-up simulation over a reef, and the result confirms the experimental data.

[1]  S. Liang,et al.  Nonlinear run-ups of regular waves on sloping structures , 2012 .

[2]  A. Arakawa,et al.  A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations , 1981 .

[3]  Guus S. Stelling,et al.  A staggered conservative scheme for every Froude number in rapidly varied shallow water flows , 2003 .

[4]  Vincenzo Casulli,et al.  Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems , 2002 .

[5]  Hunt,et al.  Design of Seawalls and Breakwaters , 1959 .

[6]  L. Perelman,et al.  Hydrostatic, quasi‐hydrostatic, and nonhydrostatic ocean modeling , 1997 .

[7]  Onno Bokhove,et al.  Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry , 2014 .

[8]  Marcel Zijlema,et al.  SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coas , 2011 .

[9]  Guus S. Stelling,et al.  NUMERICAL SIMULATION OF 3D QUASI-HYDROSTATIC, FREE-SURFACE FLOWS , 1998 .

[10]  O. Nwogu.  Alternative form of Boussinesq equations for nearshore wave propagation , 1993 .

[11]  G Andadari,et al.  Analytical and numerical studies of resonant wave run-up on a plane structure , 2019, Journal of Physics: Conference Series.

[12]  Volker Roeber,et al.  Boussinesq-type model for nearshore wave processes in fringing reef environment , 2010 .

[13]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[14]  Marcel Zijlema,et al.  Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure , 2008 .

[15]  Ira Didenkulova,et al.  New Trends in the Analytical Theory of Long Sea Wave Runup , 2009, Applied Wave Mathematics.

[16]  Gangfeng Ma,et al.  Shock-capturing non-hydrostatic model for fully dispersive surface wave processes , 2012 .

[17]  Costas E. Synolakis,et al.  The runup of solitary waves , 1987, Journal of Fluid Mechanics.

[18]  Steven A. Hughes,et al.  Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter , 2004 .

[19]  C. Synolakis,et al.  The run-up of N-waves on sloping beaches , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[20]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[21]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[22]  P. A. Madsen,et al.  A new form of the Boussinesq equations with improved linear dispersion characteristics , 1991 .

[23]  G. Wei,et al.  A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves , 1995, Journal of Fluid Mechanics.

[24]  Marcel Zijlema,et al.  An accurate and efficient finite‐difference algorithm for non‐hydrostatic free‐surface flow with application to wave propagation , 2003 .

[25]  Vincenzo Casulli,et al.  A SEMI-IMPLICIT FINITE DIFFERENCE METHOD FOR NON-HYDROSTATIC, FREE-SURFACE FLOWS , 1999 .

[26]  S. R. Pudjaprasetya,et al.  Momentum Conservative Schemes for Shallow Water Flows , 2014 .

[27]  Yefei Bai,et al.  Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation , 2012 .