Fully Nonhydrostatic Modeling of Surface Waves
Publication: Journal of Engineering Mechanics
Volume 132, Issue 4
Abstract
A fully nonhydrostatic model is tested by simulating a range of surface-wave motions, including linear dispersive waves, nonlinear Stokes waves, wave propagation over bottom topographies, and wave–current interaction. The model uses an efficient implicit method to solve the unsteady, three-dimensional, Navier-Stokes equations and the fully nonlinear free-surface boundary conditions. A new top-layer pressure treatment is incorporated to fully include the nonhydrostatic pressure effect. The model results are verified against either analytical solutions or experimental data. It is found that the model using a small number of vertical layers is capable of accurately simulating both the free-surface elevation and vertical flow structure. By further examining the model’s performance of resolving wave dispersion and nonlinearity, the model’s efficiency and accuracy are demonstrated.
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Acknowledgments
This study was supported in part by the North Temperate Lakes Long-Term Ecological Research (NSF 0217533), the Wisconsin Alumni Research Foundation (WARF 040061), and the University of Wisconsin Sea Grant (NA16RG2257). The authors thank Dr. S. Beji at the Istanbul Technical University and Dr. K. Nadaoka at the Tokyo Institute of Technology for providing the experimental data.
References
Arakawa, A. (1966). “Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part 1.” J. Comput. Phys., 1, 119–143.
Beji, S., and Battjes, J. A. (1993). “Experimental investigation of wave propagation over a bar.” Coastal Eng., 19, 151–162.
Berkhoff, J. C. W., Booy, N., and Radder, A. C. (1982). “Verification of numerical wave propagation models for simple harmonic linear water waves.” Coastal Eng., 6, 255–279.
Casulli, V. (1999). “A semi-implicit finite difference method for non-hydrostatic, free-surface flows.” Int. J. Numer. Methods Fluids, 30, 425–440.
Casulli, V., and Zanolli, P. (2002). “Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems.” Math. Comput. Modell., 36(9-10), 1131–1149.
Chen, C. J., and Chen, H. C. (1984). “Finite analytical numerical method for unsteady two-dimensional Navier-Stokes equations.” J. Comput. Phys., 53, 209–222.
Chen, Q., Kirby, J. T., Dalrymple, R. A., Kennedy, A. B., and Chawla, A. (2000). “Boussinesq modeling of wave transformation, breaking, and runup. II: 2D.” J. Waterw., Port, Coastal, Ocean Eng., 126(1), 48–56.
Chen, H. C., and Lee, S. K. (1996). “Interactive RANS/Laplace method for nonlinear free surface flows.” J. Eng. Mech., 122(2), 153–162.
Chen, Q., Madsen, P. A., Schaffer, H. A., and Basco, D. R. (1998). “Wave-current interaction based on an enhanced Boussinesq approach.” Coastal Eng., 33(1), 11–39.
Dingemans, M. W. (1994). “Comparison of computations with Boussinesq-like models and laboratory measurements.” Technical Rep. H1684.12, Delft Hydraulics.
Fenton, J. D. (1985). “A fifth-order Stokes theory for steady waves.” J. Waterw., Port, Coastal, Ocean Eng., 111(2), 216–234.
Hsiao, S.-C., Lynett, P., Hwung, H.-H., and Liu, P. L.-F. (2005). “Numerical simulations of nonlinear short waves using a multilayer model.” J. Eng. Mech., 131(3), 231–243.
Hsu, T.-J., and Liu, P. L.-F. (2004). “Toward modeling turbulent suspension of sand in the nearshore.” J. Geophys. Res., [Oceans], 109, C06018.
Huang, C.-J., Zhang, E.-C., and Lee, J.-F. (1998). “Numerical simulation of nonlinear viscous wavefields generated by piston-type wavemaker.” J. Eng. Mech., 124(10), 1110–1120.
Israeli, M., and Orszag, S. A. (1981). “Approximation of radiation boundary conditions.” J. Comput. Phys., 41(1), 115–135.
Keller, H. B. (1974). “Accurate difference methods for nonlinear two-point boundary value problems.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 11(2), 305–320.
Kennedy, A. B., Chen, Q., Kirby, J. T., and Dalrymple, R. A. (2000). “Boussinesq modeling of wave transformation, breaking, and runup. I: 1D.” J. Waterw., Port, Coastal, Ocean Eng., 126(1), 39–47.
Li, B., and Fleming, C. A. (2001). “Three-dimensional model of Navier-Stokes equations for water waves.” J. Waterw., Port, Coastal, Ocean Eng., 127(1), 16–25.
Li, T., Troch, P., and De Rouck, J. (2004). “Wave overtopping over a sea dike.” J. Comput. Phys., 198, 686–726.
Lin, P. Z., and Li, C. W. (2002). “A -coordinate three-dimensional numerical model for surface wave propagation.” Int. J. Numer. Methods Fluids, 38, 1045–1068.
Lin, P. Z., and Liu, P. L.-F. (1998). “A numerical study of breaking waves in the surf zone.” J. Fluid Mech., 359, 239–264.
Lin, P. Z., and Liu, P. L.-F. (1999). “Internal wave-maker for Navier-Stokes equations models.” J. Waterw., Port, Coastal, Ocean Eng., 125(4), 207–217.
Liu, P. L.-F. (1995). “Model equations for wave propagations from deep to shallow water.” Advances in coastal and ocean engineering, P. L.-F. Liu, ed., World Scientific, Singapore, 125–158.
Liu, P. L.-F., and Losada, I. J. (2002). “Wave propagation modeling in coastal engineering.” J. Hydraul. Res., 40(3), 229–240.
Luth, H. R., Klopman, G., and Kitou, N. (1994). “Project 13G: Kinematics of waves breaking partially on an offshore bar; LDV measurements for waves with and without a net onshore current.” Technical Rep. H1573, Delft Hydraulics.
Lynett, P., and Liu, P. L.-F. (2004). “A two-layer approach to wave modeling.” Proc. R. Soc. London, Ser. A, 460(2049), 2637–2669.
Madsen, P. A., Bingham, H. B., and Liu, H. (2002). “A new Boussinesq method for fully nonlinear waves from shallow to deep water.” J. Fluid Mech., 462, 1–30.
Madsen, P. A., Murray, R., and Sorensen, O. R. (1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics.” Coastal Eng., 15, 371–388.
Mayer, S., Garapon, A., and Sorensen, L. S. (1998). “A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics.” Int. J. Numer. Methods Fluids, 28, 293–315.
Nadaoka, K., Beji, S., and Nakagawa, Y. (1994). “A fully-dispersive nonlinear wave model and its numerical solutions.” Proc., 24th Int. Conf. Coastal Eng., ASCE, 1, 427–441.
Namin, M., Lin, B., and Falconer, R. A. (2001). “An implicit numerical algorithm for solving non-hydrostatic free-surface flow problems.” Int. J. Numer. Methods Fluids, 35, 341–356.
Nepf, H., and Monismith, S. (1994). “Wave dispersion on a sheared current.” Appl. Ocean. Res., 16(5), 313–316.
Nwogu, O. (1993). “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterw., Port, Coastal, Ocean Eng., 119(6), 618–638.
Ohyama, T., Kiota, W., and Tada, A. (1995). “Applicability of numerical models to nonlinear dispersive waves.” Coastal Eng., 24, 297–313.
Orlanski, I. (1976). “Simple boundary-condition for unbounded hyperbolic flows.” J. Comput. Phys., 21(3), 251–269.
Pacanowski, R. C., and Gnanadesikan, A. (1998). “Transient response in a z-level ocean model that resolves topography with partial cell.” Mon. Weather Rev., 126, 3248–3270.
Peregrine, D. H. (1967). “Long waves on a beach.” J. Fluid Mech., 27, 815–827.
Peregrine, D. H. (1976). “Interaction of water waves and currents.” Adv. Appl. Mech., 16, 9–117.
Stelling, G., and Zijlema, M. (2003). “An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation.” Int. J. Numer. Methods Fluids, 43, 1–23.
Tsao, S. (1959). “Behavior of surface waves on linearly varying flow.” Tr. Mosk. Fiz-Tech. Inst. Issled. Mekh. Prikl. Mat., 3, 66–84.
Vincent, C. L., and Briggs, M. J. (1989). “Refraction-diffraction of irregular waves over a mound.” J. Waterw., Port, Coastal, Ocean Eng., 115(2), 269–284.
Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. (1995). “A fully nonlinear Boussinesq model for surface waves. Part 1—Highly nonlinear unsteady waves.” J. Fluid Mech., 294, 71–92.
Wu, T. Y. (1999). “Modeling nonlinear dispersive water waves.” J. Eng. Mech., 125(7), 747–755.
Wu, T. Y. (2001). “A unified theory for modeling water waves.” Advances in applied mechanics, E. van der Giessen and T. Y. Wu, eds., Academic Press, New York, 37, 1–88.
Wu, C. H., and Yao, A. (2004). “Laboratory measurements of limiting freak waves on currents.” J. Geophys. Res., [Oceans], 109, C12002.
Yao, A., and Wu, C. H. (2005). “Incipient breaking of unsteady waves on sheared currents.” Phys. Fluids, 17, 082104.
Yoon, S. B., and Liu, P. L.-F. (1989). “Interactions of currents and weakly nonlinear water-waves in shallow-water.” J. Fluid Mech., 205, 397–419.
Yuan, H. L., and Wu, C. H. (2004). “An implicit three-dimensional fully non-hydrostatic model for free-surface flows.” Int. J. Numer. Methods Fluids, 46, 709–733.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 132 • Issue 4 • April 2006
Pages: 447 - 456
Copyright
© 2006 ASCE.
History
Received: Mar 23, 2005
Accepted: Jul 6, 2005
Published online: Apr 1, 2006
Published in print: Apr 2006
Notes
Note. Associate Editor: Nikolaos D. Katopodes
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