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Fully dispersive nonlinear water wave model in curvilinear coordinates. (English) Zbl 1116.76324
Summary: A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Thompson, J.F; Warsi, Z.U.A; Mastin, C.W, Numerical grid generation – foundations and applications, (1985), North-Holland Amsterdam · Zbl 0598.65086
[2] Hoffmann, K.A; Chiang, S.T, Computational fluid dynamics for engineers (third print), vols. 1 and 2, (1995), Engineering Education System Wichita, KS
[3] B.H. Johnson, VAHM-A vertically averaged hydrodynamic model using boundary-fitted coordinates. MP HL-80-3 US Army Engrs. Wtrwy. Experiment St., Vicksburg, Miss. 1980
[4] Spaulding, M.L, A vertically averaged circulation model using boundary-fitted coordinates, J. phys. oceanogr., 14, 973-982, (1984)
[5] J.B.T.M. Willemse, G.S. Stelling, G.K. Verbroom, Solving the shallow water equations with an orthogonal coordinate transformation. Delft Hydr. Communication No. 356. Delft Hydraulics Laboratory, Delft, The Netherlands, 1985
[6] Muin, M; Spaulding, M, Two-dimensional boundary-fitted circulation model in spherical coordinates, J. hydr. eng., September, 512-521, (1996)
[7] Sheng, Y.P, Numerical modeling of coastal and estuarine processes using boundary-fitted grids, 3rd int. symp. river sedimentation, 1426-1442, (1986)
[8] Androsov, A.A; Klevanny, K.A; Salusti, E.S; Voltzinger, N.E, Open boundary conditions for horizontal 2-D curvilinear-grid long-wave dynamics of a strait, Adv. water resour., 18, 267-276, (1995)
[9] Shi, F; Sun, W; Wei, G, A WDM method on a generalized curvilinear grid for calculation of storm surge flooding, Appl. Ocean res., 19, 275-282, (1997)
[10] Bao, X.W; Yan, J; Sun, W.X, A three-dimensional tidal wave model in boundary-fitted curvilinear grids, Estuar., coast. shelf sci., 50, 775-788, (2000)
[11] Li, Y.S; Zhan, J.M, Boussinesq-type model with boundary-fitted coordinate system, J. wtrwy. port coastal Ocean eng., 127, 3, 152-160, (2001)
[12] Shi, F; Dalrymple, R.A; Kirby, J.T; Chen, Q; Kennedy, A, A fully-nonlinear Boussinesq model in generalized curvilinear coordinates, Coast. eng., 42, 337-358, (2001)
[13] Romanenkov, D.A; Androsov, A.A; Voltzinger, N.E, Comparison of forms of the viscous shallow-water equations in the boundary-fitted coordinates, Ocean model., 3, 193-216, (2001)
[14] Sankaranarayanan, S; Spaulding, M.L, A study of the effects of grid non-orthogonality on the solution of shallow water equations in boundary-fitted coordinate sytems, J. comput. phys., 184, 299-320, (2003) · Zbl 1118.76327
[15] Engquist, B; Majda, A, Absorbing boundary conditions for the numerical simulation of waves, Math. comp., 31, 629-651, (1977) · Zbl 0367.65051
[16] Nadaoka, K; Beji, S; Nakagawa, Y, A fully dispersive weakly nonlinear model for water waves, Proc. R. soc. lond. A, 453, 303-318, (1997) · Zbl 0876.76012
[17] Arakawa, A; Lamb, V.R, Computational design of the basic dynamical processes of the UCLA general circulation model, Meth. comput. phys., 17, 174-265, (1977)
[18] Keller, H.B, Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. num. anal., 11, 305-320, (1974) · Zbl 0282.65065
[19] Lamb, H, Hydrodynamics, (1932), Dover New York, p. 738 · JFM 26.0868.02
[20] Rostafinski, W, Acoustic systems containing curved duct sections, J. acoust. soc. am., 60, 23-28, (1976)
[21] Korteweg, D.J; De Vries, G, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. mag., 39, 422-443, (1895) · JFM 26.0881.02
[22] Mei, C.C, The applied dynamics of Ocean surface waves, (1989), World Scientific Singapore · Zbl 0991.76003
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