Pressure correction method for multi-layer open flow model and wave problems

V. A. Prokofyev

Matematičeskoe modelirovanie, Tome 29 (2017) no. 6, pp. 61-88.

Résumé

This paper describes a pressure correction method for single- and multi-layer open flow models in $\sigma$-coordinates. The method does not require any complex procedures to solve the discretization of the Poisson equation and is distinguished by a high computational efficiency. The discretization of the Poisson equation is written in a matrix form, allowing its usage also in the case of basic function expansion of the depth pressure profile. Parabolic interpolation of the pressure profile is used for the free surface. The algorithm can be parallelized for GPU. The paper presents the results of algorithm verification where experimental data sensitive to the numerical dissipation of the calculation model was used. The described method can be applied to problems with discontinuous solutions. Iteration convergence is high including problems with dry-bed flooding.

Détail

Keywords: multi-layer model, free surface, fully non-hydrostatic, waves, three-dimensional flows, boundary-fitted coordinates.

Bibliographie

[1] E. Audusse, M-O. Bristeau, B. Perthame, J. Sainte-Marie, “A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation”, ESAIM: M2AN, 45:1 (2011), 169–200

[2] E. Audusse, M-O. Bristeau, M. Pelanti, J. Sainte-Marie, “Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multi-layer model. Kinetic interpretation and numeral solution”, Journal of Computational Physics, 230:9 (2011), 3453–3478

[3] V.A. Prokofyev, “Application of unified 3D hydro-thermal model of a reservoir for estimation of HPP construction influence on environment”, ICOLD, Proceedings of the International Symposium on Dams on changing word, sect. 5 (Kyoto, 2012), 69–74

[4] G.S. Stelling, M. Zijlema, “An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation”, Int. J. Numerical Methods in Fluids, 43 (2003), 1–23

[5] G. Stelling, M. Zijlema, “Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure”, Coastal Engineering, 55 (2008), 780–790

[6] M. Zijlema, G.S. Stelling, “Further experiences with computing non-hydrostatic free-surface flows involving water waves”, Int. J. Numerical Methods in Fluids, 48 (2005), 169–197

[7] Y. Yamazaki, Z. Kowalik, K.F. Cheung, “Depth-integrated, non-hydrostatic model for wave breaking and run-up”, Int. J. Numerical Methods in Fluids, 61 (2009), 473–497

[8] P. Badiei, M.M. Namin, A. Ahmadi, “A three-dimensional non-hydrostatic vertical boundary fitted model for free-surface flows”, Int. J. Numerical Methods in Fluids, 56 (2008), 607–627

[9] C.-C. Young, C.-H. Wu, W.-C. Liu, J.-T. Kuo, “A higher-order non-hydrostatic $\sigma$-model for simulating nonlinear refraction-diffraction of water waves”, Coastal Eng., 56 (2009), 919–930

[10] G. Ma, F. Shi, J.T. Kirby, “Shock-capturing non-hydrostatic model for fully dispersive surface wave processes”, Ocean Modeling, 43–44 (2012), 22–35

[11] C. Ai, S. Jin, B. Lv, “A new fully non-hydrostatic 3D free surface flow model for water wave motions”, International Journal for Numerical Methods in Fluids, 66 (2011), 1354–1370

[12] H. Yuan, C.H. Wu, “An implicit three-dimensional fully non-hydrostatic model for free-surface flow”, International Journal for Numerical Methods in Fluids, 46 (2004), 709–733

[13] S. Ullmann, Three-dimensional computation of non-hydrostatic free-surface flows, MSc. Thesis, Delft University of Technology, Delft, 2008

[14] S. Soares Frazao, Y. Zech, “Undular bores and secondary waves - Experiments and hybrid finitevolume modeling”, Journal of Hydraulic Research, 40:1 (2002), 33–43

[15] Z. Lai, C. Chen, G.W. Cowles, R.C. Beardsley, “A nonhydrostatic version of FVCOM: 1. Validation experiments”, J. Geophys. Res., 115 (2010), C11010 | DOI

[16] M. Van Reeuwijk, Efficient simulation of non-hydrostatic free-surface flow. Wave simulation with interpolation of the vertical pressure profile, MSc. Thesis, Delft Univ. of Tech., Delft, 2002

[17] Y. Bai, K.F. Cheung, “Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation”, Int. J. Numer. Methods in Fluids, 2011 | DOI

[18] N.E. Voltsinger, K.A. Klevannyi, E.N. Pelinovskii, Dlinnovolnovaia dinamika pribrezhnoi zony, Gidrometeoizdat, L., 1989, 273 pp.

[19] A.G. Kulikovskii, N.V. Pogorelov, A. Iu. Semenov, Matematicheskie voprosy chislennogo resheniia giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001, 608 pp.

[20] C.G. Mingham, D.M. Causon, “Calculation of unsteady bore diffraction using a high resolution finite volume method”, Journal of Hydraulic research, 38:1 (2000), 49–56

[21] V.A. Prokofyev, “Two-dimensional horizontal numerical model of open flow over a bed with obstacles”, Water resources, 32:3 (2005), 252–264

[22] A.I. Delis, C.P. Skeels, S.C. Ryrie, “Evolution of some approximate Riemann solvers for transient open channel flows”, Journal of Hydraulic research, 38:3 (2000), 217–231

[23] S. Beji, J.A. Battjes, “Experimental investigation of wave propagation over a bar”, Coastal Engineering, 19 (1993), 151–162

[24] J.C.W. Berkhoff, N. Booy, A.C. Radder, “Verification of numerical wave propagation models for simple harmonic linear water waves”, Coastal Engineering, 1982, no. 6, 255–279

[25] M. Louaked, L. Hanich, “TVD scheme for shallow water equations”, J. of Hydraulic research, 36:3 (1998), 363–378

[26] M.-O. Bristeau, N. Goutal, J. Sainte-Marie, “Numerical simulations of a non-hydrostatic shallow water model”, Computers Fluids, 47:1 (2011), 51–64