Abstract
We propose a three-dimensional non-hydrostatic shock-capturing numerical model for the simulation of wave propagation, transformation and breaking, which is based on an original integral formulation of the contravariant Navier–Stokes equations, devoid of Christoffel symbols, in general time-dependent curvilinear coordinates. A coordinate transformation maps the time-varying irregular physical domain that reproduces the complex geometries of coastal regions to a fixed uniform computational one. The advancing of the solution is performed by a second-order accurate strong stability preserving Runge–Kutta fractional-step method in which, at every stage of the method, a predictor velocity field is obtained by the shock-capturing scheme and a corrector velocity field is added to the previous one, to produce a non-hydrostatic divergence-free velocity field and update the water depth. The corrector velocity field is obtained by numerically solving a Poisson equation, expressed in integral contravariant form, by a multigrid technique which uses a four-colour Zebra Gauss–Seidel line-by-line method as smoother. Several test cases are used to verify the dispersion and shock-capturing properties of the proposed model in time-dependent curvilinear grids.
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References
Tonelli, M., Petti, M.: Shock-capturing Boussinesq model for irregular wave propagation. Coast. Eng. 61, 8–19 (2012)
Ortiz, P.: Shallow water flows over flooding areas by a flux-corrected finite element method. J. Hydraul. Res. 52(2), 241–252 (2014)
Gallerano, F., Cannata, G., Scarpone, S.: Bottom changes in coastal areas with complex shorelines. Eng. Appl. Comput. Fluid 11(1), 396–416 (2017)
Roeber, V., Cheung, K.F.: Boussinesq-type model for energetic breaking waves in fringing reef environments. Coast. Eng. 70, 1–20 (2012)
Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T.: A high-order adaptive time-stepping TVD solver for Boussinesq modelling of breaking waves and coastal inundation. Coast. Eng. 43–44, 36–51 (2012)
Gallerano, F., Cannata, G., Lasaponara, F.: Numerical simulation of wave transformation, breaking and runup by a contravariant fully non-linear Boussinesq equations model. J. Hydrodyn. 28(3), 379–388 (2016b)
Phillips, N.A.: A coordinate system having some special advantages for numerical forecasting. J. Meteorol. 14, 184–185 (1957)
Lin, P., Li, C.W.: A \(\upsigma \)-coordinate three-dimensional numerical model for surface wave propagation. Int. J. Numer. Methods Fluids 38, 1045–1068 (2002)
Young, C.C., Wu, C.H.: A \(\upsigma \)-coordinate non-hydrostatic model with embedded Boussinesq-type-like equations for modelling deep-water waves. Int. J. Numer. Methods Fluids 63, 1448–1470 (2010)
Bradford, S.F.: Godunov-based model for non-hydrostatic wave dynamics. J. Waterw. Port Coast. 131, 226–238 (2005)
Bradford, S.F.: Non-hydrostatic model for surf zone simulation. J. Waterw. Port Coast. 137, 163–174 (2011)
Ma, G., Shi, F., Kirby, J.T.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model. 4344, 22–35 (2012)
Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover, New York (1989)
Gallerano, F., Cannata, G., Tamburrino, M.: Upwind WENO scheme for shallow water equations in contravariant formulation. Comput. Fluids 62, 1–12 (2012)
Gallerano, F., Cannata, G., Lasaponara, F.: A new numerical model for simulations of wave transformation, breaking and longshore currents in complex coastal regions. Int. J. Numer. Methods Fluids 80(10), 571–613 (2016a)
Rosenfeld, M., Kwak, D.: Time-dependent solutions of viscous incompressible flows in moving co-ordinates. Int. J. Numer. Methods Fluids 13, 1311–1328 (1991)
Segal, A., Wesseling, P., van Kan, J., Oosterlee, C.W., Kassels, K.: Invariant discretization of the incompressible Navier–Stokes equations in boundary fitted co-ordinates. Int. J. Numer. Methods Fluids 15, 411–426 (1992)
Sharatchandra, M.C., Rhode, D.L.: New, strongly conservative finite-volume formulation for fluid flow in irregular geometries using contravariant velocity components. Numer. Heat Transf. B Fund. 26, 39–62 (1994)
Yang, H.Q., Habchi, S.D., Przekwas, A.J.: General strong conservation formulation of Navier–Stokes equations in non-orthogonal curvilinear coordinates. AIAA J. 32(5), 936–941 (1994)
Zang, Y., Street, R.L., Koseff, J.R.: A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 18–33 (1994)
Carlson, H.A., Berkooz, G., Lumley, J.L.: Direct numerical simulation of flow in a channel with complex, time-dependent wall geometries: a pseudo spectral method. J. Comput. Phys. 121, 155–175 (1995)
Xu, S., Rempfer, D., Lumlet, J.: Turbulence over a compliant surface: numerical simulation and analysis. J. Fluid Mech. 478, 11–34 (2003)
Ogawa, S., Ishiguro, T.: A method for computing flow fields around moving bodies. J. Comput. Phys. 69, 49–68 (1987)
Luo, H., Bewley, T.R.: On the contravariant form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems. J. Comput. Phys. 199, 355–375 (2004)
Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A practical Introduction, 3rd edn. Springer, Berlin (2009)
Harten, A., Lax, P.D., vanLeer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)
Vinokur, M.: An analysis of finite-difference and finite-volume formulations of conservation laws. J. Comput. Phys. 81, 1–52 (1989)
Kundu, P., Cohen, I., Dowling, D.: Fluid Mechanics, 5th edn. Academic Press, Cambridge (2011)
Zijlema, M., Stelling, G.: Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure. Coast. Eng. 55, 780–790 (2008)
Stelling, G., Zijlema, M.: 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), 1–23 (2003)
Gallerano, G., Pasero, E., Cannata, G.: A dynamic two-equation sub grid scale model. Contin. Mech. Thermodyn. 12(7), 101–123 (2005)
Rossmanith, J., Bale, D.S., LeVeque, R.J.: A wave propagation algorithm for hyperbolic systems on curved manifolds. J. Comput. Phys. 199(2), 631662 (2004)
Cannata, G., Lasaponara, F., Gallerano, F.: Non-linear shallow water equations numerical integration on curvilinear boundary-conforming grids. WSEAS Trans. Fluid Mech. 10, 13–25 (2015)
Gallerano, F., Cannata, G., DeGaudenzi, O., Scarpone, S.: Modelling bed evolution using weakly coupled phase-resolving wave model and wave-averaged sediment transport model. Coast. Eng. J. 58(3), 1650011 (2016c)
Patankar, S.V.: Numerical Heat Transfer and Fluid Flow. Mc-Graw-Hill Book Company, New York (1980)
Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education, London (2007)
Trottemberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Academic Press, New York (2001)
Casulli, V., Stelling, G.S.: Numerical simulation of 3D quasi-hydrostatic free surface flows. J. Hydraul. Eng. ASCE 124(7), 678–686 (1998)
Casulli, V.: A semi-implicit finite difference method for non-hydrostatic, free surface flow. Int. J. Numer. Methods Fluids 30, 425–440 (1999)
Beji, S., Battjes, J.A.: Experimental investigation of wave propagation over a bar. Coast. Eng. 19, 151–162 (1993)
Beji, S., Battjes, J.A.: Numerical simulation of nonlinear wave propagation over a bar. Coast. Eng. 23, 1–16 (1994)
Li, B., Fleming, C.A.: Three-dimensional model of Navier–Stokes equations for water waves. J. Waterw. Port Coast 127, 16–25 (2001)
Yuan, H., Wu, C.H.: An implicit three-dimensional fully non-hydrostatic model for free-surface flows. Int. J. Numer. Methods Fluids 46, 709–733 (2004)
Madsen, P.A., Sørensen, O.R., Schäffer, H.A.: Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves. Coast. Eng. 32, 255–287 (1997)
Stive M.J.F.: Velocity and pressure field of spilling breakers. In: Proceedings of the 17th International Conference on Coastal Engineering, vol. 1, pp. 547–566 (1980)
Tonelli, M., Petti, M.: Hybrid finite volume-finite difference scheme for 2DH improved Boussinesq equations. Coast. Eng. 56, 609–620 (2009)
Hamm L.: Directional nearshore wave propagation over a rip channel: an experiment. In: Proceedings of the 23rd International Conference of Coastal Engineering (1992)
Sørensen, O.R., Schäffer, H.A., Madsen, P.A.: Surf zone dynamics simulated by a Boussinesq type model, III. Wave-induced horizontal nearshore circulation. Coast. Eng. 50, 181–198 (1998)
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Communicated by Andreas Öchsner.
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Cannata, G., Petrelli, C., Barsi, L. et al. Numerical integration of the contravariant integral form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems for three-dimensional free surface flows. Continuum Mech. Thermodyn. 31, 491–519 (2019). https://doi.org/10.1007/s00161-018-0703-1
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DOI: https://doi.org/10.1007/s00161-018-0703-1