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The development of a Cartesian cut cell method for incompressible viscous flows. (English) Zbl 1178.76275
Summary: This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier-Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson-type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual-time iteration technique based on a slope-limited, high-order, Godunov-type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh-Taylor instability problems. Finally, a test case for wave run-up and overtopping over an impermeable sea dike is performed.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D99 Incompressible viscous fluids
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[1] Farm, AIAA Journal 32 pp 1175– (1994)
[2] ThĂ©, Numerical Heat Transfer, B: Fundamentals 26 pp 367– (1994)
[3] , . Finite volume solutions to unsteady free surface flow with application to gravity waves. Coastal Dynamics ’97. Plymouth, 1997; 118–127.
[4] Harlow, Physics of Fluids 8 pp 2182– (1967)
[5] Hirt, Journal of Computational Physics 39 pp 201– (1981)
[6] Time dependent multi-material flow with large fluid distortion. In Numerical Methods for Fluid Dynamics, (eds). Academic Press: London, 1982; 237–285.
[7] , . A solver for numerical simulation of breaking waves using a cut-cell VOF cell-staggered finite-volume approach. Technical Report, Department of Civil Engineering, Ghent University, Belgium, 2003.
[8] Ubbink, Journal of Computational Physics 153 pp 26– (1999)
[9] Unverdi, Journal of Computational Physics 100 pp 25– (1992)
[10] Sussman, Journal of Computational Physics 114 pp 146– (1994)
[11] Kelecy, Journal of Computational Physics 138 pp 939– (1997) · Zbl 0903.76058
[12] Pan, International Journal for Numerical Methods in Fluids 33 pp 203– (2000)
[13] Qian, Journal of hydraulic Engineering 129 pp 688– (2003)
[14] Qian, Proceedings of the Royal Society, Series A 462 pp 21– (2006)
[15] Yang, Aeronautical Journal 101 pp 47– (1997)
[16] Causon, Advances in Water Resources 24 pp 899– (2001)
[17] Ingram, Mathematics and Computers in Simulation 61 pp 561– (2003)
[18] Chorin, Journal of Computational Physics 2 pp 12– (1967)
[19] Rogers, AIAA Journal 28 pp 253– (1990)
[20] Rogers, AIAA Journal 29 pp 603– (1991)
[21] Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA Paper 91-1596, AIAA 10th Computational Fluid Dynamics Conference, Honolulu, HI, June 1991.
[22] Anderson, Journal of Computational Physics 128 pp 391– (1996)
[23] Pan, AIAA Journal 26 pp 163– (1988)
[24] Daly, Journal of Computational Physics 10 pp 297– (1967)
[25] Verification and Validation in Computational Science and Engineering. Hermosa Publishers: Albuquerque, 1998.
[26] Hydrodynamics and Hydromagnetics Stability. Oxford University Press: London, 1961; 428–447.
[27] , , . Wave interaction with a sea dike using a VOF finite volume method. Proceedings of the 13th International Offshore and Polar Engineering Conference, vol. 3, 2003; 325–332.
[28] Goda, Coastal Engineering Journal 41 pp 1– (1999)
[29] Hasselmann, Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe A 8 pp 95– (1973)
[30] Hedges, Maritime Engineering, Proceedings of the ICE 157 pp 113– (2004)
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