A Study of Numerical Schemes for Incompressible Fluid Flows
DOI:
https://doi.org/10.5540/tema.2005.06.01.0111Abstract
The present work is concerned with a study of numerical schemes for solving two-dimensional time-dependent incompressible free-surface fluid flow problems. The primitive variable flow equations are discretized by the finite difference method. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The diffusive terms are discretized by Implicit Backward and Crank- Nicolson schemes, and the non-linear advection terms are approximated by the high order upwind VONOS (Variable-Order Non-oscillatory Scheme) technique. In order to improved numerical stability of the schemes, the boundary conditions for the pressure field at the free surface are treated implicitly, and for the velocity field explicitly. The numerical schemes are then applied to the simulation of the Hagen-Poiseuille flow, and container filling problems. The results show that the semi-implicit techniques eliminate the stability restriction in the original explicit GENSMAC method.References
[1] S. Armfield and R. Street, The pressure accuracy of fractional-step methods for the Navier-Stokes equations on staggered grids, ANZIAM. Australian and New Zealand Industrial and Applied Mathematics, 44 (2003), 20-39.
G.K. Batchelor, “An Introduction to Fluid Dynamics”, Cambridge, 1970.
W.R. Briley and H. McDonald, An overview and generalization of implicit Navier-Stokes algorithms and approximate factorization, Comput. & Fluid, 30 (2001), 807-828.
A.F. Castelo, M.F. Tomé, M.L. Cesar, J.A. Cuminato and S. Mckee, Freeflow: An intregated simulation system for three-dimensional free surface flows, Journal of computers and visualization in science, 2 (2000), 199-210.
A.J. Chorin, A numerical method for solving incompressible viscous flow problems, Journal of Computational Physics, 2 (1967), 12-26.
V.G. Ferreira, M.F. Tomé, N. Mangiavacchi, A. Castelo, J.A. Cuminato and S. McKee, High order upwinding and the hydraulic jump, Int. J. Numer. Meth. Fluid, 39 (2002), 549-583.
P.M. Gresho, On the theory of semi-implicit projection methods for viscous incompressible flow and its implemention via a finite element method that also introduces a nearly consistent mass matrix, International Journal for Numerical Methods in Fluids, 11 (1990), 587-620.
M. Manna and A. Vacca, An efficient method for the solution of the Incompressible Navier-Stokes equations in cylindrical geometries, Journal of Computational Physics, 151 (1999), 563-584.
C.M. Oishi, “Análise e implementação de métodos implícitos no sistema FreeFlow2D”, Master’s Thesis, ICMC, University of São Paulo, 2004.
P.J. Roache, “Fundamental of Computational Fluid Dynamics”, Albuquerque, Hermosa Publishers, 1998.
M.F. Tomé and S. McKee, GENSMAC: A computational marker-and-cell method for free surface flows in general domains, J. Comput. Phys., 110 (1994), 171-186.
M.F. Tomé, S. McKee, L. Barratt, D.A. Jarvis and A.J. Patrick, An experimental and numerical investigations of container filling with viscous liquids, International Journal for Numerical Methods in Fluids, 31 (1999), 1333-1353.
S. Turek, A comparative study of some time-stepping techniques for the incompressible Naveir-Stokes equations: From fully implicit nonlinear schems to semi-implicit projection methods, International Journal for Numerical Methods in Fluids, 22 (1996), 987-1011.
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