[Show abstract][Hide abstract] ABSTRACT: In this review the conditions to be imposed on the vorticity in the calculation of two-dimensional incompressible viscous flows are discussed. Existing boundary vorticity formulas, commonly regarded as a surrogate Dirichlet boundary condition for the vorticity, are more properly interpreted as the discrete counterpart of the Neumann boundary condition for the stream function. This viewpoint helps to elucidate the algebraic equivalence of coupled numerical methods with uncoupled methods based on conditions of integral type for the vorticity. A unified understanding of several available treatments for determining correct vorticity boundary values is achieved by including in the present analysis spatial discretizations by finite differences and finite elements, coupled and uncoupled formulations of the problem as well as steady and unsteady equations. Results of some test calculations are presented to illustrate the numerical consequences of the analysis.
[Show abstract][Hide abstract] ABSTRACT: This paper provides a multigrid incremental line-Gauss-Seidel method for solving the steady Navier-Stokes equations in two and three dimensions expressed in terms of the vorticity and velocity variables. The system of parabolic and Poisson equations governing the scalar components of the vector unknowns is solved using centred finite differences on a non-staggered grid. Numerical results for the two-dimensional driven cavity problem indicate that the spatial discretization of the equation defining the value of the vorticity on the boundary is extremely critical to obtaining accurate solutions. In fact, a standard one-sided three-point second-order-accurate approximation produces very inaccurate results for moderate-to-high values of the Reynolds number unless an exceedingly fine mesh is employed. On the other hand, a compact two-point second-order-accurate discretization is found to be always satisfactory and provides accurate solutions for Reynolds number up to 3200, a target impossible heretofore using this formulation and a non-staggered grid.
International Journal for Numerical Methods in Fluids 06/2005; 13(1):49 - 59. · 1.35 Impact Factor
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