Comparison of rezone algorithms for incompressible fluid flow calculations

Abstract

THE art of computational fluid dynamics has progressed to the point that computations can be performed using mesh grids that conform with features of the flow solution or that change as the solution domain changes. Such cases include oscillating airfoils and other flow-induced motions of solid bodies, where the locations of essential features can move sinusoidally from place to place. It is the purpose of this paper to present a comparison of two designed to account for the effects of temporally changing mesjies in order to determine the one with the lesser adverse effect on the fluid flow solution. Contents The conservation of momentum (Navier-Stokes) equations can be formulated so that motion of mesh points is treated naturally in the convection term. This fact has been exploited in formulating the ICED-ALE finite difference method of Hirt et al.1 and also the quasi-Eulerian finite element method as used by Belytschko and Kennedy,2 to name just two examples. In this synoptic, numerical experiments are described which indicate that this natural approach based on conservation of momentum may not be the most accurate one. In the following paragraphs, two approaches are briefly described—the first based on conservation of momentum as mentioned above and the second on biquadratic interpolation. Then three different flow configurations are described and comparison of the algorithms for these configurations are given. The conclusion reached is that the rezone algorithm based on interpolation is to be preferred over the one based on conservation of momentum. The numerical experiments described below were performed using a computer program based on the ICED-ALE approach of Hirt et al.1 The primary differences between the program used and that of Ref. 1 are that the convection terms are evaluated before the new pressure values are found and that a direct, rather than iterative, solution method is used to find the pressure solution. In order to discuss the specific forms of the rezone which were investigated, let x(n), u(n), and pu(n) denote position, velocity, and momentum vectors, respectively, at the end of the nth time step. Here, p denotes the constant value of density. Let x(n+1) denote the new mesh location and u' and pu' the changed velocity and momentum values at Jc(w+/) based on the values uW. One rezone scheme is based on the notion that the change in momentum inside the control volume V(n) is given by the amount of momentum swept through its walls as they move from their old to their new positions. Hence pu' can be found by approximating the expression