A CONSERVATIVE ADAPTIVE PROJECTION METHOD FOR THE VARIABLE-DENSITY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

In this paper we present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three d imensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection-diffusion equ ations to predict intermediate velocities, and then project these velo cities onto a space of approximately divergence-free vector fields. Ou r treatment of the first step uses a specialized second-order upwind m ethod for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynold s number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservatio n. Our approach to adaptive refinement uses a nested hierarchy of logi cally-rectangular girds with simultaneous refinement of the girds in b oth space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fin e grids are advanced multiple steps to reach the same time as the coar se grids and the data at different levels are then synchronized. The s ingle grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical exa mples are presented to demonstrate the algorithms's accuracy and conve rgence properties, and illustrate the behavior of the method. An addit ional example demonstrates the performance of the method on a more rea listic problem, namely, a three-dimensional variable density shear lay er. (C) 1998 Academic Press.