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 dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection?diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithms's accuracy and convergence properties, and illustrate the behavior of the method. An additional example demonstrates the performance of the method on a more realistic problem, namely, a three-dimensional variable density shear layer.