An adaptive multi-grid scheme for simulation of flows

Abstract

An adaptive MG scheme has been applied to the computation of certain incompressible flows. The scheme uses a basic (low) order solver of the Navier-Stokes equations, on a system of zonal subgrids. These subgrids may be defined independently, and may contain locally refined regions. The MG scheme is used to solve efficiently the discrete equations, even on such systems of grids. Local mesh refinements are done dynamically, in regions where the estimated truncation errors are larger than the average. When a final grid system is found (with almost uniformly distributed truncation errors) the order of approximation is improved by a few additional MG cycles using a defect correction type scheme. The adaptive scheme is also used to find regions where certain simplified approximations to the governing equations (e.g. PNS and potential equations) are valid. Such approximations are than applied to produce, rapidly, solutions that are valid in these regions. In this way, boundary conditions may be applied with controlled accuracy. In regions where such approximations are not valid, the approach may produce (natural) block relaxation schemes.

The scheme has been applied to the solution of the flow in a channel with symmetric sudden enlargement. For Reynolds numbers larger than some certain value, the solution is not unique. The symmetry breaking bifurcation that occurs can be traced easily by the method.