Separated viscous flows via multigrid

Abstract

A multigrid code is developed to solve general systems of convection-diffusion equations when the diffusion terms are small, i.e., when the Reynolds number is large. Various upwinding, artificial viscosity and defect correction schemes are considered and compared. The code is applied to the Navier-Stokes equations for various flow configurations and used to study boundary layer separation from a leading edge, with ensuing formation of a downstream eddy. The asymptotic (triple deck) theory of separation is developed for this case, following Sychev, and compared to the numerical calculations at Reynolds numbers of up to 5000. Much better qualitative agreement is obtained than has been reported previously. Together with a plausible choice of two free parameters, the data can be extrapolated to infinite Reynolds number, giving quantitative agreement with triple deck theory with errors of 20% or less. The development of a region of constant vorticity is observed in the downstream eddy, and the global infinite Reynolds number limit is a Prandtl-Batchelor flow; however, when the plate is stationary, the occurrence of secondary separation suggests that the limiting flow contains an infinite sequence of eddies behind the separation point. Secondary separation can be averted by driving the plate, and in this case the limit is a single-vortex Prandtl-Batchelor flow of the type found by Moore, Saffman and Tanveer (1988); we make detailed, encouraging comparisons of the vortex sheet strength and position. By altering the boundary condition on the plate we obtain viscous eddies that approximate different members of the family of inviscid solutions. The code is also used to calculate the flow over a finite flat plate aligned with a uniform free stream; in that case, earlier conflicting results about higher-order corrections to the boundary layer are explained, and the triple deck generally believed to be established around the trailing edge is found to be consistent with the numerical results. There remains a large displacement-like effect in the boundary layer, whose exact origin is unclear.