Computing High Reynolds Number Flows by a Semi-Lagrangian Scheme

Abstract

The Semi-Lagrangian (SL) scheme developed for incompressible Navier-Stokes equations written in generalized coordinates has been explored to accurately solve the high Reynolds number flows. The instability related to the advection term of the Navier-Stokes equations is classified into linear instability and nonlinear instability. The former is controlled by the CFL condition, and the latter is due to the aliasing error. The linear instability is naturally eliminated by employment of the SL scheme. The Kawamura scheme is creatively applied to approximate the first derivatives appearing in the Hermite interpolation function to remove the nonlinear instability. The resulting numerical scheme is unconditionally stable for incompressible flows at all Reynolds numbers. Accurate numerical solutions of the unsteady flow around a 2D circular cylinder at Reynolds numbers below 100,000 have been carried out. It was found that this flow has a wide range of wave number modes. Numerous grid numbers and a small time step length must be used to guarantee the accuracy. Furthermore, a very long time average should be conducted to compare the data with the experimental measurement.