Implicit Runge-Kutta Methods to Simulate Unsteady Incompressible Flows

Abstract

A numerical method (SIMPLE DIRK Method) for unsteady incompressible viscous flow simulation is presented. The proposed method can be used to achieve arbitrarily high order of accuracy in time-discretization which is otherwise limited to second order in majority of the currently used simulation techniques. A special class of implicit Runge-Kutta methods is used for time discretization in conjunction with finite volume based SIMPLE algorithm. The algorithm was tested by solving for velocity field in a lid-driven square cavity. In the test case calculations, power law scheme was used in spatial discretization and time discretization was performed using a second-order implicit Runge-Kutta method. Time evolution of velocity profile along the cavity centerline was obtained from the proposed method and compared with that obtained from a commercial computational fluid dynamics software program, FLUENT 6.2.16. Also, steady state solution from the present method was compared with the numerical solution of Ghia, Ghia, and Shin and that of Erturk, Corke, and Goökçöl. Good agreement of the solution of the proposed method with the solutions of FLUENT; Ghia, Ghia, and Shin; and Erturk, Corke, and Goökçöl establishes the feasibility of the proposed method.