Simulation of unsteady incompressible viscous flow using higher order implicit Runge-Kutta methods - Staggered grid

A numerical method (SIMPLE DIRK Method) for transient 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 available 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 of Patankar [2] was used for 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 CFD software, FLUENT [3] using second-order implicit time discretization scheme. Steady state solution from the present method was compared with the benchmark numerical solution of Ghia et al. [4]. Good agreement of the second-order solutions of the proposed method with the second-order solutions of FLUENT [4] and Ghia et al. [4] concludes the feasibility of the proposed method.