Simulation of Unsteady Incompressible Viscous Flow Using Higher Order Implicit Runge-Kutta Methods: Staggered Grid
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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  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  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. . Good agreement of the second-order solutions of the proposed method with the second-order solutions of FLUENT  and Ghia et al.  concludes the feasibility of the proposed method. Copyright © 2007 by ASME.
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