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 the majority of currently available simulation techniques. A special class of implicit Runge-Kutta methods is used for time discretization in conjunction with the finite-volume based SIMPLE algorithm. The algorithm was tested by solving for the velocity field in a lid-driven square cavity. In the test case calculations, the 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 the velocity profile along the cavity centerline was obtained from the proposed method and compared with that obtained from a commercial computational fluid dynamics (CFD) software program, FLUENT  using a second-order implicit time discretization scheme. Steady-state solution from the present method was compared with the 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.
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