Co-Located Variables Approach Using Implicit Runge-Kutta Methods for Unsteady Incompressible Flow Simulation

Abstract

A numerical method for transient incompressible viscous flow simulation is presented with a nonstaggered grid (co-located variables) approach. A special class of implicit Runge-Kutta (RK) methods is used for time discretization in conjunction with finite-volume-based semi-implicit pressure linked equations (SIMPLE) algorithm. Owing to the use of implicit RK methods, the proposed method can be used to achieve an arbitrarily high order of accuracy in time-discretization which is otherwise limited to second order in the majority of the currently available simulation techniques. The nonstaggered grid method was tested by solving for velocity field in a lid-driven square cavity. In the test case calculations, the Power Law scheme of Patankar [9] was used in spatial discretization, and time discretization was performed using a second-order implicit Runge-Kutta method. The results from the current method were compared with the results obtained from the staggered grid method of Ijaz and Anand [10-12]. The current method produced results that were nearly equivalent to the ones obtained by using the staggered grid approach. Moreover, the current method was found to have better convergence characteristics compared to the staggered grid method when applied to the lid-driven square cavity problem.

This work was also presented at the 8th ISHMT-ASME Heat and Mass Transfer Conference, Hyderabad, India, January 3–5, 2008 [1].