Time integration methods are necessary for the solution of transient flow problems. In recent years, interest in transient flow problems has increased, leading to a need for better understanding of the costs and benefits of various time integration schemes. The present work investigates two common time integration schemes, namely the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) and the Fractional Step (FS) method.
Three two-dimensional, transient, incompressible flow problems are solved using a cell centered, finite volume code. The three test cases are laminar flow in a lid-driven skewed cavity, laminar flow over a square cylinder, and turbulent flow over a square cylinder. Turbulence is modeled using wall functions and the k - ε turbulence model with the modifications suggested by Kato and Launder. Solution efficiency as measured by the effort carried out by the flow equation solver and CPU time is examined. Accuracy of the results, generated using the SIMPLE and FS time integration schemes, is analyzed through a comparison of the results with existing experimental and/or numerical solutions.
Both the SIMPLE and FS algorithms are shown to be capable of solving benchmark flow problems with reasonable accuracy. The two schemes differ slightly in their prediction of flow evolution over time, especially when simulating very slowly changing flows. As the time step size decreases, the SIMPLE algorithm computational cost (CPU time) per time step remains approximately constant, while the FS method experiences a reduction in cost per time step. Also, the SIMPLE algorithm is numerically stable for time steps approaching infinity, while the FS scheme suffers from numerical instability if the time step size is too large. As a result, the SIMPLE algorithm is recommended to be used for transient simulations with large time steps or steady state problems while the FS scheme is better suited for small time step solutions, although both time-stepping schemes are found to be most efficient when their time steps are at their maximum stable value
