RK-SIMPLER: Explicit Time-Accurate Algorithm for Incompressible Flows

Abstract

The SIMPLE family of algorithms has popularized the pressure-based schemes for incompressible flows and is the basis of many commercial codes. The influence of pressure on velocity is of primary importance in incompressible flows. The continuity equation implicitly dictates the pressure field, yet pressure is not a variable of the mass conservation equation. The pressure term that appears in the momentum equations is often treated as a source term. Thus, there is no explicit conservation equation for pressure. This predicament, and its remedy well known by the name “pressure–velocity coupling,” is resolved in SIMPLE and its variants by obtaining an approximate pressure correction field, which is used iteratively to correct the velocity field and/or the pressure field, seeking an overall satisfaction of the conservation equations. The approximate nature of the pressure correction equation often causes convergence issues. A new algorithm is presented here, eliminating the need for the pressure correction equation, based on the fact that if the pressure field is known, the momentum equations can be solved in any number of ways to obtain the velocity field correctly. An exact equation for the pressure field is obtained by manipulating the discretized mass and momentum equations based on SIMPLER, which is the only nonlinear equation solved iteratively in the new algorithm (RK-SIMPLER). The momentum equations are cast in the form of an ordinary differential equation suitable for time integration using Runge–Kutta stages. Once the pressure field is known, the velocity field is updated explicitly every time step without iteratively solving the momentum equations. This also means that there are no subiterations within a time step. In addition, there are no corrections for the pressure or the velocity field, and hence there is no need for the approximate pressure correction equation. The RK-SIMPLER algorithm proves that, for incompressible flows, the fundamental equation is the pressure–velocity coupled exact equation for pressure and that there is no need for the nonlinear velocity equations to be solved iteratively. Also, the new algorithm presented here uses only exact equations and requires neither underrelaxation for any of the discretized quations nor subiterations for the time integration. The only approximation is that the pressure field is held constant through the Runge–Kutta update of the velocity field. The RK-SIMPLER algorithm converges well and captures the unsteady flow features for the cases analyzed. In contrast, for steady flows, the algorithm is stable but less competitive compared to unsteady flow simulations in terms of CPU time, due to the restrictions on the allowable time step.