Introducing compressibility with SIMPLE algorithm

Abstract

A comparative assessment between SIMPLE and its variant SIMPLE-C is conducted based on two-dimensional (2-D) incompressible flows, using a cell-centered finite-volume Δ-formulation on a collocated grid. The SIMPLE-C (SIMPLE Compressibility) scheme additionally combines the concept of artificial compressibility (AC) with the pressure Poisson equation, provoking the diagonal dominance of influence coefficients. An improved nonlinear momentum interpolation scheme is employed at the cell face in discretizing the continuity equation to suppress pressure oscillations. The pseudo-time step Δti remains the same in both schemes to conserve an analogous scaling with momentum and scalar nodal influence coefficients. Numerical experiments in reference to buoyancy-driven cavity flow dictate that both contrivances execute a residual smoothing enhancement, facilitating an avoidance of the velocity/pressure under-relaxation (UR). However, compared with the SIMPLE approach, included benefits of the SIMPLE-C method are the use of larger Courant numbers, enhanced robustness and convergence. Excellent consistency is obtained between results available in the literature and numerical solutions obtained by both SIMPLE and SIMPLE-C solvers. The segregated SIMPLE algorithm is finally reformulated in conjunction with a new slope/flux limiter function to predict fluid flow at all speeds. Numerical results show that the compressible variant of SIMPLE replicates correct shock speed, well-resolved shock front, contact discontinuity and rarefaction waves when compared with analytical solutions. Compressible laminar flows are computed to further support the accuracy and robustness of proposed algorithm.