Comparison of Solution Algorithm for Flow around a Square Cylinder

Numerical accuracy, numerical stability and calculation time are all important factors in the computational fluid dynamics. In this study, we compare two solution algorithms, the Simplified Marker and Cell (SMAC) method in the MAC-type methods and the Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm in the SIMPLE-type algorithms, with respect to flow around a square cylinder in constant density and unsteady-state calculations using a staggered grid to investigate the numerical accuracy, the numerical stability and the computational time. For the flow around a square cylinder, the SMAC and SIMPLE solutions are in excellent agreement at the Strouhal number, drag and lift coefficients. However, SMAC is more unstable than SIMLE with a large Courant number. The computational time of the SMAC is shorter than that of the SIMPLE with a small Courant number. NOMENCLATURE A area back back face bottom bottom face CL Lift coefficient CD Drag coefficient Cr Courant number D characteristic length Dn diffusion number, e cell surface index f frequency of vortex shedding front front face in inlet n cell surface index p pressure Re Reynolds number RC rectangular St Strouhal number s cell surface index t elapsed time top top face u velocity u velocity v velocity x coordinate y coordinate w cell surface index   under relaxation factor t time step x grid width y grid width  convergence criterion  viscosity  density  non-dimensional computational speed INTRODUCTION Solution algorithms of pressure-velocity coupling are widely used for incompressible fluid flow calculations. The solutions expend the major part of time of the Computational Fluid Dynamics (CFD), because iterative calculations are required. In the CFD, numerical accuracy, numerical stability and calculation time are all important factors. As the calculation time influences the calculation cost directly, the calculation speed is the most important factor. But at the same time, it is necessary that the numerical accuracy and numerical stability are high. Thus, in order to perform better CFD, it is required to balance these factors. In particular, we focus on coupling schemes to decrease the calculation cost. The solution algorithms solve for continuity and momentum equations, for which Marker and Cell (MAC) methods and Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) type algorithms have been widely applied. Both common parts are solving the continuity equation in implicitly, and the most significant difference between the methods is the treatment of momentum equations; the former is solved in explicitly, and the latter is in implicitly. Generally, MAC-type methods are used for unsteady-state calculations and SIMPLE-type algorithms are used for steady-state or pseudo-unsteady state calculations. However, as SIMPLE-type algorithms are widely used for