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A Non-staggered Grid, Fractional Step Method for Time-Dependent Incompressible Navier-Stokes Equations in Curvilinear Coordinates
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1994
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The non‑staggered‑grid method of Rhie and Chow, originally for steady problems, is extended to compute unsteady flows. The study presents a numerical method for solving three‑dimensional, time‑dependent incompressible Navier–Stokes equations in curvilinear coordinates. The method defines velocity and pressure at cell centers, computes fluxes at face midpoints, integrates momentum equations semi‑implicitly via approximate factorization, interpolates intermediate velocities to faces to form the pressure Poisson source terms solved iteratively with a multigrid method, and derives pressure boundary conditions from the momentum equations, ensuring mass conservation to machine accuracy. Numerical experiments, including large‑eddy simulation of a rotating and stratified upwelling flow in an irregular container, demonstrate the method’s favorable accuracy and efficiency for both steady and unsteady problems.
A numerical method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in curvilinear coordinates is presented. The non-staggered-grid method originally developed by C. M. Rhie and W. L. Chow (AIAAJ.21, 1525 (1983)) for steady state problems is extended to compute unsteady flows. In the computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the mid-point on their corresponding cell faces. The momentum equations are integrated semi-implicitly by the approximate factorization technique. The intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure Poisson equation, which is solved iteratively with a multigrid method. The compatibility condition of the pressure Poisson equation is satisfied in the same manner as in a staggered-grid method; mass conservation can be satisfied to machine accuracy. The pressure boundary condition is derived from the momentum equations. Solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method.