Numerical Solution of the Elliptic Equations for Transport of Vorticity, Heat, and Matter in Two-Dimensional Flow

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Abstract
A finite‐difference method is presented for the solution of the elliptic differential equations for the steady transport of momentum, heat, and matter in two‐dimensional domains. Special features of the method include an unsymmetrical formulation for the convection terms, which promotes convergence at some cost in accuracy; obedience to the conservation equations for all subdomains; the use of Gauss‐Seidel iteration procedure; employment of grids having nonuniform mesh; and a novel treatment of the boundary condition for vorticity. Solutions are presented for the laminar flow and heat transfer inside a square cavity with a moving top, an impinging jet, and a Couette flow with mass transfer. The influence of the Reynolds and Prandtl numbers, and of the impinging jet “free” boundary conditions is studied, and the results of the computations are shown to agree with existing physical knowledge. The influence of mesh size, mesh nonuniformity, and the vorticity wall boundary condition on convergence and accuracy is studied. It is shown that convergence may be secured for a wide range of Reynolds numbers with coarse‐meshed grids. The convergence and computation speed appear to be satisfactory for many purposes; the accuracy of the solutions is discussed, and some improvements are suggested.