A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

Abstract

SUMMARYA fourth-order compact ÿnite di erence scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes=Boussinesq equations for two-dimensional, incompressible uid ow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature.

A compact finite difference method on staggered grid for Navier–Stokes flows

“…2. Interpolate @u * =@x and @v * =@y using Equations (14) and (15). According to Reference [19], on solid boundaries u * = 0 and v * = 0.…”

Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

“…(32) with (29), (31) and (33) is a nine-point exponential high-order compact approximation on non-uniform grids for the 2D convection-diffusion equation (13). Notice that the approximations of the first-and second-partial derivatives of f may be computed by using difference operators δ x − γ x h x 2 δ 2 x , δ 2 x , δ y − γ y h y 2 δ 2 y and δ 2 y while still maintaining overall third-order accuracy on 9-point stencil.…”

“…The application of the scheme (32) with (29), (31) and (33) to Eqs. (11) and (12) obtains, respectively,…”

“…HOC difference schemes have been developed and applied to a variety of elliptic equations, the incompressible Navier-Stokes equations and Stokes problems by many authors [27,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46], because they not only provide accurate numerical results and save computational work, but also are easier to deal with boundary conditions. Although an EHOC scheme has been proposed to solve effectively the MHD duct flow problems with low-to-high Ha in [27] very recently, to the best of authors' knowledge, there is still no any report about the implementation of the HOC difference scheme on non-uniform space grids to solve the MHD duct flow problems.…”

A compact finite differences exact projection method for the Navier–Stokes equations on a staggered grid with fourth-order spatial precision

“…Recently, a number of studies have been conducted on compact finite difference schemes combined with the vorticity-stream function formulation [6][7][8]. Vorticity-stream function methods are advantageous for their lower computational cost when compared to solving the Navier-Stokes equations as a coupled system.…”