A fourth-order-accurate compact differencing scheme for steady Navier-Stokes equations

Abstract

~A fourth-order-accurate, compact differencing scheme is proposed for the solution of twodimensional steady Navier-Stokes equations in stream func t ion ($9) /vo r t i c i ty (w) form. The difference scheme is compact in the sense that in one dimension, only three nodes are required to obtain the fourth-order accuracy, in contrast to the standard fourth-order scheme that requires five nodes. The compact difference equations are derived *sing compact relations between the functions I# and wand their derivatives (h, $by, %, 9, hxx, $y, %x, and wyy). Solution of the compact difference equations is obtained by successive over-relaxation. The scheme is applied to compute the flow in a driven-cavity and the buoyancy-driven flow in a square cavity with differentially heated side walLs and insulated top and bottom walls. Higher accuracy is achieved on a coarser mesh than that required with second-order methods for achieving the same accuracy. Introduction The standard seeond-order-accurate, threepoint finite-difference discretization has formed the basis far the vast majority of numerical solutions of the equations of fluid mechanics. For complex problems involving regions of large gradients in flow-properties, the standard second-order methods become less suitable because of the increase in the number of grid points required for the desired accuracy. In some cases, particularly for three-dimensional problems, the storage requirements of the present computers may be exceeded. It appears that higher-order difference methods can be used to obtain better accuracy. However, in a conventional sense, higher-order difference formulae require additional nodes, e . g . , five-point discretization is required to achieve fourth-order accuracy. The use of additional nodes creates difficulties in computation at and near the boundary points, and the algoriehm does not retain the simple tridiagonal form. In recent years, difference methods have been proposed that are higherorder accurate and also retain the tridiagonal form. These procedures generally result in a somewhat lower truncation error than that found with a five-point discretization and can be derived from appropriate Taylor series expansions (Hermite) o r polynomial interpolation.(spline). In the former category are the Pade approximation, or so called compact scheme,' and the Mehrstellung' o r Hermitian finite-difference development . 3 In the latter group are spline collocation methods. 4 *This work was supported by McUonnell Douglas Independent Research and Development program. Laboratories: Member AIM. **Scientist, McDOnnell Douglas Research In this paper, a fourth-order-accurate compact difference scheme is formulated for numerical solution of 2-D Sfeady Navier-Stokes equations in stream function(V)/vorticity(w) form. The underlying idea behind development of the present fourth-order compact scheme is similar to the second-order method of Allen and S~uthwell;~ the coefficients of the first-order derivatives in the vorticity equation are approximated locally at a r and the difference relations are then written for the second-order derivatives yielding a difference equation containing exponential functions. Fourthorder-accurate, compact difference equations thus are derived for the Poisson equation for the stream function and the vorticity transport equation by using discrete compact relations, first suggested by Kreiss,6 between the basic function and its second derivatives in the two spatial directions. The solution of these compact difference equations is obtained by successive point-relaxation. The scheme is applied to compute the standard test cases: flow in a driven-square cavity and the buoyancy-driven flow in a square cavity with differentially heated side walls and insulated top and bottom walls. The present work represents a study of this new scheme to determine the feasibility of its use with regard to stability and convergence properties and to verify the high order of accuracy claimed. Fourth-Order-Accurate Compact Difference Relations As suggested by Kreiss6, a fourth-orderaccurate Hermitian or compact approximation to F(x) at a nodal point i can be expressed as (Fx)i = ( I + D1i-h216)Fi