Accuracy of finite-difference methods in recirculating flows

The accuracy of six numerical approximations of the convection terms in the conservation equations is examined for a steady, recirculating flow. Quadratic upwind, central, nine-point, third-order, and power-law approximations are tested as alternatives to the widely used upwind/central hybrid method. Forced flow in a heated cavity is chosen as a reasonably severe test problem. An exact analytical solution is used to evaluate truncation errors and solution errors. Expressions for the leading truncated terms, including velocity derivatives, provide insight into why errors in the convection terms dominate errors in the diffusion terms for high grid Peclet numbers. If an average solution error of less than 10 percent is desired, higher order methods are clearly superior to the first-order upwind/hybrid method. One must have at least one finite domain within a wall gradient layer to reduce flux errors to 10 percent with the second-oder central-difference method, whereas one must have at least two finite domains across the layer to achieve similar accuracy with the first-order hybrid method. Both quadratic upwind and central differencing are recommended for the numerical approximation of the convection terms.