Rotated upwind algorithms for solution of the two- and three-dimensional Euler and Navier-Stokes equations.

摘要

Rotated upwind algorithms are presented for the numerical solution of the Euler and Navier-Stokes Equations in two and three dimensions. The finite-volume algorithms are designed with the notion of aligning Roe's approximate Reimann solver in a computed preferred direction.; A baseline algorithm is developed by first comparing four rotated upwind strategies for the solution of the Euler equations in two dimensions to first-order accuracy. The four strategies are based on the combinations of two options which are a cell-edge vs. a cell-center rotation and a rotation in physical space vs. computational space. Solutions of a Mach 2 channel flow geometry are presented and compared in terms of accuracy and robustness. It is shown that the strategy of performing a cell-center rotation in computational space offers the best promise for further development.; A two-dimensional rotated upwind algorithm is then developed for both the Euler and Navier-Stokes equations. A preferred direction is computed in computational space resulting in four rotated contravariant directions. An inviscid flux computation based on Roe's approximate Riemann solver is performed in the rotated directions. Left and right state values for the Riemann solver are obtained through linear interpolation of the primitive variables. The fluxes are then transformed back onto the grid contravariant directions in a conservative manner through a coordinate transformation. The viscous fluxes are computed in a standard grid aligned manner. The solution is relaxed in time using the diagonal form of the LU-SGS scheme.; Calculation of an inviscid Mach 2 channel flow problem shows that the rotated algorithm produces more accurate results than a traditional grid aligned algorithm to both first-and second-order accuracy. Moreover, the improvements to second-order accuracy are as great as those to first-order accuracy. Viscous solutions of both a laminar and turbulent compression corner, and a turbulent shock wave impingement show that the rotated scheme improves the shock wave capturing in the inviscid portion of the flowfield to both first- and second-order accuracy. The improvements in the shock wave capturing to first-order accuracy result in improved wall pressure and skin friction distributions. However to second-order accuracy, the wall predictions are only marginally improved.; The algorithm is then extended to three-dimensions. Two series of cell-centered coordinate rotations are developed that in combination are guaranteed to align a coordinate axis in any computed preferred direction. Moreover, the orientation of the rotated system with respect to the original system is designed to simplify the interpolation and projection. The calculation of an inviscid three-dimensional shock wave surface shows the rotated algorithm to be more accurate than the grid aligned algorithm to both first- and second- order accuracy. The accuracy improvements in three dimensions are not as great as those in two dimensions. Computation of viscous flowfields in the corner of two intersecting wedges and a turbulent hypersonic inlet configuration show that the inviscid portions of the flowfield are qualitatively improved with the rotated algorithm to both first- and second-order accuracy. However, surface pressure and heat transfer predictions are only marginally improved with the rotated algorithm. (Abstract shortened by UMI.)