High-resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations

Based on flux-difference splitting, implicit high-resolution schemes are constructed for efficient computations of steady-state solution to the three-dimensional, incompressibl e Navier-Stokes equations in curvilinear coordinates. These schemes use first-order-accurate Euler backward-time differencing and second-order central differencing for the viscous shear fluxes. Up to third-order-accurate upwind differencing is achieved through a reconstruction of the solution from its cell averages. The reconstruction is accomplished by linear interpolation, where the node stencils are selected such that in regions of smooth solution the flow is highly resolved while spurious oscillations in regions of rapid changes in gradient are still suppressed. Fairly rapid convergence to steady-state solutions is attained with a vectorizable hybrid time-marching method. Flows around a sharp-edged delta wing are computed with the maximum accuracy of the upwind differencing restricted to first, second, and third order, to illustrate the effect of accuracy oh the global and on the local vortical flowfields. The results are validated with experimental data. I. Introduction T HERE are basically two options to compute threedimensional low-speed flows as steady-state solutions to the Navier-Stokes equations: numerical approximations 1) to the compressible Navier-Stokes equations for low Mach numbers (M), or 2) to the incompressible Navier-Stokes equations. In view of the numerous time-dependent methods, both explicit1 and implicit,2'4 for the compressible Navier-Stokes equations, the first option appears to be a viable choice. However, the efficiency of these schemes deteriorates considerably with decreasing subsonic Mach number due to the "stiffness" of the physical problems associated with lowspeed, viscous flow. The stiffness arises from both disparate characteristic speeds and length scales. Furthermore, the accuracy of low-speed flowfield computations with implicit approximate factorization (AF) schemes (e.g., those in Refs. 2-4) becomes questionable for low Mach numbers, since the factorization errors increase in an unbounded fashion for a decreasing Mach number.5 That suggests to remove the sound speed time scale and to assume the flow to be truly incompressible. Aziz and Heliums6 proposed to solve the Navier-Stokes equations for incompressible, fully three-dimensional flow in a vector potential-vorticity formulation. Because of its storage requirements and the necessity to solve three Poisson equations at each time level, that method failed to become popular. That leaves solving the Navier-Stokes equations for threedimensional, incompressible flow in their primitive variable formulation. Common methods for this formulation advance the solution for the velocity field by explicitly or implicitly solving the equations of motion. The updated velocity field is used to compute the corresponding pressure field by solving a Poisson equation7 or by an iterative procedure with the equation of continuity as a compatibility condition.8 Such schemes