Numerical solution of the Navier-Stokes equations for 3-D internal flows: An emerging capability

The accuracy of 3-D flow calculations is discussed. A good calculation procedure solves a set of finite difference equations reasonably quick. However, if the finite difference equations do not accurately reflect the governing differential equations, poor results will be obtained. For example, it is common practice in three dimensional flow calculations to use upwind differencing or more direct forms of numerical mixing to obtain well posed finite difference equations which are needed to maintain a stable calculation procedure. The erroneous numerical viscosity introduced can be of the same order as the natural laminar or laminar plus turbulent viscosities in the flows calculated. When numerical mixing is present a proper evaluation of viscosity or turbulence models cannot be made because the effects of numerical and physical mixing cannot be distinguished. A simple 2-D example is used to demonstrate a practical method of obtaining the finite difference form of the convection term in momentum or other conservation equations. The 2-D example introduces no numerical mixing, yet results in well posed, (i.e., strong center point coefficient) finite difference equations. The method consists of central differencing to eliminate numerical mixing and upwinded control volumes to obtain finite difference equations with strong point coefficients. An upwinding procedure for the control volumes which may be applied to three dimensional flow is given. Inviscid calculations of horseshoe vortex flow about a Rankine half body are presented.