An explicit, upwind algorithm for solving the parabolized NavierStokes equations
Abstract
An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3D Parabolized NavierStokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann Problem (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The secondorder upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vigneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting coefficient improves the stability of the scheme. The new algorithm is demonstrated for 2D and 3D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 deg. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required for the upwind PNS code are approximately equal to an explicit PNS MacCormack's code and existing implicit PNS solvers.
 Publication:

Ph.D. Thesis
 Pub Date:
 1989
 Bibcode:
 1989PhDT........30K
 Keywords:

 Algorithms;
 Cauchy Problem;
 Flow Distribution;
 Flux Vector Splitting;
 NavierStokes Equation;
 Parabolic Differential Equations;
 Ideal Gas;
 Pressure Gradients;
 Supersonic Flow;
 Fluid Mechanics and Heat Transfer