Preconditioning Methods for Low-Speed Flows

We consider the steady-state equations for a compressible uid. For low-speed ow, the system is sti because the ratio of the convective speed to the speed of sound is quite small. To overcome this di culty, we alter the time evolution of the equations but retain the same steady-state analytic equations. To achieve high numerical resolution, we also alter the arti cial viscosity of the numerical scheme, which is implemented conveniently by using other sets of variables in addition to the conservative variables. We investigate the e ect of the arti cial dissipation within this preconditioned system. We consider both the nonconservative and conservative formulations for arti cial viscosity and examine their e ect on the accuracy and convergence of the numerical solutions. The numerical results for viscous three-dimensional wing ows and two-dimensional multi-element airfoil ows indicate that e cient multigrid computations of ows with arbitrarily low Mach numbers are now possible with only minor modi cations to existing compressible Navier-Stokes codes. The conservative formulation for arti cial viscosity, coupled with the preconditioning, o ers a viable computational uid dynamics (CFD) tool for analyzing problems that contain both incompressible and compressible ow regimes. This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the rst author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.