Development of an efficient viscous preconditioning method and its application to numerical simulation of flows over airfoils

Keeping in mind the perennial need for more and more efficient and robust computational codes used in numerical optimization design of airfoils, we investigate the preconditioning of NS (Navier-Stokes) equations. We introduce the preconditioning matrices proposed by Turkel^[1] and Choi^[2] into the time derivative terms of Reynolds-Averaged NS (RANS) equations. We remove effectively the system stiffness of compressible RANS equations for nearly incompressible flows and for flow region in the boundary layer of compressible flows. For the preconditioned RANS equations, we employ a cell-centered finite-volume scheme for spatial discretization and use a multi-stage Runge-Kutta method for time marching. A FAS (Full Approximation Scheme) multigrid method is also used to effectively accelerate the convergence to steady state. We simulated the low-speed, subsonic and transonic viscous flows over RAE2822 and GAW-1 airfoils respectively and achieved excellent success; the simulation results shown in Figs. 1 through 6 in the full paper (RAE2822 airfoil) and in Figs. 7 through 9 (GAW-1 airfoil) show good agreement with experimental data and exhibit rapid convergence. The comparison of the computed results and experimental data show that both Turkel's and Choi's preconditioning methods significantly improve the efficiency and accuracy of compressible computational code when used for low-speed flows. We also show that preconditioning method is applicable to accelerating the calculation of low-speed flow region in the boundary layer of subsonic and transonic airfoils. In short, we developed an efficient and robust code for low speed, subsonic, transonic and supersonic flows that can be used in numerical optimization design of airfoils.