A Method to Construct Godunov-Type Schemes and Implicit PPV Schemes and Newton Method to Solve CFD Problems of Viscous External Aerodynamics.

The efficiency of the Newton method 1 and high resolution PPV schemes 2 to solve steady Navier-Stokes equations is assessed. Linear systems on each Newton iteration step are solved using either iterative GMRES 3 preconditioned by the incomplete LU decomposition by positions, or direct nested dissection method 1 large sparse matrix inversion procedure. Linear system coefficients are computed using numerical differentiation. As test problems, two-D viscous transonic flows about circular cylinder and NACAOOl2 airfoil are considered. An influence of a PPV scheme option, the linear system solution accuracy, Newton method initial guess choice and - associated CPU time savings are analyzed. The Newton method convergence rate at different Mach and Reynolds numbers generally proves to be linear. A significantly influenced by local supersonic zones and shocks, the allowed accuracy of the linear system solution for the Newton method convergence is found to vary depending on the concrete problem. Used with care, iterative solvers are several times 5-10 faster and economic, than direct ones, which in their turn, do not show problem parameter-dependent performance. From the three possible variants of an initial guess to compute with a high resolution PPV scheme, namely free-stream flow, first-order solution or the second order solution at different Mach andor Reynolds numbers, the most successful is the second variant. This is because otherwise the shock location is far from being exact, and at each Newton iteration step present in a PPV scheme a smooth nonlinear limiter function makes the quadratic convergence rate of the Newton method linear.