Preconditioned pseudo-compressibility methods for incompressible Navier-Stokes equations

This paper investigates the pseudo-compressibility method for the incompressible Navier-Stokes equations and the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and derives and presents the eigenvalues and eigenvectors of the Jacobian matrix of the preconditioned pseudo-compressible Navier-Stokes equations in generally cur-vilinear coordinates. Based on the finite difference discretization the cored for efficiently solving incompressible flows numerically is established. The reliability of the procedures is demonstrated by the application to the inviscid flow past a circular cylinder, the laminar flow over a flat plate, and steady low Reynolds number viscous incompressible flows past a circular cylinder. It is found that the solutions to the present algorithm are in good agreement with the exact solutions or experimental data. The effects of the pseudo-compressibility factor and the parameter brought by preconditioning in convergence characteristics of the solution are investigated systematically. The results show that the upwind Roe's scheme is superior to the second order central scheme, that the convergence rate of the pseudo-compressibility method can be effectively improved by preconditioning and that the self-adaptive pseudo-compressibility factor can modify the numerical convergence rate significantly compared to the constant form, without doing artificial tuning depending on the specific flow conditions. Further validation is also performed by numerical simulations of unsteady low Reynolds number viscous incompressible flows past a circular cylinder. The results are also found in good agreement with the existing numerical results or experimental data.