Preconditioned pseudo-compressibility methods for Three-dimensional incompressible Navier-Stokes equations

Abstract

Based on the authors' previous work [Qian Zhansen, Zhang Jinbai, Lee Chun-Hian. Preconditioned pseudo-compressibility methods for incompressible Navier-Stokes equations. SCIENCE CHINA Physics, Mechanics & Astronomy, 53, 11: 2090-2102, 2010] on the preconditioned pseudo-compressibility method for two-dimensional incompressible flows, the present work gives the theory and application of the proposed numerical scheme in three-dimensional cases. Since two free parameters exist in this algorithm, the eigenvalues and eigenvectors of the Jacobian matrix of the preconditioned pseudo-compressibility Navier-Stokes (N-S) equations in three-dimensional generally curvilinear coordinates are difficult to be obtained, consequently most of the previous proposed numerical schemes for three-dimensional conditions are central type schemes which do not use characteristic information. However, for hyperbolic problems upwind schemes are more robust and low dissipative. In the present work, we derive and present the unified form of the eigenvalues and eigenvectors of the Jacobian matrix of the preconditioned pseudo-compressible N-S equations in three-dimensional generally curvilinear coordinates when two free parameters, α and β, are included, and then the upwind Roe type finite difference scheme and corresponding numerical code are established. The reliability and convergence property of the present procedures are demonstrated by applications to both internal and external, two-and three-dimensional flows. Meanwhile, the derivation in this work reveals that when α>1 the eigenvectors of the preconditioned pseudo-compressibility N-S equations may be singular, which show the intrinsic reason of the non-robustness of this algorithm in the condition α>1.