High-Order Weighted Essentially Non-Oscillatory (WENO) scheme for Incompressible Naver-Stokes equations

Abstract

A new high-order projection method for simulating incompressible turbulent flows is presented based on the weighted essentially non-oscillatory scheme (WENO) schemes. In such a sense, this paper can be considered an extension of Zhang and Jackson’s work (Zhang & L. Jackson, 2009) in terms of solution accuracy and computational efficiency. Unlike the previous work, the present method employed the Adams-Bashforth scheme for the nonlinear convection and the Crank-Nicolson scheme for the viscous term. For the spatial discretization, the WENO scheme is employed for the convection and standard central differences are used for the viscous. By the combination of successively higher orders of WENO and CD schemes, the desired order of accuracy was achieved without appreciable extra CPU time or memory overhead. More specifically the combination of WENO3/CD2, WENO5/CD4, and WENO7/CD6 achieved the third, fifth, and seventh order of spatial accuracies respectively. A verification study was presented both in space and time by using the 2D Taylor Green vortex problem. More challenging turbulent flows are simulated by solving the 3D Taylor-Green vortex problem for successively raised Reynolds numbers of Re=1,600, 16,000, and 160,000. The results, including the evolution of total kinetic energy, enstrophy, energy spectra, and local and global vortex field, support that the proposed method can be utilized for simulating the higher Reynolds number turbulent flows.