High order low diffusion numerical schemes with the capability of shock and contact discontinuities capturing is essential to study high speed flows, turbulence, acoustics, fluid-structural interactions, combustions, etc. Developing these schemes is numerically challenging. This paper introduces several recent studies of high order accuracy shock capturing numerical algorithms for Navier-Stokes equations. It includes implicit time marching algorithms, high order weighted essentially non-oscillatory(WENO) schemes for the inviscid fluxes, high order conservative schemes for the viscous terms, and preconditioning methods for solving unified compressible and incompressible flows at all speeds. This paper is organized as follows. Section 1 gives the full Navier-Stokes equations in the Cartesian coordinates and the generalized computational coordinates, including the form of Reynolds averaged Navier-Stokes equations and spatially filtered Navier-Stokes equations for large eddy simulation. Section 2 presents the study on implicit time marching algorithms, a comparison of three different methods including the unfactored implicit Gauss-Seidel relaxation scheme, the lower-upper symmetric Gauss-Seidel method, and a new hybrid method. Section 3 describes an improvement for the fifth order WENO schemes near shock points, a generalized finite compact scheme for shock capturing, an improved seventh order WENO scheme. Section 4 introduces the high order conservative central difference schemes for viscous terms and applications. Section 5 depicts the preconditioning methods( Roe-type method and E-CUSP scheme) for solving unified compressible and incompressible flows. Each section has its independent sub-section of introduction, results, and conclusions.
|Original language||English (US)|
|Title of host publication||Navier-Stokes Equations|
|Subtitle of host publication||Properties, Description and Applications|
|Publisher||Nova Science Publishers, Inc.|
|Number of pages||67|
|State||Published - Mar 2012|
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