Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Abstract

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier-Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourthorder differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.

A study of convective flux schemes for aerospace flows

“…It is known that flux schemes may have influence in such flow solutions, as reported by Swanson, Radespiel and Turkel (1998), Zingg et al (1999), Allmaras (2002), and Bigarella (2002). The present group attributes such problems to nonphysical behavior of centered flux schemes, more precisely in the explicitly added artificial dissipation model, as reported in Bigarella, Moreira and Azevedo (2004).…”

“…This does not seem to be the approach chosen by other CUSP users (Jameson, 1995a;Jameson, 1995b;Swanson, Radespiel and Turkel, 1998;Zingg et al, 1999). In these references, the respective authors apparently define the convective flux operator similarly to the one presented in Eq.…”

Validation of Numerical Schemes and Turbulence Models Combinations for Transient Flow Around Airfoil

“…A zonal, upwind, implicit-factored algorithm was employed to solve both the mean turbulent flow, using a cell-centred, finite-volume method, and a generalized Roe's upwind fluxdifference splitted with an optional TVD operator. Zingg et al (2000) compared the accuracy of several discretization schemes for subsonic and transonic airfoil flows. Their discretization for the inviscid fluxes included: a) a second-order accurate central differences with third-order matrix numerical dissipation, b) a second-order convective upstream split scheme, c) a third-order upwindbiased differencing with Roe's flux-splitting, and d) a fourth-order central differences with third-order matrix numerical dissipation.…”

Multigrid Solution for High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations

“…Further work for the compressible Navier-Stokes equations on structured meshes is presented by Zingg et al [59], who compared the results of a fourth-order central difference discretization to a number of lower-order schemes. One of their results is shown in Figure 1-1.…”

“…In particular, numerous research efforts have been aimed at developing high-order accurate algorithms for solving partial differential equations. These efforts have led to many types of numerical schemes, including higher-order finite difference [33,59,56], finite volume [7,57], and finite element [8,20,6,40] methods for both structured and unstructured meshes. Despite these developments, in applied aerodynamics, most computational fluid dynamics (CFD) calculations are performed using methods that are at best second-order accurate.…”