On Some Numerical Dissipation Schemes

Abstract

Several schemes for introducing an arti cial dissipation into a central di erence approximation to the Euler and Navier Stokes equations are considered. The focus of the paper is on the convective u p wind and split pressure CUSP s c heme, which is designed to support single interior point discrete shock w aves. This scheme is analyzed and compared in detail with scalar dissipation and matrix dissipation MATD s c hemes. Resolution capability i s determined by solving subsonic, transonic, and hypersonic ow problems. A nite-volume discretization and a multistage time-stepping scheme with multigrid are used to compute solutions to the ow equations. Numerical solutions are also compared with either theoretical solutions or experimental data. For transonic airfoil ows the best accuracy on coarse meshes for aerodynamic coe cients is obtained with a simple MATD scheme. The coarsegrid accuracy for the original CUSP scheme is improved by modifying the limiter function used with the scheme, giving comparable accuracy to that obtained with the MATD scheme. The modi cations reduce the background dissipation and provide control over the regions where the scheme can become rst order.

A study of convective flux schemes for aerospace flows

“…The CUSP scheme implementation options are here addressed. The CUSP ctt formulation (Jameson, 1995a;Jameson, 1995b;Swanson, Radespiel and Turkel, 1998) uses constant property distributions in the cell to build the centered convective fluxes, whereas the CUSP rec option proposed in the current work uses reconstructed properties in the faces to compute such fluxes. The dissipation formulation is identical between both options, and a value of ǫ CUSP = 0.3 is chosen for the CUSP constant.…”

“…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).…”

“…The Jameson CUSP model (Jameson, 1995a;Jameson, 1995b;Swanson, Radespiel and Turkel, 1998) is inspired in earlier work on flux-vector splitting methods. It is based on a splitting of the flux function into convective and pressure contributions.…”

“…In the above scheme definitions, the k-th subscript, which indicates a variable computed in the face, has been eliminated in order to avoid overloading the equations with symbols. It is important to remark here that face properties are computed using the Roe average procedure (Roe, 1981;Swanson, Radespiel and Turkel, 1998). The centered spatial discretization of the convective fluxes, C i , in this scheme, for the present context, is defined as…”

“…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.…”