A Compressible Wall-Layer Approach Compatible with Various Turbulence Models

Abstract

Traditional wall functions have been used successfully for decades to decrease the computational cost for obtaining solutions to incompressible ∞ows with equilibrium turbulence. However, these traditional, analytic wall functions are poorly suited for more complex ∞ows. The difiusion model, a wall-layer developed by Blottner, has previously been investigated in conjunction with k-†-type, two-equation, turbulence models and the oneequation, Spalart-Allmaras model. This paper evaluates the difiusion in conjunction with k-!-type turbulence models. Wall functions have been used for decades to reduce the computing resources required to obtain reasonable engineering solutions. The traditional approach obtains analytical relations for the dependent variables at the wall-layer interface. The implementation of the traditional approach is quite simple, but accuracy sufiers when certain efiects (e.g., compressibility, real gas efiects, and non-equilibrium turbulence) are signiflcant, since these efiects are neglected to obtain the analytical forms. Because of this weakness, there is a need for continuing wall-layer development so that reasonably accurate solutions to more complex ∞ows can be obtained with fewer computational resources. Gant, Craft, et al. 1,2 developed a subgrid-type wall-layer and tested it in conjunction with k-†type turbulence models. The uses subgrids in the vicinity of the no-slip boundaries and then solves the boundary layer equations on the subgrid. a The numerical solutions to the boundary layer equations on the subgrids are matched to the exterior ∞ow solution governed by the Reynolds-Averaged Navier-Stokes (RANS) equations. The numerical solution on each subgrid has, as its outer Dirichlet boundary condition (BC), the solution values of the RANS solution (either at or interpolated to the location of the outermost subgrid point), and the RANS solution has, as its BC for the control volume faces nearest the wall, a ∞ux which is determined by the solution to the wall-layer on the subgrid between it and the wall. Since the same turbulence is used (reduced by the boundary-layer assumptions) on the subgrid as in the exterior ∞ow, consistency is provided, and accurate solutions can be obtained as long as the subgrids for the wall-layer do not extend far enough into the ∞owfleld for the boundary-layer assumptions to break down. Pressure gradient efiects are also taken into account. Although Gant et al. derived their formulations for compressible ∞ows, their published test cases were incompressible. Blottner 3,4 independently initiated a wall-layer modeling approach in 2002 which is similar to the subgrid technique of Gant et al. along with the additional assumption that the convective terms can be neglected in the inner layer. Blottner named this the \difiusion model since it retains the difiusion terms. Blottner’s assumption greatly simplifles the numerics of the subgrid solution since the elimination of convective terms permits a flnite-difierence method to be used without upwinding. Additionally, for a steady ∞ow, the