Implicit multiple-grid solution of the compressible Navier-Stokes equations using k-epsilon turbulence closure

In the present work, an algorithm for the numerical solution of the two-dimensional mass-weighted, ensemble- averaged, compressible Navier-Stokes equations is presented. Turbulence closure for the Reynolds stresses is obtained using a low-Reynolds-n umber two-equation k-s model, which includes near-wall effects. The same algorithm is used for the integration of both the Navier-Stokes and the turbulence-transport equations. The algorithm consists of a basic explicit finite-volume scheme, whose convergence is accelerated using local time-step- ping and multiple-grid techniques. Convergence is further enhanced by the use of an implicit-residual-smoothing technique, applied both in the fine and the coarser grids. The application of the numerical scheme to the solution of the k-s equations is examined in detail. Comparisons of computations with experimental data are presented for three shock-wave/turbulent boundary-layer interaction flows. Algorithm convergence rate and CPU-time require- ments are discussed. HE numerical solution of the Navier-Stokes equations for the simulation of turbulent flowfields is a subject of intense research and code development work.1 Interest is focused both on code applicability to variegated config- urations and on computational rapidity. Although direct simulation of turbulent flows2 or large-eddy numerical sim- ulations using subgrid-turbulence modeling 3'4 have appeared in the literature, current status of algorithm development and computer power indicate that for practical applications the ensemble-averaged Navier-Stokes equations should be used in conjunction with transport equations for turbulence closure.5'7 Algebraic or mixing-length models may provide satisfactory results for relatively simple flow geometries8 but require the determination of the shear-layer thickness and of the distance of each grid point from the solid boundary. Thus, they lack both geometric invariance and locality, which are highly desirable properties, from a computational viewpoint, when addressing complex flowfields with many interacting solid boundaries. Although research on Reynolds-stre ss-transport closure appears quite promising,9'10 its extension to three dimensions involves considerable increase of the number of transport equations to be solved. Also, further research con- cerning wall effects is required, the more so since the use of wall functions is considered inadequate.11 On the other hand, the use of two-equation closures, such as A>£, 12~14 is quite widespread. Notwithstanding the fact that the use of the Boussinesq hypothesis15 introduces a question- able simplification of the flow physics,16 the major drawback of the k-s model is probably the near-wall formulation which fails to reproduce correctly the effects of the solid boundary on turbulence. Several variants of the original Jones-Launder model12 have been developed in an attempt to improve upon near-wall modeling. Some ten such models have been thor- oughly examined in a critical review article by Patel et al., 17 who concluded that the models of Launder and Sharma,14