Transonic turbulent separated flow predictions using a two-layer turbulence model

Abstract

ANAVIER-STOKES code with algebraic stress model/k-e two-layer turbulence model has been developed to compute the transonic turbulent flow over an axisymmetric secantogive-cylinder-boattail (SOCBT) projectile with a flat base and over an axisymmetric bump. The implicit upwind-biased algorithm with Harten's second-order upwind total variation diminishing discretization is extended to splve the higher order turbulent transport equations of the two-layer model for high accuracy and acceptable convergent rate. Computed results are compared with both the experimental data and the results using other turbulence models. It is found that improvement of the prediction and anisotropic turbulent natures, in the recirculating region downstream of the strong shock-wave/ boundary-layer interaction flow, can be obtained using the two-layer model of turbulence. Contents In the past several years, considerable advances have been made in predictions of transonic turbulent separated flow. For the present study, the mixture of Rodi's 1 algebraic stress model (ASM) and Chien's2 k-e turbulence model is formed as the present ASM/k-e two-layer model. The two models are matched at a preselected grid line and the number of node points in the inner layer corresponds to minimum y^^ = 40 in the boundary-layer type flow region. The extra equations of turbulent quantities of (k, e, u 'u ', v ' v ', u ' v ') or (k, e) must be solved with the mass-averaged Navier-Stokes equations if an ASM or k-e turbulent model is employed. The system of governing equations with these extra variables becomes not only larger but also stiffer and is difficult to solve. Harten's second-order upwind TVD scheme3 is a good choice to discretize the convection terms and it is extended to solve the turbulent transport equations of the present two-layer model. In the present work, the mean flow equations and turbulent equations of k and e can be decoupled, so that solving 4x4 (mean flowfield equations) and 2x2 (turbulent field equations) block tridiagonal matrices can be reduced to solving a 4x4 block tridiagonal matrix and two tridiagonal matrices (k and e equations). These modifications improve the stability of the numerical computation and the details are given in Ref. 4. The nonlinear algebraic equations of ASM are solved simultaneously by Newton's method. About 15.4% more of the CPU time per iteration for the present ASM/A:-e two-layer model is required if it is compared with that of the k-e turbulence model, which consumed a CPU time of 0.89 seconds per iteration on CRAY-XMP14. During the computation, the L2 residual can be reduced more than 4.5-5 orders lower after 7000-9000 iterations arid the results can be considered as converged results. Numerical computations have been made