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International CAE Conference Program

 

Mathematical modeling of roughness effects on the unsteady transitional flows

Ali Mosahebi Mohamadi , Éric Laurendeau
École Polytechnique de Montréal

A new mathematical model is proposed and numerically investigated for the unsteady roughness-induced transitional flows. The approach is based on KW-SST transitional turbulence model, where an extra transport equation is proposed to introduce the roughness effects. The model is implemented to the Euler/Navier–Stokes Code (NSCODE), developed by the research group at École Polytechnique de Montréal. NSCODE solves the 2D Euler and Navier–Stokes equations on a multi block-structured grid. The equations are discretized using a second-order cell centered finite-volume method, stabilized with the artificial dissipation Jameson–Schmidt–Turkel (JST) scalar dissipation scheme. The gradients needed for the evaluation of the viscous fluxes are obtained at the vertices by applying the Green–Gauss formula on the dual mesh. Steady-state flow solutions are computed using the explicit multistage Runge–Kutta scheme. Convergence is accelerated by the Full Approximation Storage (FAS) multigrid scheme, where coarsening is performed by removal of every other mesh line in each direction. Restriction to the coarse grids is achieved by summing the residuals of the cells to be agglomerated and volume-averaging the conserved variables. Prolongation to the fine-grid is performed using tri-linear interpolation. On coarse levels, the dissipative fluxes are reduced to first-order accuracy to further improve robustness and speed. In addition, the coarse grid multigrid corrections are implicitly smoothed after prolongation to the fine-grid in a manner analogous to the implicit-residual smoothing scheme. A dual-time stepping approach is used for the implicit time marching of the unsteady solver, where a second order temporal discretization is employed. For closure of the Unsteady Reynolds-averaged Navier–Stokes (URANS) equations, NSCODE uses the Spalart–Allmaras (SA), Menter’s k-ω, Menter’s k-ω-SST, and on KW-SST models. The turbulence as well as transition transport equations are discretized with a first-order finite difference scheme and solved separately from the mean-flow equations using a positivity-preserving, Alternating Direction Implicit (ADI) factorization scheme. The turbulence equation(s) are solved on the fine grid and the effect of turbulence is accounted for on the coarse grids by transferring the resulting turbulent eddy viscosity. The approach is thoroughly validated through a variety of unsteady 2D transition test cases with various roughness amplitudes, showing a good agreement with previous numerical results as well as experimental data.