Numerical Evidence of Multiple Solutions for the Reynolds-Averaged Navier-Stokes Equations for High-Lift Configurations

In this paper we present evidence for the existence of multiple solutions of Reynolds– averaged Navier–Stokes equations (RANS) with the one–equation Spalart–Allmaras (SA) and two–equations Wilcox k − ω turbulence models on fixed grids in 3D and describe how they were obtained. The two major configurations considered are an ‘academic’ extruded 2D airfoil geometry and the trap wing, config–1 test case which is part of the AIAA CFD High Lift Prediction Workshop. The observed appearance of the multiple solutions seems to be closely related to smooth body separation (sometimes massive) routinely observed in flows over high lift configurations, especially near stall angles of attack. The results are obtained and cross–verified with two stabilized finite element codes (SUPG) which provide residual converged results for complex flows with second-order discretizations. In the paper, we describe the ways multiple solutions have been obtained, including such obvious ones as providing a different initial guess to the steady state solver as well as somewhat unexpected (in this context) techniques of using implicit residual smoothing while time-marching to steady state. We also discuss the phenomenon of the the so called ‘pseudo–solutions’, which we loosely define as solutions to the discrete system of equations having ‘sharp’ convergence behavior (sometimes up to 6–7 orders of the relative residual reduction) which however fail to achieve the stronger, machine–zero convergence criterion. We also present some numerical observations on the sensitivity of the obtained multiple solutions to the discretization and grid perturbations.

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