Numerical Evidence of Multiple Solutions for the Reynolds-Averaged Navier–Stokes Equations

Abstract

In this paper, evidence is presented for the existence of multiple solutions of Reynolds-averaged Navier–Stokes equations with the one-equation Spalart–Allmaras and two-equation Wilcox turbulence models on fixed grids in three dimensions and how they were obtained is described. The two major configurations considered are an “academic” extruded two-dimensional airfoil geometry and the trap wing test case. 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 (streamwise upwind Petrov–Galerkin), which provide residual converged results for complex flows with second-order discretizations. In the paper, the ways multiple solutions have been obtained are described, 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. The phenomenon of the so-called pseudosolutions is also discussed, which is loosely defined as solutions to the discrete system of equations having “sharp” convergence behavior (sometimes up to six to seven orders of the relative residual reduction), which fail to achieve the stronger, machine-zero convergence criterion. Some numerical observations are also presented on the sensitivity of the obtained multiple solutions to the discretization and grid perturbations.