RANS Equations with Reynolds Stress Closure Can Be Ill-Conditioned

Reynolds-averaged Navier-Stokes (RANS) simulations with turbulence models continue to play important roles in industrial flow simulations as high-fidelity simulations are prohibitively expensive for such flows. Commonly used linear eddy viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence (e.g., flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness and stability. Recent studies found that even substituting Reynolds stresses from DNS databases (with errors below 0.5%) into RANS equations leads to grossly inaccurate velocities. Such an observation is not only disturbing for the recently emerging data-driven Reynolds stress models but also relevant for traditional, equation-based models. This observation cannot be explained by the global matrix condition number of the discretized RANS equations. In this work, we propose a metric based on local condition numbers for a priori evaluation of the stability of Reynolds stress models. Numerical tests on turbulent channel flows at various Reynolds numbers suggest that the proposed metric can adequately explain observations in previous studies, i.e., decreased model stability with increasing Reynolds number, and better stability of the implicit treatment of Reynolds stress compared to the explicit treatment.