[Paper Review] RANS Equations with Reynolds Stress Closure Can Be Ill-Conditioned

Jinlong Wu, Heng Xiao+2 more|arXiv (Cornell University)|Mar 15, 2018

Model Reduction and Neural Networks3 citations

TL;DR

This paper proposes a local condition number metric to diagnose instability in Reynolds stress models within RANS simulations, explaining why even highly accurate Reynolds stresses from DNS yield inaccurate results. The method reveals that model stability decreases with increasing Reynolds number and that implicit treatment of Reynolds stresses is more stable than explicit treatment.

ABSTRACT

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.

Motivation & Objective

• To investigate why RANS simulations with accurate Reynolds stresses from DNS still produce grossly inaccurate results.
• To identify the root cause of instability in Reynolds stress models beyond global matrix condition numbers.
• To develop a diagnostic tool for assessing model stability prior to simulation.
• To compare the stability of explicit versus implicit treatments of Reynolds stresses in RANS solvers.

Proposed method

• Propose a local condition number metric based on the Jacobian of the RANS system to assess numerical stability at each computational point.
• Apply the metric to turbulent channel flow simulations at varying Reynolds numbers to evaluate its predictive power.
• Use DNS-derived Reynolds stresses as input to RANS equations to test model behavior under controlled, high-accuracy conditions.
• Compare stability outcomes between explicit and implicit discretization schemes of Reynolds stresses.
• Analyze the relationship between local condition numbers and observed simulation inaccuracies.
• Validate the metric's ability to explain previous observations of decreasing stability with increasing Reynolds number.

Experimental results

Research questions

• RQ1 Why do RANS simulations with highly accurate Reynolds stresses from DNS still produce inaccurate velocity predictions?
• RQ2 What causes the instability in Reynolds stress models beyond the global condition number of the discretized system?
• RQ3 How does the local condition number correlate with simulation accuracy and stability across different Reynolds numbers?
• RQ4 Why is the implicit treatment of Reynolds stresses more stable than explicit treatment in RANS simulations?
• RQ5 Can a local condition number metric reliably predict the onset of instability in Reynolds stress models?

Key findings

• The local condition number metric successfully explains the observed instability in RANS simulations using accurate DNS-derived Reynolds stresses.
• Model stability decreases significantly with increasing Reynolds number, as indicated by rising local condition numbers.
• The implicit treatment of Reynolds stresses leads to lower local condition numbers compared to explicit treatment, indicating superior stability.
• The global matrix condition number fails to capture the instability observed in simulations, highlighting the need for local metrics.
• The proposed metric provides a priori diagnostic capability for identifying unstable regions in Reynolds stress models before full simulation.
• Numerical tests confirm that the instability is not due to errors in Reynolds stress input but rather to ill-conditioning in the RANS system.