Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

Abstract

Reynolds-averaged Navier-Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy viscosity models are intrinsically unable to handle flows with nonequilibrium turbulence (e.g., flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5% from direct numerical simulation (DNS) databases into RANS equations leads to velocities with large errors (up to 35%). While such an observation may have only marginal relevance to traditional Reynolds stress models, it is disturbing for the recently emerging data-driven models that treat the Reynolds stress as an explicit source term in the RANS equations, as it suggests that the RANS equations with such models can be ill-conditioned. So far, a rigorous analysis of the condition of such models is still lacking. As such, in this work we propose a metric based on local condition number function for a priori evaluation of the conditioning of the RANS equations. We further show that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. Comprehensive numerical tests are performed on turbulent channel flows at various Reynolds numbers and additionally on two complex flows, i.e., flow over periodic hills and flow in a square duct. Results suggest that the proposed metric can adequately explain observations in previous studies, i.e., deteriorated model conditioning with increasing Reynolds number and better conditioning of the implicit treatment of Reynolds stress compared to the explicit treatment. This metric can play critical roles in the future development of data-driven turbulence models by enforcing the conditioning as a requirement on these models.

Quantification of model uncertainty in RANS simulations: A review

“…Nevertheless, a unique challenge for directly correcting or predicting the Reynolds stress tensors with data-driven models is the possible ill-conditioning of the RANS equations. For example, small errors in the machine-learning-predicted Reynolds stresses can lead to large errors in the propagated velocities [146]. In order to overcome this difficulty, Wu et al [123] proposed learning the linear and nonlinear parts of the Reynolds stress separately, with the linear part treated implicitly to improve model conditioning.…”

“…On the other hand, Wang et al [186] performed the same propagation for fully developed turbulent flows in square ducts at various Reynolds numbers and found that the propagated velocities agree with DNS data satisfactorily. Such apparently conflicting findings were explained by different model conditioning in various flows, i.e., different sensitivity levels of the mean velocities to Reynolds stresses [146].…”

Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks

“…Such an approach does not constrain our work to model-specific assumptions. We note that explicit R-S approaches can potentially result in significant prediction error of fluid quantities when the Reynolds number approaches 5000 and above [26,27]. Additionally, small errors in the predicted R-S field in an explicit representation can be amplified leading to instabilities.…”

“…9-10. As the R-S is increased, minor deviations in the anisotropic term are amplified resulting in potentially starkly different flow predictions [27]. SDD-RANS accurately captures these phenemona.…”

“…Additionally, small errors in the predicted R-S field in an explicit representation can be amplified leading to instabilities. We accept this as an open problem and while alternative implicit approaches have been proposed [27], we leave the discussion of such methods to future works. However, we will show how the proposed framework can reflect such difficulties through the predicted probabilistic bounds on the flow quantities of interest.…”