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

By Jin-Long Wu, Heng Xiao, Rui Sun, Qiqi Wang. Journal of Fluid Mechanics, 2018

195 citations recorded by the source corpus.

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 non-equilibrium 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 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 the 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.

Subfields: Computational Mechanics, Fluid Flow and Transfer Processes, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2e6e23baa903bbbf3be1f95bcdb9891c5324bf90

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In Computational Mechanics

In Fluid Flow and Transfer Processes

Heng Xiao

In Statistical and Nonlinear Physics

Citations in the corpus

Turbulence Modeling in the Age of Data (2018, methodology): “Finally, Wu et al. (2018a) utilized the fundamental connection between the governing partial diﬀerential equations and the covariance of a discrepancy ﬁeld to derive a physically consistent covariance structure.”
Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework (2017, methodology): “[22] explained such observations by pointing out that RANS equations with Reynolds stress closure models can be ill-conditioned.”
Enhancing computational fluid dynamics with machine learning (2021, unknown): “This inadequacy is due to the strict assumptions associated with the RANS model, as well as the potential ill-conditioning of the RANS equations [67].”