A 2004 comparative study by Viti, Huang and Bradshaw benchmarked five recent Reynolds-stress-transport turbulence models (Launder–Shima, Hanjalić–Jakirlić, Craft–Launder, Manceau–Hanjalić elliptic blending, and Perot's Turbulent Potential Model) on simple canonical flows, documented severe implementation-reproducibility obstacles in the published model formulations, and left three linked questions open: the models' performance in complex flowfields (separated, rotating, three-dimensional), the unmeasured computational advantages of the Turbulent Potential Model (TPM) relative to full Reynolds-stress models, and the community's lack of a formal mechanism for cross-implementation consistency of model formulations. Later literature has since established that the reproducibility mechanism now exists (the NASA/AIAA Turbulence Modeling Resource), and that differential Reynolds-stress models — including two from the study's panel — have been implementation-verified and validated in complex aerodynamic flows. However, no study has ever quantified TPM's claimed CPU-time and memory advantage over full Reynolds-stress closures in a matched comparison, and TPM remains absent from mainstream codes. The surviving open problem is a systematic, implementation-verified evaluation of the panel's five stress-transport closures in complex flowfields together with a measured comparison of the TPM's cost and robustness against a full Reynolds-stress model and against the two-equation-model cost level invoked in Perot's claim.

Background

In industrial computational fluid dynamics (CFD), turbulent flows are most often simulated with the Reynolds-averaged Navier–Stokes (RANS) equations, in which the effect of all turbulent fluctuations is represented by the Reynolds stress tensor and must be modeled. Eddy-viscosity models (such as kk-ε\varepsilon or kk-ω\omega models, which solve two transport equations for turbulence energy and a length-scale variable) assume that this tensor is aligned with the mean strain rate; they are cheap but predictably fail in flows with rotation, streamline curvature, separation, or secondary motion. Reynolds stress models (RSMs, also called second-moment closures) instead solve a transport equation for each of the six independent Reynolds stress components plus a length-scale equation, giving a more physically faithful representation at higher computational cost and with greater numerical stiffness.

In the early 2000s a family of new stress-transport closures was developed for wall-bounded and separating flows: the Launder–Shima low-Reynolds-number near-wall closure (a stress-transport formulation valid down to the wall, without wall-function boundary conditions), the Hanjalić–Jakirlić model designed for separating flows, the Craft–Launder closure designed for complex geometries, the Manceau–Hanjalić elliptic blending model (which smoothly blends near-wall and far-from-wall turbulence regimes through an elliptic blending equation), and Perot's Turbulent Potential Model (TPM). The TPM is a mathematically derived reduction of the Reynolds stress transport equations: instead of the six stress equations it solves transport equations for a scalar potential and a vector potential of the turbulent body force (the divergence of the Reynolds stress tensor), retains exact wall boundary conditions without damping functions, and was claimed to reproduce second-moment physics at a computational cost close to that of two-equation models — a claim that remained unmeasured.

Viti, Huang and Bradshaw (Third International Conference on Computational Fluid Dynamics, ICCFD3, 2004; extended journal version 2007) performed a side-by-side implementation and validation of these five models on simple benchmark flows with quality experimental data: flat-plate boundary layers with and without adverse pressure gradient, mixing layers, plane jets, and round jets. A central finding of their study was an implementation-reproducibility problem: the published equations of several models contained errors and discrepancies relative to the developers' actual codes, and independent implementation required extensive private communication with developers. They argued that model evaluation in the complex flowfields the models were designed for (separated flows, rotating and swirling flows, and three-dimensional configurations such as those studied by Hsu et al., Bardina et al., and Apsley et al.) could not be trusted until implementations were verifiably faithful.

Subsequent community infrastructure addressed part of this program. The NASA Turbulence Modeling Resource (TMR), created by C. L. Rumsey and now maintained by the American Institute of Aeronautics and Astronautics (AIAA) Turbulence Model Benchmarking Working Group, provides documented formulations of widely used RANS models — including full second-moment closures such as the SSG/LRR-ω\omega model (a differential Reynolds-stress closure combining the Speziale–Sarkar–Gatski and Launder–Reece–Rodi pressure-strain formulations with an ω\omega-equation for the turbulence length scale) — together with canonical verification cases, naming conventions, and cross-code grid-convergence studies, precisely to eliminate cross-implementation inconsistencies. Building on this, Eisfeld, Rumsey and Togiti (2016) verified the SSG/LRR-ω\omega differential RSM across different flow solvers on TMR cases and validated it on separated, curved, and secondary-flow configurations, and Cecora et al. (2015) evaluated differential RSMs — including the Hanjalić–Jakirlić model from the original panel — on aeronautical flows with axial vortices. Manceau (2015) reviewed the elliptic blending RSM's development across complex flows. In contrast, the TPM was applied to further flows only by its own developers and one external group (Pecnik and Sanz, turbine-guide-vane heat transfer), its cost claim was repeated qualitatively but never measured against a full RSM, its developer moved on to a different model family (Oriented-Eddy Collision), and it is absent from mainstream CFD codes and from the TMR.

Problem Statement

The source study (Viti, Huang and Bradshaw, 2004/2007) benchmarked five stress-transport turbulence closures — Launder–Shima, Hanjalić–Jakirlić, Craft–Launder, Manceau–Hanjalić elliptic blending, and Perot's Turbulent Potential Model (TPM) — on simple canonical flows and left three linked questions open: (i) the performance of these closures in complex flowfields (separated, rotating, and three-dimensional configurations); (ii) whether the TPM realizes its claimed computational advantages (reduced CPU time and memory) relative to full Reynolds-stress models; and (iii) how faithful, consistent implementations of published model formulations can be ensured across independent codes. Later literature has since substantially answered (iii) through the NASA/AIAA Turbulence Modeling Resource and has evaluated differential RSMs — including two of the panel's models — in complex aerodynamic flows, so the surviving open problem is: on implementations whose fidelity is independently verified against the authoritative formulation of each closure — the developer-corrected equations as documented in the cited sources, using the corrected form wherever a published equation is known to be in error, and, for a closure absent from the verification resource such as the TPM, its cited formulation — and against established verification cases, determine (a) by direct measurement, in the same solver on matched grids and with matched convergence criteria, the matched cases, grids and validation reference data being drawn from the cited complex-flow and verification sources (the Hsu et al., Bardina et al. and Apsley and Leschziner configurations and the established verification cases of the Turbulence Modeling Resource), the computational cost (CPU time per iteration, time and iteration count to converged solution, memory) and numerical robustness of the TPM relative to at least one full differential Reynolds-stress model and relative to the two-equation-model cost level invoked in Perot's claim — thereby testing by measurement the claim that the potential-form closure retains second-moment-level accuracy, on a common error metric applied uniformly to every closure and measured against the reference data, comparable to the full Reynolds-stress closures on the same cases, at a cost, measured as CPU time and memory, closer to the two-equation baseline than to the full Reynolds-stress closure's values, and with a numerical robustness, measured as convergence to the stated convergence criteria on every case within the stated iteration budget from a fixed, stated initialization, that does not fail on any case on which the full Reynolds-stress closure converges — and (b) the predictive accuracy of the study's five stress-transport closures (Launder–Shima, Hanjalić–Jakirlić, Craft–Launder, Manceau–Hanjalić elliptic blending, and TPM) in separated, rotating, and three-dimensional flowfields, compared against identified experimental or high-fidelity reference data, with each of the five closures evaluated in the separated, rotating and three-dimensional flow classes (at least one separated, one rotating, and one three-dimensional case per closure); the later differential-RSM literature covers only the Hanjalić–Jakirlić and elliptic-blending closures, and not in a side-by-side evaluation of the panel, so the following closures carry the least prior coverage: the Launder–Shima, Craft and TPM closures. These components jointly answer one question: do advanced stress-transport closures, and the mathematically reduced TPM in particular, deliver the accuracy-cost trade-off they promise in the complex flows for which they were designed?

References

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