Whether full Reynolds-stress-transport (stress-transport) turbulence closures deliver clear and systematic predictive advantages over two-equation eddy-viscosity models in flows with strong separation, rotation, streamline curvature, or three-dimensional stress anisotropy was explicitly left open by the implementation and validation study of Viti, Huang and Bradshaw (2007), as was the question of whether divergence-based formulations such as Perot's turbulent potential model deliver their claimed computational advantages (faster convergence, reduced stiffness). Later literature built the standardized benchmark infrastructure those authors called for (the NASA/AIAA Turbulence Modeling Resource) and advanced individual stress-transport closures, but produced no systematic multi-flow-class comparative verdict, and the turbulent potential model line has been dormant since 2006. This task requires an implementation-verified, replayable comparative assessment of predictive accuracy and computational cost of the stress-transport model class against two-equation eddy-viscosity models across the complex-flow regime the source study excluded.

Background

When the Navier–Stokes equations are averaged (the Reynolds-averaged Navier–Stokes, or RANS, approach underlying most engineering computational fluid dynamics, CFD), the Reynolds stress tensor — the averaged product of fluctuating velocity components that represents turbulent momentum transport — must be modeled; this is the turbulence closure problem. The dominant practical approach uses eddy-viscosity models based on the Boussinesq hypothesis: the Reynolds stresses are assumed aligned with the mean strain rate through a single scalar (isotropic) eddy viscosity, determined from transport equations for the turbulent kinetic energy k and one turbulence scale variable, such as the dissipation rate ε or the specific dissipation ω. These 'two-equation' models, exemplified by the Menter shear-stress transport (SST) k–ω model, are cheap and robust, and they dominate industrial use.

The linear, isotropic eddy-viscosity assumption is structurally wrong whenever stress anisotropy matters: it predicts zero secondary motion in non-circular ducts (where differences among the normal stresses drive corner vortices), responds poorly to system rotation and strong streamline curvature, and misrepresents strongly non-equilibrium separated flows. The alternative is the stress-transport (Reynolds-stress-transport, or second-moment) closure: instead of a constitutive relation between stresses and mean strain, a modeled transport equation is solved for each independent Reynolds-stress component plus one turbulence-scale equation, with the pressure–strain energy-redistribution term modeled explicitly. Such closures represent anisotropy, rotation, and curvature naturally, at the price of seven strongly coupled and numerically stiff equations, realizability constraints (bounds keeping predicted stresses physically admissible), and demanding near-wall treatments — which makes both their implementation fidelity and their computational cost genuine practical concerns.

A third route, due to Perot, reformulates the stress-transport idea through the 'turbulent body force': the divergence of the Reynolds stress tensor enters the mean momentum equations as an effective force, and the model instead carries transport equations for a scalar and a vector potential of that force. The resulting turbulent potential model was claimed to retain non-equilibrium capability at a cost and complexity comparable to two-equation models, with fewer unknowns, no realizability conditions, and much weaker inter-equation coupling — claimed, but never benchmarked, computational advantages.

Viti, Huang and Bradshaw (2007) reported a careful implementation and validation study of several then-recent stress-transport closures, including the Manceau–Hanjalić elliptic-blending near-wall Reynolds-stress model and the Perot turbulent potential model, in simple wall-bounded and shear-free reference flows; their earlier conference version was explicitly titled 'the Need for Standardized Procedures'. They found implementation fidelity to be the dominant difficulty, reportedly could not bring one candidate model to a usable form because its published variants were mutually inconsistent, left open whether stress-transport models give systematic advantages in genuinely complex flows, left the turbulent potential model's computational claims unquantified, and called for community benchmark infrastructure. Much of that infrastructure now exists: the NASA Langley Turbulence Modeling Resource (TMR), initiated by Rumsey and colleagues and now maintained by the AIAA Turbulence Model Benchmarking Working Group, provides documented model formulations with naming conventions for published variants, verification cases with grids and reference solutions from independently verified codes, and a validation case database — although it explicitly disclaims any attempt at broad validation of turbulence models across complex flows. What no later work supplies is the systematic, implementation-verified comparative assessment that the source called for.

Problem Statement

Determine whether full stress-transport (Reynolds-stress-transport) closures deliver clear and systematic predictive advantages (judged by a stated error metric against each benchmark case's reference or experimental data, and required to hold consistently across the covered flow classes) over two-equation eddy-viscosity models such as the SST k–ω model in the complex-flow regime from which the source study explicitly excluded itself — flows exhibiting strong separation, system rotation or streamline curvature, or three-dimensional stress anisotropy — covering at minimum one documented verification or validation case in each of these complex-flow classes (separating flow; rotating or strongly curved flow; three-dimensional anisotropic, e.g. non-circular-duct, flow); and quantify the computational properties of divergence-based stress-transport formulations — exemplified by Perot's turbulent potential model — relative to conventional stress-transport and two-equation implementations under a common solver and discretization, including convergence rate, equation stiffness, and cost per achieved accuracy. The single scientific objective joining these deliverables is to settle, with evidence a skeptical expert can replay, whether the additional complexity of the stress-transport model class pays for itself in accuracy and in computational cost, and for which flow classes.

References

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