Well-posedness of the exterior turbulent flow boundary-value problem for RANS equations with two-equation turbulence models

The exterior turbulent flow problem — the steady compressible Reynolds-Averaged Navier–Stokes (RANS) equations coupled with a two-equation $k$-$\omega$ turbulence model (Wilcox $k$-$\omega$ or Menter Shear Stress Transport (SST)) in the unbounded domain around a body, with no-slip wall conditions ($k=0$ and the singular behavior $\omega\sim C/d^2$ at the wall, $d$ the wall distance) and prescribed far-field behavior of the turbulence variables (the source gives two alternative far-field formulations and states that it cannot say which is preferred) — underlies virtually all aerodynamic CFD, yet its well-posedness has never been established. Langer and Swanson (2020) formulated the problem precisely, gave heuristic arguments and numerical examples suggesting ill-posedness in both the direct and inverse senses, showed that the common variable substitutions ($\omega=e^{\Omega}$, $k=e^{\mathcal{K}}$, $\omega=1/\tau$, $\omega=1/g^2$) are not proven equivalent to the original formulation, and demonstrated numerically that distinct eddy-viscosity fields can reproduce identical surface pressure and skin-friction data.

Background

For high-Reynolds-number aerodynamic flows, the standard simulation paradigm solves the Reynolds-Averaged Navier–Stokes (RANS) equations — the Navier–Stokes equations averaged over turbulent fluctuations, in which averaging introduces the unknown Reynolds-stress tensor that must be closed by a model. In the two-equation models considered here, the $k$-$\omega$ model of Wilcox (1988) and the Shear Stress Transport (SST) model of Menter (1994), closure is achieved through two additional transport equations: one for the turbulent kinetic energy per unit mass $k>0$ and one for the specific dissipation rate $\omega>0$ (the rate of dissipation of $k$ per unit $k$). From these, the eddy viscosity $\mu_t\ge 0$ is computed and added to the molecular viscosity in the mean-flow momentum and energy equations, representing turbulent momentum and heat transport; in the SST model $\mu_t$ further blends the $k$-$\omega$ model near walls with a $k$-$\varepsilon$-type model in the far field.

Langer and Swanson (2020) studied what they call the exterior turbulent flow problem: the steady compressible RANS equations coupled with these transport equations in the open unbounded domain exterior to a body in $\mathbb{R}^m$, $m=2,3$ — the idealization of flow past an aircraft. Solving it on a finite mesh requires boundary data at both ends: at the no-slip wall one imposes $u=0$ and $k=0$, while a three-dimensional near-wall analysis shows $\omega$ must exhibit a quadratic singularity, $\omega\sim C/d^2$ with $d$ the wall distance, whose numerical realization is delicate (practical guidance for such inflow/wall turbulence values is discussed by Spalart and Rumsey, 2007); in the far field the authors derive analytically two alternative prescriptions. Either (their formulation 1) $k$ and $\omega$ decay at infinity at power-law rates, $\omega=O(1/\|x\|_2)$ and $k=O(1/\|x\|_2^{\beta^*/\beta})$, which differ between the Wilcox and SST models; or (their formulation 2) the problem is posed on $V^+\setminus D$, the side of the singular hyperplane $E$ of the derived far-field solutions that contains the body, with $(k,\omega)\to(\infty,\infty)$ when approaching $E$ and $(k,\omega)\to(0,0)$ along rays going away from $E$. The authors state that, without knowledge of the true behavior of $k$ and $\omega$, it is impossible to say which of the two formulations is preferred, and note that the second formulation is typically closer to the application. A boundary-value problem is well-posed in Hadamard's sense if a solution exists, is unique, and depends continuously on the data. The authors argue — with analysis and examples, but explicitly without proof — that the exterior turbulent flow problem may violate this: they note it is an open question whether $k$ can even exhibit the behavior that the wall and far-field conditions on $\omega$ force upon it, they recall numerical evidence of multiple steady RANS solutions (Kamenetskiy et al., 2014), and they conjecture that the choice of boundary values for $k$ and $\omega$ may render the problem ill-posed.

They further examine variable substitutions used in production codes — $\omega=e^{\Omega}$, $k=e^{\mathcal{K}}$, $\omega=1/\tau$, $\omega=1/g^2$ — introduced to keep the turbulence variables positive during iteration and to soften the wall singularity (e.g. in discontinuous Galerkin implementations — finite-element discretizations in which the numerical solution is allowed to jump between mesh cells — such as that of Hartmann, Held and Leicht, 2011). They show that substituting $k=e^{\mathcal{K}}$ is outright incompatible with the wall condition $k=0$ (since $e^{\mathcal{K}}>0$), that $\omega=e^{\Omega}$ converts the quadratic wall singularity of $\omega$ into a logarithmic singularity of $\Omega$ but shifts the quadratic singularity into the term $(\mu_l+\sigma_\omega\mu_t)\|\mathrm{grad}\,\Omega\|_2^2$, and that the reciprocal substitutions shift the singularity from the wall to unbounded growth at infinity; and they stress that no mathematical proof of equivalence between the substituted and original boundary-value problems exists. Finally, they reinterpret turbulence modeling as an inverse problem: given noisy surface data $Y^\delta$ (e.g. pressure and skin-friction distributions on the body surface $\partial D$), reconstruct an eddy viscosity $\mu_t\ge 0$ such that the solution of the exterior turbulent flow problem reproduces the data, formally via a forward operator $F:\mu_t\mapsto Y$. A numerical experiment on the transonic RAE 2822 airfoil (a standard benchmark airfoil geometry in transonic aerodynamics) shows that two genuinely different eddy-viscosity fields (computed with and without a production limiter — a cap on the turbulent-kinetic-energy production term) yield indistinguishable surface pressure coefficients $C_p$ (normalized pressure) and skin-friction coefficients $C_f$ (normalized wall shear stress), demonstrating non-uniqueness of this reconstruction at least numerically. The corresponding data-driven reconstruction literature (Foures et al., 2014; Wu, Xiao and Paterson, 2018; Duraisamy, Iaccarino and Xiao, 2019) treats the reconstruction computationally and acknowledges its ill-posedness but supplies no well-posedness theory.

Problem Statement

Overarching objective: place the exterior turbulent flow problem — the steady compressible RANS equations coupled with a two-equation $k$-$\omega$ turbulence model (Wilcox $k$-$\omega$ or Menter SST) in the exterior domain of a body in $\mathbb{R}^m$, $m=2,3$ — on a sound mathematical footing with respect to its standard boundary prescriptions. The source leaves the following questions open.

(1) Direct well-posedness. Under the boundary prescriptions of Langer and Swanson (2020) — no-slip walls with $k=0$ and the quadratic near-wall singularity $\omega\sim C/d^2$ ($d$ the wall distance), together with the far-field prescriptions in the source's two alternative formulations (formulation 1: power-law decay at infinity, $\omega=O(1/\|x\|_2)$ and $k=O(1/\|x\|_2^{\beta^*/\beta})$ at the model-dependent rates; formulation 2: the problem posed on $V^+\setminus D$, with $(k,\omega)$ singular as the hyperplane $E$ is approached and decaying to $(0,0)$ along rays away from $E$) — is the exterior turbulent flow problem well-posed in the sense of Hadamard: does a solution exist, is it unique, and does it depend continuously on the data? The source states that it is impossible to say which of the two formulations is preferred (the second being typically closer to the application), so the question covers both formulations. A rigorous proof of well-posedness, or a rigorous proof that any Hadamard condition fails (non-existence, non-uniqueness, or discontinuous dependence on the data), is lacking. As part of this question: can $k$ in general exhibit the near-wall and far-field behavior that the boundary conditions on $\omega$ force upon it?

(2) Formulation equivalence. Are the boundary-value problems obtained by the variable substitutions $\omega=e^{\Omega}$, $\omega=1/\tau$, or $\omega=1/g^2$ equivalent to the original $k$-$\omega$ boundary-value problem when the physical boundary conditions are enforced? (The $k=e^{\mathcal{K}}$ case is already settled negatively by the source, being incompatible with the wall condition $k=0$; the remaining question is a proof of equivalence or non-equivalence for the $\omega$-substitutions, including the treatment of the shifted singularities at the wall and in the far field.)

(3) Inverse well-posedness. Consider the forward operator $F$ mapping an eddy viscosity $\mu_t\ge 0$ (constant outside a ball, so the exterior problem is meaningful) to surface data $Y\in L^2(\partial D)$ (the space of square-integrable functions on the boundary $\partial D$) such as pressure and skin-friction distributions. Is the reconstruction of $\mu_t$ from noisy data $Y^\delta$ with $\|Y^\delta-Y\|\le\delta$ well-posed — is $F$ injective on its domain and continuously invertible? If not, construct and quantify the non-uniqueness (e.g., families of distinct eddy-viscosity fields yielding the same surface data), and characterize the stability or its failure. This includes determining whether the numerical non-uniqueness demonstrated by the source (distinct eddy-viscosity fields with indistinguishable $C_p$/$C_f$ distributions) reflects non-injectivity of the continuum operator $F$.

(4) Auxiliary practical question, explicitly deferred to future work by the source: realize and analyze an automatic procedure that, given the angle of attack, partitions the outer boundary of a finite computational domain into inflow and outflow portions by determining, for each boundary control volume (a mesh cell of the finite-volume discretization touching the outer boundary), the entry and exit intersections along lines of the prescribed direction (rays), and investigate the influence of this partitioning on computed solutions and on the solution algorithm.

These items are related open questions of the source, all concerning whether the boundary-value problem actually solved by aerodynamic CFD codes is mathematically well defined. The source raises the direct question (1), the formulation question (2), and the inverse question (3) — the operator-theoretic counterpart of (1) — as separate open issues, and lists (4) as an auxiliary practical item explicitly deferred to future work. Questions (1)–(3) are the mathematical questions left open by the source, each independently decidable.

References

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