Langer and Swanson (2020) gave the first systematic formulation of the boundary-value problem for the steady, compressible Reynolds-Averaged Navier-Stokes (RANS) equations coupled with two-equation turbulence models of kk-ω\omega type (Wilcox kk-ω\omega, Menter SST) on the exterior of a body, deriving the physically motivated boundary prescriptions: k=0k=0 with a quadratic near-wall singularity of the specific dissipation rate ω\omega at no-slip walls, and prescribed freestream values with algebraic far-field decay of kk and ω\omega. They argued - explicitly without proof - that both the direct exterior problem and the inverse problem of reconstructing the eddy viscosity μt\mu_t from noisy surface data are ill-posed, and left open the equivalence of the standard positivity-preserving variable substitutions, the identifiability of μt\mu_t, and the automatic partition of truncated outer boundaries into inflow and outflow portions. The open problem is to settle the Hadamard well-posedness status of this coupled exterior problem - by a proof of existence, uniqueness, and continuous dependence under an explicit admissible class of boundary data, or by a rigorous demonstration of ill-posedness - together with the associated formulation questions. As of 2026, rigorous results exist only for unsteady relatives (notably Kolmogorov's two-equation model) on periodic or bounded domains with smooth data; the steady exterior problem with singular boundary data remains untouched.

Background

Almost all engineering computations of high-Reynolds-number turbulent flow - over wings, through turbines, around vehicles - solve the Reynolds-Averaged Navier-Stokes (RANS) equations: the Navier-Stokes equations averaged over time or an ensemble, in which the discarded turbulent fluctuations appear as an unknown Reynolds-stress tensor. Production codes close this tensor via the Boussinesq eddy-viscosity hypothesis, which postulates that the Reynolds stresses are proportional to the mean strain-rate tensor, the proportionality factor being an effective turbulent viscosity μt\mu_t. Two-equation turbulence models determine μt\mu_t from two additional transport equations: one for the turbulent kinetic energy kk (one half the mean squared velocity fluctuation) and one for a variable fixing the turbulence time scale - the dissipation rate ε\varepsilon in kk-ε\varepsilon models, or the specific dissipation rate ω∼ε/k\omega \sim \varepsilon/k in the kk-ω\omega models introduced by Wilcox and blended with kk-ε\varepsilon behavior away from walls in Menter's SST (shear-stress-transport) model that dominates aerodynamic practice; in kk-ω\omega models μt=ρk/ω\mu_t = \rho k/\omega (with ρ\rho the density).

Langer and Swanson (2020) asked what boundary conditions make the coupled steady system a well-defined boundary-value problem on the exterior domain (the unbounded fluid region surrounding the body). Local analysis forces singular data: at a no-slip wall the kinetic energy must vanish, k=0k = 0, while the ω\omega-equation forces ω(x−hn)h2→6νl/β\omega(x-hn)h^2 \to 6\nu_l/\beta as the wall distance h→0+h \to 0^+ (νl\nu_l the laminar kinematic viscosity, β\beta a model constant), with kk vanishing toward the wall like the wall distance to the power α\alpha, α≈3.23\alpha \approx 3.23 for Wilcox's constants. In the far field, freestream values k∞,ω∞k_\infty, \omega_\infty must be supplied together with algebraic decay ω=O(∥x∥−1)\omega = O(\|x\|^{-1}) and k=O(∥x∥−β∗/β)k = O(\|x\|^{-\beta^*/\beta}), with ratio of model constants β∗/β=1.2\beta^*/\beta = 1.2 for Wilcox kk-ω\omega and ≈1.086\approx 1.086 for SST. On the truncated outer boundary of a finite computational domain, turbulence data may be imposed only on the inflow portion - but which portions are inflow or outflow is unknown before the solution is computed.

Whether the resulting exterior problem is well-posed in Hadamard's sense - existence of a solution, uniqueness, and continuous dependence of the solution on the prescribed data - is open. The authors give what they call a compelling argument (although not a proof) for ill-posedness of both the direct problem and the inverse problem in which the eddy viscosity is reconstructed from noisy surface measurements such as pressure and skin-friction distributions: the coexistence of many distinct turbulence models reproducing the same surface data suggests non-uniqueness, and strong sensitivity of solutions to small data perturbations suggests discontinuous dependence. Related formulation questions are equally unresolved. Codes routinely replace kk and ω\omega by k=eKk=e^{\mathcal K}, ω=eΩ\omega=e^{\Omega} or reciprocal variables ω=1/τ\omega = 1/\tau, ω=1/g2\omega = 1/g^2 to keep the variables positive during iteration; Langer and Swanson prove that the substitution k=eKk=e^{\mathcal K} cannot be equivalent to the original problem under the wall condition k=0k=0, and show that ω=eΩ\omega=e^{\Omega} converts the quadratic wall singularity of ω\omega into a logarithmic singularity of Ω\Omega while a quadratic singularity reappears in the new source term ∥∇Ω∥2\|\nabla\Omega\|^2 - yet no proof of equivalence or non-equivalence exists for the ω\omega-substitutions. A rigorous theory exists only for relatives of the problem: the unsteady Kolmogorov two-equation model (the 1942 precursor of kk-ω\omega models, with equations for kk and a turbulence frequency ω\omega) has been analyzed on periodic or bounded domains, with global weak solutions, local strong well-posedness that stays valid where kk vanishes, and finite-time blow-up in one space dimension; for the laminar Navier-Stokes equations, identification of an unknown viscosity from boundary or modal observations is partly understood; and the data-driven field-inversion literature (reconstructing eddy-viscosity corrections from measurements by regularized optimization) proceeds entirely through regularization, presupposing rather than proving the ill-posedness that Langer and Swanson conjecture.

Problem Statement

Consider the steady, compressible RANS equations coupled with a two-equation turbulence model of kk-ω\omega type (Wilcox kk-ω\omega or Menter SST; kk denotes the turbulent kinetic energy and ω\omega the specific dissipation rate) on the exterior of a body, supplemented by the physically motivated boundary prescriptions derived by Langer and Swanson (2020): no-slip velocity at the wall with k=0k=0 and the singular near-wall condition ω(x−hn)h2→6νl/β\omega(x-hn)h^2 \to 6\nu_l/\beta as the wall distance h→0+h \to 0^+, where nn is the unit normal at the wall, νl\nu_l the laminar kinematic viscosity, and β\beta a constant of the turbulence model; prescribed freestream values of kk and ω\omega with algebraic far-field decay ω=O(∥x∥−1)\omega = O(\|x\|^{-1}) and k=O(∥x∥−β∗/β)k = O(\|x\|^{-\beta^*/\beta}), β∗\beta^* and β\beta being constants of the turbulence model; and turbulence data imposed only on the inflow portion of a truncated outer boundary. Determine whether this exterior turbulent-flow problem is well-posed in Hadamard's sense - existence of a solution, uniqueness, and continuous dependence of the solution on the prescribed data: (i) does a solution exist for admissible boundary data; (ii) is it unique; (iii) does it depend continuously on the data? A resolution is either a proof of well-posedness under an explicitly stated admissible class of boundary data and function spaces, or a rigorous demonstration of ill-posedness - non-existence for admissible data, non-uniqueness (for example, two distinct solutions of the same boundary-value problem), or discontinuous dependence on the data - with the stated boundary prescriptions respected and not silently regularized away. In either branch the admissible class must be non-degenerate: it must contain every datum that satisfies the stated wall and far-field prescriptions at an explicitly stated regularity, so that neither branch can be satisfied by a class consisting of a single datum, or otherwise chosen so that the conclusion is vacuous. The same requirement applies to the data class of the inverse problem described below: the observation configuration and the noise level δ\delta must be stated rather than chosen to force a verdict, so that uniqueness or stability is neither made trivial by an exhaustive noiseless data set nor made vacuous by a single marginal observable.

The source also leaves open the following associated formulation questions, each an independent open problem in its own right: (a) equivalence of variable substitutions - where a substitution is equivalent when it is a bijection between the admissible solution sets of the original and the substituted problems that preserves the stated singular wall and far-field conditions (in particular, the wall conditions must remain realizable in the transformed variables) - whether replacing ω\omega by ω=eΩ\omega=e^{\Omega}, ω=1/τ\omega=1/\tau, or ω=1/g2\omega=1/g^2 (the k=eKk=e^{\mathcal K} case under k=0k=0 being already settled negatively) yields a boundary-value problem equivalent to the original one when the singular wall and far-field conditions are enforced, in either direction (equivalence proof or counterexample); (b) well-posedness of the inverse problem - whether the eddy-viscosity field μt≥0\mu_t \ge 0 can be reconstructed uniquely and stably from limited, noisy data YδY^\delta whose observation operator (surface pressure and skin-friction distributions, or a stated sparse-flow alternative) and noise class ∥Yδ−Y∥≤δ\|Y^\delta - Y\| \le \delta in an explicitly stated norm are both given, with YY the data generated by the true μt\mu_t; and (c) a constructive algorithm that partitions the truncated outer boundary into inflow and outflow portions without prior knowledge of the solution, together with a justification that the partition is well defined, so that the asymptotic inflow prescriptions can be applied consistently. The common scientific objective is a complete, consistent, and mathematically controlled boundary-value formulation of the RANS/two-equation system, in both its direct and its inverse form. The Hadamard status of the direct problem (i)-(iii) under the standard above is the core question; (a)-(c) are additional formulation components the source explicitly ties to that core, each a self-contained result, not prerequisites for resolving (i)-(iii) and not resolved by it.

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