The 2016 paper of Jiang and Layton introduced a 'corrected' eddy-viscosity model for turbulence not at statistical equilibrium that appends a backscatter term β2a(w) ∂t(a(w)w)\beta^2 a(w)\,\partial_t(a(w)w), nonlinear (anisotropic) in the time derivative because the coefficient a(w)a(w) depends on ∇w\nabla w, to a Smagorinsky-type Large Eddy Simulation model. All of their analysis (a priori bounds, time-averaged dissipativity, long-term energy balance, and time-discretization error analysis) assumes the solution is strong, and the paper states that the existence of weak solutions to the new model is an open theoretical problem. Prior weak-existence results for related backscatter models (rotational Baldwin–Lomax, generalized Kelvin–Voigt, and backscatter Baldwin–Lomax variants) cover only terms that are linear in ∂tu\partial_t u with solution-independent coefficients, and at least one successor paper explicitly replaces the anisotropic term rather than analyzing it. The open problem is whether the original corrected model with the anisotropic time-derivative term admits weak solutions satisfying the natural energy inequality.

Background

Large eddy simulation (LES) is a turbulence-modeling approach in which only the large, energy-carrying eddies of a flow are computed, while the effect of the unresolved subfilter scales is represented by a closure model. The classical closure adds an eddy viscosity νT\nu_T to the molecular viscosity ν\nu in the Navier–Stokes equations, the archetype being the Smagorinsky model with νT(w)=(Csδ)2∣∇w∣\nu_T(w) = (C_s\delta)^2|\nabla w|, where CsC_s is a dimensionless constant, δ\delta the filter width, and ww the resolved velocity. Eddy-viscosity closures are strictly dissipative: they remove kinetic energy from the resolved scales at every point and time. In turbulence not at statistical equilibrium, however, energy locally flows from unresolved to resolved scales (backscatter), so such models systematically over-dissipate. Jiang and Layton (2016) derived a corrected eddy-viscosity model by modeling the fluctuation energy budget; the correction appears as an additional term in the time derivative, giving (for the Smagorinsky case)

wt+β2a(w) ∂t(a(w)w)+w⋅∇w+∇p−∇⋅([ν+νT(w)]∇w)=f,∇⋅w=0,w_t + \beta^2 a(w)\,\partial_t(a(w)w) + w\cdot\nabla w + \nabla p - \nabla\cdot\big([\nu+\nu_T(w)]\nabla w\big) = f, \qquad \nabla\cdot w = 0,

with a(w)=ν−1/2Csδ ∣∇w∣1/2a(w) = \nu^{-1/2}C_s\delta\,|\nabla w|^{1/2}, a nondimensional order-one correction parameter β\beta, and body force ff. The paper calls β2a(w) ∂t(a(w)w)\beta^2 a(w)\,\partial_t(a(w)w) an anisotropic time-derivative term: unlike a Kelvin–Voigt term −αΔwt-\alpha\Delta w_t (a linear viscoelastic damping with constant coefficient), its coefficient moves with the solution through ∇w\nabla w, so it is quasilinear in the time derivative. Their Theorem 3.2 proves a priori bounds, time-averaged dissipativity, and a long-term energy balance, assuming throughout that ww is a strong (smooth enough) solution. The natural energy for the model is the non-quadratic functional 12∥w∥2+β22∥a(w)w∥2\tfrac{1}{2}\|w\|^2 + \tfrac{\beta^2}{2}\|a(w)w\|^2, and a key estimate (their Lemma 3.1) bounds ∥a(w)w∥2\|a(w)w\|^2 by Cν−1(Csδ)2∥∇w∥L33C\nu^{-1}(C_s\delta)^2\|\nabla w\|_{L^3}^3, already suggesting the unusual function-space structure (control of cubic strain-rate integrals) any weak theory must handle. The paper explicitly states that the existence of weak solutions to the new corrected model, with the anisotropic time-derivative term, is an open theoretical problem. This matters because every analytical result in the follow-up literature (energy stability, time-discretization error estimates, variable time-step analyses) is conditional on strong solutions, so the model class currently lacks a rigorous well-posedness foundation of the kind the Leray–Hopf theory gives for the Navier–Stokes equations (a Leray–Hopf solution is a weak solution satisfying the global energy inequality).

Problem Statement

Prove or refute the existence of weak solutions to the Jiang–Layton corrected eddy-viscosity model in the Smagorinsky case: find a velocity field ww and pressure pp satisfying

wt+β2a(w) ∂t(a(w)w)+w⋅∇w+∇p−∇⋅([ν+(Csδ)2∣∇w∣] ∇w)=f,∇⋅w=0,w_t + \beta^2 a(w)\,\partial_t(a(w)w) + w\cdot\nabla w + \nabla p - \nabla\cdot\big([\nu+(C_s\delta)^2|\nabla w|]\,\nabla w\big) = f, \qquad \nabla\cdot w = 0,

on a bounded domain Ω\Omega with given initial velocity w(x,0)=w0w(x,0)=w_0 and the no-slip boundary condition w=0w=0 on ∂Ω\partial\Omega (the setting of Theorem 3.2 of the source; in the source, f∈L∞(0,∞;L2(Ω))f\in L^\infty(0,\infty;L^2(\Omega)) and w0∈L2(Ω)w_0\in L^2(\Omega)), where a(w)=ν−1/2Csδ ∣∇w∣1/2a(w) = \nu^{-1/2}C_s\delta\,|\nabla w|^{1/2}, β\beta is a fixed positive parameter, and ff is a given body force, such that the solution is weak in the Leray–Hopf sense and satisfies the natural energy inequality associated with the non-quadratic energy 12∥w∥2+β22∥a(w)w∥2\tfrac{1}{2}\|w\|^2 + \tfrac{\beta^2}{2}\|a(w)w\|^2 for almost every t≥0t\ge 0 (including t=0t=0), with the natural energy finite; that is, the initial data satisfy a(w0)w0∈L2(Ω)a(w_0)w_0\in L^2(\Omega) in addition to w0∈L2(Ω)w_0\in L^2(\Omega), and the solution satisfies w, a(w)w∈L∞(0,∞;L2(Ω))w,\, a(w)w \in L^\infty(0,\infty;L^2(\Omega)), the finiteness conclusion (3.2) proved in Theorem 3.2 of the source for strong solutions. The model equations, coefficient conventions, and energy functional are pinned exactly as in Jiang and Layton (2016); answers that change the model (for example, replacing the anisotropic term by a linear Kelvin–Voigt or rotational term with solution-independent coefficient) address a different problem and do not resolve this one.

References

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