In homogeneous isotropic turbulence, rotational invariance forces exact coefficient identities between moments of the kinetic-energy dissipation rate ε=2νSijSij\varepsilon = 2\nu S_{ij}S_{ij} and moments of the longitudinal velocity derivative ∂u1/∂x1\partial u_1/\partial x_1, the quantity accessible to single-hot-wire measurements. Boschung (2015) claimed a closed-form all-order relation ⟨εn⟩=(2ν)nCn⟨(∂u1/∂x1)2n⟩\langle\varepsilon^n\rangle = (2\nu)^n C_n\langle(\partial u_1/\partial x_1)^{2n}\rangle and validated it to sixth order in direct numerical simulation. Two independent 2026 studies, by Yang-Xu-Pumir and by Wu-Luo-Fang-Wilczek, completed and corrected this picture: the relation is exact only through fourth order; from sixth order on, longitudinal moments also involve the strain invariant tr(S3)\mathrm{tr}(S^3), Boschung's coefficients correspond to an implicit Gaussian-ensemble assumption, and the previously unconnected odd moments are exactly expressible through vortex-stretching-related invariants. With the homogeneous-isotropic (and, via Wu et al., compressible) case settled, the open problem is the extension to the flows in which dissipation is actually measured: whether and how the exact analogs of these relations carry over to anisotropic, inhomogeneous, and wall-bounded turbulence, where local isotropy demonstrably fails; quantifying the Reynolds-number dependence and resolution requirements of the strain-invariant statistics that the exact relations require as input; and establishing whether a controlled approximate probability-density-level surrogacy between dissipation and one-dimensional statistics exists, exact closed relations having been proven impossible.

Background

A central quantity in the theory of turbulence is the dissipation rate of kinetic energy, ε=2νSijSij\varepsilon = 2\nu S_{ij}S_{ij}, where ν\nu is the kinematic viscosity, Sij=(∂iuj+∂jui)/2S_{ij} = (\partial_i u_j + \partial_j u_i)/2 is the strain-rate tensor (the symmetric part of the velocity-gradient tensor), and repeated indices are summed. Fluctuations of ε\varepsilon encode the intermittency of the small scales of turbulence, and its statistical moments ⟨εn⟩\langle\varepsilon^n\rangle are standard diagnostics in theory, in direct numerical simulation (DNS), and in experiment. Measuring all nine components of the velocity-gradient tensor in a real flow is very difficult; a single hot-wire anemometer, by contrast, resolves the longitudinal velocity derivative ∂u1/∂x1\partial u_1/\partial x_1, the gradient of the velocity component along its own direction. This motivates the one-dimensional surrogate dissipation εs=15ν(∂u1/∂x1)2\varepsilon_s = 15\nu(\partial u_1/\partial x_1)^2, whose mean equals ⟨ε⟩\langle\varepsilon\rangle in homogeneous isotropic turbulence, and raises the general question of how the statistics of ε\varepsilon and of ∂u1/∂x1\partial u_1/\partial x_1 are exactly related.

In homogeneous isotropic turbulence (HIT), meaning turbulence whose statistics are invariant under translations and rotations, one-point velocity-gradient moments are isotropic tensors, which decompose into sums of products of the Kronecker delta δij\delta_{ij}. This rigid structure produces exact coefficient identities between gradient moments. The classical cases are ⟨ε⟩=15ν⟨(∂u1/∂x1)2⟩\langle\varepsilon\rangle = 15\nu\langle(\partial u_1/\partial x_1)^2\rangle and, due to Siggia (1981), ⟨ε2⟩=(2ν)2(105/4)⟨(∂u1/∂x1)4⟩\langle\varepsilon^2\rangle = (2\nu)^2(105/4)\langle(\partial u_1/\partial x_1)^4\rangle. Building on Siggia's generating-function formalism, Boschung (2015) derived the coefficient formula Cn=3n−1(2n+1)(2n+3)/2nC_n = 3^{n-1}(2n+1)(2n+3)/2^n (so C1=15/2C_1 = 15/2 and C2=105/4C_2 = 105/4) and claimed the all-order relation ⟨εn⟩=(2ν)nCn⟨(∂u1/∂x1)2n⟩\langle\varepsilon^n\rangle = (2\nu)^n C_n\langle(\partial u_1/\partial x_1)^{2n}\rangle under homogeneity, incompressibility, and local isotropy. He compared the theoretical CnC_n up to n=6n = 6 with HIT DNS data at Taylor-scale Reynolds numbers Rλ=88R_\lambda = 88 to 529529 (the Reynolds number based on the Taylor microscale, the standard microscale Reynolds number of HIT), finding deviations not exceeding ±5%\pm 5\% and growing with the order nn, which he attributed to finite grid resolution, quantified by the ratio κmaxη\kappa_{max}\eta of the largest resolved wavenumber to the inverse Kolmogorov length scale η\eta (the smallest dynamically active scale of turbulence).

Two independent 2026 studies completed and corrected this picture for the isotropic case. Both rest on the fact that, by the Cayley-Hamilton theorem (which expresses powers of a 3×33\times 3 matrix through its traces), the incompressible strain-rate tensor has only two independent invariants, T2=tr(S2)∝ε\mathcal{T}_2 = \mathrm{tr}(S^2) \propto \varepsilon and T3=tr(S3)\mathcal{T}_3 = \mathrm{tr}(S^3), the latter connected to vortex stretching, the amplification of vorticity ω=∇×u\omega = \nabla\times u by strain, with ⟨ωiSijωj⟩=−(4/3)⟨T3⟩\langle\omega_i S_{ij}\omega_j\rangle = -(4/3)\langle\mathcal{T}_3\rangle (a classical relation due to Betchov). Yang-Xu-Pumir (2026) proved that Boschung's general relation is not exact: from sixth order on, longitudinal moments contain T3\mathcal{T}_3-invariant contributions, for example ⟨(∂u1/∂x1)6⟩=(40/3003)⟨T23⟩+(128/9009)⟨T32⟩\langle(\partial u_1/\partial x_1)^6\rangle = (40/3003)\langle\mathcal{T}_2^3\rangle + (128/9009)\langle\mathcal{T}_3^2\rangle, and Boschung's CnC_n are recovered only under an implicit Gaussian-ensemble assumption that fails in turbulence (where ⟨T3⟩<0\langle\mathcal{T}_3\rangle < 0, reflecting biased strain-eigenvalue statistics); numerically the correction is small, about six percent at sixth order, so the CnC_n remain good approximations. Wu-Luo-Fang-Wilczek (2026) independently derived closed-form expressions for arbitrary-order moments of longitudinal and transverse velocity gradients, in incompressible and compressible turbulence, by orientational averaging, and state that Boschung's even-order expressions are valid only below sixth order. The previously unconnected odd moments are now exactly expressed through T3\mathcal{T}_3, for example ⟨(∂u1/∂x1)5⟩=(32/693)⟨T2T3⟩\langle(\partial u_1/\partial x_1)^5\rangle = (32/693)\langle\mathcal{T}_2\mathcal{T}_3\rangle. Wu et al. conclude that all longitudinal-gradient moments are completely determined by the joint probability density function of T2\mathcal{T}_2 and T3\mathcal{T}_3; Yang et al. show, conversely, that no exact closed relation between the probability density functions of ε\varepsilon and εs\varepsilon_s can be obtained.

All of these exact results assume homogeneity and isotropy. In the flows where dissipation is most often measured - turbulent boundary layers, channels, shear flows, the atmosphere - these assumptions fail. George et al. (2020), from dual-plane stereoscopic particle-image velocimetry (an optical technique resolving all the velocity-gradient components needed to estimate the dissipation) in a turbulent boundary layer at momentum-thickness Reynolds number Reθ=7500Re_\theta = 7500 (the Reynolds number built on the boundary-layer momentum thickness, the standard Reynolds number of boundary-layer turbulence), found that local isotropy, the assumption that the small scales of an anisotropic flow are isotropic at each point, is never valid inside the outer limit of the overlap region of the boundary layer (the region where near-wall and outer velocity scalings both apply), y/δ99≈0.1y/\delta_{99}\approx 0.1 (where δ99\delta_{99} is the 99% boundary-layer thickness, the wall-normal distance at which the mean streamwise velocity reaches 99% of its free-stream value), and that local axisymmetry and local homogeneity fail inside y+=100y^+ = 100 (distance in wall units, normalized by the viscous length scale). There even the workhorse surrogate ⟨ε⟩=15ν⟨(∂u1/∂x1)2⟩\langle\varepsilon\rangle = 15\nu\langle(\partial u_1/\partial x_1)^2\rangle loses validity, and no exact replacement relations are known. Moreover, the accuracy of the isotropic relations at finite Reynolds number is governed by invariant ratios such as ⟨T32⟩/⟨T23⟩\langle\mathcal{T}_3^2\rangle/\langle\mathcal{T}_2^3\rangle (about 1/121/12 in DNS, versus the Gaussian-ensemble value 1/181/18), whose Reynolds-number dependence has been measured only over a limited range. The problem of the exact correspondence between dissipation statistics and longitudinal-derivative statistics thus survives, in sharpened form, beyond the homogeneous-isotropic class in which it is now settled.

Problem Statement

The problem is to determine how far the program of exact relations between dissipation statistics and longitudinal-velocity-derivative statistics extends beyond the homogeneous-isotropic class in which it is now settled, and to supply the quantitative characterization that any such extension requires. It has three linked components serving this single objective. All three are required for a complete answer; they may be addressed separately, and partial progress on one of them does not by itself resolve the task.

(1) Exact relations beyond isotropy. The target is the general class of turbulent flows that are anisotropic, inhomogeneous, or wall-bounded, where local isotropy is known to fail; homogeneous shear flow, axisymmetric turbulence, and channel and boundary-layer flows are illustrative instances of that class, not a fixed list to be exhausted. For this class, establish to what extent exact analogs of the isotropic identities exist and derive those that do: decompose the one-point velocity-gradient moment tensors under the applicable symmetry group, identify the complete set of tensorial invariants and preferred-direction structures that enter each moment of ∂u1/∂x1\partial u_1/\partial x_1 (and of transverse derivatives), compute the resulting coefficient structure, and establish which combinations of invariant moments play the role that the joint statistics of T2=tr(S2)\mathcal{T}_2 = \mathrm{tr}(S^2) and T3=tr(S3)\mathcal{T}_3 = \mathrm{tr}(S^3) play in the isotropic case. Any relation so obtained must reduce to the known isotropic results - Siggia's fourth-order identity and the sixth-and-higher-order decompositions of Yang-Xu-Pumir and Wu-Luo-Fang-Wilczek - when isotropy is restored; comparison with published DNS or experimental data for the flow class considered is desirable where such data exist, but is not a requirement for a complete answer.

(2) Invariant statistics at finite Reynolds number. Quantify the Reynolds-number dependence and the DNS/experimental resolution requirements of the strain-invariant statistics that the exact relations require as input - first in the homogeneous-isotropic baseline in which these statistics are defined, in particular the ratios ⟨T32T2p⟩/⟨T2p+3⟩\langle\mathcal{T}_3^2\mathcal{T}_2^p\rangle/\langle\mathcal{T}_2^{p+3}\rangle, currently estimated near 1/121/12 only over a limited range (Rλ≤210R_\lambda \le 210), and then for each symmetry class addressed in component (1) - since these ratios determine the accuracy with which dissipation moments can be inferred from longitudinal-derivative statistics at finite Reynolds number.

(3) Probability-density-level surrogacy. Determine whether a controlled approximate relation between the probability density functions of the dissipation ε\varepsilon and one-dimensional surrogate statistics can be established with derived error bounds that are non-trivial and explicit in the Reynolds number - for example relative accuracy of order unity over the bulk of the distribution, rather than a bound that holds vacuously - exact closed relations having been shown impossible - extending to dissipation the one-dimensional surrogacy program established for enstrophy by Gotoh-Yeung.

References

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