First-principles theory of the Reynolds-number scaling of the Eulerian decompositions of the turbulent acceleration

In homogeneous isotropic turbulence, the variance of the fluid-particle (Lagrangian) acceleration, normalized by Kolmogorov scales, grows only as $R_\lambda^{0.25}$ over $R_\lambda\approx140$–$1300$, even though its local ($\partial\mathbf{u}/\partial t$) and convective ($\mathbf{u}\cdot\nabla\mathbf{u}$) contributions each grow linearly with $R_\lambda$ and nearly cancel, in accordance with the random sweeping hypothesis. The residual is dominated by the irrotational pressure-gradient contribution, while the solenoidal viscous contribution scales as $R_\lambda^{0.13}$. Neither Kolmogorov's mean-field phenomenology (which predicts a constant normalized variance) nor multifractal extensions of Eulerian intermittency to Lagrangian variables (which predict $\chi\approx0.135$) accounts for the $0.25$ exponent. The open problem is a first-principles account, from the Navier–Stokes equations, of these decomposed scalings: why the near-cancellation leaves exactly the $R_\lambda^{0.25}$ residual, what sets the empirical constants in the decomposed scaling relation and its range of validity, under what conditions the velocity–velocity-gradient decorrelation underlying the linear convective scaling holds asymptotically, and what the higher-order moments of the decomposed fields are and whether the cancellation persists at higher orders.

Background

In turbulent flow, the acceleration of a fluid particle — the Lagrangian or material acceleration — is $\mathbf{a}=D\mathbf{u}/Dt=\partial\mathbf{u}/\partial t+\mathbf{u}\cdot\nabla\mathbf{u}$, where $\mathbf{u}$ is the incompressible ($\nabla\cdot\mathbf{u}=0$) velocity. The first term, $\mathbf{a}_L\equiv\partial\mathbf{u}/\partial t$, is the local (Eulerian) acceleration, the unsteady rate of change at a fixed point; the second, $\mathbf{a}_C\equiv\mathbf{u}\cdot\nabla\mathbf{u}$, is the convective acceleration due to spatial velocity variations. Through the Navier–Stokes equations the same vector also admits a Helmholtz decomposition (splitting a vector field into a curl-free, irrotational part and a divergence-free, solenoidal part): $\mathbf{a}=\mathbf{a}_I+\mathbf{a}_S$ with $\mathbf{a}_I=-\nabla P$ the pressure-gradient contribution ($P$ is the kinematic pressure) and $\mathbf{a}_S=\nu\nabla^2\mathbf{u}$ the viscous contribution ($\nu$ is the kinematic viscosity). In statistically homogeneous, stationary turbulence the cross-correlations $\langle\mathbf{a}_I\cdot\mathbf{a}_S\rangle=\langle\mathbf{a}_I\cdot\mathbf{a}_L\rangle=\langle\mathbf{a}_S\cdot\mathbf{a}_L\rangle=0$ hold exactly, so $\langle|\mathbf{a}|^2\rangle=\langle|\mathbf{a}_C|^2\rangle-\langle|\mathbf{a}_L|^2\rangle=\langle|\mathbf{a}_I|^2\rangle+\langle|\mathbf{a}_S|^2\rangle$.

Kolmogorov's 1941 mean-field phenomenology (K41) predicts $\langle|\mathbf{a}|^2\rangle/3=a_0\langle\epsilon\rangle^{3/2}\nu^{-1/2}$ with $a_0$ a universal constant, where $\langle\epsilon\rangle$ is the mean dissipation rate; equivalently $\langle|\mathbf{a}|^2\rangle/a_K^2=3a_0$ with the Kolmogorov acceleration scale $a_K=\langle\epsilon\rangle^{3/4}\nu^{-1/4}$. Because of small-scale intermittency (the rare, intense small-scale events that make turbulence statistics non-Gaussian), $a_0$ instead grows with the Taylor-scale Reynolds number $R_\lambda$ (defined from the rms velocity fluctuation and the Taylor microscale). The multifractal formalism — which models turbulence as a union of fractal sets with locally varying scaling exponents — extends Eulerian intermittency to Lagrangian quantities through bridge relations between temporal and spatial increments, predicting $a_0\sim R_\lambda^{\chi}$ with $\chi\approx0.135$ (Borgas 1993; Biferale et al. 2004). Hill (2002), starting from exact relations between the acceleration variance and fourth-order Eulerian velocity structure functions, instead proposed $a_0\sim R_\lambda^{0.25}$. Buaria and Sreenivasan settled the question empirically: combining direct numerical simulations (DNS) of forced isotropic turbulence at $R_\lambda=140$–$1300$ (grids up to $12288^3$, with resolution $k_{\max}\eta_K\approx6$, where $k_{\max}$ is the largest resolved wavenumber and $\eta_K$ the Kolmogorov length scale) with all literature data, they found the multifractal interpolation failing at high $R_\lambda$ and $a_0\sim R_\lambda^{0.25}$ holding, with higher-order acceleration moments departing from multifractal predictions even more strongly.

The source Letter (Buaria and Sreenivasan, Phys. Rev. Fluids 8, L032601 (2023)) traced where this residual comes from. The convective-acceleration variance grows linearly, $\langle|\mathbf{a}_C|^2\rangle/a_K^2=cR_\lambda$, which follows if $\mathbf{u}$ (a large-scale quantity) and $\nabla\mathbf{u}$ (a small-scale quantity) are effectively uncorrelated — an assumption consistent with the random sweeping hypothesis (Kraichnan 1964; Tennekes 1975; Tsinober, Vedula and Yeung 2001), which states that small-scale structures are passively swept past an Eulerian observer by the large-scale velocity on a much shorter time scale than that of their own dynamics. The local-acceleration variance then obeys the three-term empirical relation $\langle|\mathbf{a}_L|^2\rangle/a_K^2=cR_\lambda-c_1R_\lambda^{0.25}-c_2R_\lambda^{0.13}$, and since $\langle\mathbf{a}_L\cdot\mathbf{a}_C\rangle=-\langle|\mathbf{a}_L|^2\rangle$, the near-perfect anti-correlation of $\mathbf{a}_L$ and $\mathbf{a}_C$ leaves $\langle|\mathbf{a}|^2\rangle\approx c_1R_\lambda^{0.25}$, dominated by the irrotational pressure gradient ($\langle|\mathbf{a}_I|^2\rangle\approx\langle|\mathbf{a}|^2\rangle$). The solenoidal viscous contribution follows $R_\lambda^{0.13}$ through the exact identity $\langle|\mathbf{a}_S|^2\rangle/a_K^2=-(35/2)S/(15)^{3/2}$ with $S$ the (negative) skewness of longitudinal velocity gradients, whose magnitude grows as $R_\lambda^{0.13}$ — the one component consistent with multifractal predictions. No existing framework predicts the residual $R_\lambda^{0.25}$: in the words of the source, the extension of Eulerian multifractals to Lagrangian quantities is "fraught with major uncertainties."

Problem Statement

For forced, statistically stationary, homogeneous and isotropic turbulence, derive from the incompressible Navier–Stokes equations — or from an equivalently predictive first-principles statistical framework — the asymptotic Reynolds-number dependence of the variances of the four Eulerian components of the fluid-particle acceleration ($\mathbf{a}_L=\partial\mathbf{u}/\partial t$, $\mathbf{a}_C=\mathbf{u}\cdot\nabla\mathbf{u}$, $\mathbf{a}_I=-\nabla P$, $\mathbf{a}_S=\nu\nabla^2\mathbf{u}$), each normalized by the Kolmogorov acceleration scale $a_K$. A complete answer must address all four components below; a partial answer covering only a subset of them is not a complete solution. The components share one objective — a predictive account of the observed decomposed scalings — and are:

(1) Explain, or legitimately refute with new asymptotics or better data, why the normalized variance of the irrotational pressure gradient (and hence of the total Lagrangian acceleration) grows as $R_\lambda^{0.25}$ over $R_\lambda\approx140$–$1300$; i.e., why the near-cancellation between the linear-in-$R_\lambda$ local and convective variances leaves precisely this residual, given that neither Kolmogorov's mean-field phenomenology (constant $a_0$) nor multifractal Lagrangian extensions ($\chi\approx0.135$) predict it.

(2) Within the same framework, predict the constants $c$, $c_1$, $c_2$ (and any residual Reynolds-number dependence of them) in $\langle|\mathbf{a}_L|^2\rangle/a_K^2=cR_\lambda-c_1R_\lambda^{0.25}-c_2R_\lambda^{0.13}$, and the range of $R_\lambda$ over which this three-term form is valid; if the $R_\lambda^{0.25}$ growth is instead legitimately refuted under (1), this component is discharged by deriving the corresponding asymptotic coefficients of the relation that replaces the three-term form, together with its range of validity.

(3) Establish, as a rigorous asymptotic statement or a controlled approximation, the conditions under which the velocity–velocity-gradient decorrelation underlying $\langle|\mathbf{a}_C|^2\rangle/a_K^2=cR_\lambda$ holds — or show where and how it fails.

(4) Determine the higher-order moments of the decomposed fields $\mathbf{a}_L$, $\mathbf{a}_C$, $\mathbf{a}_I$, $\mathbf{a}_S$ — theoretically within the framework of (1), or through adequately resolved computations — and whether the strong second-order cancellation between local and convective accelerations persists at higher orders.

Each component that names alternative routes is satisfied by any single one of them: an explanation or a legitimate refutation for (1); a rigorous asymptotic statement or a controlled approximation for (3); a theoretical treatment or adequately resolved computations for (4); and, when the $R_\lambda^{0.25}$ growth is legitimately refuted under (1), a derivation of the replacement asymptotic coefficients and range of validity for (2).

An answer that derives the exponent structure must be consistent with the exact correlation identities of homogeneous stationary turbulence and with the DNS data over $R_\lambda\approx140$–$1300$. A refutation of the $R_\lambda^{0.25}$ growth asserted in component (1) may rest on a controlled derivation or asymptotic argument, or on data beyond the current range; an answer claiming that the 0.25 scaling is a finite-Reynolds-number artifact must additionally exhibit the asymptotic regime it enters and quantify the crossover from the observed range.

References

1. Dhawal Buaria and Katepalli R. Sreenivasan, 'Lagrangian acceleration and its Eulerian decompositions in fully developed turbulence', Physical Review Fluids 8, L032601 (2023). DOI: 10.1103/PhysRevFluids.8.L032601, https://doi.org/10.1103/PhysRevFluids.8.L032601 (preprint arXiv:2210.15877, https://arxiv.org/abs/2210.15877)
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