Predictive theory of Lagrangian acceleration moments in high-Reynolds-number turbulence: the Reynolds-growing fourth-order structure-function prefactor and the failure of the dissipation-based multifractal mapping

The acceleration of fluid particles in fully developed turbulence is far more intermittent than Eulerian dissipation fluctuations, and the standard multifractal prediction built on the dimensional mapping $a\sim\epsilon^{3/4}\nu^{-1/4}$, namely $\langle a^p\rangle/a_K^p\sim R_\lambda^{\tau_{3p/4}}$, fails increasingly badly with moment order at high Reynolds number. Buaria and Sreenivasan (Phys. Rev. Lett. 128, 234502, 2022) traced the observed acceleration-variance scaling $a_0\sim R_\lambda^{0.25}$ to an unexplained Reynolds-number growth of the fourth-order Eulerian structure-function prefactor, $C_4\sim R_\lambda^{0.2}$ versus the $R_\lambda^{0.08}$ expected from multifractal and log-normal models, but their piecewise-integration framework does not extend to higher moments, and no mechanism explains why Lagrangian intermittency exceeds Eulerian dissipation intermittency. The open problem is a predictive, Navier–Stokes-faithful theory of Lagrangian acceleration moments: deriving the growth of $C_4$, predicting higher-order moment exponents, and identifying the roles of transverse velocity gradients, nonlocal pressure effects, and temporal correlations along particle trajectories. Follow-up work through January 2026 has constrained the mechanism (pressure-gradient dominance of the variance, a Lagrangian–transverse-Eulerian spectral correspondence, and shell-model counter-evidence) but closes none of these components.

Background

Turbulence can be described from two complementary viewpoints: the Eulerian frame, in which the velocity field $\mathbf{u}(\mathbf{x},t)$ is observed at fixed spatial points, and the Lagrangian frame, which follows fluid particles along trajectories $\mathbf{X}(t)$. The Lagrangian acceleration is the instantaneous acceleration of such a fluid particle, $\mathbf{a}=d\mathbf{u}/dt=\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}$; it is among the most intermittent quantities measured in turbulence, with extremely heavy-tailed probability distributions. Its natural scale follows the Kolmogorov (Heisenberg–Yaglom) dimensional estimate $a_K\sim\langle\epsilon\rangle^{3/4}\nu^{-1/4}$, where $\epsilon$ is the kinetic-energy dissipation rate per unit mass, $\nu$ is the kinematic viscosity, and angular brackets denote ensemble (volume) averages. The relevant control parameter is the Taylor-scale Reynolds number $R_\lambda=u'\lambda/\nu$ (based on the root-mean-square velocity $u'$ and the Taylor microscale $\lambda$), with well-resolved direct numerical simulations (DNS) of isotropic turbulence now reaching $R_\lambda\approx1300$.

Small-scale intermittency denotes the increasingly violent, rare fluctuations of velocity increments and gradients at smaller scales, which defeat mean-field (Kolmogorov-1941, K41) scaling. An $n$-th order structure function is the moment $\langle(\delta u_r)^n\rangle$ of a velocity increment $\delta u_r$ across a spatial separation $r$; in the inertial range (scales between the large-scale length $L$ and the Kolmogorov dissipation length $\eta_K$) it scales as $r^{\xi_n}$ with anomalous exponents $\xi_n<n/3$, while in the dissipation range it behaves as $r^n$ with a prefactor that itself grows with $R_\lambda$. The multifractal formalism models $\delta u_r\sim(r/L)^h$ as locally Hölder continuous with exponent $h$, the set of points with exponent $h$ having fractal dimension $D(h)$; a steepest-descent evaluation then predicts the scaling of dissipation moments $\langle\epsilon^q\rangle/\langle\epsilon\rangle^q\sim R_\lambda^{\tau_q}$.

Biferale et al. (Phys. Rev. Lett. 93, 064502, 2004) extended this framework to Lagrangian statistics by combining the mapping $a\sim\epsilon^{3/4}\nu^{-1/4}$ with the multifractal dissipation moments, predicting $\langle a^p\rangle/a_K^p\sim R_\lambda^{\zeta_p}$ with $\zeta_p=\tau_{3p/4}$. This treats Lagrangian intermittency as a naive extension of Eulerian dissipation intermittency. An earlier interpolation formula for the normalized acceleration variance $a_0=\langle a^2\rangle/a_K^2\sim R_\lambda^{\chi}$ with $\chi\approx0.135$ (Sawford et al., Phys. Fluids 15, 3478, 2003) fit the then-available data, and Hill (J. Fluid Mech. 452, 361, 2002) had proposed $\chi=0.25$ by a different route.

Buaria and Sreenivasan (Phys. Rev. Lett. 128, 234502, 2022) tested these predictions with new, very well-resolved DNS at $R_\lambda=140$–$1300$ combined with literature data. The variance data depart from the interpolation formula and follow $a_0\sim R_\lambda^{0.25}$; higher moments deviate from the multifractal predictions $\zeta_p=\tau_{3p/4}$ far more strongly, and replacing dissipation $\epsilon$ by enstrophy (squared vorticity, the more intermittent transverse-gradient quantity) does not repair the disagreement. As a constructive step they used the exact identity $\langle|a|^2\rangle\simeq\langle|\nabla p|^2\rangle$ (the viscous contribution being small), with the pressure-gradient variance given by an integral over fourth-order Eulerian longitudinal, transverse, and mixed structure functions, and evaluated it piecewise across the dissipation and inertial ranges. This yields $a_0\sim R_\lambda^{(2\alpha-\alpha\xi_4+2\beta)/(4-\xi_4)}$, where the dissipation flatness $F\sim R_\lambda^{\alpha}$ ($\alpha\approx0.387$), the fourth-order exponent $\xi_4\approx1.3$, and the fourth-order structure-function prefactor $C_4\sim R_\lambda^{\beta}$. The measured $\beta\approx0.2$ substantially exceeds the value $\approx0.08$ predicted by multifractal and log-normal models, and it is this excess that drives the stronger variance scaling; with $\beta=0.08$ one would recover essentially the old (wrong) exponent. The authors state that this framework 'does not seem amenable for generalization to higher-order moments' and that 'the intermittency of Lagrangian quantities remains an open problem, even more compellingly than before.'

Subsequent work has sharpened where a solution must act. The acceleration variance is dominated by the irrotational pressure-gradient contribution (scaling as $R_\lambda^{0.25}$), the convective and local Eulerian parts individually scale as $R_\lambda^1$ and nearly cancel (consistent with the random-sweeping idea that small scales are advected past a probe by large-scale motion), and Lagrangian inertial-range exponents saturate at $\approx2$ for moment orders $p\gtrsim8$, matching the transverse — not longitudinal — Eulerian multifractal spectrum (via the frozen-flow identification of temporal and spatial increments). Meanwhile, shell models of turbulence, which lack coherent vortical structures, are accurately described by the standard multifractal acceleration prediction. A theory must therefore be tensorial (distinguishing longitudinal from transverse gradients), account for the nonlocal pressure (a Poisson solve over all scales), and address temporal correlations along trajectories – none of which the scalar multifractal mapping captures.

Problem Statement

Construct a predictive, physically grounded theory of the moments $\langle a^p\rangle$ of the Lagrangian fluid-particle acceleration in Navier–Stokes turbulence as functions of the moment order $p$ and the Taylor-scale Reynolds number $R_\lambda$ — one that does not rest on the naive dissipation-based mapping $a\sim\epsilon^{3/4}\nu^{-1/4}$ whose moment consequences fail quantitatively. A complete answer should accomplish three jointly supporting objectives: (i) derive, rather than fit, the Reynolds-number growth of the fourth-order Eulerian structure-function prefactor $C_4\sim R_\lambda^{\beta}$ — the measured $\beta\approx0.2$ versus $\approx0.08$ from multifractal and log-normal models — which is the main driver of the acceleration-variance scaling $a_0\sim R_\lambda^{0.25}$ for $R_\lambda\gtrsim200$; (ii) predict the higher-order Lagrangian acceleration moment exponents, which the piecewise-integration treatment of the variance does not reach and which deviate from $\zeta_p=\tau_{3p/4}$ increasingly strongly with $p$; and (iii) identify the mechanism by which Lagrangian intermittency is stronger than Eulerian dissipation intermittency — in particular the respective roles of transverse velocity gradients, nonlocal pressure effects, and temporal correlations along particle trajectories. These three objectives are jointly required for a complete answer: $\beta$ in (i) must be deduced from quantities the theory itself predicts, not fitted to the measured acceleration or structure-function data; the three roles named in (iii) must each be addressed; and a result that establishes only one of the objectives is a partial solution rather than a resolution. The theory's predictions must be falsifiable against existing DNS and experimental data (up to $R_\lambda\approx1300$ in DNS, with experiments now reaching $R_\lambda\approx6000$) and must state at least one nontrivial prediction testable beyond the data used to construct it.

References

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