Beyond the multifractal bridge: a predictive theory of Lagrangian acceleration statistics and the origin of fourth-order structure-function prefactor growth in high-Reynolds-number turbulence

Fluid particles in turbulence experience acceleration fluctuations far more intermittent than predicted by extending Eulerian multifractal intermittency to the Lagrangian frame. Buaria and Sreenivasan (Phys. Rev. Lett. 128, 234502, 2022) showed from well-resolved DNS up to Taylor-scale Reynolds number $R_\lambda\approx1300$ that the normalized acceleration variance grows as $a_0\sim R_\lambda^{0.25}$ (not $R_\lambda^{0.135}$ as multifractal models predict), that deviations grow with moment order, and that the driver is an unexplained growth of the inertial-range fourth-order structure-function prefactor $C_4\sim R_\lambda^{\beta}$ with $\beta\approx0.2$ versus the predicted $\beta\approx0.08$. The open problem is a Navier-Stokes-grounded theory that (i) explains the Reynolds-number growth of $C_4$ and $a_0$, (ii) predicts higher-order Lagrangian acceleration moments, and (iii) identifies the mechanism by which Lagrangian intermittency exceeds Eulerian dissipation intermittency. Follow-up work through 2026 has built the Eulerian longitudinal-transverse multifractal scaffolding and identified pressure-gradient dominance, but explicitly leaves the Lagrangian/acceleration extension open.

Background

In turbulence, the acceleration of an idealized fluid particle (the Lagrangian acceleration, $a=D\mathbf{u}/Dt$, the material derivative of the velocity field) is dominated by the pressure-gradient term of the Navier-Stokes equation and is among the most intermittent observable quantities in the field: its probability density has extremely heavy tails. Kolmogorov's 1941 dimensional theory (K41) implies that any acceleration component has variance $\langle a^2\rangle=a_0\langle\varepsilon\rangle^{3/2}\nu^{-1/2}$, the Heisenberg-Yaglom relation, where $\varepsilon$ is the mean energy dissipation rate, $\nu$ the kinematic viscosity, and $a_0$ was originally expected to be a universal constant. Experiments and direct numerical simulation (DNS) instead find that $a_0$ grows with the Taylor-scale Reynolds number $R_\lambda$ (the Reynolds number based on the velocity rms and the Taylor microscale, the standard measure of the separation between large and small scales).

The dominant theoretical framework, the multifractal model, attributes intermittency to a distribution of local Hölder exponents governing the singularities of the velocity field. For Lagrangian statistics it was extended through the bridge relation $a\sim\varepsilon^{3/4}\nu^{-1/4}$, i.e. identifying the temporal derivative along a trajectory with spatial small-scale quantities; the $p$-th absolute acceleration moment then scales as $\langle|a|^p\rangle\sim\langle\varepsilon^{3p/4}\rangle\nu^{-p/4}$, so that, in terms of the dissipation moments $\langle\varepsilon^q\rangle/\langle\varepsilon\rangle^q\sim R_\lambda^{\tau_q}$ given by Eulerian multifractal spectra (p-model, She-Leveque, or Kolmogorov's 1962 log-normal model), $\langle|a|^p\rangle/\langle|a|^p\rangle_K\sim R_\lambda^{\zeta_p}$ with $\zeta_p=\tau_{3p/4}$. This yields $a_0\sim R_\lambda^{0.135}$, embodied in the widely used Sawford empirical interpolation for $a_0(R_\lambda)$. An alternative dimensional route to $a_0\sim R_\lambda^{0.25}$ had earlier been proposed by Hill (2002), but it rests on assumed relations between $a_0$ and the gradient flatness that the newer data contradict; the source paper concludes that the agreement of the resulting exponent was fortuitous.

Buaria and Sreenivasan (2022) combined literature data with new well-resolved DNS of forced isotropic turbulence for $R_\lambda=140$–$1300$ and found: (a) $a_0\sim R_\lambda^{0.25}$ for $R_\lambda\gtrsim200$, invalidating the multifractal prediction and the interpolation formula; (b) the deviations grow with moment order — for the K41-normalized moments the DNS gives $\zeta_2=0.25$, $\zeta_4=1.60$, $\zeta_6=3.95$ against predicted $0.135$–$0.14$, $0.94$–$1.13$, $2.06$–$2.95$ (equivalently, the variance-normalized moments scale as $R_\lambda^{1.10}$ and $R_\lambda^{3.20}$ against predicted $0.67$–$0.85$ and $1.66$–$2.53$); and (c) replacing dissipation by enstrophy $\Omega=|\boldsymbol{\omega}|^2$ (with $\boldsymbol{\omega}$ the vorticity, meant to capture the stronger intermittency of transverse velocity gradients) is insufficient. Their constructive result starts from the exact relation $\langle|\mathbf{a}|^2\rangle=\langle|\nabla p|^2\rangle+\nu^2\langle|\nabla^2\mathbf{u}|^2\rangle$ together with an exact formula expressing the pressure-gradient variance as a weighted radial integral of fourth-order Eulerian structure functions — the longitudinal $D_{1111}(r)$ (moments of longitudinal velocity increments over separation $r$) plus mixed transverse combinations — which is well approximated by the $D_{1111}$ term alone. Modeling $D_{1111}(r)$ piecewise across the dissipation and inertial ranges, with the gradient flatness $F=\langle(\partial u/\partial x)^4\rangle/\langle(\partial u/\partial x)^2\rangle^2$ (the dimensionless fourth-to-second moment ratio of the streamwise velocity gradient, the standard measure of dissipation-range intermittency) scaling as $F\sim R_\lambda^{\alpha}$ ($\alpha\approx0.387$), the inertial-range prefactor $C_4\sim R_\lambda^{\beta}$, and the inertial exponent $\xi_4\approx1.3$, piecewise integration gives $a_0\sim R_\lambda^{(2\alpha-\alpha\xi_4+2\beta)/(4-\xi_4)}$, which reproduces the observed $R_\lambda^{0.25}$ — but only because the measured $\beta\approx0.2$ is about 2.5 times the value $\beta\approx0.08$ that multifractal and log-normal models predict. The paper's conclusion is that the intermittency of Lagrangian quantities remains an open problem, and that the piecewise framework 'does not seem amenable for generalization to higher-order moments'.

Problem Statement

Provide a predictive theory of Lagrangian acceleration statistics in Navier-Stokes turbulence at high Reynolds number — equivalently, resolve the three jointly linked open issues stated by Buaria and Sreenivasan (Phys. Rev. Lett. 128, 234502 (2022)) as one objective. The common objective is a framework, derived from the Navier-Stokes equations or controlled approximations to them, that predicts the Reynolds-number dependence of Lagrangian acceleration moments from Eulerian statistics or other first principles, without ad hoc fitting. Concretely, the answer must: (1) explain the observed growth of the normalized acceleration variance $a_0\sim R_\lambda^{0.25}$ for $R_\lambda\gtrsim200$ by accounting for the growth of the inertial-range fourth-order longitudinal structure-function prefactor, $C_4\sim R_\lambda^{\beta}$ with $\beta\approx0.2$ in current DNS up to $R_\lambda\approx1300$, which exceeds the $\beta\approx0.08$ predicted by Eulerian multifractal and log-normal models; (2) extend the prediction to higher-order absolute moments $\langle|a|^p\rangle$ for $p>2$, whose measured K41-normalized exponents ($\zeta_2=0.25$, $\zeta_4=1.60$, $\zeta_6=3.95$ against bridge predictions $0.135$–$0.14$, $0.94$–$1.13$, $2.06$–$2.95$) deviate from the multifractal bridge $\zeta_p=\tau_{3p/4}$ increasingly with order, noting that the piecewise fourth-order-structure-function integral used for the variance does not generalize in any known way to higher orders; and (3) identify the mechanism by which Lagrangian intermittency is stronger than Eulerian dissipation-range intermittency: explain why the bridge $a\sim\varepsilon^{3/4}\nu^{-1/4}$ fails and why its enstrophy variant is insufficient, and determine the role of each of the following in that mechanism — transverse velocity gradients, the nonlocal pressure Hessian (the matrix of second spatial derivatives of the pressure field, which governs the pressure-gradient contribution to the acceleration), and temporal correlations along fluid trajectories. Any claimed theory must make quantitative, falsifiable predictions for $a_0(R_\lambda)$, the exponents $\zeta_p$, and the Reynolds-dependence of $C_4$, testable against existing and future DNS/experimental data up to $R_\lambda\approx1300$ and beyond.

References

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