Making the persistent-acceleration decomposition of Lagrangian turbulence predictive: theoretical models for the coarse-grained-acceleration PDF $f(\alpha)$ and conditional correlations $C_{\alpha}(\tau)$, and the high-Reynolds-number asymptote of the coarse-graining time $\Theta$

Bentkamp, Lalescu and Wilczek (Nat. Commun. 10, 3550, 2019) showed that conditioning Lagrangian tracer statistics in homogeneous isotropic turbulence on the squared particle acceleration coarse-grained over a viscous time scale $\Theta$ decomposes the strongly intermittent single-particle statistics into close-to-Gaussian subensembles. Their characteristic-functional framework reproduces acceleration, velocity-increment, and dispersion statistics across $R_{\lambda}\approx210$–$509$, but only by taking the PDF $f(\alpha)$ of the coarse-grained squared acceleration and the conditional velocity autocorrelations $C_{\alpha}(\tau)$ directly from direct numerical simulation (DNS) data — the framework is descriptive, not predictive. The open problem is to supply theoretically grounded, non-DNS-fed models for $f(\alpha)$ and $C_{\alpha}(\tau)$ that make the framework fully predictive, together with the auxiliary open issue the same authors raise: whether the optimal coarse-graining time $\Theta$ approaches a Reynolds-number-independent asymptote at very high $R_{\lambda}$.

Background

In the Lagrangian description of turbulence one follows individual fluid tracers — idealized point particles advected with the flow — rather than the velocity field at fixed points. Single-particle Lagrangian statistics govern turbulent dispersion and mixing and are paradigmatic examples of intermittency: tracer accelerations $\mathbf{a}(t)$ display extended quiescent periods interrupted by violent, short-lived bursts when the particle passes through intense small-scale structures such as vorticity filaments. As a consequence, the probability density function (PDF) of a single acceleration component has very heavy (nearly exponential-to-stretched-exponential) tails, with a flatness — the normalized fourth moment $\langle a^4\rangle/\langle a^2\rangle^2$, equal to 3 for a Gaussian — far above the Gaussian value, and this non-Gaussianity relaxes only slowly with time scale. Quantitative studies rely on direct numerical simulation (DNS, i.e., fully resolved numerical solution of the Navier–Stokes equations) and on high-resolution particle-tracking experiments, typically reported as functions of the Taylor-scale Reynolds number $R_{\lambda}$ (the standard dimensionless measure of the intensity of turbulence, formed from the root-mean-square velocity fluctuation, the mean energy dissipation rate, and the kinematic viscosity; larger $R_{\lambda}$ means a wider separation between the largest and smallest flow scales), with $\tau_{\eta}$ the Kolmogorov viscous time scale setting the smallest temporal scale of the flow.

Classical reductionist attempts — multifractal models (models in which the local scaling exponents of the small-scale statistics form a whole spectrum of values rather than a single number, reflecting the spotty spatial structure of intense dissipation), superstatistics (modeling non-Gaussian statistics as a superposition of simpler distributions with a fluctuating parameter), and refined-similarity formulations in terms of locally averaged dissipation — had not produced a practically useful decomposition of the full Lagrangian ensemble into simple subensembles. Bentkamp, Lalescu and Wilczek (2019) introduced the persistent acceleration: the squared acceleration coarse-grained over a viscous time window, $\alpha(t)\equiv\int_{-\infty}^{\infty}\mathrm{d}\tau\,F_{\Theta}(\tau)\,\mathbf{a}^2(t+\tau),$ with $F_{\Theta}$ a Gaussian filter of standard deviation $\Theta$ (in practice $\Theta\approx2$–$3\,\tau_{\eta}$). Conditioning on $\alpha$ partitions trajectories into subensembles of different local small-scale activity, and — remarkably — the conditional single-component acceleration statistics within each subensemble are close to Gaussian, with the conditional variance growing roughly linearly in $\alpha$.

This observation yields a closed kinematic framework. The complete statistical information of a one-component Lagrangian velocity time series $u(t)$ is encoded in its characteristic functional $\phi[\vartheta]=\langle\exp(\mathrm{i}\int\mathrm{d}t\,\vartheta(t)u(t))\rangle$ (the generating object of the functional formalism of turbulence, from which all multi-time statistics follow by functional differentiation). The authors show that $\phi$ is well approximated by a probabilistic superposition of Gaussian characteristic functionals conditioned on $\alpha$, $\phi[\vartheta]=\int_0^\infty\mathrm{d}\alpha\,f(\alpha)\,\phi_{\alpha}^{\mathrm{G}}[\vartheta]$, where the Gaussian subensemble functional is fully determined by the conditional single-component velocity autocorrelation function $C_{\alpha}(\tau)=\langle u(t_0-\tau/2)\,u(t_0+\tau/2)\mid\alpha(t_0)=\alpha\rangle$. Projecting this functional gives explicit superposition formulas for the velocity PDF (mixture of Gaussians with variances $C_{\alpha}(0)$), the acceleration PDF (variances $-C_{\alpha}''(0)$), and the velocity-increment PDFs (variances given by the conditional second-order structure function), and by construction the framework is consistent with the kinematic evolution equations of these PDFs. Fed with DNS-extracted $f(\alpha)$ (extremely broad, spanning more than six orders of magnitude, with an approximately log-normal right tail) and $C_{\alpha}(\tau)$ (larger variance and faster decorrelation at larger $\alpha$, with a secondary oscillatory maximum at the largest $\alpha$, attributed to particle trapping in vortical structures), the framework accurately reproduces acceleration, increment, velocity, correlation, and dispersion statistics at $R_{\lambda}\approx210$–$509$.

The framework is thus descriptive by construction: both ingredients, the PDF $f(\alpha)$ and the conditional correlations $C_{\alpha}(\tau)$, are extracted from DNS. The authors state explicitly that only when theoretical models for these two ingredients become available does the framework become fully predictive — which they identify as an exciting direction for future work. A further issue is raised: the only free parameter, the coarse-graining time $\Theta$, shows data-set-dependent optimal values (near $3\,\tau_{\eta}$ at $R_{\lambda}\approx210$ and $350$, near $2\,\tau_{\eta}$ at $R_{\lambda}\approx509$) without a clear trend, with large-scale intermittency suspected to play a role; whether $\Theta$ reaches an asymptotic, Reynolds-number-independent viscous value at very high $R_{\lambda}$ is posed as a question for the next generation of experiments and simulations. In addition, while second-order and many finite-dimensional statistics are captured essentially exactly (some by design), the authors document moderate deviations in higher-order measures, most visibly the acceleration flatness, which is systematically overestimated for too small $\Theta$ and underestimated for too large $\Theta$.

Problem Statement

Turn the persistent-acceleration decomposition of Lagrangian turbulence from a descriptive into a predictive framework. Specifically, for homogeneous isotropic turbulence:

1. (Core) Derive or construct theoretically grounded models — not extracted or calibrated from DNS of the target statistics — for (i) the PDF $f(\alpha)$ of the coarse-grained squared Lagrangian acceleration $\alpha(t)=\int_{-\infty}^{\infty}\mathrm{d}\tau\,F_{\Theta}(\tau)\,\mathbf{a}^2(t+\tau)$ (with $F_{\Theta}$ a Gaussian filter of width $\Theta\sim$ a few $\tau_{\eta}$), including its broad, near-log-normal right tail and its Reynolds-number dependence, and (ii) the conditional single-component velocity autocorrelation functions $C_{\alpha}(\tau)$, including the growth of $C_{\alpha}(0)$ with $\alpha$, the faster decorrelation at larger $\alpha$, and the secondary oscillatory maximum at the largest $\alpha$. A successful answer must, when inserted into the superposition formulas $f(u)=\int\mathrm{d}\alpha\,f(\alpha)\,g(u;C_{\alpha}(0))$, $f(a)=\int\mathrm{d}\alpha\,f(\alpha)\,g(a;-C_{\alpha}''(0))$, and the corresponding increment and dispersion expressions — where $g(x;\sigma^2)$ denotes a zero-mean Gaussian probability density of variance $\sigma^2$, consistent with the near-Gaussianity of the conditional subensembles — reproduce measured (DNS or experimental) single-particle statistics — acceleration PDF and flatness, velocity-increment PDFs across time lags, velocity and acceleration autocorrelations, and mean-squared displacement — quantitatively and across at least two Reynolds numbers, with all model parameters fixed a priori from theory or from measurements independent of the validation data.

2. (Auxiliary, raised by the same authors) Determine, empirically and/or theoretically, whether the optimal coarse-graining time $\Theta$ approaches an asymptotic, Reynolds-number-independent value (in units of the Kolmogorov time $\tau_{\eta}$) at very high $R_{\lambda}$, and in particular disentangle the roles of Reynolds number and large-scale intermittency in the observed data-set dependence of the optimal $\Theta$.

Both components are required for a complete answer. The two are scientifically linked aspects of one objective: they supply the missing ingredients and settle the only free parameter of the same characteristic-functional decomposition, and a complete answer establishes the first fully predictive, self-consistent statistical theory of Lagrangian single-particle turbulence within this framework. (A natural further generalization — extending the decomposition to multi-particle statistics or to non-isotropic, inhomogeneous flows — is not part of the author-declared problem and is not required.)

References

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