Dynamic multiscaling of multi-time multi-scale correlations in three-dimensional Navier–Stokes turbulence and its universality

In fully developed three-dimensional turbulence, multi-time multi-scale correlation functions of velocity increments are predicted to exhibit dynamic multiscaling: each statistical order is characterized by its own decorrelation time, with dynamic exponents connected to the anomalous equal-time scaling exponents by bridge relations that, for two-time correlations, predict the same scaling independently of how the moment is split between the two measurement times. The only direct measurement in three-dimensional Navier–Stokes turbulence (Biferale, Calzavarini, and Toschi, 2011, using quasi-Lagrangian probes in direct numerical simulation at Taylor-scale Reynolds numbers 141–176) found evidence for dynamic multiscaling and consistency with bridge relations, but with finite-Reynolds-number and finite-size effects, an unresolved dependence on the moment-splitting parameter, unquantified residual sweeping at large scale separations, and no experimental confirmation. Later work confirmed the multiscaling framework in two-dimensional turbulence, Burgers turbulence, shell models, and magnetohydrodynamic and rotating analogues; multi-time scaling laws were derived from hidden scale symmetry in a shell model; and Eulerian spatio-temporal correlations were quantified by functional-renormalization-group and DNS studies. However, no study has established or refuted dynamic multiscaling of quasi-Lagrangian multi-time multi-scale correlations in three-dimensional turbulence at high Reynolds number, resolved the observed moment-splitting dependence of the measured exponents against the $q$-independence predicted by the bridge relations, or validated the bridge relations experimentally via particle tracking. The problem is to settle these questions.

Background

Fully developed turbulence transfers energy from large scales of size $L$ (where it is injected) down to the Kolmogorov dissipative scale $\eta$ (where viscosity converts it to heat); the range in between, where the dynamics is believed to be universal, is the inertial range. Equal-time statistics are quantified by structure functions $S_p(r)=\langle|\delta u_r|^p\rangle$, where $\delta u_r$ is the velocity difference across a distance $r$; Kolmogorov's 1941 (K41) dimensional theory predicts $S_p\sim r^{p/3}$, while experiments and simulations find anomalous scaling $S_p\sim r^{\zeta_p}$ with exponents $\zeta_p\neq p/3$ — the phenomenon of intermittency.

The temporal generalization — how fast velocity fluctuations at a given scale decorrelate in time — is far less understood. In the Eulerian frame (measurements at fixed points), temporal decorrelation is dominated by sweeping: small eddies are passively advected by larger ones, which ties decorrelation times linearly to scale ($T_E\sim r$, a trivial dynamic exponent) and masks the intrinsic dynamics of the energy transfer. To remove sweeping one measures in the quasi-Lagrangian frame, a reference frame attached to the trajectory of a fluid particle, introduced by Belinicher and L'vov (1987); such measurements have so far been realized only in direct numerical simulations (DNS) of the Navier–Stokes equations.

The source study (Biferale, Calzavarini, and Toschi, 2011) measured multi-time, multi-scale correlation functions $C^{(q,p-q)}_{R,r}(\tau)=\frac{\langle(\delta u_R(t))^q(\delta u_r(t+\tau))^{p-q}\rangle}{\langle(\delta u_R(t))^q(\delta u_r(t))^{p-q}\rangle},$ where $R$ and $r$ are two spatial separations evaluated at two different times along quasi-Lagrangian trajectories, the moment-splitting parameter $q$ fixes how the order-$p$ moment is distributed between the two times, and the two-time moment is normalized by the corresponding equal-time moment, together with their integral correlation times $T^{(q,p-q)}(R,r)=\int_0^\infty C\,d\tau$ of the normalized correlation. Dynamic multiscaling, proposed for hydrodynamic turbulence by L'vov, Podivilov, and Procaccia (1997), is the prediction that these times scale as $T\sim r^{z^{(q)}(p)}$ with not one but an infinite hierarchy of dynamic exponents $z^{(q)}(p)$ — one for each order $p$ — connected to the equal-time anomalous exponents $\zeta_p$ by bridge relations, which for these two-time correlations predict the same scaling independently of the splitting $q$. Before 2011 the phenomenon and its bridge relations had been established only in simplified systems: shell models of turbulence (discrete toy models mimicking the cascade over a geometric sequence of scales; Biferale, Boffetta, Celani, and Toschi 1999), where Mitra and Pandit (2004) further showed that different ways of extracting time scales from time-dependent structure functions — integral, derivative, and exit-time protocols — yield genuinely different dynamic exponent families, the 'varieties' of dynamic multiscaling, each with its own class of bridge relations, while the $q$-dependence of the two-time structure-function exponents is suppressed and should not affect them, and Ray, Mitra, and Pandit (2008) showed these bridge relations are linear and shared between decaying and statistically steady turbulence.

The 2011 study — DNS at Taylor-scale Reynolds numbers $Re_\lambda=141$ and $176$ (a Reynolds number built on the Taylor microscale), with probe stations riding tracer trajectories — found: (i) the quasi-Lagrangian construction removes sweeping (Eulerian integral times scale as $\sim r$, quasi-Lagrangian ones as $\sim r^{2/3}$); (ii) evidence for dynamic multiscaling, with normalized exponents $z(p)/z(2)$ deviating from the K41 value of 1 in the direction of the bridge-relation predictions ($z(4)/z(2)=0.74/0.67$ and $z(6)/z(2)=0.78/0.67$, i.e., above 1); (iii) bridge relations valid within numerical accuracy, though the predicted intermittent corrections are small and the error bars large; and (iv) a delayed-peak structure of the multi-scale correlations supporting locality of the cascade in space and time. What the study could not do — and what remains open — was reach clean inertial-range power laws: pure scaling was hindered by finite-Reynolds-number and finite-size effects, the measured $q=1$ versus $q=p-1$ exponents differed by amounts comparable to the error bars although the bridge relations predict the same scaling independently of $q$, residual sweeping and finite-size effects at large scale separations were left unquantified, the increasingly wild temporal fluctuations at small scales were noted but their consequences left unexamined, and no experimental validation was attempted, although the authors observed that the method 'needs only the knowledge of the velocity at just a few points along a Lagrangian trajectory', suggesting current particle-tracking set-ups might already access it.

Problem Statement

Establish whether the temporal dynamics of the energy cascade in three-dimensional Navier–Stokes turbulence exhibits universal dynamic multiscaling — an infinite hierarchy of order-dependent dynamic exponents connected to the equal-time anomalous exponents by bridge relations. Concretely, the task is to extend the quasi-Lagrangian multi-time multi-scale correlation measurements of Biferale, Calzavarini, and Toschi (2011) along the following lines of inquiry:

(a) Determine, for the orders $p=4$ to $8$ measured by the source at both of its splitting choices $q=1$ and $q=p-1$, the dynamic exponents $z^{(q)}(p)$ of the integral correlation times $T^{(q,p-q)}(R,r)$ at sufficiently high Reynolds number to obtain clean inertial-range power laws, quantifying finite-Reynolds-number corrections and the small intermittent deviations from the dimensional K41-type predictions.

(b) Test the bridge relations connecting the dynamic exponents to independently measured equal-time exponents $\zeta_p$ of the same flow, and establish whether the slight dependence on the moment-splitting parameter $q$ observed at $Re_\lambda\simeq141$–$176$ is a finite-statistics artifact — the bridge relations predict the same scaling independently of $q$ — or a genuine deviation from that prediction, i.e., whether the dynamic multiscaling is universal.

(c) Quantify and eliminate residual sweeping and finite-size effects, in particular at large scale separations $R$ in the multi-scale correlations, where the source study could not disentangle them from inertial-range scaling.

(d) Realize the measurement experimentally via Lagrangian particle tracking (the source notes the approach needs only the velocity 'at just a few points along a Lagrangian trajectory') and test the DNS findings and the bridge relations in laboratory turbulence.

An answer resolves the task when it decisively establishes or refutes, for these quasi-Lagrangian correlations, the dynamic-multiscaling hierarchy and its bridge relations, including whether the observed dependence on the moment-splitting parameter $q$ is a finite-statistics artifact or a genuine deviation from the predicted $q$-independence; the required coverage is the orders $p=4$ to $8$ at both splitting choices $q=1$ and $q=p-1$ — demonstrating those exponents and their consistency with the bridge relations and with $q$-independence resolves the task, and an argument covering every order of the infinite hierarchy is neither required nor implied; items (a)–(c) constitute the required core, item (d) is a suggested extension, and no single item by itself resolves the task.

The applied sub-question raised in passing by the source — whether the growth of temporal fluctuations at small scales affects time-marching stability criteria and temporal subgrid parametrizations — is a downstream question and is not part of the required core.

References

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