Dynamic-multiscaling universality between decaying and statistically steady turbulence beyond Kraichnan and shell models

Ray, Mitra and Pandit (New J. Phys. 10, 033003, 2008) showed, analytically for the Kraichnan passive-scalar model and its shell-model analogue and numerically for GOY shell models of fluid and passive-scalar turbulence, that in decaying turbulence the time-dependent structure functions factorize into a part depending on the measurement time-origin $t_0$ and a universal decay part, and that after a normalization that removes $t_0$-dependence the dynamic-multiscaling exponents and their linear bridge relations to equal-time exponents coincide with those of statistically steady turbulence — a property they call dynamic universality. Whether this universality holds for genuine Navier–Stokes turbulence and Navier–Stokes-adverted passive scalars remains open: the factorization has been proven only for linear or white-in-time advecting velocities, no direct numerical simulation (DNS) or experiment has resolved the hierarchy of dynamic-multiscaling exponents in decaying full-fluid turbulence, the systematic uncertainties and parameter ranges are unquantified, and the dependence on dissipation mechanism, Schmidt number and forcing is unknown. The problem is to establish or refute this dynamic universality for real fluids through rigorous theory, first-principles DNS, or controlled experiments on quasi-Lagrangian multi-time structure functions.

Background

In homogeneous, isotropic turbulence the order-$p$ equal-time velocity structure function $S_p(r)\sim r^{\zeta_p}$ measures velocity differences across a separation $r$ in the inertial range between the forcing scale $L$ and the dissipation scale. Kolmogorov's 1941 dimensional analysis (K41) predicts a single exponent $\zeta_p=p/3$, but experiments and simulations find a nonlinear, convex $\zeta_p$ — multiscaling — which is the signature of intermittency. The temporal analogue of this question asks how multi-time, multi-scale correlation functions decay in time, in the same spirit as dynamic scaling near a critical point in condensed-matter systems. In a fixed (Eulerian) frame the answer is trivial: large eddies sweep smaller ones past the probe, so time lags scale linearly with $r$ and every dynamic exponent equals 1 (the sweeping effect). Nontrivial temporal dynamics is instead accessed in the quasi-Lagrangian frame introduced by Belinicher and L'vov, in which the velocity field is measured in a coordinate system carried along by a fluid particle, thereby removing sweeping. Even then, a single order $p$ does not possess a single dynamic exponent: from the order-$p$ time-dependent structure function $F_p(r,\{t_1,\ldots,t_p\})$, one extracts degree-$M$, order-$p$ integral and derivative time scales $T^{I}_{p,M}$ and $T^{D}_{p,M}$ (from moments of, and short-time derivatives of, the decay of $F_p$), each scaling as $r^{z^{I,D}_{p,M}}$ — an infinite hierarchy of dynamic-multiscaling exponents. For statistically steady (forced) turbulence, Mitra and Pandit (Phys. Rev. Lett. 93, 024501 (2004); Phys. Rev. Lett. 95, 144501 (2005)) showed in shell models that these exponents are tied to the equal-time exponents by linear bridge relations, for example $z^{I}_{p,M}=1+(\zeta_{p-M}-\zeta_p)/M$ and $z^{D}_{p,M}=1+(\zeta_p-\zeta_{p+M})/M$. The shell models used are the GOY model — a chain of complex amplitudes $u_n$ at geometrically spaced wavenumbers $k_n$, coupled shell-to-shell so as to mimic the turbulent energy cascade — and its passive-scalar analogue; the Kraichnan model replaces the advecting velocity by a Gaussian field, white in time, with spatial covariance growing as $r^{\xi}$, which makes the advection problem exactly solvable.

In decaying turbulence there is no stationary ensemble: observables depend on $t_0$, the time after the flow is prepared at which measurement begins. Ray, Mitra and Pandit (2008) systematized this case for four models: the Kraichnan passive-scalar model (A), its shell analogue (B), the GOY fluid shell model (C), and the GOY-advected passive-scalar shell model (D). They proved analytically for A and B, and verified numerically for B, C and D, that the decaying time-dependent structure function factorizes as $F_p(r,t_0,t)=A_p(r,t_0)\,F^{f}_p(r,t)$, where $A_p$ carries all dependence on the time origin and $F^{f}_p$ is the statistically steady counterpart; consequently the normalized quantity $Q_p(r,t_0,t)=F_p(r,t_0,t)/F_p(r,t_0,0)$ is independent of $t_0$. After this normalization, the dynamic-multiscaling exponents $z^{I,D}_{p,M}$ and the bridge relations agreed, within statistical error bars, between decaying and statistically steady ensembles — strong evidence for dynamic universality.

The evidence, however, stops short of the Navier–Stokes equations themselves. Models A and B are linear in the advected field with white-in-time advecting velocity, and C and D are shell models, not partial differential equations. For genuine Navier–Stokes turbulence nothing comparable is known. At the time, the only full-fluid Lagrangian dynamic exponent extracted from direct numerical simulation (DNS) was of order $p=2$ (Kaneda, Ishihara and Gotoh, Phys. Fluids 11, 2154 (1999)); Lagrangian particle-tracking experiments (La Porta et al., Nature 409, 1017 (2001); Mordant et al., Phys. Rev. Lett. 89, 254502 (2002)) measure single-trajectory statistics, and the statistics of velocity time increments along trajectories (Chevillard et al., Phys. Rev. Lett. 95, 064501 (2005)) are, as the 2008 paper emphasizes, distinct from the spatiotemporal structure functions required for dynamic multiscaling. The measurement the 2008 paper calls for — tracking at least pairs of Lagrangian particles with a controlled initial separation $r$ so that quasi-Lagrangian multi-time structure functions and their normalized variants can be computed — had not been performed in DNS at hierarchy-resolving orders, nor in any laboratory flow.

Problem Statement

Establish, or refute, the dynamic-multiscaling universality of Ray, Mitra and Pandit (2008) for genuine Navier–Stokes turbulence and Navier–Stokes-adverted passive scalars. The universality claim — proved only for linear or white-in-time advecting-velocity models and for shell models — is that in decaying, homogeneous, isotropic turbulence the quasi-Lagrangian time-dependent structure function factorizes as $F_p(r,t_0,t)=A_p(r,t_0)\,F^{f}_p(r,t)$, with $F^{f}_p$ the statistically steady counterpart, and the normalized ratio $Q_p(r,t_0,t)=F_p(r,t_0,t)/F_p(r,t_0,0)$ independent of the measurement origin $t_0$, and that the resulting degree-$M$, order-$p$ dynamic-multiscaling exponents $z^{I,D}_{p,M}$ (from integral and derivative time scales) and their linear bridge relations to the equal-time exponents $\zeta_p$ coincide with those of statistically steady turbulence. Three linked lines of work serve this single objective; they are aspects of one programme, not cumulative requirements (see the resolution rule at the end of this section): (1) a rigorous analytical derivation of the factorization and $t_0$-independence for nonlinear Navier–Stokes dynamics or fully realistic quasi-Lagrangian fields, going beyond the exactly solvable Kraichnan-type and shell models; (2) first-principles direct numerical simulations of the full Navier–Stokes equations, or controlled laboratory measurements, of Lagrangian or quasi-Lagrangian time-dependent structure functions in decaying turbulence — for example by tracking pairs of particles with controlled initial separations $r$ — with statistical averaging sufficient to resolve the hierarchy of exponents $z^{I,D}_{p,M}$, and their comparison with the statistically steady values; and (3) a validation layer: quantification of the systematic uncertainties (inertial-range fitting windows, finite-size effects, the onset of the trivial $E_{\mathrm{tot}}(t)\sim t^{-2}$ decay regime when the integral scale reaches the system size) and delineation of the ranges of Reynolds number, scale separation and times over which the universality holds, together with its dependence on the dissipation mechanism, Schmidt number and forcing, beyond the existing hyperviscous shell-model study. A resolution by any route answers the problem: a rigorous factorization proof for a genuinely nonlinear Navier–Stokes setting; a decisive DNS or experimental comparison showing exponent equality within quantified systematics; or a demonstrated breakdown, i.e. exponent differences between decaying and steady ensembles beyond the quantified uncertainties.

References

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