Large-order saturation of longitudinal structure-function exponents in three-dimensional turbulence

The theory of Sreenivasan and Yakhot (Phys. Rev. Fluids 6, 104604, 2021) derives the anomalous scaling exponents of longitudinal velocity-increment structure functions in three-dimensional turbulence from a novel closure of the Hopf equation, predicting that the exponents saturate at large moment order, approaching a constant value of about 7.3, with two striking corollaries: a smallest dynamically relevant scale $\eta_\infty=L\,Re^{-1}$ far below the Kolmogorov scale, and velocity excursions of order the large-scale velocity across those smallest scales. The authors themselves state that no experiment or simulation has compellingly demonstrated saturation for longitudinal structure functions, and that the predicted value rests on uncontrolled closure inputs (near-constant amplitude ratios, a spatially constant mean dissipation rate, and white-in-time Gaussian forcing). The problem has remained open and has been sharpened by recent work: 2023 direct numerical simulations at Taylor-scale Reynolds numbers up to 1300 confirm saturation only for transverse (about 2.1) and Lagrangian (about 2) exponents, leaving the longitudinal prediction untested; a 2024 follow-up by the same authors and collaborators builds the saturation into an 'asymptotic fourth state' while candidly acknowledging the forcing-transfer gap; and a 2026 experiment reports the first experimental evidence for longitudinal saturation - but at a value of about 2.2 in shear turbulence, explicitly incompatible with the predicted 7.3. Settling whether longitudinal exponents saturate, at what value, and on what dynamical basis - together with the attendant smallest-scale claim $L\,Re^{-1}$ - remains an open problem of turbulence theory.

Background

Turbulence research characterizes the statistics of velocity differences across scales through structure functions. For a velocity component $u$ measured at two points separated by distance $r$, the longitudinal increment is $\delta_r u=u(\mathbf{x}+r\mathbf{e})-u(\mathbf{x})$ with the separation taken along the same direction as the measured component, and the $n$-th order structure function is $S_n(r)=\langle(\delta_r u)^n\rangle\propto r^{\zeta_n}$ within the inertial range - the range of scales between the energy-injection scale $L$ and the dissipation scale. Kolmogorov's 1941 dimensional theory predicts 'normal' scaling $\zeta_n=n/3$ (with the exact third-order result $\zeta_3=1$, the 4/5 law), but experiments and simulations have long shown anomalous scaling: $\zeta_n$ grows more slowly than $n/3$ because the small scales are intermittent, dominated by rare intense events. In the multifractal description, the velocity field is locally smooth with Hoelder exponent $h$ on sets of fractal dimension $D(h)$, giving $\zeta_p=\min_h(ph+3-D(h))$; if the smallest such exponent is $h_{\min}=0$, the exponents saturate at large order, $\zeta_p\to\zeta_\infty=3-D(0)$, which obeys the bound $\zeta_\infty\le 3$ provided the fractal dimension is non-negative, $D(0)\ge 0$; a negative dimension $D(0)<0$ would instead give $\zeta_\infty>3$, for which this bound does not apply.

Saturation at large order is well documented in related systems in which pressure does not oppose gradient steepening: Burgers turbulence (the Navier-Stokes equations with the pressure term removed, whose solutions develop discontinuous shocks) and passive scalars advected by turbulence, whose 'steep cliffs' produce saturated exponents (Iyer et al. 2018). In three-dimensional turbulence itself, direct numerical simulations (DNS) at Taylor-scale Reynolds numbers $R_\lambda$ (the Reynolds number formed with the root-mean-square velocity fluctuation and the Taylor microscale, the standard intensity measure for homogeneous turbulence) up to about 1300 show that the exponents of transverse increments (separation perpendicular to the measured component) saturate at about 2-2.1 for orders $p\gtrsim 10$, and that Lagrangian (along-trajectory) exponents saturate at about 2 for orders $p\gtrsim 8$; experimental determinations of the transverse saturation value have ranged from about 2 to 2.8. For longitudinal increments the question has remained open.

Sreenivasan and Yakhot (2021) derived expressions for the longitudinal $\zeta_n$ at all moment orders by analyzing the Navier-Stokes equations through the Hopf equation - the exact functional equation for the probability density of velocity differences, whose closure is the central difficulty of the subject. Their closure combines scale-elimination (producing a scale-dependent turbulent viscosity $\nu_T\sim r\,\delta_r u$), matching between dissipation and integral scales, and the empirically observed near-constancy of amplitude ratios $(2n-1)A_{2n-2,2}/A_{2n,0}\approx 4/3$ between coefficients of the closed equations; crucially, retaining the time derivative of the pressure-gradient equation converts normal scaling into anomalous scaling. The resulting interpolation, $\zeta_n\approx 7.3\,n/(n+19)$, agrees with available data up to the highest reliably measured longitudinal orders at the time ($n\approx 10$-12) and saturates at $\zeta_\infty\approx 7.3$. If saturation is real, two far-reaching consequences follow: the smallest dynamically relevant scale for extreme moments is $\eta_\infty=L\,Re^{-1}$, a factor $Re^{1/4}$ smaller than the Kolmogorov scale $\eta\sim L\,Re^{-3/4}$ (the dissipation scale, below which viscosity smooths out the velocity field), so that velocity excursions across even the smallest scales can be of order the large-scale velocity $u_0$; and the dominant structures must be long and thin, of length $L$ and thickness of order $L/Re$. In the multifractal reading, $\zeta_\infty\approx 7.3$ corresponds to a negative dimension $D(0)=3-7.3=-4.3$, i.e., singularities rarer than isolated points that exist only in an ensemble sense.

The authors themselves state the outstanding issues: no experiment or simulation has compellingly demonstrated saturation for longitudinal structure functions; the quantitative saturation value depends on the assumed amplitude ratios; the matching and coarse-graining steps lack rigorous control; the derivation assumes a spatially constant mean dissipation rate $\varepsilon$ whose intermittent fluctuations may alter the predictions; the theory is formulated for white-in-time Gaussian large-scale forcing, so its transfer to physically forced Navier-Stokes turbulence involves, in their own later words, 'a considerable jump in physical reasoning'; and resolving the predicted extreme scales in DNS is beyond current computational capability.

Problem Statement

Determine the asymptotic behavior at large moment order of the longitudinal inertial-range structure-function exponents $\zeta_n$ of three-dimensional turbulence, and in particular settle the Sreenivasan-Yakhot prediction that they saturate. All facets below jointly answer one question: is the predicted large-order saturation of the longitudinal exponents real, at what value does it occur, and on what dynamical basis? A resolution requires the first two facets, the second being discharged by a saturation value derived from the equations or, if the exponents are shown not to saturate, by a controlled derivation of the large-$n$ growth law, and the third facet is required only when the exponents are shown to saturate, a refutation of saturation settling it automatically.

First, empirically: establish, by experiment or by adequately resolved and statistically converged direct numerical simulation at sufficiently high Reynolds number, whether $\zeta_n$ for longitudinal increments saturates as $n\to\infty$, and determine the saturation value if it exists - in particular decide between the predicted value of about 7.3, the values near 2-2.2 suggested by transverse and Lagrangian measurements and by the 2026 shear-flow experiment, and the absence of saturation (unbounded growth, however slow). The evidence must control the three recognized bottlenecks that the source paper and subsequent work identify: statistical convergence of high-order moments dominated by rare tail events, spatial resolution below the Kolmogorov scale, and Reynolds-number sufficiency of the inertial range; and it must address whether high-order statistics measured in shear flows transfer to homogeneous isotropic turbulence.

Second, theoretically: place the saturation value on controlled footing. Justify, correct, or replace the closure inputs that currently fix it - the matching and coarse-graining steps (point-splitting, the turbulent-viscosity estimate), the assumed near-constancy of the amplitude ratios $(2n-1)A_{2n-2,2}/A_{2n,0}\approx 4/3$, the constant mean dissipation assumption (including the effect of the intermittency of $\varepsilon$), and the passage from white-in-time Gaussian large-scale forcing to physically forced Navier-Stokes turbulence - so that the saturation value is derived from the equations rather than parameterized, and reconcile the result with the experimental and numerical data.

Third, as consequences that stand or fall with the saturation: settle whether the smallest dynamically relevant scale for extreme moments is indeed $\eta_\infty=L\,Re^{-1}$, a factor $Re^{1/4}$ below the Kolmogorov scale, with velocity excursions of order the large-scale velocity across those scales, and establish what this implies for resolving extreme events in direct numerical simulation and for the geometry of the most singular structures of turbulence.

References

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