Saturation and Multifractality of Lagrangian and Eulerian Scaling Exponents in Three-Dimensional Turbulence

Inertial-range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at ≈ 2 . 1 for moment orders p ≥ 10 , significantly differing from the longitudinal exponents (which are predicted to saturate at ≈ 7 . 3 for p ≥ 30 from a recent theory). The Lagrangian scaling exponents likewise saturate at ≈ 2 for p ≥ 8 . The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well-known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implications of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration.

Turbulent flows in nature and engineering comprise a hierarchy of eddies, with smaller eddies coexisting within larger ones and extracting energy from them.To understand the deformation and rotation of smaller eddies, the key mechanisms driving energy transfers, it is essential to examine the velocity increments across a smaller eddy of size r ≪ L (say), where L is the large-eddy size [1][2][3].The longitudinal velocity increment δu r = u(x+r)−u(x) corresponds to the case when the velocity component u(x) is in the direction of separation r.For velocity v(x) taken orthogonal to r, transverse velocity increment δv r = v(x + r) − v(x) is obtained.
The motivation to study the small eddies (and hence velocity increments) stems from their purported universality, postulated by Kolmogorov (1941) [1]-K41 henceforth-which has since become the backbone of turbulence theory and modeling [3,4].Building upon K41, one surmises that moments of increments (δu r ) p , called structure functions, follow a universal power-law scaling in the so-called inertial-range: where η is the viscous cutoff scale.Establishing such a simple scaling enables dramatic simplification in studying a wide range of turbulent flows, and thus, structure functions have been of persistent interest and a cornerstone of turbulence theory [2,3,5,6].K41 originally postulated ζ p = p/3; this result is known to be exact for p = 3, i.e., ζ 3 = 1, but extensive studies from [7] to [8] (and others in between) have clearly established nonlinear deviations of ζ p from p/3 for p = 3.This so-called anomalous scaling is attributed to the intermittency of interscale energy transfer processes (see, e.g., [2,3,5,6]).
Since turbulence can also be fundamentally explored from a Lagrangian viewpoint [2,[9][10][11][12], forceful argu-ments can be similarly made for Lagrangian velocity increments δu τ = u(t + τ ) − u(t) over time lag τ , measured along fluid-particle trajectories, and Lagrangian structure functions |δu τ | p defined therefrom [13].Extension of K41 phenomenology to Lagrangian increments gives: where the temporal inertial-range is defined using T L , the Lagrangian integral time and τ η , the time-scale of viscous dissipation [2].Since Lagrangian trajectories trace the underlying Eulerian field, it is natural to expect that a relation between Lagrangian and Eulerian exponents can be obtained.Using K41, one obtains ζ L p = p/2 [2]; but, experimental and numerical studies again show nonlinear deviations from this prediction [14][15][16][17].Several attempts have been made [18][19][20] to quantify these deviations in terms of Eulerian intermittency, but they remain deficient for at least two reasons.First, the temporal scaling range in turbulence is substantially more restrictive than spatial scaling range [2,3], making it difficult to robustly extract the Lagrangian scaling exponents.Second, past attempts have overwhelmingly focused on characterizing Lagrangian intermittency from longitudinal Eulerian intermittency, assuming that longitudinal and transverse exponents are identical, despite counter-evidence [21][22][23][24][25][26].
In this Letter, presenting new data from direct numerical simulations (DNS) of isotropic turbulence at higher Reynolds numbers, we address both these challenges.We extract both Lagrangian and Eulerian scaling exponents.Our Eulerian results reaffirm recent results [8].We then demonstrate an excellent correspondence between Lagrangian exponents and transverse Eulerian exponents, using as basis the same multifractal spectrum; this is different from the multifractal spectrum for longitudinal FIG. 1. Inertial-range scaling exponents for longitudinal and transverse Eulerian structure functions, the former from [8,40] and the latter from the present data (consistent with [8]).Various theoretical predictions [1,[40][41][42] are also shown.The transverse exponents depart from all predictions and saturate.
Direct Numerical Simulations: The description of DNS is necessarily brief here because they have been described in many recent works [28][29][30][31][32][33].The simulations correspond to the canonical setup of forced stationary isotropic turbulence in a triply periodic domain and are carried out using the highly accurate Fourier pseudospectral methods in space and second-order Runge-Kutta integration in time; the large scales are numerically forced to achieve statistical stationarity [34,35].A key feature of the present data is that we have achieved a wide range of Taylor-scale Reynolds number R λ , going from 140 − 1300 (on grids of up to 12288 3 points) while maintaining excellent small-scale resolution [29,36].For Lagrangian statistics, a large population of fluid particles is tracked together with the Eulerian field.For R λ ≤ 650, up to 64M particles are tracked for each case, whereas for R λ = 1300, 256M particles are tracked (with M= 1024 2 ) [37][38][39], providing ample statistics for convergence.
Saturation of transverse exponents: Anomalous scaling confers upon each moment order a separate and independent significance, instead of a mutual dependence (such as ζ p = p/3 based on K41).Multifractals have enjoyed considerable success in describing this behavior [3,6], but lack any direct connection to Navier-Stokes equations.Further, recent DNS at high R λ have shown noticeable departures of ζ p from multifractal predictions for high orders [8].Instead, starting from Navier-Stokes equations, a recent theory [40] was able to provide an improved prediction for ζ p .Additionally, this theory also predicts that longitudinal exponents saturate with the moment-order, i.e., lim symmetry.)Multifractal models based on phenomenological considerations do not differentiate between longitudinal and transverse exponents, i.e. ζ tr 2p = ζ 2p , and general arguments have also been advanced to the same end [43,44].However, several studies have persistently pointed out that the two sets of exponents are different [21][22][23][24][25][26]; recent work at high R λ [8] has confirmed the differences, also showing that transverse exponents saturate: ζ tr ∞ ≈ 2.1 for p ≥ 10.Incidentally, this saturation is very different from ζ ∞ ≈ 7.3 (for p ≥ 30) predicted for longitudinal exponents in [40].
These findings are summarized in Fig. 1, showing the longitudinal and transverse exponents.Also included are K41 prediction, multifractal results [41,42] and the result from [40].Important considerations go into establishing the reliability of high-order exponents with respect to statistical convergence, adequacy of grid resolution, and R λ -dependence.This discussion can be found in [8] and will not be repeated here.Instead, we focus on ζ tr p , which clearly depart from ζ p and saturate for p ≥ 10.The implication of different longitudinal and transverse exponents for small-scale universality is discussed later; we first demonstrate how ζ tr p is directly related to the Lagrangian exponents.
Lagrangian exponents from DNS: Robust extraction of scaling exponents requires sufficient scale separation to allow a proper inertial-range to exist.The Eulerian spatial scale separation for the highest R λ = 1300 is L/η ≈ 2500 [8], while the temporal range is T L /τ K ≈ 105 [45], thus making it inherently difficult to obtain a proper Lagrangian inertial-range [46,47].This difficulty is highlighted in Fig. 2, which shows the log local slope of S L p (τ ) at various R λ , for p = 2 and 4 in panels (a) and (b), respectively; although there is a suggestion of a plateau for the fourth-order, the local slopes of the curves are still changing with R λ .This is in contrast to the corresponding Eulerian result for p = 2, shown in Fig. 3, where a clear inertial-range emerges with R λ .
Because of this difficulty, Lagrangian exponents cannot be directly extracted even at the highest R λ available.However, by using extended self-similarity [48], we can obtain the exponents with respect to the second-order [17].Fig. 4 shows the ratio local slope of S L (τ ) to that of S L 2 (τ ).Evidently, a conspicuous plateau emerges for different orders in the same scaling range, seemingly independent of R λ Thus, we can extract the ratios ζ L p /ζ 2 , which also was the practice in earlier works [15][16][17].The justification for using ζ L 2 as the reference comes from the expectation S L 2 ∼ ǫ τ [2]; since the mean dissipation appears linearly, the result ζ L 2 = 1 is free of intermittency (akin to ζ 3 = 1 for Eulerian exponents [49]).
Extending the procedure in Fig. 4, the ratios are extracted for upto p = 10 and shown in Fig. 5.We also include earlier results from both experiments and DNS [15][16][17]20], obtained at comparatively lower R λ .Overall, the current results at higher R λ are in excellent agreement with prior results (which had larger error bars).A remarkable result, endemic to all cases, is that the Lagrangian exponents saturate for p 8, similar to the transverse Eulerian exponents in Fig. 1.The data in Fig. 5 are also compared with various predictions, which we discuss next.
The multifractal framework: Evidently, the data in Fig. 5 strongly deviate from K41.Following [3,19], we will consider the well known multifractal model for relating Eulerian and Lagrangian exponents.The key concept in multifractals is that the (Eulerian) velocity increment δu r over a scale r is Hölder continuous, i.e., δu r ∼ r h , where h is the local Hölder exponent with the multifracal spectrum D(h) [3,50].From this local scaling, Eulerian structure functions are readily derived by integrating over all possible h, as (δu r ) p ∼ h r ph+3−D(h) dh.Using steepest-descent argument for r ≪ L gives The Lagrangian extension of multifractals relies on the phenomenological assumption that spatial and temporal separations are interchangeable: r ∼ τ δu r , akin to frozen flow hypothesis, with δu r ∼ δu τ [19].This stipulation gives δu τ ∼ τ h/(1−h) , resulting in the Lagrangian exponents Thus, Lagrangian exponents can be directly predicted using the Eulerian multifractal spectrum D(h).Since past works have predominantly focused on Eulerian longitudinal exponents, with the implicit assumption that transverse exponents are same, the D(h) of the longitudinal exponents has been used to infer Lagrangian exponents.However, such predictions do not work as we see next.
Lagrangian exponents from the transverse multifractal spectrum: As we saw, none of the multifractal predictions for Lagrangian exponents using Eulerian longitudinal exponents agree with the data.In contrast, the prediction corresponding to transverse Eulerian exponent (green dot-dashed line in Fig. 5) closely follows the measured results, particularly capturing the saturation at high orders.Note, the predicted saturation value ζ L ∞ ≈ 2.1, is the same for both transverse Eulerian and Lagrangian exponents, The actual Lagrangian data saturate at a very slightly smaller value.We believe this minor difference (of only 5%) stems from the fact that even at R λ = 1300, the temporal inertial-range is underdeveloped, and the intermittency-free result of ζ L 2 = 1 is not unambiguously realized.Since Lagrangian 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 FIG.6.The multifractal spectra for various models.The vertical dashed lines at 1  3 log 2 (0.7)(≈ 0.17) and 1 9 mark the minimum h allowed for p-model [41] She-Leveque [42], respectively, which preclude saturation; whereas D(h = 0) ≈ 3 − 2.1 = 0.9 marks saturation for transverse exponents at exponents shown in Fig. 5 are extracted as ratios ζ L p /ζ L 2 , this minor discrepancy in the saturation values could be explained by small departures from the expectation of ζ L 2 = 1.Given this and also possible statistical uncertainties (at highest orders), the close correspondence between the transverse Eulerian exponents and Lagrangian exponents is quite remarkable.
It is worth noting that Lagrangian exponents saturate for slightly smaller p than for transverse Eulerian exponents.This readily follows from Eqs. ( 3)-(4) as a kinematic effect.For Eulerian exponents, ζ 3 = 1 is exact, corresponding to h ≈ 1  3 , D(h) ≈ 3, which conforms to the intermittency-free K41 result [3].This gives ζ L 2 = 1 as the corresponding Lagrangian result for h ≈ 1 3 , D(h) ≈ 3.This argument can be extended to higher orders to show that Lagrangian exponents at order p correspond to transverse exponents at order 3p/2.Thus, it follows that Lagrangian exponents saturate at smaller p.A similar correspondence can also be provided for other Lagrangian statistics, for instance, the second-moment of acceleration (the temporal velocity-gradient) corresponds to the third-moment of spatial velocity-gradients [2,27].
Discussion: Two significant results emerge from our work: (a) scaling exponents saturate for both transverse Eulerian and Lagrangian structure functions; and (b) the saturation of Lagrangian exponents is characterized solely by the transverse Eulerian exponents (and not the longitudinal, as previously believed).Given that the transverse exponents are smaller for large p, this seems reasonable from the steepest-descent argument [3].
The saturation of scaling exponents is extreme form of anomalous behavior, but is not uncommon; it holds for Burgers equation [51], passive scalar turbulence [52][53][54].However, its prevalence in velocity field has become apparent only recently [8,40].The theory of [40] predicts that Eulerian longitudinal exponents saturate as well, although at very high moment orders that cannot be yet validated.In contrast, both transverse Eulerian exponents and Lagrangian exponents saturate and at the same value of ≈ 2. Further, using a simple physical correspondence based on frozen flow hypothesis, they are related through the same multifractal spectrum (which differs from known spectrum for longitudinal Eulerian exponents).Interestingly, the saturation exponent of 2 implies a fractal co-dimension of 1 [3,6], suggesting that the saturation likely comes from localized (very) thin vortex filaments, which are known to be prevalent at the smallest scales [34,36,55].
Our results also bring forth some important questions.First is the extension of the multifractals from inertial-to dissipative-range, i.e., describing the scaling of velocitygradients.Such an extension relies on the phenomenological criterion that the local Reynolds number, describing the dissipative cutoff, is i.e., δu r r/ν = 1 [3,40,56].As highlighted in recent works [36,57], this is valid for longitudinal increments, but not for transverse increments, essentially because of how vorticity and strainrate interact in turbulence.It can thus be expected that the extension of multifractals to dissipation-range works for longitudinal velocity-gradients, but not for transverse velocity-gradients.Since the current results suggest that Lagrangian intermittency is linked to transverse Eulerian intermittency, it follows that the extension to acceleration statistics would be an issue, as confirmed by our recent studies [27,58].In addition, acceleration components are strongly correlated in turbulence [58,59], which is a feature of Navier-Stokes dynamics that is not accounted for by multifractals.
A second question concerns the meaning of universality given the longitudinal and transverse exponents behave differently.One strategy could be to consider a joint multifractal spectrum for longitudinal and transverse increments.It might be possible to set appropriate conditions on both to enable the inertial-range universality and the transition from the inertial-to dissipation-range.Essentially, addressing the discrepancy between longitudinal and transverse intermittency presents a critical and pressing problem in turbulence theory.

2 FIG. 3 .
FIG.3.Local slopes for the Eulerian second-order structure functions at different R λ .In contrast to Lagrangian data in Fig.2, a clear inertial-range emerges with Reynolds number.