Reynolds number scaling of velocity increments in isotropic turbulence

Using the largest database of isotropic turbulence available to date, generated by the direct numerical simulation (DNS) of the Navier-Stokes equations on an 8192 3 periodic box, we show that the longitudinal and transverse velocity increments scale identically in the inertial range. By examining the DNS data at several Reynolds numbers, we infer that the contradictory results of the past on the inertial-range universality are artifacts of low Reynolds number and residual anisotropy. We further show that both longitudinal and transverse velocity increments scale on locally averaged dissipation rate, just as postulated by Kolmogorov’s reﬁned similarity hypothesis, and that, in isotropic turbulence, a single independent scaling adequately describes ﬂuid turbulence in the inertial range.


I. INTRODUCTION
This Rapid Communication resolves the previously contradictory results on inertial-range universality, to be described immediately below, of two orthogonal velocity increments in isotropic turbulence. Velocity increments are differences of velocities between two spatial positions separated by a fixed distance, and are of fundamental importance in turbulence research [1][2][3]. In isotropic turbulence, velocity increments in parallel and perpendicular directions to the separation distance r, known as the longitudinal (δu r ) and transverse (δv r ) increments, respectively, describe the velocity increment statistics completely. These statistics are usually in the form of moments of δu r and δv r , and are known, respectively, as the longitudinal and transverse structure functions. The order of the structure function is the moment order of the velocity increments. The power-law scaling of both the longitudinal and transverse structure functions with respect to r should be the same for inertial-range universality to hold, where the inertial range is the range of separation distances r which are small compared to the large scale of turbulence but large compared to the viscous cut-off scale. The scaling exponents for second-order structure functions of δu r and δv r are indeed the same, but higher-order quantities can scale differently in principle [4], so there is no a priori assurance of principle that universality holds. The majority of the empirical evidence available, both experimental and numerical, shows that longitudinal and transverse structure functions indeed scale differently at higher orders [5][6][7][8][9][10][11][12][13], though some studies suggest otherwise [14,15]. Predicated on the consensus that the longitudinal and transverse structure functions scale differently, it was argued in Ref. [8] that the longitudinal increments are closely associated with the energy dissipation, while the transverse increments are linked more closely to enstrophy (squared turbulent vorticity). This argument seemed eminently plausible at the time because the evidence then available also suggested that, while the mean values of dissipation and enstrophy were equal, they differed in higher-order moments. Thus, theoretical arguments [16][17][18] which have advanced the view that the two types of structure functions, on the one hand, and energy dissipation and enstrophy, on the other, must scale similarly have been without clear empirical support. Here, we show, using the direct numerical simulation (DNS) data at a variety of Reynolds numbers including the largest simulations available to date, on a box of 8192 3 grid points corresponding to a Taylor-scale Reynolds number R λ of 1300 [19], that the theoretical arguments are essentially correct, and that previously observed differences are artifacts of low Reynolds number, remnant shear, etc. In conjunction, we show that the so-called refined similarity hypothesis of Kolmogorov [20], holds for both the longitudinal and transverse increments in the inertial range. This hypothesis links the inertial range dynamics with that of the dissipative range, which is where the energy dissipation essentially occurs.
To be specific, we provide the following definitions before proceeding further. Let = ν 2 (∂u i /∂x j + ∂u j /∂x i ) 2 denote the turbulent energy dissipation rate per unit mass, where ν is the kinematic viscosity of the fluid. The dissipation scale η = (ν 3 / ) 1/4 , where · denotes a suitable average. The longitudinal velocity increment is defined as where the velocity component u and the separation distance r are both in the same direction, x. The transverse velocity increment, δv r ≡ v(x + r) − v(x), where the separation distance r (still in the direction x) is normal to the velocity component v. Denote the pth-order longitudinal and transverse structure functions as S p L (r) ≡ (δu r ) p and S p T (r) ≡ (δv r ) p , respectively. All odd moments of δv r vanish in isotropic turbulence (but not those of δu r ). Assume, following the spirit of Kolmogorov [1], that S p L (r) ∼ r ζ p L and S p T (r) ∼ r ζ p T . Since T under isotropy, we aim first to obtain ζ p T and ζ p L for p 4. We obtain scaling exponents of longitudinal and transverse structure functions, ζ p L and ζ p T , respectively, from a large DNS database of isotropic turbulence in a periodic box, spanning a wide range of Reynolds numbers [21], the largest of them being on a grid size of 8192 3 , corresponding to a Taylor-scale Reynolds number R λ of 1300 [19]. A statistically steady state is obtained by forcing the low Fourier modes. For details see Refs. [19,22] and the references cited therein. Averages over multiple large-eddy turnover times were used in the analysis.

II. RESULTS AND DISCUSSION
In locally isotropic turbulence, the following relations must hold at scale r:   The normalized third-order longitudinal velocity structure function S 3 L (r)/r is used to determine the extent of the inertial range: it is the range of scales for which −S 3 L (r)/r is equal to the exact theoretical value of 4/5 [2]. This quantity is plotted in Fig. 2 for the 8192 3 simulation (corresponding to R λ = 1300), as a function of the spatial separation, r. Denoting the inertial range pragmatically as the scale range within which the exact value of 4/5 is satisfied to better than an accuracy of 5%, we can see from the figure that it corresponds to 50 < r/η < 400, approximately a decade in extent. Figure 3 shows the local slopes of the power-law scaling relations, in the inertial range (and to either side of it), computed from the 8192 3 simulation for orders p = 4, 6, and 8. Slopes for both longitudinal and transverse structure functions are shown. It is evident from the figure that they are essentially the same in the inertial range, for each given p, at least for p up to 8.
The longitudinal and transverse scaling exponents obtained from least-squares fits at two different Reynolds numbers, in  the respective inertial ranges, are given in Table I for orders p = 2-8. The error bars have been obtained from the temporal variations in the fitted slopes from independent stationary state snapshots. The differences between ς p L and ς p T shrink as we go to higher Reynolds numbers, indicating that the mismatch between the exponents at lower R λ is the result of a superposition of leading and subleading power laws with coinciding exponents between longitudinal and transverse statistics, but with different prefactors.
To make this point even clearer, one can plot the quantity (δv r ) p against (δu r ) p , parametrized by various values of r. If the slope of such graphs is unity in the inertial range, we can say that the two families of structure functions scale the same. The corresponding plots are shown in Fig. 4, for p = 6 and p = 8, for two different Reynolds numbers. Inertial ranges have been identified in the two graphs by the ( ) symbol. We find that at the low Reynolds number of R λ = 240, (δv r ) 6 Fig. 4. These results demonstrate that ζ p L and ζ p T approach each other as the Reynolds number increases, and that the differences between them, observed previously, must be attributed to the finite Reynolds number effect. This is the first major conclusion of the Rapid Communication.
An additional supporting comment is useful. The use of local slopes or the extended self-similarity (ESS) technique [23] yields very similar results. In the ESS method, the scaling exponents are determined by plotting various orders of transverse and longitudinal structure functions against their respective second-order moments and determining their slopes in the inertial range.
We now turn to the so-called refined similarity hypothesis (RSH) of Kolmogorov [20], which links statistically the fluctuation of the velocity increments of a known scale, say r, in the inertial range to the dissipation fluctuations averaged over a volume of O(r 3 ). That is, the vector velocity increment  [ u(r)] at separation vector r is statistically related to the three-dimensional average of dissipation at scale |r| = r, r ≡ V (r) dV , with the "local" averaging performed over the volume V (r) ∼ O(r 3 ) [20,24], through the relation where the dimensionless velocity vector V(r) is independent of r as well as r in the inertial range, and hence universal [20]. It is convenient to rewrite Eq. (3) in terms of two scalar functions V L and V T as When p is odd, isotropy stipulates that V p T = 0. However, since V p L = 0 for odd p (>1), in the inertial range, it necessarily follows that the functions V L and V T will themselves be different (δu r is skewed and δv r is not) for any given r L. So the focus of universality is that V L and V T will both statistically asymptote to independent functions, though different from one another in the inertial range. The universality of V(r) is a direct consequence of the fact that the longitudinal and transverse structure functions have the same scaling in the inertial range. It then follows naturally that a rigorous examination of the universality of V L and V T serves as useful checks for the result that ζ p L = ζ p T at all even p, in the inertial range. There is a second point to be made. If we had followed the ideas of Ref. [8] that the transverse velocity is closely connected to enstrophy, the logical consequence would be to replace the locally averaged energy dissipation in Eq. (5) by the locally averaged enstrophy. However, it has been shown in Ref. [19] that the probability density functions (PDFs) of the energy dissipation and enstrophy are the same for large amplitudes at high Reynolds numbers (while this is not true at low Reynolds numbers considered in Ref. [8]). It thus appears logical to believe that the locally averaged dissipation ought to appear in both Eq. (4) and Eq. (5). Our results verify this assertion as well.
Past verifications of RSH have focused, largely for reasons of convenience, on longitudinal velocity increments [22,[25][26][27][28][29][30]. If r p ∼ r τ p in the inertial range, it follows from Eq. (4) that Now, because δu r and δv r both scale the same way (i.e., ζ p L and ζ p T are the same for even orders), the following relation between ζ p T and τ p/3 should also hold good for any even p in the inertial range, so we can write We first address the universality of V L and V T . In Fig. 6, we plot the PDF of V L and V T for r in the inertial range at R λ = 1300. The respective PDFs of V L and V T collapse for all separations in the inertial range indicating that they are independent of r and r , as expected from arguments of universality. The conclusion from Fig. 6 is that K62 holds for both longitudinal and transverse increments in the inertial range. As already mentioned, the main difference between the PDFs of V L and V T is that the former is negatively skewed with V 3 L = −4/5 [20], while the latter is symmetric around the origin with zero skewness. Figure 7(a) shows the joint probability density function (JPDF) of V L and V T at an inertial range separation for R λ = 1300; (b) and (c) of the figure show slivers of the JPDF to reveal two distinct cross sections.
It is easily seen that the PDFs of V L and V T do not have particularly extended tails, and are hence nonintermittent. It is thus far easier to work with the normalized velocity increments V L and V T statistically than with the velocity increments themselves which are highly intermittent and extremely sensitive to finite sampling effects [3,5].
Since it is now clear that V L and V T are both independent of r and r (and of each other), it follows from Eqs. (6) and  (7)] at R λ = 1300. Symbols ( ) and ( ) correspond to the longitudinal and transverse cases, respectively. If RSH is correct, the measured differences between ζ p L and τ p/3 and that between ζ p T and τ p/3 for various values of p must equal p/3, which is the dashed line. The correspondence is very good. Vertical bars indicate the extent of 95% confidence intervals (which are significant mainly at higher orders). (7), that ζ p L − τ p/3 and ζ p T − τ p/3 must both be equal to p/3. In Fig. 8 we verify this expectation from numerical results for p up to 10. The exponents τ p/3 of the three-dimensional local averages of dissipation are obtained from least-squares fits in the inertial range. With increasing p, both ζ p L − τ p/3 and ζ p T − τ p/3 coincide with p/3, showing that Eqs. (6) and (7) are valid.
A final comment is in order. In isotropic turbulence, the incompressibility condition says that second-order statistics are defined by a single scalar function. In fact, when we invoke RSH in the inertial range, it is easy to show that V 2 T / V 2 L = 4/3. We have indeed verified that this is quite accurately the case in the inertial range. Such relationships between higher-order moments of V L and V T are not straightforward, but computations suggest that they too approach, in the inertial range, constants whose numerical values increase with the order of the moment.

III. CONCLUSION
We have used DNS data at high Reynolds numbers to assess the past assertion that the longitudinal and transverse velocity structure functions scale differently in the inertial range. This difference was thought to be connected to the notion that the longitudinal velocity increments are connected to dissipation and the transverse velocity increments are connected to enstrophy, while the dissipation and enstrophy themselves have different PDFs. An implication is that the Kolmogorov RSH would have to be postulated independently, and differently, for the longitudinal and transverse velocity increments.
In Ref. [19], it has been shown, using the highest Reynolds numbers attained in the 8192 3 simulations, that dissipation and enstrophy essentially scale the same and that any differences seen in the past are due to lower finite Reynolds numbers of the past simulations. It is then natural to revisit the previous work in some detail.
Using the high Reynolds number simulations of [19], we have shown that, in isotropic turbulence at high enough Reynolds numbers, the longitudinal and transverse velocity increments scale the same. Furthermore, a single RSH theory, which connects the statistics of velocity increments to the locally averaged dissipation rate, subsumes both longitudinal and transverse cases. That is, a single scaling group, δu r and r , with a single set of independent exponents, is enough to describe intermittent physics in fluid turbulence in the inertial range of isotropic turbulence.
These findings suggest the possibility of a deeper connection between the symmetric part of the strain rate, i.e., the dissipation physics, and the antisymmetric part of the strain rate, i.e., vortex dynamics, than previously portended. It is possible, however, that this deeper connection is violated in shear flows. We recall that, among the past data showing differences between ζ p L and ζ p T was a very high Reynolds number measurement in the atmospheric boundary layer [7]. We believe that it is the remnant shear effect in that flow that may have introduced some nontrivial aspects that break the deeper connection just mentioned.