Derivative moments in turbulent shear flows

We propose a generalized perspective on the behavior of high-order derivative moments in turbulent shear flows by taking account of the roles of small-scale intermittency and mean shear, in addition to the Reynolds number. Two asymptotic regimes are discussed with respect to shear effects. By these means, some existing disagreements on the Reynolds number dependence of derivative moments can be explained. That odd-order moments of transverse velocity derivatives tend not vanish as expected from elementary scaling considerations does not necessarily imply that small-scale anisotropy persists at all Reynolds numbers.Comment: 11 pages, 7 Postscript figure

Local properties of extended self-similarity in 3D turbulence

Using a generalization of extended self-similarity we have studied local scaling properties of 3D turbulence in a direct numerical simulation. We have found that these properties are consistent with lognormal-like behavior of energy dissipation fluctuations with moderate amplitudes for space scales $r$ beginning from Kolmogorov length $\eta$ up to the largest scales, and in the whole range of the Reynolds numbers: $50 \leq R_{\lambda} \leq 459$. The locally determined intermittency exponent $\mu(r)$ varies with $r$; it has a maximum at scale $r=14 \eta$, independent of $R_{\lambda}$.Comment: 4 pages, 5 figure

Anisotropic Homogeneous Turbulence: hierarchy and intermittency of scaling exponents in the anisotropic sectors

We present the first measurements of anisotropic statistical fluctuations in perfectly homogeneous turbulent flows. We address both problems of intermittency in anisotropic sectors and hierarchical ordering of anisotropies on a direct numerical simulation of a three dimensional random Kolmogorov flow. We achieved an homogeneous and anisotropic statistical ensemble by randomly shifting the forcing phases. We observe high intermittency as a function of the order of the velocity correlation within each fixed anisotropic sector and a hierarchical organization of scaling exponents at fixed order of the velocity correlation at changing the anisotropic sector.Comment: 6 pages, 3 eps figure

Statistics of Dissipation and Enstrophy Induced by a Set of Burgers Vortices

Dissipation and enstropy statistics are calculated for an ensemble of modified Burgers vortices in equilibrium under uniform straining. Different best-fit, finite-range scaling exponents are found for locally-averaged dissipation and enstrophy, in agreement with existing numerical simulations and experiments. However, the ratios of dissipation and enstropy moments supported by axisymmetric vortices of any profile are finite. Therefore the asymptotic scaling exponents for dissipation and enstrophy induced by such vortices are equal in the limit of infinite Reynolds number.Comment: Revtex (4 pages) with 4 postscript figures included via psfi

Longitudinal Structure Functions in Decaying and Forced Turbulence

In order to reliably compute the longitudinal structure functions in decaying and forced turbulence, local isotropy is examined with the aid of the isotropic expression of the incompressible conditions for the second and third order structure functions. Furthermore, the Karman-Howarth-Kolmogorov relation is investigated to examine the effects of external forcing and temporally decreasing of the second order structure function. On the basis of these investigations, the scaling range and exponents $\zeta_n$ of the longitudinal structure functions are determined for decaying and forced turbulence with the aid of the extended-self-similarity (ESS) method. We find that $\zeta_n$'s are smaller, for $n \geq 4$, in decaying turbulence than in forced turbulence. The reasons for this discrepancy are discussed. Analysis of the local slopes of the structure functions is used to justify the ESS method.Comment: 15 pages, 16 figure

Dynamical equations for high-order structure functions, and a comparison of a mean field theory with experiments in three-dimensional turbulence

Two recent publications [V. Yakhot, Phys. Rev. E {\bf 63}, 026307, (2001) and R.J. Hill, J. Fluid Mech. {\bf 434}, 379, (2001)] derive, through two different approaches that have the Navier-Stokes equations as the common starting point, a set of steady-state dynamic equations for structure functions of arbitrary order in hydrodynamic turbulence. These equations are not closed. Yakhot proposed a "mean field theory" to close the equations for locally isotropic turbulence, and obtained scaling exponents of structure functions and an expression for the tails of the probability density function of transverse velocity increments. At high Reynolds numbers, we present some relevant experimental data on pressure and dissipation terms that are needed to provide closure, as well as on aspects predicted by the theory. Comparison between the theory and the data shows varying levels of agreement, and reveals gaps inherent to the implementation of the theory.Comment: 16 pages, 23 figure

More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models

Turbulence anisotropy and the SO(3) description

We study strongly turbulent windtunnel flows with controlled anisotropy. Using a recent formalism based on angular momentum and the irreducible representations of the SO(3) rotation group, we attempt to extract this anisotropy from the angular dependence of second-order structure functions. Our instrumentation allows a measurement of both the separation and the angle dependence of the structure function. In axisymmetric turbulence which has a weak anisotropy, this more extended information produces ambiguous results. In more strongly anisotropic shear turbulence, the SO(3) description enables one to find the anisotropy scaling exponent. The key quality of the SO(3) description is that structure functions are a mixture of algebraic functions of the scale with exponents ordered such that the contribution of anisotropies diminishes at small scales. However, we find that in third-order structure functions of homogeneous shear turbulence the anisotropic contribution is always large and of the same order of magnitude as the isotropic part. Our results concern the minimum instrumentation needed to determine the parameters of the SO(3) description, and raise several questions about its ability to describe the angle dependence of high-order structure functions