High order Lagrangian velocity statistics in turbulence

We report measurements of the Lagrangian velocity structure functions of orders 1 through 10 in a high Reynolds number (Taylor microscale Reynolds numbers of up to R(lambda) = 815 ) turbulence experiment. Passive tracer particles are tracked optically in three dimensions and in time, and velocities are calculated from the particle tracks. The structure function anomalous scaling exponents are measured both directly and using extended self-similarity and are found to be more intermittent than their Eulerian counterparts. Classical Kolmogorov inertial range scaling is also found for all structure function orders at times that trend downward as the order increases. The temporal shift of this classical scaling behavior is observed to saturate as the structure function order increases at times shorter than the Kolmogorov time scale.     

Pressure spectrum in homogeneous turbulence

The pressure spectrum in homogeneous steady turbulence is studied using direct numerical simulation with resolution up to 1024(3) and the Reynolds number R(lambda) between 38 and 478. The energy spectrum is found to have a finite inertial range with the Kolmogorov constant K = 1.65+/-0.05 followed by a bump at large wave numbers. The pressure spectrum in the inertial range is found to be approximately P(k) = B(p)epsilon;(4/3)k(-7/3) with B(p) = 8.0+/-0.5, and followed by a bump of nearly k(-5/3) at higher wave numbers. Universality and a new scaling of the pressure spectrum are discussed.     

Warm cascades and anomalous scaling in a diffusion model of turbulence

A phenomenological turbulence model in which the energy spectrum obeys a nonlinear diffusion equation is analyzed. The general steady state contains a nonlinear mixture of the constant-flux Kolmogorov and fluxless thermodynamic components. Such "warm cascade" solutions describe a bottleneck phenomenon of spectrum stagnation near the dissipative scale. Transient self-similar solutions describing a finite-time formation of steady cascades are analyzed and found to exhibit nontrivial scaling behavior.     

间歇湍流的分数阶动力学

间歇湍流意味着湍流涡旋并不充满空间,其维数介于2和3之间.湍流扩散为超扩散,且概率密度分布具有长尾特征.本文将流体力学的Navier-Stokes(NS)方程中的黏性项用分数阶的拉普拉斯算子表达.分析表明,分数阶拉普拉斯的阶数α和间歇湍流的维数D相联系.对于均匀各向同性的Kolmogorov湍流α=2,即用整数阶NS方程描述.而对于间歇性湍流,一定用分数阶的NS方程来描述.对于Kolmogorov湍流,扩散方差正比于t3,即Richardson扩散.而对于间歇性湍流,扩散方差要比Richardson扩散更强.     

Imprint of large-scale flows on turbulence

We investigate the locality of interactions in hydrodynamic turbulence using data from a direct numerical simulation on a grid of 1024(3) points; the flow is forced with the Taylor-Green vortex. An inertial range for the energy is obtained in which the flux is constant and the spectrum follows an approximate Kolmogorov law. Nonlinear triadic interactions are dominated by their nonlocal components, involving widely separated scales. The resulting nonlinear transfer itself is local at each scale but the step in the energy cascade is independent of that scale and directly related to the integral scale of the flow. Interactions with large scales represent 20% of the total energy flux. Possible explanations for the deviation from self-similar models, the link between these findings and intermittency, and their consequences for modeling of turbulent flows are briefly discussed.     

A shell-model approach to fractal-induced turbulence

We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension exists below which the average rate of inter-scale energy transfer <T _n> is a decreasing function of the wavenumber k_n and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, <T _n> is an increasing function of k_n and the structure function scaling exponents deviate significantly from Kolmogorov values. Received 25 June 2001 / Received in final form 5 April 2002 Published online 19 July 2002     

Inertial clustering of particles in high-Reynolds-number turbulence

We report experimental evidence of spatial clustering of dense particles in homogenous, isotropic turbulence at high Reynolds numbers. The dissipation-scale clustering becomes stronger as the Stokes number increases and is found to exhibit similarity with respect to the droplet Stokes number over a range of experimental conditions (particle diameter and turbulent energy dissipation rate). These findings are in qualitative agreement with recent theoretical and computational studies of inertial particle clustering in turbulence. Because of the large Reynolds numbers a broad scaling range of particle clustering due to turbulent mixing is present, and the inertial clustering can clearly be distinguished from that due to mixing of fluid particles.     

On the role of pressure in elasto-inertial turbulence

The dynamics of elasto-inertial turbulence is investigated numerically from the perspective of the coupling between polymer dynamics and flow structures. In particular, direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 are used to study the formation and dynamics of elastic instabilities and their effects on the flow. Based on the splitting of the pressure into inertial and polymeric contributions, it is shown that the polymeric pressure is a non-negligible component of the total pressure fluctuations, although the rapid inertial part dominates. Unlike Newtonian flows, the slow inertial part is almost negligible in elasto-inertial turbulence. Statistics on the different terms of the Reynolds stress transport equation also illustrate the energy transfers between polymers and turbulence and the redistributive role of pressure. Finally, the trains of cylindrical structures around sheets of high polymer extension that are characteristics of elasto-inertial turbulence are shown to be correlated with the polymeric pressure fluctuations.     

Compressible turbulence: the cascade and its locality

We prove that interscale transfer of kinetic energy in compressible turbulence is dominated by local interactions. In particular, our results preclude direct transfer of kinetic energy from large-scales to dissipation scales, such as into shocks, in high Reynolds number turbulence as is commonly believed. Our assumptions on the scaling of structure functions are weak and enjoy compelling empirical support. Under a stronger assumption on pressure dilatation cospectrum, we show that mean kinetic and internal energy budgets statistically decouple beyond a transitional conversion range. Our analysis establishes the existence of an ensuing inertial range over which mean subgrid scale kinetic energy flux becomes constant, independent of scale. Over this inertial range, mean kinetic energy cascades locally and in a conservative fashion despite not being an invariant.     

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X(Sα(t)), the subordinator Sα(t) is termed as the inverse-time α-stable subordinator and the process X(τ) satisfies the corresponding time homogeneous Ito stochastic differential equation.     

On the scaling of three-dimensional homogeneous and isotropic turbulence

In this paper we investigate the scaling properties of three-dimensional isotropic and homogeneous turbulence. We analyze a new form of scaling (extended self-similarity) recently introduced in the literature. We found that anomalous scaling of the velocity structure functions is clearly detectable even at a moderate and low Reynolds number and it extends over a much wider range of scales with respect to the inertial range.     

Effect of symmetry on volume conserving surface models

We study the effect of symmetry on volume conserving models without deposition and evaporation. By using the master equation approach, we identify two types of stochastic continuum equation with a conservative noise, depending on the symmetry of hopping rate in diffusion rules. In the model with symmetric hopping rate, a Laplacian term is essentially absent from the continuum equation. The dynamic scaling of this model is thus determined by the nonlinear fourth order equation with a conservative noise. When the symmetry is broken, a Laplacian term may be present, so the asymptotic scaling behavior is governed by the Laplacian term with nonzero coefficient. We verify this result by investigating a simple discrete model analytically.     

Geometry and scaling laws of excursion and iso-sets of enstrophy and dissipation in isotropic turbulence

Motivated by interest in the geometry of high intensity events of turbulent flows, we examine the spatial correlation functions of sets where turbulent events are particularly intense. These sets are defined using indicator functions on excursion and iso-value sets. Their geometric scaling properties are analysed by examining possible power-law decay of their radial correlation function. We apply the analysis to enstrophy, dissipation and velocity gradient invariants Q and R and their joint spatial distributions, using data from a direct numerical simulation of isotropic turbulence at Re_λ ≈ 430. While no fractal scaling is found in the inertial range using box-counting in the finite Reynolds number flow considered here, power-law scaling in the inertial range is found in the radial correlation functions. Thus, a geometric characterisation in terms of these sets’ correlation dimension is possible. Strong dependence on the enstrophy and dissipation threshold is found, consistent with multifractal behaviour. Nevertheless, the lack of scaling of the box-counting analysis precludes direct quantitative comparisons with earlier work based on multifractal formalism. Surprising trends, such as a lower correlation dimension for strong dissipation events compared to strong enstrophy events, are observed and interpreted in terms of spatial coherence of vortices in the flow.     

Local coupling of shell models leads to anomalous scaling

We consider cascade models of turbulence which are obtained by restricting the Navier-Stokes equation to local interactions. By combining the results of the method of extended self-similarity and a novel subgrid model, we investigate the inertial range fluctuations of the cascade. Significant corrections to the classical scaling exponents are found. The dynamics of our local Navier-Stokes models is described accurately by a simple set of Langevin equations proposed earlier as a model of turbulence [20]. This allows for a prediction of the intermittency exponents without adjustable parameters. Excellent agreement with numerical simulations is found.     

The maximal randomness principle in d-dimensional turbulence

A set of self-consistent equations in one-loop approximation in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the maximal randomness principle of the incompressible velocity field with stationary energy spectral flux, is obtained. Thanks to the applied principle the model statistics becomes essentially non Gaussian. The set of equations does not possess the infrared and ultraviolet divergences near the obtained Kolmogorov spectral exponents. The solution of these equations leads to the Kolmogorov exponents, but its amplitude proportional to the Kolmogorov constantC _k is negative for Euclidean dimensiond=3. Systematic investigation is made of (inertial) steady state scaling solutions for dimensions 2<d<2.55695,where constantC _k (d) becomes positive. Considered in this way, the model stability is discussed in the context of widely studied fractal aspects of turbulence.We have greatly benefited from discussions with Dr. Altaisky from JINR Dubna. The authors (M.H. and M.S.) are grateful to D. I. Kazakov and to director D. V. Shirkov for hospitality at the Laboratory of Theoretical Physics, JINR, Dubna.This work was supported by Fundamental Research Russian Fund, International Scientific Fund (grant R-63000) and by Slovak Grant Agency for Science (grant 2/550/93).     

Nonuniversality of Critical Exponents in a Fractional Quenched Kardar–Parisi–Zhang Equation

In this article we address the problem of the depinning transition for driven interfaces in random media. We introduce a fractional Kardar–Parisi–Zhang equation with quenched noise, in which the normal diffusion term is replaced by a fractional Laplacian accounting for long-range interactions through quenched disorder. The critical values of the external driving force and nonlinear term coefficient evidently depend on the system size at the depinning transition. For a fixed value of the external driving force, the fractional order much determines the value of the nonlinear term coefficient that leads to a depinned interface. Near the depinning threshold, the critical exponent obtained numerically is nonuniversal, and weakly depends on the fractional order.     

Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross-Pitaevskii equation with a small-scale dissipation

The energy spectrum of superfluid turbulence is studied numerically by solving the Gross-Pitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law.     

A new assessment of the second-order moment of Lagrangian velocity increments in turbulence

The behaviour of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov 4/5-law into a relation predicting the linear (in time) scaling for the second-order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterisation of the Lagrangian structure functions based on Batchelor model is introduced and tested on data for 3d turbulence, and for 2d turbulence in the inverse cascade regime. Such parameterisation supports the idea, previously suggested, that both Eulerian and Lagrangian data are consistent with a linear scaling plus finite-Reynolds number effects affecting the small- and large timescales. When large-time saturation effects are properly accounted for, compensated plots show a detectable plateau already at the available Reynolds number. Furthermore, this parameterisation allows us to make quantitative predictions on the Reynolds number value for which Lagrangian structure functions are expected to display a scaling region. Finally, we show that this is also sufficient to predict the anomalous dependency of the normalised root mean squared acceleration as a function of the Reynolds number, without fitting parameters.     

Spectral slope and Kolmogorov constant of MHD turbulence

The spectral slope of strong MHD turbulence has recently been a matter of controversy. While the Goldreich-Sridhar model predicts a -5/3 slope, shallower slopes have been observed in numerics. We argue that earlier numerics were affected by driving due to a diffuse locality of energy transfer. Our highest-resolution simulation (3072(2)×1024) exhibited the asymptotic -5/3 scaling. We also discover that the dynamic alignment, proposed in models with -3/2 slope, saturates and cannot modify the asymptotic, high Reynolds number slope. From the observed -5/3 scaling we measure the Kolmogorov constant C(KA)=3.27±0.07 for Alfvénic turbulence and C(K)=4.2±0.2 for full MHD turbulence, which is higher than the hydrodynamic value of 1.64. This larger C(K) indicates inefficient energy transfer in MHD turbulence, which is in agreement with diffuse locality.     

Experimental study on the local similarity scaling of the turbulence spectrum in the turbulent boundary layer

The streamwise fluctuating velocity in the turbulent boundary layer is measured under approximately medium Reynolds Number by hot wire in order to investigate the scaling properties of the overlapped turbulent spectrum among energy-containing area, inertial subrange and dissipation range based on FFT analysis. The experiment indicates that the high Reynolds flow reported before is not indispensable to produce −1 scaling. So far as the measured position is provided with much higher spatial resolution and enough closing to the wall, −1 scaling is determinate to exist when approaching medium Reynolds. The scaling ranges are supposed to begin at inner scale and end in outer scale, which reveals the local similarity of the energy spectrum over the energy-containing eddies near the wall. In the logarithmic area (y ⁺ > 130), −5/3 scaling occurs in the energy spectrum, while moving away from the wall with Reynolds numbers increasing, the inertial subrange extends to the lower wavenumbers. On the condition k ₁ η ≫ 0.1, the curves of the turbulence spectrum in the logarithmic layer are superposed, which expresses the similarity of turbulence energy distributed in Komogorov scaling area and exhibits local isotropy characteristics by virtue of the viscous dissipation. Supported by the National Natural Science Foundation of China (Grant Nos. 10832001 and 10872145), the Program for New Century Excellent Talents in Universities of Education Ministry of China, and the Plan of Tianjin Science and Technology Development (Grant No. 06TXTJJC13800)