Local multifractal thermodynamics of 3D turbulence

Using results of a direct numerical simulation (DNS) of 3D turbulence we show that the observed generalized scaling (i.e. scaling moments versus moments of different orders) is consistent with a lognormal-like distribution of turbulent energy dissipation fluctuations with moderate amplitudes for all space scales available in this DNS (beginning from the molecular viscosity scale up to largest ones). Local multifractal thermodynamics has been developed to interpret the data obtained using the generalized scaling, and a new interval of space scales with inverse cascade of generalized energy has been found between dissipative and inertial intervals of scales for sufficiently large values of the Reynolds number. Received 21 July 2000     

Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing

In this paper, we characterise the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence, restricted in a vertical plane, under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) coexists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate k^?3 scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent potential energy (PE) also approximately follows a k^?3 power-law and its flux is directed to small scales. For moderate stratification, there is no VSHF and the KE of the turbulent flow exhibits Bolgiano–Obukhov scaling that transitions from a shallow k^?11/5 form at large scales, to a steeper approximate k^?3 scaling at small scales. The entire range of scales shows a strong forward enstrophy flux, and interestingly, large (small) scales show an inverse (forward) KE flux. The PE flux in this regime is directed to small scales, and the PE spectrum is characterised by an approximate k^?1.64 scaling. Finally, for weak stratification, KE is transferred upscale and its spectrum closely follows a k^?2.5 scaling, while PE exhibits a forward transfer and its spectrum shows an approximate k^?1.6 power-law. For all stratification strengths, the total energy always flows from large to small scales and almost all the spectral indicies are well explained by accounting for the scale-dependent nature of the corresponding flux.     

Levels of complexity in turbulent time series for weakly and high Reynolds number

We use the detrended fluctuation analysis (DFA), the detrended cross correlation analysis (DCCA) and the magnitude and sign decomposition analysis to study the fluctuations in the turbulent time series and to probe long-term nonlinear levels of complexity in weakly and high turbulent flow. The DFA analysis indicate that there is a time scaling region in the fluctuation function, segregating regimes with different scaling exponents. We discuss that this time scaling region is related to inertial range in turbulent flows. The DCCA exponent implies the presence of power-law cross correlations. In addition, we conclude its multifractality for high Reynold’s number in inertial range. Further, we find that turbulent time series exhibit complex features by magnitude and sign scaling exponents.     

Multifractal dimension of Lagrangian turbulence

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than -1 is accounted for, the models are shown to agree well with the measured spectrum.     

Lagrangian velocity statistics in turbulent flows: effects of dissipation

We use the multifractal formalism to describe the effects of dissipation on Lagrangian velocity statistics in turbulent flows. We analyze high Reynolds number experiments and direct numerical simulation data. We show that this approach reproduces the shape evolution of velocity increment probability density functions from Gaussian to stretched exponentials as the time lag decreases from integral to dissipative time scales. A quantitative understanding of the departure from scaling exhibited by the magnitude cumulants, early in the inertial range, is obtained with a free parameter function D(h) which plays the role of the singularity spectrum in the asymptotic limit of infinite Reynolds number. We observe that numerical and experimental data are accurately described by a unique quadratic D(h) spectrum which is found to extend from h(min) approximately 0.18 to h(max) approximately 1.     

Statistics of three-dimensional lagrangian turbulence

We consider a superstatistical model for a Lagrangian tracer particle in a high-Reynolds-number turbulent flow. The analytical model predictions are in excellent agreement with recent experimental data for flow between counter-rotating disks. In particular, the predicted Lagrangian scaling exponents zeta_{j} agree well with the measured exponents reported in H. Xu et al. [Phys. Rev. Lett.10.1103/PhysRevLett.96.114503 96, 114503 (2006)]. The model also correctly predicts the shape of acceleration probability densities, correlation functions, statistical dependencies between components, and explains the fact that enstrophy lags behind dissipation.     

Topological evolution near the turbulent/non-turbulent interface in turbulent mixing layer

The topological evolution near the turbulent/non-turbulent interface (TNTI) in turbulent mixing layer is studied by means of statistical analysis of the invariants of velocity gradient tensor (VGT) based on direct numerical simulation data. The dynamics of topological evolution is investigated in terms of the source terms of the evolution equations for the invariants, including the pressure effect term, viscous effect term and interaction term among the invariants. It is found that the local topology of fluid particles at the TNTI evolves from non-focal region to focal region in the plane of the second (Q) and the third (R) invariants of the VGT. The topological evolution is mainly associated with the pressure effect term in the TNTI region. According to the analysis of the evolution of enstrophy and dissipation, the enstrophy increase and the dissipation decrease are revealed in the TNTI region, which are caused by viscous vorticity diffusion near the TNTI. A weak correlation between the strain rate and the rotation rate is found in the TNTI region which is related to the reduction of enstrophy production. The alignments between vorticity and strain near the TNTI are investigated and a strong alignment of the vorticity with the extensive eigenvector direction is identified in the TNTI region.     

The turbulent/nonturbulent interface in penetrative convection

The effect of buoyancy on the turbulent/nonturbulent interface (TNTI) and viscous superlayer are studied by performing direct numerical simulation of penetrative convection. In this flow, rising turbulent thermals alternate with unmixed fluid entrained from above, forming a TNTI between the turbulent and irrotational flow regions. We detect the TNTI using a broad range of enstrophy iso-levels, from the very low levels of the outer fringes of the turbulent flow region to high levels located in the turbulent flow region. We study the local entrainment velocity v_n by which the TNTI propagates outwards relative to the fluid flow while entraining unmixed fluid into the turbulent region. The relative entrainment velocity is decomposed into a viscous, an inertial and a baroclinic torque term, respectively. For low enstrophy levels we find a viscous superlayer (VSL) where viscous diffusion dominates, while inertial and baroclinic torque terms are small. It is only for higher iso-levels in the buffer region of the TNTI, which extends from the edge of the VSL to the threshold for which v_n = 0, that the inertial enstrophy production term plays a significant role. Penetrative convection does not feature a turbulent core where v_n > 0 (i.e. inward moving enstrophy isosurfaces) that has been previously identified in other entraining flows such as jets or gravity currents. Surprisingly, the baroclinic torque remains inactive throughout the whole range of enstrophy iso-levels. The smallness of the baroclinic torque against viscous effects in the TNTI is supported by a dimensional argument which predicts that at high Reynolds number the baroclinic torque term will be negligible.     

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.     

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)     

High-repetition rate measurements of temperature and thermal dissipation in a non-premixed turbulent jet flame

High-repetition rate laser Rayleigh scattering is used to study the temperature fluctuations, power spectra, gradients, and thermal dissipation rate characteristics of a non-premixed turbulent jet flame at a Reynolds number of 15,200. The radial temperature gradient is measured by a two-point technique, whereas the axial gradient is measured from the temperature time-series combined with Taylor’s hypothesis. The temperature power spectra along the jet centerline exhibit only a small inertial subrange, probably because of the low local Reynolds number (Re_δ ≈ 2000), although a larger inertial subrange is present in the spectra at off-centerline locations. Scaling the frequency by the estimated Batchelor frequency improves the collapse of the dissipation region of the spectra, but this collapse is not as good as is obtained in non-reacting jets. Probability density functions of the thermal dissipation are shown to deviate from lognormal in the low-dissipation portion of the distribution when only one component of the gradient is used. In contrast, nearly log-normal distributions are obtained along the centerline when both axial and radial components are included, even for locations where the axial gradient is not resolved. The thermal dissipation PDFs measured off the centerline deviate from log-normal owing to large-scale intermittency. At one-half the visible flame length, the radial profile of the mean thermal dissipation exhibits a peak off the centerline, whereas farther downstream the peak dissipation occurs on the centerline. The mean thermal dissipation on centerline is observed to increase linearly with downstream distance, reach a peak at the location of maximum mean centerline temperature, and then decrease for farther downstream locations. Many of these observed trends are not consistent with equivalent non-reacting turbulent jet measurements, and thus indicate the importance of understanding how heat release modifies the turbulence structure of jet flames.     

Path Lengths in Turbulence

By tracking tracer particles at high speeds and for long times, we study the geometric statistics of Lagrangian trajectories in an intensely turbulent laboratory flow. In particular, we consider the distinction between the displacement of particles from their initial positions and the total distance they travel. The difference of these two quantities shows power-law scaling in the inertial range. By comparing them with simulations of a chaotic but non-turbulent flow and a Lagrangian Stochastic model, we suggest that our results are a signature of turbulence.     

A practical method to experimentally evaluate the Hausdorff dimension: an alternative phase-transition-based methodology

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.     

Multi-scale properties of large eddy simulations: correlations between resolved-scale velocity-field increments and subgrid-scale quantities

We provide analytical and numerical results concerning multi-scale correlations between the resolved velocity field and the subgrid-scale (SGS) stress-tensor in large eddy simulations (LES). Following previous studies for Navier–Stokes equations, we derive the exact hierarchy of LES equations governing the spatio-temporal evolution of velocity structure functions of any order. The aim is to assess the influence of the subgrid model on the inertial range intermittency. We provide a series of predictions, within the multifractal theory, for the scaling of correlation involving the SGS stress and we compare them against numerical results from high-resolution Smagorinsky LES and from a-priori filtered data generated from direct numerical simulations (DNS). We find that LES data generally agree very well with filtered DNS results and with the multifractal prediction for all leading terms in the balance equations. Discrepancies are measured for some of the sub-leading terms involving cross-correlation between resolved velocity increments and the SGS tensor or the SGS energy transfer, suggesting that there must be room to improve the SGS modelisation to further extend the inertial range properties for any fixed LES resolution.     

A priori assessment of the subgrid scale viscous/scalar dissipation closures in compressible turbulence

An appraisal is made of several subgrid scale (SGS) viscous/scalar dissipation closures via a priori analysis of direct numerical simulation data in a temporally evolving compressible mixing layer. The effects of the filter width, the compressibility level and the Schmidt number are studied for several models. Based on the scaling of SGS kinetic energy, a new formulation for SGS viscous dissipation is proposed. This yields the best overall prediction of the SGS viscous dissipation within the inertial subrange. An SGS scalar dissipation model based on the proportionality of the turbulent time scale with the scalar mixing time scale also performs the best for the filter widths in the inertial subrange. Two dynamic methods are implemented for the determination of the model coefficients. The one based on the global equilibrium of dissipation and production is shown to be more satisfactory than the conventional dynamic model.     

A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.     

Statistical properties of visibility graph of energy dissipation rates in three-dimensional fully developed turbulence

We study the statistical properties of complex networks constructed from time series of energy dissipation rates in three-dimensional fully developed turbulence using the visibility algorithm. The degree distribution is found to have a power-law tail with the tail exponent α=3.0. The exponential relation between the number of the boxes N_B and the box size l_B based on the edge-covering box-counting method illustrates that the network is not self-similar, which is also confirmed by the hub-hub attraction according to the visibility algorithm. In addition, it is found that the skeleton of the visibility network exhibits excellent allometric scaling with the scaling exponent η=1.163±0.005.     

Scaling properties of the temperature field in convective turbulence

We report the scaling properties of temperature in turbulent convection in water. In the central region of the convection cell, we find that the peak frequency of the temperature dissipation spectra may be identified as the "Bolgiano frequency," with respect to which the temperature power spectra are universal functions; and that the usual inertial range is taken up entirely by the buoyancy subrange, so that a "high frequency" scaling subrange emerges only through an extended-self-similarity-type analysis. Moreover, the buoyancy subrange assumes the value of 2/5 predicted for the Bolgiano-Obukhov scaling only in the central region of the cell; in the mixing zone the exponent for the high frequency scaling exponent has a value of 2/3.     

A speculative study of 23-order fractional Laplacian modeling of turbulence: some thoughts and conjectures

This study makes the first attempt to use the 23-order fractional Laplacian modeling of Kolmogorov -53 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and molecular Brownian diffusivity are considered to be the bifractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 23-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Levy 23 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions.