Yves Gagne

Velocity probability density functions of high Reynolds number turbulence

Abstract This paper deals with the probability density function (PDF) of velocity differences between two points separated by distance r . Measurements of PDFs were made, for r lying in the inertial range, for two different flows: in a jet with R λ = 852 and in a wind tunnel with R λ = 2720. These PDFs have a characteristic, non-Gaussian, shape with “exponential” tails. Following Kolmogorovs general ideas of log-normality, a new model for the PDF is developed which contains two parameters determined by experiments. This empirical model agrees with the experimental results that the tails of the PDF deviate from a truly exponential behaviour, in particular for small r . In addition, the model leads to the general scaling law 〈( Δ ln e ) 2 〉 ∼ ( r / r 0 ) - β differenr from Kolmogorovs third hypothesis 〈( Δ ln e ) 2 〉 ∼ -μ ln( r / r 0 ) restricted to the inertial range only ( Δ ( x ) is x - 〈 x 〉). We develop also a formalism, based on an extremum principle, which is consistent with both the log-normality of e and the above mentioned power law. In this formalism, β can be interpreted as the codimension of dissipative structures and asymptotically varies as β = β 1 / ln Rλ .

Turbulent transport of material particles: an experimental study of finite size effects.

We present experimental Lagrangian statistics of finite sized, neutrally bouyant, particles transported in an isotropic turbulent flow. The particles diameter is varied over turbulent inertial scales. Finite size effects are shown not to be trivially related to velocity intermittency. The global shape of the particles acceleration probability density functions is not found to depend significantly on its size while the particles acceleration variance decreases as it becomes larger in quantitative agreement with the classical k(-7/3) scaling for the spectrum of Eulerian pressure fluctuations in the carrier flow.

Log-similarity for turbulent flows?

Abstract We critically examine a recently proposed representation for turbulent energy spectra (called here log-similarity) which is predicted both by the multifractal model and the variational approach of the small scale intermittency. Within this representation, spectra are homothetic in the log-log plot, the homothety factor depending on the Taylor scale based Reynolds number Rλ. Experimental results on energy spectra and velocity structure functions of order 2, 3 and 6 measured in different flows at several Rλ show that this factor is β(Rλ), a measurable exponent which characterises intermittency in the inertial range. Experimentally, β displays the existence of a transition in the neighbourhood of Rλ ≅ 400 and for Rλ tending to infinity, β behaves as 1/In(Rλ/R∗).

Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: Comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylors microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

Intense vortical structures in grid‐generated turbulence

This paper presents a set of experiments aimed at investigating the features and the statistical frequency of intense vortical structures (sometimes called ‘‘filaments’’, or ‘‘worms’’) as manifested by a migrating bubble technique in a mean shear free, homogeneous, isotropic, stationary turbulence generated by oscillating grids in a water tank for Rλ reaching up to 300. It is found that the nucleation of filaments at the surface of the walls of the tank, where boundary layers are liable to destabilize is much more frequent than in the homogeneous bulk of the tank where one filament is typically detected each hundred large scale turnover time. This distinction between the wall surface and the bulk activity, supplemented with the fact that the size of the filaments and their lifetime compare with the length and time‐scales of the largest structures of the flow leads us to formulate an elementary model explaining the origin and the geometrical features of these intense vortical structures in turbulent flows ...

Wavelet analysis of fully developed turbulence data and measurement of scaling exponents

Wavelet analysis can be used to measure directly the scaling exponents characterizing the local multifractal behaviour of the velocity field at inertial-range scales, without recourse to dissipation-type quantities. Preliminary results using data from the Modane S1 wind tunnel indicate that the most frequent exponents are close to the Kolmogorov 1941 value of 1/3. Violent rare events are however found with negative exponents of −0.1 or less.

Intermittency and Reynolds number

Hot wire measurements of longitudinal and transverse increments are performed in three different types of flows on a large range of Reynolds numbers (100≲Rλ≲3000). An improved technique based on cumulant expansion of velocity structure functions is used to estimate the spreading of the pdfs and to study their scaling properties in the inertial range. Thus, the rate of intermittency depth through the scales of flow, called here β(Rλ), is experimentally introduced, and it is shown that β(Rλ) has a universal behavior on a very large Reynolds numbers range.

Reynolds dependence of third-order velocity structure functions

We study the experimental dependence of the third-order velocity structure function on the Taylor based Reynolds number, obtained in different flow types over the range 72⩽Rλ⩽2260. As expected, when the Reynolds number is increasing, the third-order velocity structure functions (plotted in a compensated way) converge very slowly to a possible −4/5 plateau value according to the Kolmogorov 41 theory. Actually, each of these normalized third-order functions exhibits a maximum, at a scale close to the Taylor microscale λ. In this Brief Communication, we show that experimental data are in good agreement with the recent predictions of Qian and Lundgren. We also suggest that, from an experimental point of view, a log-similar plot suits very well to study carefully the behavior of the third-order velocity structure functions with the flow Reynolds number.

Second-order structure function in fully developed turbulence.

We relate the second-order structure function of a time series with the power spectrum of the original variable, taking an assumption of statistical stationarity. With this approach, we find that the structure function is strongly influenced by the large scales. The large-scale contribution and the contribution range are, respectively, 79% and 1.4 decades for a Kolmogorov -5/3 power spectrum. We show numerically that a single scale influence range, over smaller scales is about 2 decades. We argue that the structure function is not a good method to extract the scaling exponents when the data possess large energetic scales. An alternative methodology, the arbitrary order Hilbert spectral analysis which may constrain this influence within 0.3 decade, is proposed to characterize the scaling property directly in an amplitude-frequency space. An analysis of passive scalar (temperature) turbulence time series is presented to show the influence of large-scale structures in real turbulence and the efficiency of the Hilbert-based methodology. The corresponding scaling exponents ζ(θ)(q) provided by the Hilbert-based approach indicate that the passive scalar turbulence field may be less intermittent than what was previously believed.

Accurate estimation of third-order moments from turbulence measurements

Abstract. Politano and Pouquets law, a generalization of Kolmogorovs four-fifths law to incompressible MHD, makes it possible to measure the energy cascade rate in incompressible MHD turbulence by means of third-order moments. In hydrodynamics, accurate measurement of third-order moments requires large amounts of data because the probability distributions of velocity-differences are nearly symmetric and the third-order moments are relatively small. Measurements of the energy cascade rate in solar wind turbulence have recently been performed for the first time, but without careful consideration of the accuracy or statistical uncertainty of the required third-order moments. This paper investigates the statistical convergence of third-order moments as a function of the sample size N . It is shown that the accuracy of the third-moment || ) 3 > depends on the number of correlation lengths spanned by the data set and a method of estimating the statistical uncertainty of the third-moment is developed. The technique is illustrated using both wind tunnel data and solar wind data.