Hilbert-Huang transform is a method that has been introduced recently to decompose nonlinear, nonstationary time series into a sum of different modes, each one having a characteristic frequency. Here we show the first successful application of this approach to homogeneous turbulence time series. We associate each mode to dissipation, inertial range and integral scales. We then generalize this approach in order to characterize the scaling intermittency of turbulence in the inertial range, in an amplitude-frequency space. The new method is first validated using fractional Brownian motion simulations. We then obtain a 2D amplitude-frequency representation of the pdf of turbulent fluctuations with a scaling trend, and we show how multifractal exponents can be retrieved using this approach. We also find that the log-Poisson distribution fits the velocity amplitude pdf better than the lognormal distribution. p-6
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 Taylor's 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.
We report an experimental investigation of the longitudinal space-time cross-correlation function of the velocity field, C(r, τ ), in a cylindrical turbulent Rayleigh-Bénard convection cell using the particle image velocimetry (PIV) technique. We show that while the Taylor's frozen-flow hypothesis does not hold in turbulent thermal convection, the recent elliptic model advanced for turbulent shear flows (He & Zhang 2006) is valid for the present velocity field for all over the cell, i.e., the isocorrelation contours of the measured C(r, τ ) have a shape of elliptical curves and hence C(r, τ ) can be related to C(r E , 0) via r 2 E = (r − U τ ) 2 + V 2 τ 2 with U and V being two characteristic velocities. We further show that the fitted U is proportional to the mean velocity of the flow, but the values of V are larger than the theoretical predictions. Specifically, we focus on two representative regions in the cell: the region near the cell sidewall and the cell's central region. It is found that U and V are approximately the same near the sidewall, while U ≃ 0 at cell center.
International audienceWe consider here surf zone turbulence measurements, recorded in the Eastern English Channel using a sonic anemometer. In order to characterize the intermittent properties of their fluctuations at many time scales, we analyze the experimental time series using the Empirical Mode Decomposition (EMD) method. The series is decomposed into a sum of modes, each one narrow-banded, and we show that some modes are associated with the energy containing wave-breaking scales, and other modes are associated with small-scale intermittent fluctuations. We use the EMD approach in association with a newly developed method based on Hilbert spectral analysis, representing the probability density function in an amplitude-frequency space. We then characterize the fluctuations in a stochastic framework using a cumulant generating function for all scales, and compare the results obtained from direct and classical structure function analysis, to EMD-Hilbert spectral analysis results, showing that the former method saturates at large scales, whereas the latter method is more precise in its scale approach. These results show the strength of the new EMD-hilbert spectral analysis method for data presenting a strong forcing such as found in shallow water, wave dominated situations
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