Detrended structure-function in fully developed turbulence

Abstract

The classical structure-function (SF) method in fully developed turbulence or for scaling processes in general is influenced by large-scale energetic structures, known as the infrared effect. Therefore, the extracted scaling exponents ζ(n) might be biased due to this effect. In this paper, a detrended structure-function (DSF) method is proposed to extract scaling exponents by constraining the influence of large-scale structures. This is accomplished by removing a first-order polynomial fitting within a window size ℓ before calculating the velocity increment. By doing so, the scales larger than ℓ, i.e. r ≥ ℓ, are expected to be removed or constrained. The detrending process is equivalent to a high-pass filter in a physical domain. Meanwhile, the intermittency nature is retained. We first validate the DSF method by using a synthesised fractional Brownian motion for mono-fractal processes and a lognormal process for multifractal random walk processes. The numerical results show comparable scaling exponents ζ(n) and singularity spectra D(h) for the original SFs and DSFs. When applying the DSF to a turbulent velocity obtained from a high Reynolds number wind tunnel experiment with Re_λ ≃ 720, the third-order DSF demonstrates a clear inertial range with on the range 10 < ℓ/η < 1000, corresponding to a wavenumber range 0.001 < kη < 0.1. This inertial range is consistent with the one predicted by the Fourier power spectrum. The directly measured scaling exponents ζ(n) (respectively, singularity spectrum D(h)) agree very well with a lognormal model with an intermittent parameter μ = 0.33. Due to large-scale effects, the results provided by the SFs are biased. The method proposed here is general and can be applied to different dynamics systems in which the concepts of multiscaling and multifractal are relevant.

Keywords:

• fully developed turbulence
• intermittency
• detrended structure-function

Acknowledgements

Y. Huang thanks Prof. F.G. Schmitt for useful comments and suggestions. We thank Prof. Meneveau for sharing his experimental velocity database, which is available for download at C. Meneveau’s web page: http://www.me.jhu.edu/meneveau/datasets.html. We thank the two anonymous referees for their useful comments and suggestions.

Additional information

Funding

This work is sponsored by the National Natural Science Foundation of China [grant number 11072139], [grant number 11032007], [grant number 11161160554], [grant number 11272196], [grant number 11202122], [grant number 11332006]; ‘Pu Jiang’ Project of Shanghai [grant number 12PJ1403500]; Innovative Program of Shanghai Municipal Education Commission [grant number 11ZZ87]; the Shanghai Program for Innovative Research Team in Universities.