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 infrared effect. Therefore, the extracted scaling exponents $\zeta(n)$ 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 $1$st-order polynomial fitting within a window size $\ell$ before calculating the velocity increment. By doing so, the scales larger than $\ell$, i.e., $r\ge \ell$, are expected to be removed or constrained. The detrending process is equivalent to be a high-pass filter in physical domain. Meanwhile the intermittency nature is retained. We first validate the DSF method by using a synthesized fractional Brownian motion for mono-fractal processes and a lognormal process for multifractal random walk processes. The numerical results show comparable scaling exponents $\zeta(n)$ 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 $Re_{\lambda}\simeq 720$, the 3rd-order DSF demonstrates a clear inertial range with $\mathcal{B}_3(\ell)\simeq 4/5\epsilon \ell$ on the range $10<\ell/\eta<1000$, corresponding to a wavenumber range $0.001<k\eta<0.1$. This inertial range is consistent with the one predicted by the Fourier power spectrum. The directly measured scaling exponents $\zeta(n)$ (resp. singularity spectrum $D(h)$) agree very well with a lognormal model with an intermittent parameter $\mu=0.33$. Due to large-scale effects, the results provided by the SFs are biased.