Asymmetry of velocity increments in turbulence

We use well-resolved direct numerical simulations of high-Reynolds-number turbulence to study a fundamental statistical property of turbulence—the asymmetry of velocity increments—with likely implications on important dynamics. This property, ignored by existing small-scale phenomenological models, manifests most prominently in the nonmonotonic trend of velocity increment moments (or structure functions) with the moment order, and in differences between ordinary and absolute structure functions for a given separation distance. We show that high-order structure functions arise nearly entirely from the negative side of the probability density of velocity increments, essentially removing the ambiguity between ordinary and absolute moments, and provide a plausible dynamical interpretation of this result.

Figure 1

Integrands of odd-order moments of the nondimensional longitudinal increment $V={\delta }_{r}u/{\left(r\epsilon \right)}^{1/3}$ for orders 3, 7, and 11 are shown in panels (a), (b), and (c), respectively, for inertial range separation $r/\eta =125$ from ${4096}^3$ DNS at $R_{\lambda }=650$ in a cube of edge-length $L_0$. The average energy dissipation rate $\epsilon$ equals the interscale energy transfer rate. Panel (d) is the ratio of contributions from the negative and positive parts of the PDF of ${\delta }_{r}u$, with increasing moment order. Panels (e), (f), and (g) are instantaneous line traces along the $x$ direction, contributing to moments $\langle V^3\rangle$, $\langle V^7\rangle$, and $\langle V^{11}\rangle$, respectively. The reference line at zero is given in all panels. Panel (h) shows the energy flux at three different scales in the inertial range, demonstrating that the large excursions are located around the same spatial positions at all inertial-range scales. This suggests the presence of coherent structures responsible for the large amplitudes of forward energy cascade.

Figure 2

Longitudinal structure functions $|\langle {\left({\delta }_{r}u\right)}^n\rangle {|}^{1/n}$ and $\langle |{\delta }_{r}u{{|}^n\rangle }^{1/n}$ as functions of order $n$ for a ${4096}^3$ cube, $R_{\lambda }=650$, with the scale-separation fixed at $r/\eta \approx 125$. Inset shows their ratio for odd orders $n=3,5,7$, and 9 as functions of the Taylor microscale Reynolds number $R_{\lambda }$. Error bars (too small to stand out) are the standard deviation of temporal variations of the moments in the statistically steady state.

Figure 3

(a) Scaling exponents ${\zeta }_n/n$ and ${\xi }_n/n$ for ordinary and absolute structure functions, as a function of order $n$, for the longitudinal case. As before, the error bars, too small to stand out, represent the standard deviation of local slopes in inertial-range power laws. (b) Ratio of the scaling exponents ${\zeta }_n$ of the longitudinal structure function and ${\zeta }_n^{-}$, which is the corresponding exponent of its negative part, for varying moment order, $n$. Horizontal line at unity is given for reference. Error bars represent the standard deviation of the ratio of local slopes of the respective inertial range power laws.

Figure 4

A typical instantaneous trace of the $x$-component velocity along the $x$ direction normalized by the root-mean-square velocity as a function of $x/L_0$, where $L_0$ is the box length (with 8192 grid points to a side), and $R_{\lambda }=1300$. The trace is dominated by steep negative cliffs (which contribute to ${\delta }_r{u}^{-}$) and gentler positive cliffs (which contribute to ${\delta }_r{u}^{+})$ connected by ramp-like structures. Inset (a) expands the region $x/L_0\in (0.2,0.3)$ (marked by the rectangle) in terms of $x/\eta$ to reveal the negative-cliff structure while inset (b) expands $x/L_0\in (0.70,0.72)$ (marked by the rectangle) in terms of $x/\eta$ to highlight the positive-cliff structure. Negative cliffs are consistently much deeper than the positive ones. Inset (c) shows the normalized velocity difference ${\delta }_{r}u=u(x+r)-u\left(x\right)$ for $r/\eta =175$ as a function of $x/L_0$. Horizontal dashed lines at zero is given for reference.