Dual scaling and the n-thirds law in grid turbulence

Abstract

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$, i.e. a scaling based on the Kolmogorov scales ($u_K$, $\eta$) and another based on ($u'$, $L$) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$. As $Re_\lambda$ increases, the scaling based on ($u'$, $L$) extends to increasingly smaller values of $r/L$ while the scaling based on ($u_K$, $\eta$) extends to increasingly larger values of $r/\eta$. The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$. The use of an empirical model for $\overline {{(\delta u)}^n}$, which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$, shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$. This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$. The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$.