Boundedness of the velocity derivative skewness in various turbulent flows

Abstract

The variation of $S$, the velocity derivative skewness, with the Taylor microscale Reynolds number $\mathit{Re}_{{\it\lambda}}$ is examined for different turbulent flows by considering the locally isotropic form of the transport equation for the mean energy dissipation rate $\overline{{\it\epsilon}}_{iso}$. In each flow, the equation can be expressed in the form $S+2G/\mathit{Re}_{{\it\lambda}}=C/\mathit{Re}_{{\it\lambda}}$, where $G$ is a non-dimensional rate of destruction of $\overline{{\it\epsilon}}_{iso}$ and $C$ is a flow-dependent constant. Since $2G/\mathit{Re}_{{\it\lambda}}$ is found to be very nearly constant for $\mathit{Re}_{{\it\lambda}}\geqslant 70$, $S$ should approach a universal constant when $\mathit{Re}_{{\it\lambda}}$ is sufficiently large, but the way this constant is approached is flow dependent. For example, the approach is slow in grid turbulence and rapid along the axis of a round jet. For all the flows considered, the approach is reasonably well supported by experimental and numerical data. The constancy of $S$ at large $\mathit{Re}_{{\it\lambda}}$ has obvious ramifications for small-scale turbulence research since it violates the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) but is consistent with the original similarity hypothesis (Kolmogorov, Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303).