Reappraisal of the velocity derivative flatness factor in various turbulent flows

Abstract

We first analytically show, starting with the Navier–Stokes equations, that the value of the derivative flatness is controlled by pressure diffusion of energy, viscous destructive effects and large-scale effects (decay and/or production). The latter two terms tend to zero when the Taylor-microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. We argue that the pressure-diffusion term should also tend to a constant at large $Re_{\unicode[STIX]{x1D706}}$. Available data for the velocity derivative flatness, $F$, in different turbulent flows are re-examined and interpreted in the light of the finite-Reynolds-number effect. It is found that $F$ can differ from flow to flow at moderate $Re_{\unicode[STIX]{x1D706}}$; for a given flow, $F$ may also depend on the initial conditions. The data for $F$ in various flows, e.g. along the axis in the far field of plane and circular jets, and grid turbulence, show that it approaches a constant, with a value slightly larger than 10, when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. This behaviour for $F$ is supported, at least qualitatively, by our analytical considerations. The constancy of $F$ at large $Re_{\unicode[STIX]{x1D706}}$ violates the refined similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) to account for the intermittency of the energy dissipation rate. It is not, however, inconsistent with Kolmogorov’s original similarity hypothesis (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), although we contend that the power-law relation $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ (Kolmogorov 1962), which is widely accepted in the literature, has in reality been almost invariably used to ‘model’ the finite-Reynolds-number effect for the laboratory data and has been strongly influenced by the weighting given to the atmospheric surface layer data. The inclusion of the latter data has misled previous investigations of how $F$ varies with $Re_{\unicode[STIX]{x1D706}}$.