A general self-preservation analysis for decaying homogeneous isotropic turbulence

Abstract

A general framework of self-preservation (SP) is established, based on the transport equation of the second-order longitudinal velocity structure function in decaying homogeneous isotropic turbulence (HIT). The analysis introduces the skewness of the longitudinal velocity increment, $S(r,t)$ ($r$ and $t$ are space increment and time), as an SP controlling parameter. The present SP framework allows a critical appraisal of the specific assumptions that have been made in previous SP analyses. It is shown that SP is achieved when $S(r,t)$ varies in a self-similar manner, i.e. $S=c(t){\it\phi}(r/l)$ where $l$ is a scaling length, and $c(t)$ and ${\it\phi}(r/l)$ are dimensionless functions of time and $(r/l)$, respectively. When $c(t)$ is constant, $l$ can be identified with the Kolmogorov length scale ${\it\eta}$, even when the Reynolds number is relatively small. On the other hand, the Taylor microscale ${\it\lambda}$ is a relevant SP length scale only when certain conditions are met. The decay law for the turbulent kinetic energy ($k$) ensuing from the present SP is a generalization of the existing laws and can be expressed as $k\sim (t-t_{0})^{n}+B$, where $B$ is a constant representing the energy of the motions whose scales are excluded from the SP range of scales. When $B=0$, SP is achieved at all scales of motion and ${\it\lambda}$ becomes a relevant scaling length together with ${\it\eta}$. The analysis underlines the relation between the initial conditions and the power-law exponent $n$ and also provides a link between them. In particular, an expression relating $n$ to the initial values of the scaling length and velocity is developed. Finally, the present SP analysis is consistent with both experimental grid turbulence data and the eddy-damped quasi-normal Markovian numerical simulation of decaying HIT by Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53).