A general self-preservation analysis for decaying homogeneous isotropic turbulence

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)phi(r/l) where l is a scaling length, and c(t) and 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 eta, even when the Reynolds number is relatively small. On the other hand, the Taylor microscale 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 similar to (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 lambda becomes a relevant scaling length together with 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