Relation between the moments of longitudinal velocity derivatives and of dissipation in turbulence

TL;DR

This paper investigates the relationship between moments of velocity derivatives and energy dissipation in turbulence, revealing that higher moments involve additional strain tensor contributions beyond simple proportionality.

Contribution

It introduces a refined analysis of the moments of energy dissipation, accounting for strain tensor invariants, and compares theoretical assumptions with actual turbulence data.

Findings

Higher moments of dissipation are not simply proportional to velocity derivative moments.

Additional contributions involve the invariant of the strain tensor, tr(S^3).

The Gaussian distribution assumption for strain tensor invariants closely matches actual turbulence data.

Abstract

In homogeneous and isotropic turbulence, measurements of the longitudinal velocity derivative, $\partial_1 u_1$, make it possible to estimate a surrogate of the rate of energy dissipation per unit mass, $\epsilon$: $\epsilon_s = 15 \nu (\partial_1 u_1)^2 $, where $\nu$ is the fluid viscosity, in the sense that the averages of $\epsilon$ and $\epsilon_s$ are equal. We show here that the $n^{th}$ moments of the fluctuations $\epsilon$ and $\epsilon_s$, for $n > 2$, are not exactly proportional to each other, and that the expression for the moment $\langle \epsilon_s^n \rangle$ for $ n \ge 3$ involves in addition to a term proportional to $\langle \epsilon^n \rangle$, other contributions involving the invariant of the strain tensor, $\SSs$: ${\rm tr}( \SSs^3)$. The contribution of this term depends on the distribution of the dimensionless ratio $\mathcal{R} \equiv {\rm tr}(\SSs^3)/{\rm tr}(\SSs^2)^{3/2}$. We find, however, that the relation obtained by assuming that $\mathcal{R}$ is uniformly distributed in the interval $-1/\sqrt{6} \le \mathcal{R} \le 1/\sqrt{6}$, which is obtained when the matrix $\SSs$ has a Gaussian distribution, differs by no more than a few percents from the exact distribution.