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