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, $\epsilon$: ${\epsilon }_s=15\nu {\left({\partial }_1{u}_1\right)}^2$, where $\nu$ is the fluid viscosity, in the sense that the averages of $\epsilon$ and ${\epsilon }_s$ are equal. We show here that the $n\text{th}$ 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, $\mathbf{S}$: $\mathrm{tr}\left({\mathbf{S}}^3\right)$. The contribution of this term depends on the distribution of the dimensionless ratio $\mathcal{R}\equiv \mathrm{tr}\left({\mathbf{S}}^3\right)/\mathrm{tr}{\left({\mathbf{S}}^2\right)}^{3/2}$. 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 $\mathbf{S}$ has a Gaussian distribution, differs by no more than a few percent from the exact distribution.

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• Homogeneous turbulence
• Isotropic turbulence
• Three-dimensional turbulence
• Turbulence
• Turbulence modeling