arXiv:2601.16436v1 Announce Type: new
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.