Relation between the moments of longitudinal velocity derivatives and of dissipation in turbulence
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arXiv
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2026
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| author | Yang, Ping-Fan Xu, Haitao Pumir, Alain |
| author_facet | Yang, Ping-Fan Xu, Haitao Pumir, Alain |
| contents | 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, $ε$: $ε_s = 15 ν(\partial_1 u_1)^2 $, where $ν$ is the fluid viscosity, in the sense that the averages of $ε$ and $ε_s$ are equal. We show here that the $n^{th}$ moments of the fluctuations $ε$ and $ε_s$, for $n > 2$, are not exactly proportional to each other, and that the expression for the moment $\langle ε_s^n \rangle$ for $ n \ge 3$ involves in addition to a term proportional to $\langle ε^n \rangle$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16436 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relation between the moments of longitudinal velocity derivatives and of dissipation in turbulence Yang, Ping-Fan Xu, Haitao Pumir, Alain Fluid Dynamics 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, $ε$: $ε_s = 15 ν(\partial_1 u_1)^2 $, where $ν$ is the fluid viscosity, in the sense that the averages of $ε$ and $ε_s$ are equal. We show here that the $n^{th}$ moments of the fluctuations $ε$ and $ε_s$, for $n > 2$, are not exactly proportional to each other, and that the expression for the moment $\langle ε_s^n \rangle$ for $ n \ge 3$ involves in addition to a term proportional to $\langle ε^n \rangle$, 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. |
| title | Relation between the moments of longitudinal velocity derivatives and of dissipation in turbulence |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2601.16436 |