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

TL;DR

This paper investigates how the moments of the energy-dissipation surrogate $\epsilon_s = 15 \nu (\partial_1 u_1)^2$ relate to the true dissipation $\epsilon$ in homogeneous isotropic turbulence, focusing on even moments of the longitudinal velocity derivative. The authors express $\langle (S_{11})^{2n} \rangle$ as a sum of terms $\langle T_2^{m} T_3^{2p} \rangle$ with $m+3p=n$, and determine the coefficients $A^{(n)}_{m,p}$ by exploiting eigenvalue decompositions of the rate-of-strain tensor and angular averaging; they obtain an exact closed form for the leading coefficient $A^{(n)}_{n,0}$ and provide explicit values for higher-order mixed terms up to $n=8$. They show that, in general, moments of $\epsilon_s$ involve contributions from $T_3$ and its powers, which implies that simple proportional relations between $\langle \epsilon^n \rangle$ and $\langle \epsilon_s^n \rangle$ (as proposed by Boschung 2015) are not exact; a Gaussian/uniform-$\mathcal{R}$ assumption yields the expected $1/18$ (and $1/180$) weights for certain mixed terms, but DNS data reveal small deviations due to a biased eigenvalue distribution of the strain tensor. Despite these deviations, the Gaussian-based approximations remain accurate to within a few percent, making them useful in modeling, while the work clarifies the role of $\mathcal{R}$-driven corrections in intermittency and dissipation statistics.

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, $ε$: $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.

Figures (2)

• Figure 1: The distribution of (a) $\mathcal{R}$ and (b) $|\mathcal{R}|$, conditioned on several values of $\mathcal{T}_2/\langle \mathcal{T}_2 \rangle$, for the runs described in Appendix \ref{['appendixC']} (see Table \ref{['tab:dns']}), the convention chosen being such that the color becomes brigher when $R_\lambda$ increases. At very small values of $\mathcal{T}_2/\langle \mathcal{T}_2 \rangle$, the distribution of $\mathcal{R}$ is almost uniform, but increasingly deviates from a uniform distribution when $\mathcal{T}_2 /\langle \mathcal{T}_2 \rangle$ becomes larger. The values of $\mathcal{T}_2 /\langle \mathcal{T}_2 \rangle$ indicated on the legend were obtained by averaging over all values in the interval $1/2^{4}$ and $2^{4}$ around the nominal value.
• Figure 2: The ratio $\langle \mathcal{T}_2^3 \rangle / \langle \mathcal{T}_3^2 \rangle$ at several values of $R_\lambda$, corresponding to the runs listed in Appendix \ref{['appendixC']} (see Table \ref{['tab:dns']}).