Refined similarity hypothesis using 3D local averages

The refined similarity hypotheses of Kolmogorov, regarded as an important ingredient of intermittent turbulence, has been tested in the past using one-dimensional data and plausible surrogates of energy dissipation. We employ data from direct numerical simulations, at the microscale Reynolds number $R_λ\sim 650$, on a periodic box of $4096^3$ grid points to test the hypotheses using 3D averages. In particular, we study the small-scale properties of the stochastic variable $V = Δu(r)/(r ε_r)^{1/3}$, where $Δu(r)$ is the longitudinal velocity increment and $ε_r$ is the dissipation rate averaged over a three-dimensional volume of linear size $r$. We show that $V$ is universal in the inertial subrange. In the dissipation range, the statistics of $V$ are shown to depend solely on a local Reynolds number.

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