Far Dissipation Range of Turbulence

Abstract

The very small scales of isotropic, Navier-Stokes turbulence at Reynolds number ${\mathcal{R}}_{\lambda }\approx 15$ are studied by high-resolution direct numerical simulation (DNS) and by integration of the direct-interaction (DIA) equations. The DNS follows the tail of the energy spectrum over more than thirty decades of magnitude. The energy spectrum in the far-dissipation range $5k_d<k<10k_d$ is well-fitted by $k^{\alpha }\exp (-ck/k_d)$, where $k_d$ is the Kolmogorov dissipation wavenumber, $\alpha \approx 3.3$ and $c\approx 7.1$. For values of $m$ that emphasize the far-dissipation range, the fields $(-{\nabla }^2{)}^m\mathbf{u}$ exhibit strong spatial intermittency, associated with gentle spatial variations of the lower-$k$ part of the velocity field. DIA analysis gives a prefactor $k^3$ and an exponential decay more rapid than DNS. Averaging over an ensemble of DIA solutions, suggested by the observed intermittency, removes some of the discrepancy.

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