# MFE

## Seminar

### Rescaled vorticity moments in the 3D Navier-Stokes equations

Gibbon, J (Imperial College London)
Monday 02 December 2013, 10:15-11:00

Seminar Room 1, Newton Institute

#### Abstract

Co-authors: D. D. Donzis (Texas A and M), A. Gupta (University of Rome Tor Vergata), R. M. Kerr (University of Warwick), R. Pandit (Indian Institute of Science Bangalore), D. Vincenzi (CNRS, Universite de Nice)

The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations is addressed using a new set of variables whose evolution has been calculated through three sets of numerical simulations. These variables are defined on a periodic box $[0,\,L]^{3}$ such that $D_{m}(t) = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}$ where $\alpha_{m}= 2m/(4m-3)$ \& the set of frequencies $\Omega_{m}$ for $1 \leq m \leq \infty$ are defined by $\Omega_{m}(t) = \left(L^{-3}\I |\mbox{\boldmath$\omega$}|^{2m}dV\right)^{1/2m}$\,; the fixed frequency $\varpi_{0} = \nu L^{-2}$. All three simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1} < D_{m}$. Moreover, the $D_{m}$ squeeze together such that $D_{m+1}/D_{m}\nearrow 1$ as $m$ increases. This regime is shown to connected to the depletion of nonlinearity. The first simulation is of very anisotropic decaying turbulence\,; the second pair is of decaying isotropic turbulence from random initial conditions \& of forced isotropic turbulence at constant Grashof number\,; the third $4096^{3}$ simulation is of very high Reynolds number forced, stationary, isotropic turbulence.

#### Video

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