Turbulence on a Fractal Fourier Set

Abstract

A novel investigation of the nature of intermittency in incompressible, homogeneous, and isotropic turbulence is performed by a numerical study of the Navier-Stokes equations constrained on a fractal Fourier set. The robustness of the energy transfer and of the vortex stretching mechanisms is tested by changing the fractal dimension D from the original three dimensional case to a strongly decimated system with D=2.5, where only about 3% of the Fourier modes interact. This is a unique methodology to probe the statistical properties of the turbulent energy cascade, without breaking any of the original symmetries of the equations. While the direct energy cascade persists, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set E(k)∼k(-5/3+3-D) explains the results. At small scales, the intermittency of the vorticity field is observed to be quasisingular as a function of the fractal mode reduction, leading to an almost Gaussian statistics already at D∼2.98. These effects must be connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism.

On the vortex dynamics in fractal Fourier turbulence

“…Finally, the idea of changing the "effective dimension" between D = 2 and D = 3 has been explored within shell models of turbulence, by modifying the conserved quantities of the system [24]. More recently, in [17], fractally Fourier decimated NavierStokes equations were studied for the first time in the range 2.5 ≤ D ≤ 3. Two main results emerged: (i) average Table 1.…”

“…In this paper, we propose to further investigate the relation between intermittency and vortex stretching by a novel approach to three dimensional turbulence. This consists in numerically solving the Navier-Stokes equations on a multiscale sub-set of Fourier modes (also dubbed Fourier skeleton), belonging to a fractal set of dimension a Contact author: a.lanotte@isac.cnr.it D ≤ 3 [16,17]. For D = 3, the original problem is recovered.…”

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

“…[9]). Subsequently this method was used in several other studies [10][11][12][13] to understand triadic interactions inter alia intermittency, equilibrium solutions and turbulence.…”

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

“…This approach allows us to decimate the number of triad interactions in Fourier space as a function of the wavenumbers involved as well as to consider the problem in non-integer, fractal dimensions D. In Ref. [20] the first results for a set of simulations of the decimated, three-dimensional (3d) Navier-Stokes equation have been reported with the intriguing conclusion that fractal Fourier decimation leads, rather quickly (i.e., for a very small reduction of the Fourier modes D 3), to vanishing intermittency.…”