Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence

Abstract

Visual manifestations of intermittency in computations of three dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension $D=1,2,3$ and, in the light of the occurrence of thin sets, to discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

arXiv comment1 figure containing 2 sub-figures

References in corpus (10)

• The multifractal nature of turbulent energy dissipation
• Inverse energy cascade in three-dimensional isotropic turbulence
• Turbulence in non-integer dimensions by fractal Fourier decimation
• Turbulence on a Fractal Fourier set
• Lagrangian Statistics for Navier-Stokes Turbulence under Fourier-mode reduction: Fractal and Homogeneous Decimations
• Fully developed turbulence and the multifractal conjecture
• Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers Equation
• On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations
• On the Vortex Dynamics in Fractal Fourier Turbulence
• Weak and strong solutions of the $3D$ Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales

Cited by in corpus (6)

• Extreme dissipation and intermittency in turbulence at very high Reynolds numbers
• A correspondence between the multifractal model of turbulence and the Navier-Stokes equations
• Identifying the multifractal set on which energy dissipates in a turbulent Navier-Stokes fluid
• How close are shell models to the 3D Navier-Stokes equations?
• Phase transitions in the fractional three-dimensional Navier-Stokes equations
• An analytical and computational study of the incompressible Toner-Tu Equations