Development of high vorticity structures and geometrical properties of the vortex line representation

Abstract

The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling ${\omega }_{max}\sim {\ell }^{-2/3}$ between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D.S. Agafontsev et al, Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes' development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with respect to conservation of the Cauchy invariants and describes compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to $1536^3$ nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling ${\omega }_{max}\sim {\ell }^{-2/3}$.

30 pages, 6 figures

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• From two-dimensional to three-dimensional turbulence through two-dimensional three-component flows
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Cited by in corpus (8)

• Statistical Equilibrium of Circulating Fluids
• Clebsch Confinement and Instantons in Turbulence
• A matching theory to characterize sound emission during vortex reconnection in quantum fluids
• Capillary Flotation in a System of Two Immiscible Bose-Einstein Condensates
• Vortex Sheet Turbulence as Solvable String Theory
• Compressible vortex structures and their role in the onset of hydrodynamic turbulence
• Statistical properties of the velocity field for the 3D hydrodynamic turbulence onset
• Stability of tangential discontinuity for the vortex pancakes