Development of high vorticity structures in incompressible 3D Euler equations

Abstract

We perform the systematic numerical study of high vorticity structures that develop in the 3D incompressible Euler equations from generic large-scale initial conditions. We observe that a multitude of high vorticity structures appear in the form of thin vorticity sheets (pancakes). Our analysis reveals the self-similarity of the pancakes evolution, which is governed by two different exponents $e^{-t/T_{\ell }}$ and $e^{t/T_{\omega }}$ describing compression in the transverse direction and the vorticity growth respectively, with the universal ratio $T_{\ell }/T_{\omega }\approx 2/3$. We relate development of these structures to the gradual formation of the Kolmogorov energy spectrum $E_k\propto k^{-5/3}$, which we observe in a fully inviscid system. With the spectral analysis we demonstrate that the energy transfer to small scales is performed through the pancake structures, which accumulate in the Kolmogorov interval of scales and evolve according to the scaling law ${\omega }_{\max }\propto {\ell }^{-2/3}$ for the local vorticity maximums ${\omega }_{\max }$ and the transverse pancake scales $\ell$.

31 pages, 18 figures

References in corpus (4)

• Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods
• Incompressible Euler Equations: the blow-up problem and related results
• Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics
• Renormalization group formalism for incompressible Euler equations and the blowup problem

Cited by in corpus (13)

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• Development of high vorticity structures and geometrical properties of the vortex line representation
• Development of high vorticity in incompressible 3D Euler equations: influence of initial conditions
• Slipping flows and their breaking
• Compressible vortex structures and their role in the onset of hydrodynamic turbulence
• Lagrangian dynamics and regularity of the spin Euler equation
• Optimal renormalization of multi-scale systems
• Notes about collapse in magnetohydrodynamics
• Statistical properties of the velocity field for the 3D hydrodynamic turbulence onset
• Stability of tangential discontinuity for the vortex pancakes