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 $\ell$.

arXiv comment31 pages, 18 figures

References in corpus (13)

• Computing Nearly Singular Solutions Using Pseudo-Spectral Methods
• Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows
• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• 3D Euler about a 2D Symmetry Plane
• Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods
• Transient vortex events in the initial value problem for turbulence
• Incompressible Euler Equations: the blow-up problem and related results
• Blowup as a driving mechanism of turbulence in shell models
• Renormalization and universality of blowup in hydrodynamic flows
• Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics
• Statistical features of freely decaying two-dimensional hydrodynamic turbulence
• Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations
• Renormalization group formalism for incompressible Euler equations and the blowup problem

Cited by in corpus (13)

• Spontaneously stochastic solutions in one-dimensional inviscid systems
• Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice
• Spontaneous stochasticity of velocity in turbulence models
• Asymptotic solution for high vorticity regions in incompressible 3D Euler equations
• 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