Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations

Abstract

Incompressible 3D Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudo-spectral method in anisotropic grids of up to 972 × 2048 × 4096.

The new analytical solution of the 3D Navier-Stokes equation for compressible medium clarifies the sixth Millennium Prize problem

“…In [7], under formulating the problem of the existence of a solution of the 3D NS equation, it has been offered to impose restrictions to considering only cases of solutions with the zero divergence of the velocity field. Therewith in [7] noted is importance of treatment of those particular 3D flows, for which the effect of stretching of vortex filaments in a finite time may lead to a limitation on the existence of solutions of the NS equation in the small (im Kleinen) only.…”

“…Therewith in [7] noted is importance of treatment of those particular 3D flows, for which the effect of stretching of vortex filaments in a finite time may lead to a limitation on the existence of solutions of the NS equation in the small (im Kleinen) only. The reached conclusion that there are smooth divergent solutions of the 3D NS equation at the expense of considering even small viscosity bears witness to an admissibility of a positive solution of the problem of the existence of smooth divergent-free solutions on an unbounded interval of time as well.…”

“…Only weak nonstationary solutions describing, for example, dynamics and interactions between singular vortex objects in the two-dimensional (2D) and three-dimensional (3D) ideal incompressible medium are known [3,5,6]. At the same time, for the 3D ideal medium flows there are some conceptual ideas on a possibility of the existence of nonstationary solutions of the EulerHelmholtz (EH) equation only on an unbounded interval of time 0 ≤ t < t 0 (see [1,3,6,7] and the references given therein). This time value for incompressible medium is determined exclusively by the 3D effect of vortex filament stretching, which may lead to an explosive unbounded growth of the enstrophy (the integral of the squared a lot of interesting results, but at the same time led to a new, still unsolved, problem of the closure in the description of different moments of the vortex field, an approximated solution ofIntroduction 1.…”

“…The proper understanding of many processes in nature and engineering systems is closely connected with the existence of the fundamental and applied problem of turbulence, which remains unsolved for more than a century due to the absence of exact analytical nonstationary smooth vortex solutions of the Navier -Stokes (NS) equation. The development of the statistical approach to its solution gave vorticity over space) in finite time t 0 [1,3,6,7]. On the other hand, known are the exact stationary modes of flows of the viscous incompressible medium in the form of the Burgers and Sullivan [3] vortices for which this, potentially dangerous with respect to appearance of singularity, effect of the vortex filament stretching is accurately compensated by the effect of the viscosity.…”

Statistical Properties of the Velocity Field for the 3D Hydrodynamic Turbulence Onset

“…Kolmogorov's arguments are based on the assumptions of statistical homogeneity and isotropy of the flow and also locality of nonlinear interaction at the scales of the inertial interval. Then, the dynamics at these scales can be described by the Euler equations and the emergence of the Kolmogorov's relations may be expected before the viscous scales get excited [4][5][6][7].In particular, as we demonstrated in our previous papers [8,9], the power-law energy spectrum with close to Kolmogorov's scaling can be observed in a fully inviscid flow when its dynamics is dominated by the pancake-like high-vorticity structures [10][11][12]. Such structures generate strongly anisotropic vorticity field in the Fourier space, concentrated in "jets" extended in the directions perpendicular to the pancakes.…”