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

Abstract

We study the statistical correlation functions for the three-dimensional hydrodynamic turbulence onset when the dynamics is dominated by the pancake-like high-vorticity structures. With extensive numerical simulations, we systematically examine the two-points structure functions (moments) of velocity. We observe formation of the power-law scaling for both the longitudinal and the transversal moments in the same interval of scales as for the energy spectrum. The scaling exponents for the velocity structure functions demonstrate the same key properties as for the stationary turbulence case. In particular, the exponents depend on the order of the moment non-trivially, indicating the intermittency and the anomalous scaling, and the longitudinal exponents turn out to be slightly larger than the transversal ones. When the energy spectrum has power-law scaling close to the Kolmogorov's one, the longitudinal third-order moment shows close to linear scaling with the distance, in line with the Kolmogorov's 4/5-law despite the strong anisotropy. PACS numbers: 47.27.Cn, 47.27.De, 47.27.ek 1. Despite great practical importance, there are only a few exact results known for turbulence theory. The basic result is the Kolmogorov's 4/5-law [1-3], which for the inertial interval of scales r is written aswhere δv is the longitudinal variation of velocity, ε is the mean energy dissipation in unit mass, and ... denotes ensemble-averaging. Using dimensional analysis, Kolmogorov also found relations for the second-order structure functions, δv 2 ∝ ε 2/3 r 2/3 , and the energy spectrum, E k ∝ ε 2/3 k −5/3 . 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. These jets, occupying only a small fraction of the entire spectral space, dominate in the energy spectrum, leading to formation of the power-law interval E k ∝ k −α with the exponent α close to 5/3 and expanding with time to smaller scales. Moreover, the power-law scaling extends significantly longer * dmitrij@itp.ac.ru if the emerging jets align close to the same direction, increasing the anisotropy of the flow.In this paper we continue these studies and present numerical evidence that, despite the strong anisotropy, the 4/5-law may also be satisfied before the viscous scales get excited. With numerical simulations of the threedimensional Eule...

Compressible vortex structures and their role in the onset of hydrodynamic turbulence

“…The inset shows the spectrum in semi-logarithmic scales. [39] transverse moments of order n are calculated as the corresponding integral sums over all points on the sphere r and all nodes x,…”

Stability of tangential discontinuity for the vortex pancakes

“…which would seem to indicate a double exponential amplification of the perturbation. However, in numerical experiments [10][11][12][13]15] we have not observed this type of instability.…”

“…Second, in the numerical experiments [10][11][12][13]15] we have not seen any characteristic signs of the KH instability. In the simulations, we have used the pseudospectral Runge-Kutta fourth-order method, in which all the spatial derivatives are calculated via the fast Fourier transform, and the adaptive grid, which resolves optimally the perpendicular direction of the main vorticity pancake and is adjusted automatically based on the analysis of the Fourier spectrum of the solution [10].…”

“…The number of pancakes increases with time, and they provide the leading contribution to the energy spectrum. In particular, for some initial conditions [11,15], we have observed formation of the Kolmogorov spectrum E k ∝ k −5/3 and the powerlaw scalings for the longitudinal and transverse structure functions of the velocity, in a fully inviscid flow. Taking into account the exponential growth of the vorticity maximum and the exponential decrease of the pancake thickness, the maximum increment of the KH instability for the pancake structure should be characterized by the exponential time dependency,…”

“…The similar results have been obtained for other pancake structures developing from the initial flow I 1 . Numerical experiments performed in [13] for the initial flow I 2 , as well as in [11,15] for several dozens of random initial flows in grids limited by 1024 3 nodes, have not revealed any signs of the KH instability as well.…”