

Isotropization of twodimensional hydrodynamic turbulence
Abstract:
The direct cascade of twodimensional hydrodynamic turbulence in a square box with periodic boundary conditions along both coordinates in the frame of the solution of equation for vorticity are numerically investigated in the presence of pumping and viscouslike damping. Equation for vorticity was solved numerically using pseudospectral Fourier method, while integration in time was performed with the use of a hybrid RungeKutta/Crank?Nicholson thirdorder scheme. The initial conditions were chosen as random sets of Gaussian shape vortices with zero mean vorticity and randomly distributed over the entire domain. Simulations were performed at the Computer Center of the Novosibirsk State University (with the use of the NVIDIA CUDA technology). The spatial resolution was up to 16384x16384.
Numerical results are shown that the formation of a powerlaw dependence on wave number k in the Kraichnantype spectrum of turbulence formed owing quasishocks of vorticity is a very quick process. If at an early stage (at the time of order of the inverse of the pumping increment), the development of turbulence is about the same scenario as in the case of a freely decaying turbulence. Formed quasisingular distribution of divorticity, which in kspace correspond to jets  Fourier transforms of quasishocks, leading to a strong turbulence anisotropy. In the next much slower stage, the structure of quasishocks lines is complicated. The distances between quasishocks lines are reduced, and the spectrum becomes more isotropic. It is also important to note that the probability distribution function of vorticity at these times there is a formation of exponential tail at large arguments, which can be extrapolated as a linear dependence of vorticity in accordance with the theoretical predictions. The probability distribution function for divorticity also has two specific areas: the first  the distribution function is close to Poisson PDF, the second (large value of divorticity)  distribution function is exponential behavior with more distinct linear dependence than similar for vorticity. Both of these observations suggest that direct cascade of turbulence at large times loses anisotropy due to a tendency to breaking. In our opinion, there are at least two possible reasons of turbulence isotropization. The first reason may be related to the pumping area, where, in spite of the strong dissipation at low k, formed largescale vortices (it's some remains not killed until the end of inverse cascade), which, due to its rotation, contribute into the system of vorticity quasishocks additional stretching of divorticity lines, and on the other hand  makes the system of significant lines of the divorticity field more complicated. We observed that the isotropic spectrum is carried out by time, more than time of Kraichnantype enstrophy transferring, when pumping wave reaches the viscous region. As is known, the direct cascade of turbulence is a nonlocal (or rather  weakly nonlocal), that is accompanied by the appearance of logarithmic corrections to the Kraichnan spectrum. Locality of turbulence means that the main nonlinear interaction is interaction between scale of the same order. Interaction greatly different scales is strongly depressed. In this situation, both boundary to the inertial interval areas  pumping and viscous dissipation, as isotropic sources, in our opinion, are responsible for the turbulence isotropization of direct cascade. Authors
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