Certain problems in the theory of developed hydrodynamical turbulence

Abstract

The classical phenomenology of turbulence, the scaling concept and universal behavior of the inertial range of fine scales of turbulence in the limit of very high Reynolds numbers, Re→∞, is briefly discussed. The profound topological properties of inviscid Euler flows are reviewed in relevant details. The topology of vorticity lines and streamlines is argued to be of central importance to both large- and small-scale turbulence in viscous flows. Thereby a helical fluctuation is regarded as a structural unit intrinsic to the nature of turbulence. The concept of turbulence as a hierarchy of unstable, quasi-Euler flows, or helical fluctuations (structures) naturally implicates the intrinsic existence of turbulent “coherent” structures, as well as the intermittence of small scales of turbulence. The possibility of upscale instability of 3D turbulent (3D inverse, “cascade”) flows with subsequent growth of helical structures is pointed out. The maxima of the energy rate dissipation, nonlinear coupling and enstrophy (vorticity square) as a result of topological constraints caused by helicity fluctuations, are argued to be inevitably confined to fractal sets of dimensions less than three. Comparison with experimental data on turbulent “coherent” structures in laboratory and geophysical flows, and with large-scale numerical simulations is discussed. This implicity favors the relevance of the concept. In particular, probably all “coherent” structures possess a significant coherent helicity, or consist of helical cells with the opposite signs of helicity. The generality and universility of the effect of alignment of velocity and vorticity vectors is further elaborated by numerical simulations of isotropic homogeneous turbulence. The phenomenology of fractal models of turbulence is reviewed. The relevance of hyperbolic, multifractal models, as most probably realized in turbulent flows as Re→∞, is discussed. A relation with the helical concept is noted. The path integral method as a dynamical approach to turbulence, together with the fractal phenomenology, are utilized to derive a model dynamical equation for the nonintermittent averaged component of the velocity field. The resulting equation, related idealogically to the Reynolds stress equation, contains a linear term turbulent viscosity, a nonlinear term turbulent viscosity (modified nonlinear coupling), and a stirring force through the inertial range. The model equation, compared with the Reynolds equation, is closed by the properties of the fractal model. The model equation can be solved asymptotically in the limit of small scales and times and simultaneously Re→∞, by means of the perturbative modified renormalization group method. The results converge to all orders of the perturbation theory. This is in distinct contrast with the original Navier-Stokes equation, where the perturbation methods, including the perturbation renormalization group approach, and closures of any kind, are bound to fail, as they are intrinsically incapable of describing the inhomogeneity of turbulence and intermittence. When path integral and the model dynamical equation are solved for the hyperbolic fractal model, the results are compared with the latest available experimental data on the high-order velocity structure functions, up to the eighteenth order. The theoretical and experimental figures appear to be in striking proximity to each other. Among other predictions of the theory are: a nonmonotonous, weak divergence of skewness as a function of Re, the presence of oscillating in space nonanalytical corrections to the velocity structure functions, the propagating nature of topological fluctuations, etc. The alignment of velocity and vorticity vectors in turbulent flows is theoretically justified by the developed dynamical approach in conjunction with the hyperbolic fractal properties. It is also shown that the helicity fluctuations are subsequent to the basic process of stretching of vorticity lines and their breaks and reconnections by viscous effects. It appears that the generation of enstrophy is marginally dominant at the fractal. On the other hand, the average energy dissipation is dominated by the bulk of turbulence, while the fluctuations of the rate of energy dissipation are determined by its absolute maximum at the fractal. The predictions are tested against the experimental results of recent numerical simulations and available laboratory data. Further numerical and analytical studies are suggested. In particular, a possibility to utilize the above theoretical deductions with respect to the fine structure of turbulence in order to derive a model equation adequately describing the large-scale inhomogeneous flows (Large Eddy Simulation) is briefly discussed.