Hydrodynamics of Turbulent Flows

Modeling of turbulence in the presence of large structures implies sharing the Euler equations (for process of description of large-scale dynamics [1-3]) and Navier–Stokes type equations (to describe the high-frequency background surrounding large structures). The main energy of the turbulent flow contain large vortices; its also determine the structure of the flow. Distribution vortices of this type does not reflect the random structure of perturbations, but corresponds to the physical the laws of hydrodynamics, when the inertial terms in the Navier–Stokes equations predominate over the stresses, due to viscosity. Then the pair of forces arising from the pressure field and dynamic forces, associated with the velocity field creates a vortex structure. birth process large eddies and the flow dynamics after that should be described by the Euler equations.

However, the Euler equations in the numerical implementation of such a description require a significant modification of the form, including, in particular, taking into account the numerical viscosity based on phenomenological assumptions (including empirical coefficients). This actually means ignoring the physical mechanisms of turbulence. by introducing artificial mathematical assumptions.

Therefore, it is necessary to consider the possibilities of describing the conditions implementation of vortex structures in a hydrodynamic flow from the point of view of molecular-kinetic approach based on equations of the Boltzmann type and more general, obtained with certain simplifications of the form of correlation functions from the BBGKI chain. This approach helps to establish genesis conditions and coherent (large- and meso-scale) structures in a liquid, and vortex high-frequency regimes for large values of the Reynolds parameter. Many authors have turned to the kinetic construction of algorithms and numerical simulation based on them fluid dynamics. We especially note the papers [4-5], where the question of describing turbulence with the inclusion of additional members of the BBGKY chain is touched upon; papers [6-7], in which the direct solution the Boltzmann equations, vortex structures and small-scale turbulence are studied; papers [8-9], where the Taylor–Couette problem was studied based on formalism of kinetic equations.

In this paper, author analyzes the properties of the stationary Boltzmann-type kinetic equation for the possibility of existence of its solutions of a special form (associated with closed cycles on the manifold of system states). These solutions can be correlated with vortex structures in a fluid. The analysis uses the Lyapunov–Schmidt formalism and corollaries theory of branching of solutions of systems of differential equations. The results of this analysis can be applied to describe the appearance of coherent meso- and macrostructures in shear currents.

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