Abstract
Kolmogorov seems to have been the first to recognize that a two-equation model of turbulence might be used as the basis of turbulent flow prediction. Nowadays, a whole hierarchy of phenomenological two-equation models of turbulence is in place.
The structure of their governing equations is similar to the Navier-Stokes equations for incompressible fluids, the difference is that the viscosity is not constant but depends on two scalar quantities that measure the effect of turbulence: the average of the kinetic energy of velocity fluctuations (i.e. the turbulent energy) and the measure related to the length scales of turbulence. For these two scalar quantities two additional evolutionary convection-diffusion equations are added to the generalized Navier-Stokes system.
Although Kolmogorov's model has so far been almost unnoticed, it exhibits interesting features. First of all, in contrast to other two-equation models of turbulence, there is no source term in the equation for the frequency.
Consequently, nonhomogeneous Dirichlet boundary conditions for the quantities measuring the effect of turbulence are assigned to a part of the boundary. Second, the structure of the governing equations is such that one can find an "equivalent" reformulation of the equation for turbulent energy that eliminates the presence of the energy dissipation acting as the source in the original equation for turbulent energy and which is merely an L-1 quantity.
Third, the material coefficients such as the viscosity and turbulent diffusivities may degenerate, and thus the a priori control of the derivatives of the quantities involved is unclear. We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorov's two-equation model of turbulence.
The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stick-slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications. (C) 2019 Elsevier Ltd.
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