Shell model of turbulent viscosity for the boundary layer

Abstract

The problem of numerical simulation of developed turbulent flows is usually reduced to the choice of one or another closure of the mean field equations. It is hardly possible to find a universal solution to this issue; nevertheless, the development of an approach based on general principles remains an active research topic. This paper proposes a model of turbulent viscosity described in terms of the characteristics of velocity field fluctuations calculated on the basis of shell models. These models reproduce correctly the scale distribution of the turbulent energy and the spectral energy fluxes for hydrodynamic flows of various physical nature. Shell models are constructed using symmetry properties and conservation laws of the complete system of equations, as well as an assumption on homogeneity and isotropy of turbulence. Phenomenological relations implying specific spectral laws are not involved. In the developed approach, we did an attempt to determine the turbulent viscosity, while maintaining the universality and flexibility of shell models. The resulting mathematical formulation is a set of models for large (mean field equation) and small (cascade model) scales, as well as closing relations. The model implements energy conjugation of variables of different scales, which provides a nonlinear relation of fields at different levels. Taking into account the influence of the mean field on the energy distribution of turbulent fluctuations is a distinctive feature of the proposed approach. Numerical solutions are obtained for the flow in a plane infinite channel at various Reynolds numbers. It is shown that the obtained results are consistent with modern concepts of the logarithmic profile of the velocity field in the boundary layer. The physical meaning of model parameters is substantiated. Asymptotic solutions that qualitatively correspond to the Prandtl model are found.