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Approximating the larger eddies in fluid motion. III: The Boussinesq model for turbulent fluctuations. (English) Zbl 1078.76553
Summary: In 1877 Boussinesq (and others) put forward the basic analogy between the mixing effects of turbulent fluctuations and molecular diffusion: \(-\nabla\cdot\overline{(u'u')}\sim -\nabla\cdot\bigl(\nu_{\text{T}}(\nabla\bar{u}+\nabla\bar{u}^t)\bigr)\). This assumption lies at the heart of essentially all turbulence models and subgridscale models. By revisiting the original arguments of Boussinesq, Saint-Venant, Kelvin, Reynolds and others, we give three new approximations for the turbulent viscosity coefficient \(\nu_T\) in terms of the mean flow based on approximation for the distribution of kinetic energy in \(u'\) in terms of the mean flow \(\bar{u}\). We prove existence of weak solutions for the resulting system (NSE plus the proposed subgridscale term). Finite difference implementations of the new eddy viscosity/subgrid-scale model are transparent. We show how it can be implemented in finite element procedures and prove that its action is no larger than that of the popular Smagorinski-subgrid-scale model.
Part II, cf. G. P. Galdi and W. J. Layton, Math. Models Methods Appl. Sci. 10, No. 3, 343–350 (2000; Zbl 1077.76522). Further parts have been reviewed in (V) Zbl 1042.76537.

76F65 Direct numerical and large eddy simulation of turbulence
76M20 Finite difference methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics