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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: $-\nabla ·\overline{\left({u}^{\text{'}}{u}^{\text{'}}\right)}\sim -\nabla ·\left({\nu }_{\text{T}}\left(\nabla \overline{u}+\nabla {\overline{u}}^{t}\right)\right)$. 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 ${u}^{\text{'}}$ in terms of the mean flow $\overline{u}$. 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.

##### MSC:
 76F65 Direct numerical and large eddy simulation of turbulence 76M20 Finite difference methods (fluid mechanics) 76M10 Finite element methods (fluid mechanics)
##### Keywords:
turbulent viscosity coefficient; existence; weak solutions