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Zbl 1078.76553
Iliescu, T.; Layton, W. J.
Approximating the larger eddies in fluid motion. III: The Boussinesq model for turbulent fluctuations. (English)
[J] An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 44, No. 2, 245-261 (1998). ISSN 1221-8421; ISSN 0041-9109/e

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. {\it G. P. Galdi} and {\it 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 2000:: *76F65 Direct numerical and large eddy simulation of turbulence
76M20 Finite difference methods
76M10 Finite element methods

Keywords: turbulent viscosity coefficient; existence; weak solutions

Citations: Zbl 1077.76522; Zbl 1042.76537

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