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Impact of the variations of the mixing length in a first order turbulent closure system. (English) Zbl 1040.35057

The authors study the turbulent circulation model. The equations are derived from Navier-Stokes turbulent kinetic energy system. Related results can be found in [T. Gallouët and R. Herbin, Appl. Math. Lett. 17, 49–55 (1994; Zbl 0791.35043) and R. Lewandowski, Nonlinear Anal., Theory Methods Appl. 28, 393-417 (1997; Zbl 0863.35077)]. A very simplified example of turbulence model is studied, but the attention is focused on the nonlinearities linked to the turbulent eddy viscosity \(\nu_t\). The mixing length \(\ell\) acts as a parameter which controls the turbulent part in \(\nu_t\). The main theoretical results obtained concern the uniqueness of the solution for bounded eddy viscosities and for small values of \(\ell\). A convergence theorem is proved for the turbulent kinematic energy, and the numerical results allow the conjecture that this classical turbulence model obtained with one degree of closure regularizes the solution.

MSC:

35Q30 Navier-Stokes equations
76F65 Direct numerical and large eddy simulation of turbulence
76M10 Finite element methods applied to problems in fluid mechanics
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References:

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