Theoretical and Numerical Local Null Controllability of a Ladyzhenskaya–Smagorinsky Model of Turbulence

This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove...

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Bibliographic Details
Published in:Journal of mathematical fluid mechanics 2015-12, Vol.17 (4), p.669-698
Main Authors: Fernández-Cara, E., Límaco, J., de Menezes, S. B.
Format: Article
Language:English
Subjects:
Classical and Continuum Physics
Fluid- and Aerodynamics
Mathematical Methods in Physics
Physics
Physics and Astronomy
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Summary:This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, with distributed controls locally supported in space. The proof relies on rather well known arguments. However, some specific difficulties are found here because of the occurrence of nonlocal nonlinear terms. We also present an iterative algorithm of the quasi-Newton kind that provides a sequence of states and controls that converge towards a solution to the control problem. Finally, we give the details of a numerical approximation and we illustrate the behavior of the algorithm with a numerical experiment.
ISSN:1422-6928
1422-6952
DOI:10.1007/s00021-015-0232-7