Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations

Pavel I. Plotnikov; Jan Sokolowski

doi:10.3934/eect.2013.2.495

Article Contents

2013, Volume 2, Issue 3: 495-516. Doi: 10.3934/eect.2013.2.495

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Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations

Pavel I. Plotnikov^1, and
Jan Sokolowski^2,

1.
Lavryentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Lavryentyev pr. 15, Novosibirsk 630090
2.
Institut Élie Cartan Nancy, UMR7502 Université Lorraine, CNRS, INRIA, Laboratoire de Mathématiques, 54506 Vandoeuvre-lès-Nancy Cedex

Received: February 28, 2013

Revised: April 30, 2013

Published: August 2013

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Abstract

The flow around a rigid obstacle is governed by the compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain in two spatial dimensions with a compact obstacle in its interior. The flight of the airflow is characterized by the work shape functional, to be minimized over a family of admissible obstacles. The lift of the airfoil is a given function of temporal variable and should be maintain closed to the flight scenario. The continuity of the work functional with respect to the shape of obstacle in two spatial dimensions is shown for a wide class of admissible obstacles compact with respect to the Kuratowski-Mosco convergence.
The dependence of small perturbations of approximate solutions to the governing equations with respect to the boundary variations of obstacles is analyzed for the nonstationary state equation.

Keywords:

Mathematics Subject Classification: Primary: 35Q35, 49Q10; Secondary: 35A35, 35D30, 35Q30, 49J20, 76N10.

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