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

Abstract : The flow around a rigid obstacle is governed by compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain 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 given in function of time and should be closed to the flight scenario. Therefore, the minimization for a given lift of the work functional with respect to the shape of obstacle in two spatial dimensions is considered. The shape optimization problems for the compressible Navier-Stokes equations with the nonhomogeneous Dirichlet conditions in a bounded domain with an obstacle are considered in the monograph (Plotnikov, Sokolowski, Springer, 2012) in the general case of three spatial dimensions. In the present paper, for the purposes of wellposednes of shape optimization problems, there is no restrictions on the regularity of admissible obstacles, which are simply connected compact subsets of a given, bounded hold-all domain. The presented results are derived in the general framework established in (Plotnikov, Sokolowski, Springer, 2012). It means that in two spatial dimensions we obtain the same optimal shape existence result for the compressible Navier-Stokes equations as it is for the Laplacian (Sverak, JMPA, 1993). However, the complete existence proof is substantially more complex compared to the Laplacian, and it is given with all details in (Plotnikov, Sokolowski, Springer, 2012) for the general case of three spatial dimensions. In the present paper the general theory of shape optimization developed in (Plotnikov, Sokolowski, Springer, 2012) is adapted to the particular case of two spatial dimensions. The shape optimization problem of work minimization over a class of admissible obstacles is introduced. The continuity of the work functional with respect to the obstacle in two spatial dimensions is shown for a wide class of admissible obstacles. The dependence of local solutions to the governing equations with respect to the boundary variations of obstacles is analyzed. The shape derivatives (Sokolowski, Zolesio, Springer, 1992) of solutions to the compressible Navier-Stokes equations are derived. The shape gradient (Sokolowski, Zolesio, Springer, 1992) of the work functional is obtained. The framework for numerical methods of shape optimization (Plotnikov, Sokolowski, Zochowski, MMAR'09, 2009) is established for nonstationary, compressible Navier-Stokes equations.
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[Research Report] Université de lorraine. 2013
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Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. [Research Report] Université de lorraine. 2013. 〈hal-00807319〉

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