Adjoint wall functions: a new concept for use in aerodynamic shape optimization.

*(English)*Zbl 1346.76059Summary: The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the \(k-\varepsilon \) turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the \(k-\varepsilon \) one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.

##### MSC:

76G25 | General aerodynamics and subsonic flows |

76N25 | Flow control and optimization for compressible fluids and gas dynamics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76F65 | Direct numerical and large eddy simulation of turbulence |

49J99 | Existence theories in calculus of variations and optimal control |

65K10 | Numerical optimization and variational techniques |

##### Keywords:

continuous adjoint method; sensitivity derivatives; optimization; Navier-Stokes equations; turbulence; wall functions
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\textit{A. S. Zymaris} et al., J. Comput. Phys. 229, No. 13, 5228--5245 (2010; Zbl 1346.76059)

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