Shape Optimization by an Adjoint Solver based on a near-wall Turbulence Model

The aim of this paper is to present the adjoint equations for shape optimization derived from steady incompressible Navier-Stokes (N-S) equations and an objective functional. These adjoint Navier-Stokes equations have a similar form as the N-S equations, while the source terms and the boundary conditions depend on the chosen objective. Additionally, the gradient of the targeted objective with respect to the design variables is calculated. Based on this, a modification of the geometry is computed to arrive at an improved objective value. In order to find out, whether a more sophisticated approach is needed, the adjoint equations are derived by using two different approaches. The first approach is based on the frozen turbulence assumption and the second approach, which is advanced in this paper, is derived from the near wall k − ζ − f turbulence model. Furthermore, it is important to note that the adjoint method based on the k − ε turbulence model usually requires the use of adjoint wall functions. Hence, the work performed here proposes a direct integration of primal and adjoint k − ζ − f equations towards the wall as a final target instead of using the previous approach, which requires adjoint wall functions. A derivation of the adjoint equations based on the k − ζ − f model is given in the paper. Preliminary results show that there are benefits in solving these additional equations despite increased computation time.