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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. For the purpose of finding out, whether a more sophisticated approach is necessary or not, the adjoint equations are derived by using two different approaches. One set of equations uses the frozen turbulence assumption as shown in [1]. Differences between present derivations and those given in [1] are provided in this paper. This is further extended by including a turbulence model following [2]. The results show differences between the two approaches. In some cases, even the sign of the shape gradients may be different in turbulent flow regions. The second and more advanced approach, which includes adjoint turbulence model equations, reaches a similar or better objective value. Therefore, there are benefits in solving these additional equations despite increasing computation time.
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