# Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach
## 4 Results
### Flow and Adjoint Field  
In some engineering applications, the adjoint field is used as a sensitivity map within the design process. For the design of complex geometries, like cars, the solution of the adjoint field offers the information where the surface should be changed to achieve a more aerodynamical efficient behavior. For this type of application, the adjoint field is only solved once and the surface is not adopted automatically within an optimization. However, it is also possible to compute gradients from the adjoint field, which then are used for optimization. This is also done in this work. To get an impression of the adjoint field, it is shown for the initial geometry at a cut through the blade, showing the airfoil geometry, at an exemplary radial position of 3 m of the span. In Figure 3, cuts through the blade are colored by the velocity (left) and adjoint velocity (right). In the left plot, typical flow around rotor blades can be seen. The stagnation point is below the leading edge, which shows a positive angle of attack. From the stagnation point onwards along the first part of the suction side, the flow is accelerating and finally the flow is slightly separated at the last third of the suction side. The adjoint velocity in the right figure shows an upstream movement, which is also expected from the theory. It is high along the suction side, where the flow gradients are also large. Then, the adjoint velocity moves upstream from the stagnation point on and follows a path around the angle of attack. The trailing edge of an airfoil is a discontinuity that impairs the convergence of the adjoint field. The flow is separated in the rear part of the blade, thus turbulent effects are dominant in this region. At this part of the blade, the frozen turbulence approach decreases the convergence behavior of the adjoint field. This can also be seen at the adjoint velocity plot, showing a peak at the trailing edge. A used workaround for this problem in literature is to exclude the cells close to discontinuities from the evaluated gradients [16]. This would imply an additional processing of the gradients. For this work, such an exclusion of certain blade areas is not done, as this could overlap with the effects of the chosen projection of gradients. Testing this projection was one of the main goals in this work, which is why overlapping effects are avoided. However, it could easily be added for further applications of the optimization tool using the existing utilities available in OpenFOAM. Figure 3: Magnitude (Mag) of the velocity \(U\) and adjoint velocity \(Ua\) around the blade at a cut at 3 m of the span. In order to show the flow around the blades in its rotational component, the Q-criterion in terms of the vorticity was used in Figure 4.  
The shown flow structures then are colored by the magnitude of the flow velocity and the adjoint velocity, respectively. The computations were performed for both the flow and adjoint velocity. As the adjoint field moves upstream, both views of the rotor are shown for the adjoint field. One view is in the downstream direction, according to the view for the flow field and one is in the upstream direction according to the propagation of the adjoint field. In Figure 4a, the tip and root vortex of the rotor can be seen, as well as low velocities close to the stagnation point and increasing velocities towards the tip of the blade. The tip and root vortex impair the flow field convergence at the root and tip area of the blade, respectively. The computed flow variables are less stable and reach inferior convergence in stationary simulations due to the instationary flow phenomena. As the gradients are evaluated along the movable part of the blade, which are according to Table 1 starting at 2 m spanwise and reach up to 4.75 m of the span, the influence of the vortices is supposed to be small for the gradient computation.