Optimal shape determination of a body located in incompressible viscous fluid flow

The purpose of this study is to present a formulation and numerical results of a shape optimization of a body located in the incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of fluid forces is introduced. The performance function should be minimized satisfying the state equation. Based on the adjoint equation, the gradient of the performance function can be derived. The weighted gradient method is successfully utilized as the minimization algorithm. The bubble function finite element method originated by authors’ group is used for the spatial discretization. For the control variables, the coordinate of the body is employed. The Delaunay triangulation is employed for the discretization with smoothing technique for the gradient. At the moderate Reynolds number flow, the drag minimization of a wing shape body located in the unsteady Navier–Stokes flow is carried out. Starting from a circular cylinder, the streamline shape has been obtained, which is similar to the NACA wing shape. However, the shape shows front-edged and rear-round type. It is clarified that the Reynolds number Re = 250 flow corresponds to the critical state at which the minimum drag force shape changes from the front-edge shape to the front-round shape.