General Methodologies for Incompressible Flow Design Problems

Two design methodologies based on Incomplete-Gradient adjoint approaches for ow problems governed by the incompressible Navier-Stokes (NS) equations are presented. The main features of the algorithms is that they avoid solving the adjoint equations, saving an important amount of CPU time. Furthermore, the methodologies are general in the sense that they do not depend on the geometry representation, and all the gridpoints on the surface to be optimized can be chosen as design parameters. The partial derivatives of the ow equations with respect to the design parameters are computed by nite di erences. In this way, this computation is independent of the numerical scheme employed to obtain the ow solution and the type of mesh. Once the sensitivities and the direction of movement have been computed, the new solid surface is obtained with a pseudo-shell approach in such a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Furthermore, this surface parametrization allows to impose the problem geometrical restrictions in a very easy manner. The volume mesh is updated to x the new boundary using an innovative level approach, which allows to compute the sensitivity contribution of the interior mesh points by using nite di erences in a very fast manner. The methodologies can deal with multi-objective function problems, and ow restrictions such as constant lift, etc. Some 2D and 3D numerical examples are shown.