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A Mixed Adjoint Formulation for Incompressible Turbulent Problems

A design methodology based on a mixed adjoint approach for flow problems governed by the Incompressible Turbulent Navier Stokes equations is deduced and tested. The main feature of the algorithm is that instead of solving an exact discrete adjoint equation, it solves a fastconverging low-order adjoint formulation, saving an important amount of CPU time, and giving a smoothed approximation to the real gradient. It has been shown that this type of smoothed gradients is very convenient to avoid possible diverging cycles in the whole design process, and to reduce the total optimization cost. The boundary conditions for the discrete adjoint formulation are inferred at the continuous level. In this way, the formulation is mixed. Furthermore, the methodology is general in the sense that it does 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 flow equations with respect to the mesh movements are computed by finite differences. Hence, this computation is independent of the numerical scheme employed to obtain the flow solution and of the mesh type. Once the sensitivities and the direction of movement have been computed, the new solid surface is obtained with an improved pseudo-shell approach in such a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Moreover, this surface parametrization allows to impose geometrical restrictions in a very easy manner. The volume mesh is updated to fix the new boundary using an innovative level approach for the highly stretched elements close to the solid boundary (boundary layer mesh), and a quasi-incompressible elastic movement scheme for the rest of them. Such type of combined mesh movement algorithm allows to compute the sensitivity contribution of the interior mesh points by using finite differences in a very fast manner, and avoids expensive remeshing procedures during the whole design process. The methodologies can deal with multi-objective function problems. Some numerical examples are presented to demonstrate the methodology behaviour.