Adjoint system method in shape optimization of some typical fluid flow patterns

In this paper a shape optimization approach based on the Hadamard geometric optimization method is developed. Four case studies representing typical fluid flow patterns in fluid dynamics, i.e. flow around an obstacle, flow in a 90. or 180. elbow pipe and flow in a dyadic tree, are considered. Low velocities are imposed at the inlet of each case study in order to operate in laminar flow regime. The objective is to determine the shape that minimizes the energy dissipated by viscous friction subjected to the Navier-Stokes equations and to iso-volumic constraint. The required gradients of the performance index and constraint with respect to the shape are computed by means of the adjoint system method. The momentum equations are implemented and solved using the OpenFOAM CFD software, and the solver ''adjointShapeOptimizationFoam'' is modified in order to compute the solution of the resulting optimization problems. The optimal shapes obtained in the four case studies are in very good agreement with the available literature works. Moreover, they allow a significant reduction of the dissipated energy ranging from 10.8 to 53.3 %.