Numerical shape optimization in Fluid Mechanics at low Reynolds number

In this paper, a simple and robust numerical shape optimization approach is presented. This approach is based on the Hadamard geometric optimization method and tested on 3 two-dimensional case studies representing standard flows in fluid dynamics, namely the flow around an obstacle, in a 90 elbow pipe and in a dyadic tree. This viscous flows are driven by the stationary Navier-Stokes equations without turbulence model. 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 of the 3 aforementioned case studies that minimizes the energy dissipation in the fluid due to the work of viscous forces under a volume 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 Navier-Stokes equations and the adjoint system are implemented and solved by using the finite volume method within OpenFOAM CFD software. The solver "adjointShapeOptimizationFoam" is modified in order to implement the optimization algorithm and determine the best shape in each of the three considered case studies. The optimal shapes obtained in the three 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 %. Therefore, a decrease of the pressure losses in each case is also achieved in the same proportion.