Shape design optimization for viscous flows in a channel with a bump and an obstacle

A shape design optimization problem for viscous flows in an open channel with a bump and an obstacle are investigated. An analytical expression for the shape design sensitivity involving different cost functionals is derived using the adjoint method and the material derivative concept. A channel flow problem with a bump as a moving boundary is taken as an example. The shape of the bump, represented by Bezier curves of order 3, is optimized in order to minimize the vortices in the flow field. Numerical discretizations of the primal (flow) and adjoint problems are achieved using the Galerkin FEM method. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. Striking differences are found for the optimal shape control corresponding to the 3 different cost functionals, which constitute different quantifications of vorticity.