An optimization approach for stokes flow in a channel with a cylindrical block by the BEM

Abstract

The Stokes flow problem in a channel with a cylindrical block is solved by using an optimization approach. The stream function—vorticity formulation is adopted to characterize the flow problem. As the stream function value on the cylindrical block is a priori unknown it is treated as the decision parameter. The objective function to be minimized is taken as the square of the Gauss condition of the vorticity function. The sensitivity analysis of optimization is done by the adjoint variable method, necessitating the solution of an adjoint problem besides the primary (flow) problem. For the discretization of all the equations the boundary element method is utilized. The minimization process is performed by a nonlinear programming method with the convergence within a given tolerance achieved in a few iteration steps. Besides the stream function and vorticity, the pressure distribution is obtained afterwards by solving Laplace's equation. The numerical results are provided for various flow functions in terms of, e.g. streamlines, equivorticity, and constant pressure lines within the flow field.