Optimal shape design for fishways in rivers

Fishways are hydraulic structures that enable fish to overcome obstructions to their spawning and other migrations in rivers. In this paper we first introduce a mathematical formulation of the optimal design problem for a vertical slot fishway, where the state system is given by the shallow water equations determining the height of water and its velocity, the design variables are the geometry of the slots, and the objective function is related to the existence of rest areas for fish and a water velocity suitable for fish leaping and swimming capabilities. We also obtain an expression for the gradient of the objective function via the adjoint system. From the numerical point of view, we present a characteristic-Galerkin method for solving the shallow water equations, and an optimization algorithm for the computation of the optimal design variables. Finally, we give numerical results obtained for a standard 10 pools channel.