The continuous adjoint method for the design of hydraulic turbomachines

This article presents the development and application of the continuous adjoint method for designing/optimizing the shape of hydraulic turbomachines. The Reynolds-averaged flow equations are solved in the rotating reference frame and the terms arising from the differentiation of the Coriolis and centripetal forces are taken into account in the formulation of the adjoint equations. The objective functions presented in this article can be used for achieving (a) the optimal collaboration of the runner impeller with the draft tube, by controlling the meridional and circumferential velocity profiles at the exit of the runner, (b) the operation at the desired hydraulic head and/or (c) the cavitation suppression. All of them are used to improve an existing Francis runner. It is important to note that the objective function related to cavitation is, by definition, non-differentiable and a way to effectively handle it, is proposed. The continuous adjoint method is presented in its most general form and could readily be adapted to other objective functions in the field of hydraulic turbomachines.