Validation of mathematical models for predicting the swirling flow and the vortex rope in a Francis turbine operated at partial discharge

The vortex rope in a hydro turbine draft tube is one the main and strong sources of pulsations in non-optimal modes of hydro turbine operation. We examine the case of a Francis turbine model operated at partial discharge, where a strong precessing vortex rope is developed in the discharge cone downstream the runner. Available experimental data provide the circumferentially averaged axial and circumferential velocity profiles, as well as the vortex rope geometry, precessing frequency, and the level of pressure fluctuation at the wall. The mathematical models presented in this paper can correctly recover all this information without actually computing the full three-dimensional unsteady flow in the hydraulic turbine. As a result, we provide valuable mathematical tools for assessing the turbine behaviour at off-design operating regimes in the early stages of runner design, with computational effort several orders of magnitude less than the current approaches of simulating the complex turbine flow.