Approximated Analytical Model and Validation of Helical Vortex Rope inside Draft Tube of Francis Turbine

When a Francis hydraulic turbine operates at off-design regimes and particular at partial discharge, the residual swirl downstream runner leads to flow instability of the turbine, with development of a heliacal vortex rope and associated large pressure fluctuations on the draft tube. It is not only notoriously difficult and expensive to be computed by using full 3D turbulent unsteady flow models, but also can not meet the requirement of robust, tractable and accurate prior assessment of the flow unsteadiness level with a wide operating range before actually knowing the detail runner geometry. We established a quasi-analytical model of the vortex rope in the discharge cone based on the theory of processing helical vortex tube in swirl flow. The axial and circumferential velocity profiles can be quickly calculated with this averaged velocities model, and were validated by comparing the numerical results with the experimental results presented in the references. It can be further applied to optimize the flow path design in order to reduce the vortex rope and expanding the operating range for a Francis turbine.