Numerical simulations and analysis of the vortex rope formation in a draft tube cone of a Francis turbine operating at part-load conditions are carried out. Steady simulations are performed using a 2-D axisymmetric computational grid and unsteady simulations are carried out using a 3-D computational grid for a simplified axisymmetric draft tube. Two part-load operating conditions with same head and different flow rates are considered. The flow rates of these two operating points correspond to 91% of the flow rate at best efficiency point (case I) and 70% of the flow rate at best efficiency point (case II). Steady, axisymmetric simulations show the formation of a central stagnant region in the draft tube which becomes larger as flow rate decreases. This region results in flow blockage and reduction of the pressure recovery coefficient. It is shown that the pressure recovery coefficient is reduced by 46% by decreasing the flow rate from 91% of the best efficiency point (case I) to 70% of the best efficiency point (case II) while loss coefficient becomes 5 times larger. Present unsteady, three-dimensional simulations correctly predict the overall shape of the vortex rope and the calculated vortex rope frequency differs only 5% from experimental data. It is shown that the vortex rope is formed due to the roll-up of the shear layer at the interface between the low-velocity inner region (results from the crown cone wake) and highly swirling outer flow. Finally a flow control technique which uses a water jet injected from the runner crown tip along the axis is investigated. The jet accelerates the flow near the centerline (stagnant region) and decreases the relative velocity and thereby the shear between low-velocity inner region and high-velocity outer flow and hence prevents the vortex rope formation. The jet discharge is optimized for minimum overall losses. The optimized jet decreases total losses by 13% for case I and the vortex rope is eliminated. The fraction of water used for the optimized jet is less than 0.5% of the turbine discharge. As shown in the present study, this control technique can suppress severe pressure fluctuations resulting from the vortex rope formation.

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