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Flow in the draft tube of a hydraulic turbine operating under off-design conditions is very complex. The instability of the swirling flow may lead to the formation of a helical precessing vortex called the “vortex rope”. The vortex rope causes efficiency reduction, severe pressure fluctuation, and even structural vibration. The primary objectives of the present study are to model and analyze the vortex rope formation using high fidelity numerical simulations. In particular, this work aims to understand the fundamental physical processes governing the formation of the vortex rope, and to investigate the capability of turbulence models to simulate this complex flow. Furthermore, mitigation of the vortex rope formation is addressed. Specifically, a vortex rope control technique, which includes injection of water from the runner crown tip to the inlet of the draft tube, is numerically studied. A systematic approach is considered in this study starting from the simplest and advancing towards the most complicated test case. First, steady simulations are carried out for axisymmetric and three-dimensional grids in a simplified axisymmetric geometry. It is shown that steady simulations with Reynolds-averaged Navier-Stokes (RANS) models cannot resolve the vortex rope, and give identical symmetric results for both the axisymmetric and three-dimensional flow geometries. These RANS simulations underpredict the axial velocity by at least 14%, and turbulent kinetic energy (TKE) by at least 40%, near the center of the draft tube even quite close to the design condition. Moving farther from the design point, models fail in giving the correct levels of the axial velocity in the draft tube. This is attributed to the underprediction of TKE production and diffusion near the center of the draft tube where the vortex rope forms. Hence, a new RANS model taking into account the extra production and diffusion of TKE due to vortex rope formation is developed, which can successfully predict the mean flow velocity with as much as 37% improvements in comparison with the realizable k-ε model. Then, unsteady simulations are performed, where it is concluded that Unsteady RANS (URANS) models cannot capture the self-induced unsteadiness of the vortex rope, but instead give steady solutions. The hybrid URANS/large eddy simulation (LES) models are proposed to be used in unsteady simulations of the vortex rope. Specifically, a new hybrid URANS/LES model in the framework of partially-averaged Navier-Stokes (PANS) modeling is developed. This new model is one of the main contributions of the present study. The newly developed PANS model is used in unsteady numerical simulations of two turbulent swirling flows containing vortex rope formation and breakdown, namely swirling flow through an abrupt expansion and the flow in the FLINDT draft tube, a model-scale draft tube of a Francis turbine. The present PANS model accurately predicts time-averaged and root-mean-square (rms) velocities in the case of the abrupt expansion, while it is shown to be superior to the delayed detached eddy simulation (DDES) and shear stress transport (SST) k-ω models. Predictions of the reattachment length using the present model shows 14% and 23% improvements compared to the DDES and the SST k-ω models, respectively. For the case of the FLINDT draft tube, four test cases covering a wide range of operating conditions from 70% to 110% of the flow rate at the best efficiency point (BEP) are considered, and numerical results of PANS simulations are compared with those from RANS/URANS simulations and experimental data. It is shown that RANS and PANS both can predict the flow behavior close to the BEP operating condition. However, RANS results deviate considerably from the experimental data as the operating condition moves away from the BEP. The pressure recovery factor predicted by the RANS model shows more than 13% and 58% overprediction when the flow rate decreases to 91% and 70% of the flow rate at BEP respectively. Predictions can be improved dramatically using the present unsteady PANS simulations. Specifically, the pressure recovery factor is predicted by less than 4% and 6% deviation for these two operating conditions. Furthermore, transient features of the flow that cannot be resolved using RANS/URANS simulations, e.g., vortex rope formation and precession, is well captured using PANS simulations. The frequency of the vortex rope precession, which causes severe fluctuations and vibrations, is well predicted by only about 2.7% deviation from the experimental data. Finally, the physical mechanism behind the formation of the vortex rope is analyzed, and it is confirmed that the development of the vortex rope is associated with formation of a stagnant region at the center of the draft tube. Based on this observation, a vortex rope elimination method consisting of water jet injection to the draft tube is introduced and numerically assessed. It is shown that a small fraction of water (a few percent of the total flow rate) centrally injected to the inlet of the draft tube can eliminate the stagnant region and mitigate the formation of the vortex rope. This results in improvement of the draft tube performance and reduction of hydraulic losses. Specifically in the case of the simplified FLINDT draft tube, the loss coefficient can be reduced by as much as 50% and 14% when the turbine operates with 91% and 70% of the BEP flow rate, respectively. In addition, reduction (by about 1/3 in the case with 70% of BEP flow rate) of strong pressure fluctuations leads to more reliable operation of the turbine.