## 1 Introduction  
A water turbine is a prime mover. It is important fluid equipment that converts the pressure energy of water into turbine mechanical energy or electrical energy by using water pressure to drive the runner rotation. It can be used as a driving component in many fields such as residual pressure recovery, seawater desalination, power supply systems, long-distance pipeline systems, and so on. The flow in the draft tube of a water turbine is a complex hydrodynamic phenomenon [1]. It is characterized by severe flow instability, and there is a vortex strip in the flow, which will cause draft tube surging in severe cases [2]. Self-induced flow instability and vortex strips can lead to many adverse operating characteristics, including reduced efficiency, noise, vibration, pressure pulsation, and unstable output. Therefore, the flow state of the draft tube is of great significance for the efficient and stable operation of the water turbine [3]. In an ideal state, the inlet water flow of the draft tube is non-rotational, that is, the normal velocity of the water flowing out of the runner into the draft tube is zero [4]. However, due to the structure of the water turbine, during actual operation, the flow has circulation. The flow entering the draft tube has the same rotation direction as the runner, which will form a vortex [5]. For a water turbine, an axial eccentric vortex strip is formed in the center area of the draft tube, accompanied by low-frequency, high-amplitude pressure fluctuations, which dominate the entire flow channel [6]. For a pump acting as a turbine, an approximately axisymmetric twisting vortex will form in the draft tube [7]. The phenomenon of swirling flow, unstable fluid detachment, backflow, and complex vortices in the draft tube can all cause obvious hydraulic losses. The impact on hydraulic efficiency cannot be ignored [8], which is also prone to flow-induced vibration and noise [9]. According to previous research, efficiency, and stability are important performance indicators of the draft tube, which are related to the utilization of water energy and the normal operation of the whole machine. Changing the downstream flow structure of the runner is one of the main measures taken to reduce hydraulic losses and improve flow stability in the draft tube. The measures employed to change the downstream flow structure of the runner include installing baffles or fins, changing the runner cone structure, and changing the draft tube structure. The installation of baffles in the draft tube can collect the flow at the outlet of the runner and correct the flow direction, thereby disrupting the development of strong swirling flow and reducing the amplitude of pressure pulsation [10]. The installation of fins in the draft tube can induce a small extra rope. The interaction between the main vortex strip and the extra rope changes the flow field inside the draft tube pipe [11], which can significantly mitigate the formation of the vortex strip, restrict the generation of bubble vortex strip, lower vortex intensity, reduce cavitation rate, and decrease pressure pulsation [12]. By optimizing the shape and length of the fins, the maximum unsteady pressure can be reduced by about 41% [13]. The longer fins extending to the elbow exhibit lower vortex strip strength, which can achieve maximum pressure recovery in the draft tube and significantly reduce vortex intensity by about 94%. Longer fins suppress low-frequency pressure peaks by about 60%, significantly reducing instability and vibration, while shorter fins also moderately minimize it [14]. Unconventional fins that are away from the wall, closer to the diffuser axis, were subsequently shown to be able to reliably suppress the unsteady vortex strip [15]. Changing the runner cone structure is mainly achieved by changing the profile [16; 17; 18] and drilling or slotting holes [19; 20].