Introduction

Part load (PL) conditions are designated by lower flowrate (Q) operating conditions. The downstream swirl instabilities at this condition contribute to the higher losses in the draft tube (DT).⁽¹⁾ Because of the significant runner exit circumferential velocity at PL condition, a helical precessing vortex is generated called vortex rope. In the runner rotation direction, these vortex ropes rotate with a precession frequency of about 0.2 to 0.4 times the rotational frequency of the runner.⁽²⁾ The vortex rope is a significant contributor to the DT flow instabilities which are responsible for vibration and pounding noise.⁽³⁾ The turbine may fail catastrophically because of the DT surge when the vibrational frequency matches the natural frequency of the structure.

In the Francis turbine cavitation is a very common phenomenon that results in noise and vibration in the draft tubes. At PL condition cavitation can take place inside t he v ortex rope which further increases the amplitude of the pressure fluctuations at the DT wall. Hence the turbine may fail due to resonance as well as due to pitting of the material.⁽⁴⁾

The nature of the DT flow induces the low-frequency instabilities which disrupt the stable and safe operation of the turbine so, to suppress this DT surge various techniques have been developed. There are three main control approaches for dealing with the pressure changes caused by the cavitating vortex rope; stabilizer fins mounted on the DT wall, air or water injection, and runner cone extension.⁽⁵⁾ These strategies have virtues in terms of reducing the surge, but they also have flaws. Given other physical design restrictions on turbines, extending the runner cone can be prohibitive. Insufficient air supply keeps the vortex rope unhindered and the natural frequency may decrease which leads to the resonance.⁽⁶⁾ Muntean et al.⁽⁷⁾ discovered that when the turbine works at a reduced discharge, the dynamic behaviour is severely harmed by the air injection control mechanism. Also, air or water admission frequently necessitates the use of more electricity.

Installing fins in the entrance cone of a draft tube is a popular method of reducing pressure surges. It is an intrusive type of mitigation strategy which decreases the high swirl content in the outlying zone of the DT since it is mounted on the DT circumference parallelly in the flow direction. Nishi et al.⁽⁸⁾ used wall pressure pulses to evaluate the influence of fins on the half load surging and suggested that triangular fins are more effective than square ones. Mounted fins, on the other hands, were discovered to be prone to significant efficiency losses, cavitation erosion, and the ability to enhance structural vibrations.

Taking into account the combination of characteristics, some researchers used both fins and air admission approaches, called hybrid control, to alleviate the draft tube surge.(^9,10) In this hybrid control, holes on the surface of a fin or on the runner crown are used to allow air in. Kim et al.⁽¹⁰⁾ investigated numerically the effect of anti-swirl fins along with air injection from runner crown for which they considered four fins. Cylindrical holes are protruded from the surfaces of the fins inside the DT in their model. At PL condition without considering cavitation, they discovered that the control methodology significantly reduces the pressure pulsation and the operation range is broadened for the Francis turbine. But, the intrusion portion of the injection holes obstructs the incoming flow and some complex flow phenomenon takes place which was not elaborated by the researchers.

Conclusively, the utilization of fins along with the air injection technique greatly mitigate the DT synchronous pressure pulsations. The obstruction created by the injection holes in the DT flow induces flow complexity and randomness near the hole surface which could be detrimental to the safe working of the Francis turbine. Therefore, a thorough investigation is needed for the explanation of the physics behind these complex flows. To address this important issue of the control techniques, the current study investigates numerically the effect of air injection hole protrusion inside the DT on the pressure pulsation. The operating condition is selected at PL condition with cavitation for the same geometrical model used by Kim et al.⁽¹⁰⁾. The anti-swirl fins with and without air injection holes are considered for the study but the air injection technique is not taken into account. A transient numerical investigation is carried out for the accurate prediction of the flow complexities. The experimental investigation is also performed for the numerical validation.

The present numerical study utilizes the three-dimensional (3-D) geometry of the Francis turbine model. The turbine model consists of stationary components; spiral casing, stay vanes, guide vanes and a draft tube as well as rotating component; runner. The basic function of the draft tube is to guide the runner exit flow and convert the excess kinetic energy (KE) into static pressure. The DT has two diffusers for better conversion. The full geometrical model is shown in Fig. 1 in which a zoomed-in view of the anti-swirl fins is also depicted. There are four fins mounted on the DT out of which two are long fins placed opposite to each other and two fins are short fins that are also placed diagonally. At each fin, one air injection hole (red colour in Fig. 1) is generated which is protruded inside the DT and acts as an obstruction in the flow. The holes are situated at a location of 0.09 D₂ from the DT inlet. In general, the flow gets obstructed by the fin’s geometry but due to the intrusive nature of the holes the incoming flow gets more deflected and flow exhibits unsteadiness near the holes.

Fig. 1 
3D geometrical model of the Francis turbine

The operating condition for the turbine is set at a lower flowrate for which Q/Q_BEP=0.74 (BEP-Best efficiency point) that lies between the range of flowrate which exhibits severe downstream instabilities. For the cavitating vortex rope, cavitation condition is also incorporated in this study. The cavitation phenomenon is quantified by a non-dimensional quantity called Thoma number (σ). The σ depends on the suction head of the turbine, so by regulating pressure at the DT outlet, the suction head is created for which the cavitation initiates. This condition is termed cavitation inception and the value of σ=0.266 in the current investigation. Hence, the pressure pulsation with and without holes is transiently investigated at PL condition with cavitation inception point. The specifications of the Francis turbine model are tabulated in Table 1. The coefficients definitions in Table 1 can be referenced in ref.⁽¹¹⁾

The internal flow physics is predicted using the commercial software ANSYS CFX in the present numerical study. Since the flow in the turbine is turbulent and transient in nature, the physics of this flow is captured by solving unsteady Reynolds averaged Navier Stokes (URANS) equations assuming the flow is incompressible. The closure problem of the URANS equations is treated by the scale adaptive simulation shear stress transport (SAS-SST) model which precisely captures the vortical flows in the Francis turbine.⁽¹²⁾ To model the cavitation a widely accepted two-phase Rayleigh-Plesset model is employed.⁽¹³⁾ The numerical solution needs discrete points so grids are generated on the turbine model. For spiral casing and DT, a tetrahedral mesh is generated while for other components hexahedral mesh is employed. The runner is a spinning component so the y⁺ value for this is kept below 5. The mesh independency is handled by a technique called Grid c onvergence i ndex (GCI) f or which three meshes are generated; fine, medium and coarse.⁽¹⁴⁾ For these meshes discretization errors are calculated. For fine mesh with 8.95×10⁶ grid elements, the GCI values of efficiency and flowrate are 0.0091 and 0.0093. Since these values are below 0.01 so the fine mesh is considered for further investigations. The generated grids on the turbine model are depicted in Fig. 2.

Fig. 2 
Computational grid on the turbine model

The physical components of the turbine are merged with the interfaces. The rotating and stationary domains are connected with General Grid Interface (GGI). The boundary conditions are specified in Table 2. The boundary condition for the inlet of the spiral casing is mass flow rate and the static pressure is applied at the DT outlet for the cavitation inception. No-slip and automatic wall functions are applied at the turbine’s wall. The unsteady cavitating flow is solved by initializing the solutions by steady cavitating solutions which are also initialized by steady non-cavitating solutions. Hence, the transient investigation has better convergence. The time step is calculated for the 1.5° of runner rotation and five full runner revolutions are considered for the solution.

The present study investigates the effect of protrusion of air injection holes in the DT flow which are attached to the fins. These holes obstruct the incoming f low and induce some random complex ities. The accurate calculations of the performances and internal flow characteristics of the turbine with and without holes are performed using the unsteady numerical methodology. The effect of these holes was investigated at PL condition incorporating the cavitation inception point at σ=0.266. It was found that the protrusion of the air injection holes reduces the efficiency by about 1% compared to the without holes case by creating some complex flows near the holes. The non-uniformity caused by these holes is the result of longer and wider vortex rope which is minimized by removing the holes. Also, significant low-pressure zones are attached to the fin’s holes which are reduced for the case of without holes. The maximum swirl intensity inside the DT is reduced by about 41% for the DT without holes which will lead to the stable operation of the turbine. Finally, random pressure pulsations at a higher frequency are the results of the hole’s protrusion which is remarkably suppressed for without holes case. Furthermore, this study can be utilized for the optimization of the holes and fin’s shape.