The effect of damping grid on vortex rope and pressure fluctuation in the draft tube of Francis turbine

The pressure fluctuation caused by vortex bands is the main cause of vibration in Francis turbines under off-design conditions. Therefore, adopting effective strategies to affect the vortex rope and suppress the pressure fluctuation caused by the vortex rope is great significance for the stable operation of the Francis turbine. In this study, the damping grid were proposed to optimize the flow state of the fluid in the draft tube, and unsteady numerical simulations were performed to determine their mechanism of action. And a detailed analysis was conducted on the test results to understand the impact of damping grids on the Francis turbine vortex rope. The results show that the pressure fluctuation of draft tube under off-design condition is mainly caused by the low-frequency fluctuation caused by the vortex rope. The damping grid can effectively affect the generation of vortex rope and reduce the low-frequency pressure fluctuation in the Francis turbine.


Introduction
On a global scale, energy conservation and emission reduction have become recognized as effective solutions to reduce environmental issues, which inevitably requires people to reduce their dependence on fossil fuels.Therefore, utilizing renewable energy such as wind, light, and water energy in people's lives is a good choice [1,2].In this context, hydropower stations are widely used to meet the growing demand for electricity.The Francis turbine plays an important role in hydroelectric power generation, with high reliability, environmental friendliness, flexible regulation, and high efficiency [3].Because Francis turbines can respond to operating conditions in real-time and have strong flexibility, they can greatly enhance the stability of the power grid.However, due to natural environment, electrical load, and other reasons, Francis turbines often operate at off-design operating points, which may cause strong pressure fluctuation, noise, cavitation, and other phenomena.Even the unit may generate strong vibrations, causing fatigue damage to the Francis turbine.Now, the development of Francis turbines is towards large capacity and size, which greatly reduces the stiffness of the turbines and enhances their instability.A large number of studies have proved that the spiral vortex belt generated by draft tube under off design conditions is an important factor causing these phenomena [4,5].
In order to suppress and eliminate the vortex rope of draft tube, scholars have conducted a lot of research on the mechanism of the generation and development of the vortex rope of draft tube, the relationship between the vortex rope and pressure fluctuation [6,7], and adopted some methods to suppress the vortex rope of draft tube.Zhang et al. [8] used genetic algorithm combined with fuzzy logic to optimize the trailing edge of the runner blade, improving the efficiency of the unit and reducing pressure pulsation under small guide vane opening.Zhou et al. [9,10] introduced a 0°-16.8°angledraft tube diffuser into the draft tube, and the pressure fluctuation at different positions will have different changes, but the overall pressure fluctuation shows a decreasing trend.Shrestha et al. [11,12] used J-grooves in the draft tube, installation of diversion plate at elbow, different extension scheme of runner cone, runner cone with spiral grooves , air supply or water jet structure to suppress the generation of vortex rope.However, there are few studies on the installation of damping grids in the taper pipe through Computational fluid dynamics.Therefore, this paper analyzes the internal flow characteristics of the scheme of installing damping grids in the taper pipe.

Francis turbine model
The model of a Francis turbine is obtained by proportionally reducing the turbine with a runner diameter of D1=7.1m and a speed of nr=93.75r/min to investigate the effect of using damping grids on the performance.The runner diameter of the Francis turbine model is D1m=352mm.As shown in Figure 1, it is composed of five parts, namely volute casing, guide vanes, stay vanes, runner and draft tube.There are 13 runner blades, 24 guide vanes and stay vanes, and the rated working condition specific speed is ns=276.1.The off-design operating design point of the Francis turbine model is at Hm = 20 m for the head, Qm = 0.411 m 3 /s for the water flow rate and the rotational speed is Nm = 1110 r/min.

Damping grid
To suppress the generation of vortex bands, a damping grid is installed below the impeller.The threedimensional model with a damping grid is shown in Figure 3

The effect of damping grid on efficiency
Installing a damping grid at the draft tube will have a certain impact on the efficiency of the water turbine.Under rated working condition, the efficiency of the water turbine with a damping grid installed decreases by 0.6%.Under off-design operating design point, the impact on the efficiency of the hydraulic turbine slightly increases, with a decrease of 1.05% in efficiency.Under rated working condition, the water flow distribution in the draft tube is relatively uniform, and the resistance loss formed by the damping grid is relatively small.Under off-design operating design point, due to the turbulence of the water flow in the draft tube and the high intensity of vortices, the resistance loss formed by the damping grid is significant, resulting in a decrease in efficiency.

The influence of damping grid on vortex rope
Figure 4 shows the vortex rope formed by an iso-surface of 0.55 times the outlet pressure.Under the draft tube without damping grid, the vortex rope in the draft tube is spiral and thick.The spiral vortex ropes to rotate around the draft tube, causing strong pressure fluctuations in the draft tube.In the draft tube with damping grid installed, the damping grid effectively restrains the vortex rope in the draft tube, causing the structure of the vortex rope to be damaged and dispersed.The shape of the vortex rope is significantly shorter and thinner than when it was without damping grid.

Pressure fluctuation
When the Francis turbine operates under off-design operating points, the internal pressure of the unit changes sharply, causing instability in the operation of the unit and seriously affecting the safe operation of the Francis turbine electromechanical station.Therefore, studying the pressure fluctuation characteristics of Francis turbines has great engineering significance, and the research results can provide reference for the safe operation of power plants.The amplitude of pressure pulsation is often defined as Peaks to troughs, which measures the fluctuation of pressure within a time period relative to the average pressure within that time period.Therefore equation in the pressure fluctuation was adopted as follows (1) ,where is the relative amplitude, P is the pressure, P is the average pressure,  is the density, g is the gravitational acceleration, and H is the head.
-/ 100% 1000 Transform time-domain signals into frequency-domain signals using Fast Fourier Transform (FFT) for easy analysis.The frequency is normalized by the runner rotation frequency (fn).As shown in Figure 6, the main frequency of the draft tube volute position is 0.167fn, and its main source is the vibration caused by the vortex rope of the draft tube.The pressure pulsation in the bladeless zone mainly consists of two components, namely 13 times the blade speed frequency and low-frequency pressure fluctuation.The draft tube is located in the middle of the taper pipe (DT01), and the amplitude of the pressure fluctuation is the highest.The low-frequency pressure fluctuation is transmitted to the upstream and downstream.
Compared to not installing damping grid, installing damping grid significantly reduces the pressure fluctuation in the flow channel.The low-frequency pressure fluctuation inlet of volute casing decreases by 27%, the vaneless area (VL01) decreases by 19%, and the Draft tube DT01 and DT02 positions are reduced by 54% and 82% respectively.

Conclusions
Transient numerical simulation was conducted on the off-design operating points of the Francis turbine, and the internal flow characteristic of the Francis turbine was studied.The influence of introducing damping grid on the pressure pulsation of the Francis turbine was analyzed.Based on these results, the following conclusions can be summarized: High amplitude low-frequency pressure pulsation occurs in the Francis turbine under this offdesign operating points, which is generated by the vortex rope of draft tube and affects the whole flow channel.
Installing damping grid in the draft tube can significantly reduce the low-frequency pressure fluctuation in the entire flow channel, and ensure the safe operation of the Francis turbine.

Figure 1 .
Figure 1.Full flow passage of Francis turbine model for CFD analysis.

2. 2 .
Numerical method Computational fluid dynamics (CFD) analysis is a very useful tool for analyzing the performance of various hydraulic machinery.This article uses ANSYS CFX to analyze the flow characteristics of a Francis turbine with or without a damping grid installed.The turbulence model uses RNG k-ε, and the cavitation model uses Rayleigh Plesset model for simulation.The outlet of draft tube adopts 0pa as the boundary condition, and the inlet adopts mass flow rate of 0.411m3/s as the boundary condition.

Figure 2 . 2 .
shows the numerical mesh, with approximately 5 million elements.At this time, the damping grid is not installed at the draft tube.(a) Runner and details of the runner (b) Stay/guide vanes and details of the vanes (a) Draft tube (b) Volute casing Figure Mesh of each component.
(a).The damping grid is composed of 10 interlocking damping bars, with a width of 5 millimeters and a height of 50 millimeters, arranged at intervals of 80 millimeters.The position of the damping grid is discretized using ANSYS Mesh.At this time, the number of grids in the damping grid is approximately 11 million, as shown in Figure3(b).

Figure 3 .
Draft tube with damping grid.

Figure 4 .
Figure 4. Influence of damping grid on vortex rope of draft tube.

Figure 5 .
Figure 5. Schematic diagram of the monitoring points within the Francis turbine.As shown in Figure 5, in order to monitor the internal pressure fluctuation of the Francis turbine, two monitoring points are arranged at the inlet of the volute casing (CS01) and the b vaneless area (VL01).One monitoring point is arranged at the taper pipe section of Draft tube(DT01), and one monitoring point is arranged at the draft tube elbow section(DT02).The amplitude of pressure pulsation is often defined as Peaks to troughs, which measures the fluctuation of pressure within a time period relative to the average pressure within that time period.Therefore equation in the pressure fluctuation was adopted as follows(1) ,where is the relative

Figure 6 .
Figure 6.Pressure fluctuation of the monitoring points within the Francis turbine.As shown in Figure6, the main frequency of the draft tube volute position is 0.167fn, and its main source is the vibration caused by the vortex rope of the draft tube.The pressure pulsation in the bladeless zone mainly consists of two components, namely 13 times the blade speed frequency and low-frequency pressure fluctuation.The draft tube is located in the middle of the taper pipe (DT01),