Effect of different draft tube designs on the phase resonance in a pump-turbine based on compressible CFD simulation

An investigation into projects with significant noise issues during full-load turbine operation revealed a distinct draft tube elbow design difference compared to more successful projects. This prompted an initiative to explore the impact of draft tube shape on pressure pulsations within the pump-turbine during full-load operation. Various draft tube designs were tested on the same prototype-scale pump-turbine unit using 3D compressible transient simulations. Results indicated increased pressure pulsation amplitude in the spiral casing for the flat elbow draft tube, accompanied by phase shifts that rendered pressure pulsations close to zero at certain locations, converting direct pulsations into indirect ones. Moreover, phase variations in the opposing runner channel aligned with multiples of the rotation period, suggesting possible cancellation of pressure pulsation waves. These improvements were confirmed through contour analysis, which showed reduced negative pressure areas in the draft tube, minimized pressure differences, and more uniform and stable streamlines. This study analyses the influence of draft tube shape on internal flow field pressure fluctuations, offering theoretical insights for enhancing pump turbine stability and guiding further research.


Introduction
Pumped storage power station plays an important role in grid flexible operation and energy structure development; therefore, it is often desirable to have wider operating range for the pump turbine and the pump turbine may be operating at the far off-design operating condition.So some accidents in recent years make the society pay more attention to its safety and reliable operation.Before the accident, the direct manifestation is abnormal vibration and noise, which is phase resonance (PR) and processing vortex rope (PVR) caused by rotor-stator interference (RSI) inside the unit.There will be low frequency vortex with pressure pulsation in the draft tube when the turbine is not running in the full-load operation, which will lead to the vibration of the unit.In addition, if the frequency of upstream rotating water flow is equal to its rotation frequency, which will produce a variety of resonant excitation forces, and even cause serious PR, which will bring bad influence to the safe and reliable operation of the unit.
The flow mechanism and PVR in the draft tube have been studied in detail.When the turbine unit is operating under partial load, the flow rate is relatively small.The water flow entering the runner obtains a rotational speed under the action of centrifugal force, and then the water flow in the draft tube has a peripheral velocity component.Insufficient load with a peripheral velocity component will result in a vacuum at the center of the draft tube inlet, creating conditions for backflow in the draft tube, and vortex pressure pulsation may occur.In addition, the synchronous pressure pulsation will be generated in the draft tube due to the rotation of the water flow, and the pulsation frequency is equal to the rotation frequency of the rotating water flow.Generally speaking, the amplitude of the pressure pulsation in the draft tube is small compared with the amplitude of the pressure pulsation in the vaneless region and the runner channel, but this complex water body resonance, if the characteristic frequencies of any two resonances are coincident, serious phase resonance phenomenon will be caused.
[1]- [5] Dörfler et al. [6] monitored the changes of pressure pulsation amplitude and dominant frequency inside the draft tube.Based on the mathematical vortex motion theory, Nishi [7] used a simple and effective vortex model to estimate the change of pressure pulsation.It is characterized by replacing the traditional surface vortex model with a three-dimensional vortex model, predicting and calculating the change of the flow rate field in the straight cone section of the draft tube from a three-dimensional perspective.The application of the discrete vortex method in the tail pipe of the hydraulic turbine was also discussed [8], [9].Arispe et al. [10] studied the parameter problem of the elbow section of the draft tube based on the initial geometry.By modifying the generatrix, logarithmic helical curve, circular arc curve and hyperbolic helical curve were used to define the longitudinal elbow profile.Through this method, the optimal curve is found, the geometry of the elbow section of the draft tube is determined, and the efficiency of the turbine is improved.
Table 1 summarizes the information of pumped-storage power plants that have been put into operation at home and abroad.Comparing the basic parameters of the four pumped-storage power stations, the basic parameters of power stations A, B and C and D are similar, but the vibration and noise of stations A, C are obviously better than those of stations B, D. After investigating the threedimensional modeling of these four power stations, it was found that the elbow sections of draft tubes A and C are circular, while the elbow sections of draft tubes B and D are flat, see figure 1.Therefore, it is speculated that the flat elbow-shaped draft tube may have a positive effect on the weakening and elimination of vibration noise.Investigate the pump turbines of the pumped storage power station in operation, pumped storage station A, B, C and D, focusing on the design of the draft tube, the differences in design are mainly in the variation law of the cross-section area along the centerline of the draft tube as shown in figure 2 and the shape of the flow cross section.The direct difference of draft tube is the shape of elbow, which can be divided into two types: flat elbow and round elbow as shown in figure 1.At present, the research on the pressure pulsation characteristics of the internal flow of the pumpturbine on draft tubes with different elbow shapes is not comprehensive.In this paper, considering the weak compressibility of water, a pump-turbine unit is used as the model for the original size modeling, and the circular elbow and flat elbow draft tubes are assembled for simulation analysis.The influence of the turbine draft tube vortex, pressure pulsation and phase resonance provide a reference for the stable operation of the unit.

Water weak compressibility
The weak compressibility of water was introduced through a user-defined expression in ANSYS-CFX as follows [11], [12]: Where the subscript 0 denotes a reference quantity, ρ is the density and p is the pressure.The sound speed a0 is assumed to be constant.A comparative study on the the influence of the compressibility effects has been done before and results could be found in [13].

Setting of time step and sound speed
From a previous exploration of the step size settings in unsteady simulations and the sound speed in compressible mode simulations, it was concluded that the selection of appropriate timestep is critical in order to study the complex process of wave reflection and superposition, and it is recommended to take ∆t = 1/32 or 1/40 blade passing time (bpt) as the appropriate timestep from a comparative study [13].
Considering one of the conditions for resonance to occur is that the wavelength λ of the sound waves in the water is close to the prototype scale.The circumferential length of the spiral casing of the model in this study is Lsp=25.2m.From the ratio of Lsp/2λ, it is calculated that the sound speed a0=1200m/s is reasonable [13].

3D geometry model and mesh
The model of the pump-turbine is built according to the prototype scale, which is mainly composed of seven parts: spiral casing inlet extension, spiral casing, stay vanes, guide vanes, impeller, draft tube and draft tube extension.The basic geometric parameters are shown in the following table 2. In order to study the influence of different draft tubes on the pressure pulsation inside the whole passage, the unit is equipped with two different draft tube DT1 and DT2 for numerical simulation.Threedimensional models and elbow sections of draft tube DT1 and DT2 are shown in the figure 3 and figure 4.
ANSYS-ICEM and TurboGrid software are used for division and combination of different components of the computation domain.Considering that the rotor-stator interference occurs in the vaneless region between guide vanes and the runner, it is particularly important to ensure the quality of the boundary layer under the limitation of computing resources.After quality inspection of the generated meshes, it is found that the value of y+ in the vaneless region is small and the boundary layer meshes are fine, which can meet the accuracy requirements of calculation.The total number of meshes and drawing software of each part are shown in table 3 and the meshes of all the components are presented in figure 5.

Compressibility function and boundary conditions
Considering the weak compressibility of water, the acoustic reflection is set to 'nonreflection'.The study focuses on the unsteady characteristics of the pressure when the machine is under full-load operation in the turbine mode, at the rated discharge.The initial condition of transient simulation is the solution of the steady state.The computed physical time was 3.0 s, corresponding to 28. 6

Monitoring points and sections
In order to understand the travelling of pressure waves caused by rotor-stator interaction, a great number of monitoring points has been set up in each component as shown in figure 6(a).In the spiral casing, they were located at the centerline, with SP00 close to the nose vane and SP21 at the spiral casing inlet.Inside the stay vanes and guide vanes, there were 4 groups of monitors located at 4 directions (indicated with "E"," N", "W" and "S", respectively) and starting from the impeller inlet (marked with VL_*01) up to the inlet of distributor (marked with VL_*14); see figure 6(b).Inside the impeller, 4 sets of monitoring points in the middle of impeller passage and on the surface of hub inside was added, namely A01 to A14 located in channel 1, B01 to B14 located in channel 3, C01 to C14 located in channel 5, D01 to D14 located in channel 7, as shown in figure 6(b).Within the draft tube, 4 sets of monitors have been laid on the inner side wall (indicated with "I"), the outer side wall (indicated with "O"), the left side wall (indicated with "L") and the right side wall (indicated with "R").figure 6(c) shows the specific location of monitors inside the draft tube.

Simulation results and discussions
The pressure pulsation coefficient discussed hereafter is calculated with formula below: (2) Where p is the transient pressure, the average pressure, ρ the local density, and U=ωr the blade peripheral velocity at its maximum radius.

Pressure pulsation inside the draft tube
Generally speaking, the pressure pulsation inside the draft tube of the pump-turbine are very low and could be neglected in terms of its amplitudes.Under the full load condition, the water flow from the upstream component into the draft tube has a large rotational speed, thus forming a negative pressure area at the central axis of the draft tube inlet.figure 7 shows the contour of pressure at the centerline plane of both draft tubes, where the blue area is the negative pressure area and could be called the processing vortex rope (PVR).From the contour plot, little difference could be found.

Pressure pulsation inside the spiral casing
The pressure pulsation inside the spiral casing of the original design with DT1 was already studied previously, a certain kind of phase resonance (PR) phenomenon was found and the details of the results could be found in [13].Although the draft tube is located downstream the impeller and is far away from the spiral case, figure 9 compared the signal of pressure pulsation inside the spiral casing with a significant difference observed.The amplitudes are quite similar for both designs, but the phase difference among all the monitors (with the phase lag due to wave travelling added) Δφsp,x differs very much between both designs.Figure 10 presents the phase differences among all the monitors of the spiral circumferences for both designs.For DT1, it can be seen that the pressure fluctuations of all the monitoring points in the spiral casing have almost the same phase, and this is the so-called direct PR.But for DT2, the pressure fluctuations of all the monitors in the spiral casing have quite a large phase differences, which lead to higher amplitudes at some locations and lower amplitude at other locations.

Pressure pulsation inside the impeller channel
It is quite difficult to understand why DT2 has such an effect on the pressure pulsation at far upstream the impeller.Obviously the rotor stator interaction (RSI) provides the source of pressure pulsation.These pressure pulsations travel upstream through the distributor channels into the spiral casing.They also travel downstream through the rotating impeller into the draft tube.
In the stationary domain close to the rotating impeller such as distributor and draft tube, one would observe the flow behavior at the blade passing frequency (BPF) and its harmonics (2BPF, 3BPF and so on).But inside the rotating impeller, one would observe the flow behavior at gate passing frequency (1GPF) and its harmonics (2GPF, 3GPF and so on).
Figure 11 shows the distribution of the pressure pulsation coefficients of these three frequencies, figure 11 (a) and (b) respectively correspond to the pump-turbine equipped with DT1 and DT2.By comparing (a) and (b) of 5.18, it can be seen that the two different draft tube models have almost no influence on the pressure pulsation coefficients at characteristic frequencies 1GPF and 1BPF, but have a great influence at characteristic frequencies 2BPF.In terms of numerical value, the pressure pulsation amplitude of 2BPF in the runner channel of the pump turbine equipped with DT2 is 2-4 times that of the pump turbine equipped with DT1, which indicates that the pressure pulsation at 2BPF travels downstream through the rotating impeller into the DT2 more than DT1.Considering the reflection and superposition of pressure waves, there is a situation where pressure waves cancel each other, so the influence of pressure pulsation cannot be directly determined from the amplitude alone.Groups A and C of monitoring points in the runner channel are opposite, and the positions of monitoring points in Groups B and D are opposite.Groups A and C of monitoring points in the runner channel are opposite, and the positions of monitoring points in Groups B and D are opposite.The trend of the phase of the monitoring points in the opposite group is consistent regardless of the pulse frequency, and there is a constant phase difference between the opposite groups.In figure 12, the phase difference of pressure pulsation measured at the monitoring points in groups A and C is 6°, and that at groups B and D is 16°.While in the simulation of the model equipped with draft tube DT2, the phase difference of pressure pulsation measured at monitoring points in groups A and C is 89°, and that at groups B and D is 142°, which could explain better the phase difference in figure 10.

Conclusions
In the study for projects with strong noise observed, an obvious difference is found on its draft tube designs.Then with acoustic models available in CFX software, a 3D compressible CFD simulation is performed on a pump turbine in prototype scale in order to study the effect of different draft tube designs.A comparative study is done in order to understand the travel of pressure pulsation inside the unit operating of both designs.The following conclusions could be drawn: 1) Although the pressure pulsation inside the draft tube of the pump-turbine are very low and could be neglected in terms of its amplitudes, but different draft tube design may have an important effect on the dynamic behavior of the flow inside.In this study it is found the signals of DT1 show a synchronous nature, while that of DT2 show an asynchronous nature.
2) The pressure pulsation inside the spiral casing at 2BPF, coming out from the distributor channels and travelling upstream from the nose vane, of DT1 has almost the constant phase without difference, which is a direct phase resonance.While for DT2, the pressure pulsation travel with much larger phase differences, resulted in higher amplitude at some location and almost zero amplitude at some specific location, which could be termed as indirect resonance.3) In the impeller channel, the pressure pulsation amplitude of DT2 is 2-4 times higher than that of DT1, which indicates that the pressure pulsation at 2BPF travels downstream through the rotating impeller into the DT2 more than DT1, and DT2 has a larger damping effect.
(a) Elbow of station D. (b) Elbow of station C.

Figure 2 .
Figure 2. Curves of different draft tube cross-sectional areas.

Figure 6 .
Figure 6.Setup of the monitors in the whole flow passage (a), inside the distributor and impeller channel (b) and in the draft tube (c).

Figure 7 .
Figure 7. Pressure contour at inlet of the different draft tubes.

Figure 8
Figure8compared the signals of pressure in both draft tube at all the monitoring points inside the draft tube during one revolution, an obvious difference could be seen that the signals from DT1 show a synchronous nature, but the signals from DT2 show an asynchronous and more stable nature than DT1.So even their amplitudes are neglectable, but their phases present different features with both designs, so the draft tube design still has an important effect on the dynamic behaviour of the flow inside.

Figure 8 .
Figure 8. Signals of pressure pulsation in draft tube.

Figure 11 .
Figure 11.Comparison of the amplitudes of characteristic frequency in the runner channel.

Figure 12 .
Figure 12.Phase difference of characteristic frequency in runner channel.

Table 1 .
Comparison of parameters and working conditions of stations.

Table 2 .
Basic data of the pump-turbine model.

Table 4 .
impeller revolutions.Constant timestep ∆t is 1/40 bpt.Settings of boundary conditions are shown in the table 4. Settings of boundary conditions.