On the Vaneless Space Vortex Structures in a Kaplan Turbine Model Operating at Speed No Load

: The growing installation of intermittent renewable energy sources is forcing hydraulic turbines to work more frequently at speed no load when dangerous vaneless space vortex structures and stochastic flow phenomena can occur. An experimental campaign has been performed in a reduced-scale Kaplan turbine model at speed no load for different combinations of guide vane and runner blade angles under non-cavitation and cavitation conditions. Several simultaneous vaneless space vortex structures, all of them inducing torsional rotor vibrations, have been observed. Nonetheless, only one of them has been found to dominate over the rest depending on the blade and guide vane angles. Off-board pressures, torques and vibrations as well as on-board strains have been measured to characterize their nature, intensity, dynamic behavior and induced structural response. Their precession frequencies have been found to depend on the discharge factor, the number of vortices and their location inside the vaneless space. Under cavitation conditions, their dynamic behavior has not been significantly altered but the induced structural response has increased at the low-pressure side of the turbine.


Introduction
The increase in the installation of new intermittent renewable energy sources, such as solar and wind power, must be complemented with other energy sources, such as hydropower, able to stabilize the grid, storage energy and supply peak load.Nevertheless, these conditions force turbines to be operated for longer periods of time in off-design and transient conditions [1][2][3][4].For instance, a hydraulic turbine under a grid stabilization mode works 4% of the time at speed no load (SNL) and 24% at deep part load (DPL), while it only works 1% at SNL and 0% at DPL under a base load mode [5].Deleterious hydrodynamic phenomena, such as vaneless space vortex (VSV) structures and stochastic flow phenomena, occur at SNL and DPL, which may cause high-pressure fluctuations, damage the runner blades [3] and impose more regular refurbishment requirements [6,7].
At SNL, turbines rotate at nominal speed without generating power while the discharge is not zero.The hydraulic energy is dissipated through the generation of large-scale unsteady vortex structures that break down into smaller vortices until the energy dissipation occurs at the smallest scales [8].
The flow at SNL is primarily stochastic and provokes high dynamic stresses on runner blades of Francis turbines [5,[8][9][10][11][12][13][14][15].Most investigations focused on determining the relative fatigue damage inflicted on runner blades when operated at SNL or DPL compared to that induced at best efficiency point (BEP).SNL and DPL operation have been concluded to generate most fatigue damage due to stochastic flow phenomena [5,8,[10][11][12][13][14][15].Other researchers have found that in addition to stochastic excitations, specific Francis turbines were confirmed to be generated due to mainly backflow.In the case of the propeller turbine, large VSV structures were found in the numerical simulations although vibrations on the turbine prototype were within acceptable levels.
As introduced previously, the VSV structures are hydraulic phenomena of periodic nature and, consequently, they generate periodic pressure pulsations and they exert periodic forces on the turbine.Their analysis in the frequency domain is then appropriate.In the current work, the time domain signals were transformed to the frequency domain through Discrete Fourier Transforms (DFT) solved by means of Fast Fourier Transform (FFT) algorithms.The FFT is an appropriate and efficient approach to solve DFT reducing the computation effort significantly.For instance, the computation effort to solve DFT of signals with more than 512 discrete samples is reduced to a 1% using the FFT evaluation relative to the direct evaluation [34].
Most literature related to VSV structures in Kaplan turbines focused on their characterization when generated in a particular SNL condition.However, in order to deepen the insight on these hydraulic structures, there is the need to also study their induced structural responses on different components of the turbine as well as their dynamic evolution with the SNL condition and with the presence of cavitation.In this sense, an experimental test campaign measuring strains, torques, pressure and acceleration on on-board and off-board components of a Kaplan turbine model was carried out when operated in a wide range of SNL conditions with and without cavitation.

Test Stand Description and Instrumentation
A 1:3.875 reduced-scale replica of the Porjus Kaplan U9 prototype was selected for this study.Figure 1 shows the shaft line below the generator, a hydraulic bearing supporting the bottom of the shaft, the penstock, the spiral casing and the entire draft tube.The runner has 6 blades and a reference diameter, D, of 400 mm.The distributor is composed of 18 unequally distributed stay vanes and 20 equally distributed guide vanes.The vaneless space presents a height, B 0 , of 165.6 mm and a width, V 0 , of 480 mm as shown in the bottom of Figure 1.
The experimental setup comprised a large number of sensors installed on the rotor (on-board) and on fixed parts of the test stand (off-board) to study the dynamic response of the turbine in off-design operating conditions as presented in detail by Roig et al. [35].Among them, the following sensors shown in Figure 2 were selected for the present study: • 1 accelerometer on the draft tube cone wall in radial direction (A0y).
• 1 pressure transducer on the vaneless space in axial direction (PVS A ).
• 1 strain gauge on the pressure side of a runner blade (SB R ).
• 1 strain gauge on the shaft measuring the torque strain (SS T ).

Measurement Program
Several combinations of guide vane angles, α, and runner blade angles, β, were selected as presented in Table 1.It must be noted that β= −17 0 and β= 10 0 correspond to closed and open runner positions, respectively.Hence, the selected angles covered all the possible opening range in SNL conditions and they corresponded to the same cam curve with a speed factor, N 11 , of: where H is the head and n is the rotational speed.As shown in Table 1, the turbine was tested at eight different discharges without cavitation.Moreover, three conditions were also tested with cavitation.A sufficient time was waited for each test to reach stable and steady conditions prior to taking continuous measurements of approximately 5 min.The corresponding discharge factor, Q 11 , and Thoma number, σ, have been calculated using Equations ( 2) and (3): where Q, NPSH, P va , g and z r are the discharge, the net positive suction head, the vapor pressure, the gravity and the height of the runner, respectively, and The experimental setup comprised a large number of sensors installed on (on-board) and on fixed parts of the test stand (off-board) to study the dynamic of the turbine in off-design operating conditions as presented in detail by Roig Among them, the following sensors shown in Figure 2 were selected for the pres

Measurement Program
Several combinations of guide vane angles, α, and runner blade angles, β, we lected as presented in Table 1.It must be noted that β = −17⁰ and β = 10⁰ corresp closed and open runner positions, respectively.Hence, the selected angles covered possible opening range in SNL conditions and they corresponded to the same cam with a speed factor,  , of: where H is the head and n is the rotational speed.As shown in Table 1, the turbin tested at eight different discharges without cavitation.Moreover, three conditions also tested with cavitation.A sufficient time was waited for each test to reach stab steady conditions prior to taking continuous measurements of approximately 5 mi corresponding discharge factor,  , and Thoma number, , have been calculated Equations ( 2) and (3):  Compressed air or vacuum were applied on the water free surface in the downstream tank to change the NPSH.Then, a manometer and a mass flow meter were used to measure P 2 and V 2 , respectively.

Methodology
The recorded time signals were converted to the frequency domain by means of the Fast Fourier Transform (FFT) with a Hanning window.This has permitted to identify the different frequencies associated with the VSV structures and to quantify their root mean square (RMS) amplitude levels.Each signal was divided in five segments of 60 s each to average results and calculate standard deviations.The FFT was directly applied to all signals except for those of the accelerometers which were firstly integrated to obtain the vibration velocity.
Since the VSV structures are periodic phenomena dependent on the shaft rotating frequency, all the spectra were plotted as a function of the reduced frequency, f * = f / f n , where f n = 14.5 Hz is the turbine rotating frequency.

Identification of the VSV Structures
A VSV structure is a hydraulic phenomenon originated in the vaneless space which is formed by a certain number of vortices, z D , ideally distributed equidistant to each other.The entire structure precesses at a particular frequency, f 0 , resulting in all the vortices following a circular orbit around the rotor.Based on the work of P ůlpitel et al. [28], the frequency of a particular VSV structure, f S , detected directly with an off-board sensor depends on the number of vortices of the structure, z D , and its precession frequency, f 0 : The frequency detected with an on-board sensor, f R , depends on z D , f 0 and f n , as shown in Equation ( 6), if the turbine rotates in the same direction as the precession of the VSV structure: But if the turbine rotates in opposite direction to the precession of the VSV structure, the frequency detected with an on-board sensor, f R , would depend on z D , f 0 and f n as follows: Combining Equations ( 5) and (6) or Equations ( 5) and ( 7), z D can be isolated to obtain Equation (8) or Equation (9), respectively.Hence, z D can be determined only knowing f S , f R and f n .
Figure 3 presents the spectra of the guide vane torque (TGV), the pressure (PVS A ), the blade strain (SB R ) and the shaft torque strain (SS T ) for α equal to 3.98 • .The TGV and PVS A off-board spectra show two frequency peaks, each one corresponding to a particular VSV structure.Similarly, the SB R and SS T on-board spectra also show two frequency peaks, corresponding to the same particular structures when detected from a rotating frame of reference.The frequency peaks measured off-board, f S , and on-board, f R , can then be combined according to Equation (8) to calculate the number of vortices z D forming the VSV structure.For instance, considering the first frequency peaks at f * S = 1.25 and f * R = 0.77, z D equates to 2. Then, considering the second frequency peaks at f * S = 1.71 and at f * R = 1.28, z D equates to 3.These results indicate that the first frequency peak is associated with a VSV structure formed by two vortices and the second peak with a VSV structure formed by three vortices which precess in the same direction as the turbine rotation.Throughout this article, the VSV structures with two vortices and three vortices will be named as 2V and 3V, respectively.
* = 1.28,  equates to 3.These results indicate that the first frequency peak is asso ated with a VSV structure formed by two vortices and the second peak with a VSV stru ture formed by three vortices which precess in the same direction as the turbine rotatio Throughout this article, the VSV structures with two vortices and three vortices will named as 2V and 3V, respectively.In addition to these dominant 2V and 3V structures, two additional frequency pea with significantly lower amplitudes can also be distinguished in the spectra shown in F ure 4. For the frequency peaks highlighted in orange,  equates to 4, and the structu will be referred to as 4V.For the frequency peaks highlighted in purple, inducing l strains on the blades,  equates 6, and the structure will be referred to as 6VGV.Similar for the frequency peaks highlighted in green, inducing low torques on the guide van again  equates 6, and the structure will be referred to as 6VB.Surprisingly, these tw structures present the same number of vortices.Nevertheless, the 6VGV presents high  In addition to these dominant 2V and 3V structures, two additional frequency peaks with significantly lower amplitudes can also be distinguished in the spectra shown in Figure 4.For the frequency peaks highlighted in orange, z D equates to 4, and the structure will be referred to as 4V.For the frequency peaks highlighted in purple, inducing low strains on the blades, z D equates 6, and the structure will be referred to as 6V GV .Similarly, for the frequency peaks highlighted in green, inducing low torques on the guide vanes, again z D equates 6, and the structure will be referred to as 6V B .Surprisingly, these two structures present the same number of vortices.Nevertheless, the 6V GV presents higher amplitudes than the 6V B in the TGV spectra, suggesting that it occurs closer to the guide vanes.Meanwhile, 6V B presents higher amplitude than the 6V GV in the SB R spectra, confirming that it occurs closer to the runner blades.
No integer values for z D could be found when combining the frequencies observed, according to Equation ( 9), at f * S on the TGV spectra and at f * R on the SB R spectra, which confirms that there are not VSV structures precessing in the opposite direction to the turbine rotation.

Precessing Frequencies of the VSV Structures
Once z D is calculated with Equation ( 8), the precession frequency, f 0 , of each structure can be derived from Equation (5) or Equation (6).As seen in Figure 5, the computed reduced precession frequencies, f * 0 , corresponding to 2V and 3V follow approximately lineal and parallel trends when plotted as a function of Q 11 .The 3V presents lower precession frequencies than the 2V, suggesting that the higher the number of vortices per structure is, the lower f 0 is.
No integer values for  could be found when combining the frequencies observed according to Equation ( 9), at  * on the TGV spectra and at  * on the SBR spectra, which confirms that there are not VSV structures precessing in the opposite direction to the turbine rotation.   of all the VSV structures are primarily governed by the SNL condition, and in turn and  .Similarly, other authors also found that the frequency of the VSV struct pended on the blade opening [27].Moreover, per equal  , the 6VB apparently c the runner blades presents always higher precession frequencies than those of t apparently closer to the guide vanes.

Excitation Type of the VSV Structures
Based on Equation ( 10), the VSV structure induces an axial or a torsional v when  = 0, and a bending or whirling vibration when  = ±1 [35].This result d on  , the integers  and  and the number of runner blades,  .

𝑁 = 𝑘 • 𝑧 − 𝑗 • 𝑧
( = 0, ±1) Values of N were calculated through different combinations of k and j for 2V 6VGV and 6VB.None of these combinations resulted in N = ±1 but mainly in N < > +1 with certain specific combinations resulting in N = 0, as depicted in Table 2. that all the VSV structures induce a torsional rotor vibration and that their frequen lutions are similar suggests that all of them may be related to each other, with the 3V being the main structures and the 4V, 6VGV and 6VB secondary structures.The 4V and 6V GV follow the same trend as the 2V and 3V when plotted as a function of Q 11 but presenting lower precession frequencies, which confirms that the structures with higher number of vortices precess slower.On the contrary, the 6V B , apparently closer to the runner blades than the guide vanes, follow a different trend to the 2V and 3V, suggesting that the location of the VSV structures influences the frequency evolution.
Based on these two distinct frequency evolutions, one followed by 2V, 3V, 4V and 6V GV and another one followed by 6V B , it can be concluded that the precession frequencies of all the VSV structures are primarily governed by the SNL condition, and in turn by Q 11 , and z D .Similarly, other authors also found that the frequency of the VSV structures depended on the blade opening [27].Moreover, per equal z D , the 6V B apparently closer to the runner blades presents always higher precession frequencies than those of the 6V GV apparently closer to the guide vanes.

Excitation Type of the VSV Structures
Based on Equation (10), the VSV structure induces an axial or a torsional vibration when N = 0, and a bending or whirling vibration when N = ±1 [35].This result depends on z D , the integers j and k and the number of runner blades, z R .
Values of N were calculated through different combinations of k and j for 2V, 3V, 4V, 6V GV and 6V B .None of these combinations resulted in N = ±1 but mainly in N < −1 or N > +1 with certain specific combinations resulting in N = 0, as depicted in Table 2.The fact that all the VSV structures induce a torsional rotor vibration and that their frequency evolutions are similar suggests that all of them may be related to each other, with the 2V and 3V being the main structures and the 4V, 6V GV and 6V B secondary structures.
Table 2. k and j combinations for each VSV structure giving a N = 0.

Dynamic Behavior of the 2V and 3V Structures
The following sections are only focused on the 2V and 3V structures because, as seen in Figure 4, they present higher intensities than the rest.Figure 6a,b present a zoom around 2V and 3V in the PVS A spectra as well as the amplitudes of their induced pressure fluctuations, respectively.At small guide vanes angles from 2.6 • to 4.97 • , the 2V peaks appear at f * 0 from 1.09 to 1.32 and the 3V peaks from 1.53 to 1.79, which increase in amplitude and frequency linearly.The fact that the 3V presents higher and sharper frequency peaks than the 2V, which are lower and wider, suggests that 3V could be the dominant structure at low discharges.At 6.8 • , the amplitude of 2V drops drastically and 3V keeps its dominance showing a sharp amplitude increase.higher and sharper frequency peaks than the 2V, which are lower and wider, suggests that 3V could be the dominant structure at low discharges.At 6.8°, the amplitude of 2V drops drastically and 3V keeps its dominance showing a sharp amplitude increase.At 7.35°, whereas no pressure fluctuations induced by 2V are detected, the 3V peak shows a  * of 1.95 and the highest amplitude equal to 0.0185 bar.Suddenly, the dominance switches and only the pressure fluctuations induced by 2V are measured at 7.86° and 8.13° with  * values of 1.46 and 1.47 and maximum amplitudes around 0.021 bar.
The present findings agree with the observations of other authors who reported sudden decreases and increases in amplitude of different VSV structures when changing the SNL condition [18,33].The novelty of the present work is the coexistence of VSV structures with different dynamic behaviors, amplitudes and locations.The simultaneity and the various similarities observed in the frequency evolutions and induced rotor vibrations between different VSV structures supports the hypothesis that they are not independent as considered by previous studies but related to each other.Depending on the discharge factor as well as the guide vane and runner blade angles: (i) both 2V and 3V coexist, (ii) 2V dominates over 3V or (iii) 3V dominates over 2V.

Induced Responses of the 2V and 3V Structures
Figure 7 shows the amplitude of the 2V and 3V frequency peaks detected in the SBR, SST and TGV spectra.A first observation is that the amplitude evolutions resemble those observed on the pressure signals, which confirms that they are induced by the same phenomena detected from different locations.Additionally, it must be noted that although most sensors do not detect the 2V at 7.35° and the 3V at 7.86° and 8.13°, the SBR and TGV spectra show small peaks confirming the coexistence of both structures at all SNL conditions and reinforcing the hypothesis that they are not independent.Among all strain measurements, the blade ones (SBR) present higher values than the shaft torque ones (SST) with a maximum of 0.55 µm/m induced by the 2V at 8.13° and 0.9 µm/m by the 3V at 7.35°.The present findings agree with the observations of other authors who reported sudden decreases and increases in amplitude of different VSV structures when changing the SNL condition [18,33].The novelty of the present work is the coexistence of VSV structures with different dynamic behaviors, amplitudes and locations.The simultaneity and the various similarities observed in the frequency evolutions and induced rotor vibrations between different VSV structures supports the hypothesis that they are not independent as considered by previous studies but related to each other.Depending on the discharge factor as well as the guide vane and runner blade angles: (i) both 2V and 3V coexist, (ii) 2V dominates over 3V or (iii) 3V dominates over 2V.

Induced Responses of the 2V and 3V Structures
Figure 7 shows the amplitude of the 2V and 3V frequency peaks detected in the SB R , SS T and TGV spectra.A first observation is that the amplitude evolutions resemble those observed on the pressure signals, which confirms that they are induced by the same phenomena detected from different locations.Additionally, it must be noted that although most sensors do not detect the 2V at 7.35 • and the 3V at 7.86 • and 8.13 • , the SB R and TGV spectra show small peaks confirming the coexistence of both structures at all SNL conditions and reinforcing the hypothesis that they are not independent.Among all strain measurements, the blade ones (SB R ) present higher values than the shaft torque ones (SS T ) with a maximum of 0.55 µm/m induced by the 2V at 8.13 • and 0.9 µm/m by the 3V at 7.

Induced Responses of the 2V and 3V Structures
Figure 7 shows the amplitude of the 2V and 3V frequency peaks detected in the SBR, SST and TGV spectra.A first observation is that the amplitude evolutions resemble those observed on the pressure signals, which confirms that they are induced by the same phenomena detected from different locations.Additionally, it must be noted that although most sensors do not detect the 2V at 7.35° and the 3V at 7.86° and 8.13°, the SBR and TGV spectra show small peaks confirming the coexistence of both structures at all SNL conditions and reinforcing the hypothesis that they are not independent.Among all strain measurements, the blade ones (SBR) present higher values than the shaft torque ones (SST) with a maximum of 0.55 µm/m induced by the 2V at 8.13° and 0.9 µm/m by the 3V at 7.35°.Similarly, the TGV measurements also present a maximum of 0.16 Nm induced by 2V at 7.86° and 0.21 Nm by 3V at 7.35°.It must be noted that the maximum induced responses by 3V present higher values than 2V, unlike as observed in the pressures.Equation (11) [28,36] permits to calculate the frequency of a VSV structure, f S_R , when measuring its induced vibration transmitted from the rotating rotor with an off-board sensor, using f S , k, Z R , f n and j: Table 3 presents the values of f S_R for 2V and 3V calculated using Equation (11) when α is 2.6 • , 3.98 • , 6.8 • and 7.35 • .Figure 8a,b present the spectra of the vibration velocity measured on the draft tube cone (A0y) and the amplitudes of the peaks induced by 3V at 2.6 • , 3.98 • , 6.8 • and 7.35 • , respectively.It must be noticed that 2V is not detected by A0y, suggesting that 2V is not well transmitted to the turbine supporting structure.However, 3V is detected at the same frequencies listed in Table 3.The amplitude evolution of the 3V peak correlates with those previously found for the pressures, strains and torques, as presented in Figures 6b, 7a, 7b and 7c, respectively.In this case, the frequency peaks associated with 3V present lower amplitudes than other sources of excitation, such as the mechanical unbalance at f * = 1 and their harmonics, as shown in Figure 8a.These results differ from the vibrations measured in a Kaplan turbine prototype by P ůlpitel et al. [28] showing high peaks dominated by the VSV structures but agrees with the results found by Nennemann et al. [33] concluding that the vibrations derived from VSV structures have low intensity compared with other sources of excitation.

Cavitation Effects on 2V and 3V Structures
Figure 9 permits to compare the TGV, PVS A , SB R , SS T and A0y spectra when the turbine is operated under non-cavitation and cavitation conditions.A first observation is that the frequencies of the 2V and the 3V under non-cavitation conditions are approximately equal to those measured under cavitation conditions.For instance, the 2V peaks show exactly the same frequency for all SNL conditions and the 3V peaks only present a slight frequency decrease of less than 0.25% at 7.35 • .The VSV structures originate at the vaneless space and thus above the runner blades, at the high-pressure side of the turbine.However, cavitation usually occurs at the low-pressure side, below the runner blades.The frequency of the VSV structures may not be altered under cavitation condition because the amount of cavitation generated at the vaneless space might be low.
Appl.Sci.2023, 13, 13285 Equation (11) [28,36] permits to calculate the frequency of a VSV structure measuring its induced vibration transmitted from the rotating rotor with an off sor, using  , ,  ,  and : 3 presents the values of  _ for 2V and 3V calculated using Equatio α is 2.6°, 3.98°, 6.8° and 7.35°.8a,b present the spectra of the vibration velocity measured on th cone (A0y) and the amplitudes of the peaks induced by 3V at 2.6°, 3.98°, 6.8 respectively.It must be noticed that 2V is not detected by A0y, suggesting th well transmitted to the turbine supporting structure.However, 3V is detected frequencies listed in Table 3.The amplitude evolution of the 3V peak correlate previously found for the pressures, strains and torques, as presented in Figur and 7c, respectively.In this case, the frequency peaks associated with 3V p amplitudes than other sources of excitation, such as the mechanical unbalan and their harmonics, as shown in Figure 8a.These results differ from the vibr ured in a Kaplan turbine prototype by Půlpitel et al. [28] showing high peak by the VSV structures but agrees with the results found by Nennemann et cluding that the vibrations derived from VSV structures have low intensit with other sources of excitation. (a)

Cavitation Effects on 2V and 3V Structures
Figure 9 permits to compare the TGV, PVSA, SBR, SST and A0y spectra bine is operated under non-cavitation and cavitation conditions.A first obs the frequencies of the 2V and the 3V under non-cavitation conditions are equal to those measured under cavitation conditions.For instance, the 2 exactly the same frequency for all SNL conditions and the 3V peaks only p frequency decrease of less than 0.25% at 7.35°.The VSV structures originate space and thus above the runner blades, at the high-pressure side of the tur cavitation usually occurs at the low-pressure side, below the runner blades of the VSV structures may not be altered under cavitation condition becau of cavitation generated at the vaneless space might be low.
The amplitudes of the 2V peaks at 3.98° under cavitation conditions are equal to those found under non-cavitation conditions.However, the 2V indu under cavitation conditions are consistently lower than those under non-cavita at 8.13°.Similarly, the amplitudes of the 3V peaks at 3.98° and 7.35° under ca consistently lower than those under non-cavitation for most of the signals.A a low degree of cavitation generated in the vaneless space, the intensity of the decrease when the turbine is operated under cavitation conditions.
Exceptionally, A0y shows similar 3V amplitudes under cavitation and conditions at 3.98° and a 3V amplitude increase with cavitation at 7.35°.T increase cannot be explained considering a lineal transmission from the hy tion in the vaneless space to the turbine supporting structure.A0y is mount of the draft tube cone and thus at the low-pressure side of the turbine.I amount of cavitation generated is high and thus it probably has a signifi the modal and dynamic response of the draft tube cone.This unexpected crease could happen due to a change in the natural frequency of the draf sulting in a resonance with the 3V excitation, or of the modal properties o cone, increasing the dynamic response at the 3V frequency.The amplitudes of the 2V peaks at 3.98 • under cavitation conditions are approximately equal to those found under non-cavitation conditions.However, the 2V induced amplitudes under cavitation conditions are consistently lower than those under non-cavitation conditions at 8.13 • .Similarly, the amplitudes of the 3V peaks at 3.98 • and 7.35 • under cavitation are also consistently lower than those under non-cavitation for most of the signals.
Although there is a low degree of cavitation generated in the vaneless space, the intensity of the VSV structures decrease when the turbine is operated under cavitation conditions.
Exceptionally, A0y shows similar 3V amplitudes under cavitation and non-cavitation conditions at 3.98 • and a 3V amplitude increase with cavitation at 7.35 • .This amplitude increase cannot be explained considering a lineal transmission from the hydraulic excitation in the vaneless space to the turbine supporting structure.A0y is mounted on the walls of the draft tube cone and thus at the low-pressure side of the turbine.In this part, the amount of cavitation generated is high and thus it probably has a significant impact on the modal and dynamic response of the draft tube cone.This unexpected amplitude increase could happen due to a change in the natural frequency of the draft tube cone, resulting in a resonance with the 3V excitation, or of the modal properties of the draft tube cone, increasing the dynamic response at the 3V frequency.

Conclusions
Five simultaneous VSV structures with different number of vortices and induc torsional rotor vibrations were found in a reduced-scale Kaplan turbine model opera at SNL.Each structure precessed at a particular frequency in the turbine rotation direct which depended on the turbine discharge factor, its number of vortices and its locat More specifically, the precession frequencies decreased with an increasing number of tices and increased for closer distances of the structures to the runner blades.Among the VSV structures observed, those formed by two and three vortices presented the h est pressure fluctuations, strains and torques.The similarities observed in the freque evolutions and in the induced rotor vibrations among the five different VSV structu suggest that there is some interrelation among them.
When the turbine was operated under cavitation conditions, the precession frequ cies of the dominant VSV structures were not significantly altered.Their induced

Conclusions
Five simultaneous VSV structures with different number of vortices and inducing torsional rotor vibrations were found in a reduced-scale Kaplan turbine model operating at SNL.Each structure precessed at a particular frequency in the turbine rotation direction, which depended on the turbine discharge factor, its number of vortices and its location.More specifically, the precession frequencies decreased with an increasing number of vortices and increased for closer distances of the structures to the runner blades.Among all the VSV structures observed, those formed by two and three vortices presented the highest pressure fluctuations, strains and torques.The similarities observed in the frequency evolutions and in the induced rotor vibrations among the five different VSV structures suggest that there is some interrelation among them.
When the turbine was operated under cavitation conditions, the precession frequencies of the dominant VSV structures were not significantly altered.Their induced responses at different parts of the turbine test stand slightly decreased with the exception of the draft tube cone where they increased significantly.
Then, the runner and the whole rotor should be designed having their natural frequencies far away from the VSV excitation frequencies and their harmonics to avoid dangerous resonances at SNL.Additionally, the machine should operate in the SNL condition presenting the less intense VSV structures in order to minimize the wear and tear generated at this operating point.
As a future work, numerical simulations replicating the experimental tests should be performed for a better understanding of this particular flow field.The numerical results will help to understand the morphology of these structures and the mechanism driving the change in precession frequency and intensity observed experimentally.

Figure 1 .
Figure 1.Schematic of the test stand with Section A-A and heights Z2 and Zr indicated (t matic of the vaneless space with dimensions (B0, V0 and D) (bottom left).Top view of sec (bottom right).

Figure 1 .
Figure 1.Schematic of the test stand with Section A-A and heights Z 2 and Z r indicated (top).Schematic of the vaneless space with dimensions (B 0 , V 0 and D) (bottom left).Top view of section A-A (bottom right).

Figure 2 .
Figure 2. Position of the sensors mounted on the Kaplan turbine model.

Figure 2 .
Figure 2. Position of the sensors mounted on the Kaplan turbine model.

Figure 4
Figure 4 presents the TGV and SBR spectra showing the frequency peaks correspon ing to 2V and 3V at all SNL conditions.Although the frequencies  and  correspon ing to 2V and 3V change with the SNL condition, their combination, according to Equati (8), always results in constant values of  = 2 and  = 3, respectively.In addition to these dominant 2V and 3V structures, two additional frequency pea with significantly lower amplitudes can also be distinguished in the spectra shown in F ure 4. For the frequency peaks highlighted in orange,  equates to 4, and the structu will be referred to as 4V.For the frequency peaks highlighted in purple, inducing l strains on the blades,  equates 6, and the structure will be referred to as 6VGV.Similar for the frequency peaks highlighted in green, inducing low torques on the guide van again  equates 6, and the structure will be referred to as 6VB.Surprisingly, these tw structures present the same number of vortices.Nevertheless, the 6VGV presents high

Figure 3 .
Figure 3. Spectra of the guide vane torque (TGV), the pressure (PVS A ), the blade strain (SB R ) and the shaft torque strain (SS T ) for α equal to 3.98 • .

Figure 4
Figure 4 presents the TGV and SB R spectra showing the frequency peaks corresponding to 2V and 3V at all SNL conditions.Although the frequencies f S and f R corresponding to 2V and 3V change with the SNL condition, their combination, according to Equation (8), always results in constant values of z D = 2 and z D = 3, respectively.In addition to these dominant 2V and 3V structures, two additional frequency peaks with significantly lower amplitudes can also be distinguished in the spectra shown in Figure4.For the frequency peaks highlighted in orange, z D equates to 4, and the structure will be referred to as 4V.For the frequency peaks highlighted in purple, inducing low strains on the blades, z D equates 6, and the structure will be referred to as 6V GV .Similarly, for the frequency peaks highlighted in green, inducing low torques on the guide vanes, again z D equates 6, and the structure will be referred to as 6V B .Surprisingly, these two structures present the same number of vortices.Nevertheless, the 6V GV presents higher amplitudes than the 6V B in the TGV spectra, suggesting that it occurs closer to the guide vanes.Meanwhile, 6V B presents higher amplitude than the 6V GV in the SB R spectra, confirming that it occurs closer to the runner blades.No integer values for z D could be found when combining the frequencies observed, according to Equation (9), at f * S on the TGV spectra and at f * R on the SB R spectra, which confirms that there are not VSV structures precessing in the opposite direction to the turbine rotation.

Figure 4 .
Figure 4. Spectra of (a) TGV and of (b) SBR at all SNL conditions.

Figure 4 .
Figure 4. Spectra of (a) TGV and of (b) SB R at all SNL conditions.

Figure 5 .
Figure 5. Reduced precessing frequencies corresponding to each VSV structure at all SN tions.

Figure 5 .
Figure 5. Reduced precessing frequencies corresponding to each VSV structure at all SNL conditions.
Figure6a,b present a zoom around 2V and 3V in the PVS A spectra as well as the amplitudes of their induced pressure fluctuations, respectively.At small guide vanes angles from 2.6 • to 4.97 • , the 2V peaks appear at f * 0 from 1.09 to 1.32 and the 3V peaks from 1.53 to 1.79, which increase in amplitude and frequency linearly.The fact that the 3V presents higher and sharper frequency peaks than the 2V, which are lower and wider, suggests that 3V could be the dominant structure at low discharges.At 6.8 • , the amplitude of 2V drops drastically and 3V keeps its dominance showing a sharp amplitude increase.At 7.35 • , whereas no pressure fluctuations induced by 2V are detected, the 3V peak shows a f * 0 of 1.95 and the highest amplitude equal to 0.0185 bar.Suddenly, the dominance switches and only the pressure fluctuations induced by 2V are measured at 7.86 • and 8.13 • with f * 0

Figure 6 .
Figure 6.(a) Spectra and (b) amplitudes of the frequency peaks of 2V and 3V, on PVSA at all SNL conditions.

Figure 6 .
Figure 6.(a) Spectra and (b) amplitudes of the frequency peaks of 2V and 3V, on PVS A at all SNL conditions.

Figure 6 .
Figure7shows the amplitude of the 2V and 3V frequency peaks detected in the SB R , SS T and TGV spectra.A first observation is that the amplitude evolutions resemble those observed on the pressure signals, which confirms that they are induced by the same phenomena detected from different locations.Additionally, it must be noted that although most sensors do not detect the 2V at 7.35 • and the 3V at 7.86 • and 8.13 • , the SB R and TGV spectra show small peaks confirming the coexistence of both structures at all SNL conditions and reinforcing the hypothesis that they are not independent.Among all strain measurements, the blade ones (SB R ) present higher values than the shaft torque ones (SS T ) with a maximum of 0.55 µm/m induced by the 2V at 8.13 • and 0.9 µm/m by the 3V at 7.35 • .Similarly, the TGV measurements also present a maximum of 0.16 Nm induced by 2V at 7.86 • and 0.21 Nm by 3V at 7.35 • .It must be noted that the maximum induced responses by 3V present higher values than 2V, unlike as observed in the pressures.

Figure 7 .
Figure 7. Amplitudes of the frequency peaks of 2V and 3V on (a) SB R , (b) SS T and (c) TGV, at all SNL conditions.

Figure 7 .
Figure 7. Amplitudes of the frequency peaks of 2V and 3V on (a) SBR, (b) SST and (c) T conditions.

Table 1 .
Guide vane and runner blade angles, discharge coefficient, Thoma number and cavitation presence

Table 3 .
Reduced frequencies calculated with Equation (11) that should be detected from an off-board sensor measuring the vibrations transmitted from the rotor for 2V and 3V.

Table 3 .
Reduced frequencies calculated with Equation (11) that should be detected board sensor measuring the vibrations transmitted from the rotor for 2V and 3V.