Experimental Investigation of Part Load Vortex Rope Mitigation With Rod Protrusion in an Axial Turbine

The present paper investigates the rotating vortex rope (RVR) mitigation on an axial turbine model by the radial protrusion of four cylindrical rods into the draft tube. RVR mitigation is of particular interest due to the unfavorable pressure pulsations it induces in the hydraulic circuit that can affect turbine life and performance. The protrusion lengths, which were the sameamong the four rods,were variedaccording to a predeﬁned sequence. The experiments were performed under four part-load regimes ranging from upper part load to deep part load.Time-resolvedpressuremeasurementswereconductedattwosectionsonthedrafttubewallalongwithhigh-speedvideographyandefﬁciencymeasurementtoinvestigatetheeffect ofthemitigationtechniqueontheRVRcharacteristicsandturbineperformance.Therecordedpressuredataweredecomposedandstudiedthroughspectralanalyses,phase-averaging,andstatisticalanalysesoftheRVRfrequencyandpeak-to-peakpressureamplitudedistributions.Theresultsshoweddifferentlevelsofpressureamplitudemitigation rangingfromapproximately10 % to 85 % depending on the operating condition, protrusion length, and the method of analysis. The hydraulic efﬁciency of the turbine decreased by a maximum of 3.5 % that of the best efﬁciency point (BEP) with the implementation of the mitigation technique. The variations in the obtained mitigation levels and efﬁciencies depending on protrusion length and operating condition indicate the need for the implementation of a feedback-loop controller. Thus, the protrusion length can be actively optimized based on the desired mitigation target. [DOI: 10.1115/1.4064610]


Introduction
With the increased share of intermittent renewable energy sources such as wind and solar in electricity generation, the role of hydropower is shifting toward an effective tool in balancing the power grid.This necessitates more flexibility in the operation of hydraulic turbines as they are expected to operate at a wider range than the optimal condition-they were originally designed for known as the best efficiency point (BEP).The extended employment of hydraulic turbines under off-design conditions exposes the machines to adverse effects on efficiency, lifespan, and structural integrity.Among these operations, the part-load (PL) condition is particularly critical due to intense low-frequency pressure fluctuations that introduce increased turbine vibrations [1].These fluctuations may lead to the occurrence of rotor resonance [2].Thus, the urge to avoid such conditions limits the turbine operating range, restricts the machine's flexibility, and jeopardizes grid stability.
Under PL operation, a significant residual swirl enters the turbine draft tube.Swirl deceleration in the conical draft tube results in flow separation and the formation of a quasi-stagnant zone in the center.Global instability analyses of the draft tube flow indicate that the RVR forms because of swirling flow instability, with the interface between the quasi-stagnant zone and its surrounding flow being the source of instability [3,4].An RVR frequency ranging from 0.15 to 0.4 of the runner frequency has been reported in the literature, depending on the specific setup and PL conditions investigated [5][6][7][8].The RVR precession is the main source of low-frequency pressure oscillations under PL, and its amplitude has been reported to be up to 10 times larger than those induced by the runner rotation [8].The RVR-induced pressure oscillations can be decomposed into asynchronous, and synchronous modes traveling in the circumferential, and axial directions, respectively [9].The rotational and axial nature of the two modes causes the former to be a local phenomenon in the draft tube while the latter propagates throughout the conduit in the form of a standing wave [10].The RVR precession reduces the turbine efficiency, and the pressure waves induced can cause power swings, wear, and tear over time [11][12][13].Thus, it is crucial to find remedies to mitigate the impact of this phenomenon, especially given the increased frequency of PL operations in hydropower.
The techniques proposed for RVR mitigation are categorized into fluid-based, and geometrical methods [14].The fluid-based methods include air injection, water injection, or a combination of both, mostly inside the draft tube [15].Aeration can take place axially (through the runner), peripherally (discharge ring), or from the wicket gates in the spiral casing.Air injection from the runner cone has been reported to be less costly in energy with the most favorable impact on efficiency at low airflow rates [16].Numerical investigations by Qian et al. [17] showed that pressure pulsations decreased in the draft tube with air injection, and the cross-sectional distribution of the pressure became more uniform.Lou et al. [18] showed that low airflow rates introduced from the shaft of a Francis turbine lead to swirl deceleration hence alleviating the pressure pulsations in the draft tube at PL.At higher airflow rates, the vortex structure shifted from a spiral form to a stable cylindrical form, further reducing the oscillations, while the drop in efficiency increased.The quasi-stagnant region in the draft tube is known to be the source of instability that leads to RVR formation under PL [19].Hence, extensive investigations have been performed to inject water from the shaft to introduce momentum into this region [20][21][22][23][24][25].Moreover, peripheral injection is another method aimed at reducing the swirl close to the draft tube wall [26].Experimental investigation of cone water injection in a swirl generator showed significant rotating mode mitigation, pressure recovery improvement, and notable plunging mode aggravation for injection rates above 11.5% [21].The high flow rates required to effectively mitigate the RVR is the limiting factor in the applicability of fluid-based methods.In addition, flow injection from the runner cone demands complex and extensive modifications to the existing power plants.Javadi et al. [22] investigated circumferential jet injections from the sides of the runner cone in a swirl generator.The RVR-induced pressure fluctuations were successfully mitigated, and the volume of the RVR was reduced for an injection rate below 3%.Bosioc et al. [24] showed in their velocity measurements along the conical diffuser of a swirl generator that water injection from the runner crown caused the increase of axial flow in the draft tube center.Thus, the deficit of axial momentum in the quasi-stagnant region is eliminated, and the RVR is mitigated.
Unlike the fluid-based methods, geometry-based RVR mitigation methods are various in their design and diverse in their impact.Some of the more widely studied methods in this realm include the installation of fins, baffles, j-grooves, and guide vanes in the draft tube [27][28][29][30][31][32][33][34].More recently, more novel techniques have been proposed, such as the inclination of the draft tube, adjusting the diffuser outlet cross section using a diaphragm, triggering RVR dissipation with the installation of a perforated conical obstacle inside the draft tube, and the installation of a free-spinning runner downstream of the main runner cone [35][36][37][38][39][40].Baffles, fins, and j-grooves reduce the swirl in the periphery of the stagnant region by redirecting the flow toward the center of the draft tube [29,32].This causes an increase of momentum in the central low-pressure zone, contributing to a more uniform pressure distribution across the draft tube cross section due to the stagnant zone shrinkage.Implementing a diaphragm at the draft tube outlet led to significant mitigation of the rotating mode and improvement of the pressure recovery.In contrast, the plunging mode was aggravated [35].At higher shutter rates of the diaphragm mechanism, the vortex form shifted from a spiral into a bubble shape that was perceived to be less harmful.Joy et al. [33] numerically studied RVR mitigation using different arrangements of guide vanes installed in the draft tube and achieved optimum mitigation of 95%.They observed that the installation of the guide vanes restricts the RVR motion to the center of the draft tube.In another study, the impact of the draft tube guide vanes with variable angles was experimentally investigated on the turbine performance [34].The results showed a minimal efficiency loss of below 0.6% while achieving complete mitigation of the RVR.
The methods investigated so far are aimed directly at either manipulating the swirling flow around the RVR or the stagnant region in the center of the draft tube.The present authors have previously introduced a new method that can combine the effects mentioned in the sense that it can be both directed toward the tangential flow region and stagnant region.Oscillating and stationary protrusion of rods into the draft tube flow field were investigated, and a maximum RVR mitigation of 77% was obtained on a reduced turbine model with a runner diameter of 100 mm [41,42].In these investigations, four cylindrical rods that could be traversed in the radial direction were installed in the draft tube.For protrusions less than 60% of the draft tube inlet radius, the amplitude of the RVR rotating mode was reduced due to the swirl deceleration in the draft tube.However, with the increase of protrusion, significant mitigation of the RVR was observed both in plunging and rotating modes for the stationary rod protrusion experiments.Similarly, an optimal protrusion length was found at 80% of the draft tube radius, where the highest mitigation materialized in both modes.It was stipulated that the disruption of the RVR and stagnant region interface caused the significant mitigation.The proposed technique can be adjusted and optimized based on the operating conditions and the corresponding status of the RVR with the protrusion length.In addition, unless the draft tube is embedded in concrete, this method requires only minimal changes to the draft tube without the need to modify the turbine pit components.In this paper, the proposed method is further investigated on a larger scale, IEC-certified model turbine at a wider range of operating conditions.RVR mitigation employing stationary rod protrusion was conducted at four PL operating conditions to examine the impact of the method on the turbine performance and draft tube pressure pulsations.The results were studied based on time-resolved pressure measurements on the draft tube wall at two axial sections.The effect of rod protrusion on the RVR periodic behavior was studied based on a statistical analysis of the pressure measurements.Thus, a better understanding of the underlying mechanisms could be obtained by combining pressure data analyses with high-speed videography (HSV).[7,8,10,43,44].The model turbine runner has a diameter of D n ¼ 0.4 m and is equipped with six adjustable blades.The distributor comprises 20 guide vanes and 18 stay vanes.The model is installed in a closed-loop circuit between two pressure tanks as shown in Fig. 1.The tank pressures can be adjusted separately, which thus makes it possible to conduct experiments under cavitating conditions.The model turbine is geometrically and hydraulically similar to the prototype (in accordance with IEC 60193 [45]) and has  The operation of Kaplan turbines is controlled through a doubleregulation mechanism by adjusting both the angle of the guide vanes and the runner blades.Runner blade pitch angle is adjusted according to the guide vane opening to offer the best efficiency (no swirl) under the different heads.However, this regulation is performed slowly to avoid the unfavorable flow phenomena including RVR formation under PL.Thus, the addition of a new degree-of-freedom to the system may facilitate a faster regulation through the prevention or alleviation of the RVR formation.The complex blade pitch regulation mechanism is also prone to malfunctions, and it is rather common for Kaplan turbines to operate at a fixed blade angle.Therefore, the experiments in this study were conducted on the same propeller curve with a constant blade angle of b ¼ 0.8 deg.Hence, the turbine operation was resemblant to propeller turbines.The fixed blade angle in the present model turbine produces flow conditions similar to a single-regulated turbine as shown in Ref. [8].

Material and Methods
Four different PL conditions were investigated with guide vane opening angles of a ¼ 22 deg, 20 deg, 16 deg, and 14 deg.Hence, RVR mitigation was examined under the operating conditions of its presence ranging from the upper PL to deep PL.The head and runner rotational frequency were kept constant in all measurements at H ¼ 7.5 m and n ¼ 870 RPM (f n ¼ 14.5 Hz), with uncertainties equal to 0.02% and 0.39% of the average values, respectively.The flowrate was controlled using a pump for every operating condition to keep the head constant according to the following equation: where q, DP, g, A 1 , and A 2 denote the water density, pressure difference between the spiral casing inlet and draft tube outlet, gravitational acceleration, waterway cross section at the spiral casing inlet, and cross section at the draft tube outlet, respectively.The parameters corresponding to each operating condition are presented in Table 1.
and n ed ¼ nD n = ffiffiffiffiffiffi gH p represent the discharge factor and speed factor of the turbine, respectively.
2.2 Mitigation System.Four rods with a diameter of D rod ¼ 0.15 D n were installed at an axial location 0.36 D n downstream of the end of the hub and equally separated around the draft tube periphery.Previous downscaled model experiments, the results of which are partly reported in Ref. [42], showed that the use of four rods can provide effective RVR mitigation.Thus, the experiments in the present paper were performed using four rods to examine the concept of RVR mitigation using rod protrusion on a larger model scale.A schematic arrangement of the rods is shown in Fig. 2. To insert the rods radially into the draft tube, four Kollmorgen AKM2G-33H servomotors were used.The rotational motion of the servomotors was translated into linear motion using four PC32 linear actuators from Thomson Linear.The actuators offer a nominal stroke of up to 200 mm with an accuracy of 0.01 mm.All the actuators were moved simultaneously to provide equal protrusion lengths for all the rods.The protrusion setup used in these experiments could provide protrusion lengths of up to L Ã ¼ L=R DT ¼ 0.79, where R DT is the draft tube radius at the position of the rod axis (R DT ¼ 0.226 m).A LabVIEW program was developed to control the rod motion, and the protrusion length could be monitored from the servomotor feedback signals.Thus, the protrusion lengths of interest were fed to the program, and the rods were automatically inserted into the draft tube according to the defined sequence that is described in Sec.2.3.

Instruments and Measurements.
For the pressure measurements, a total of eight piezo-electric transducers (113B28 PCB) were used with a range, resolution, and rise time of 344.7 kPa, 7 Pa, and 1 ls, respectively.The transducers were flush-mounted on the draft tube wall at two sections along the draft tube axis.Section A was located 0.25 D n upstream of the runner cone lower end while the recordings at section B were performed 0.75 D n downstream of the runner cone lower end.Section A included the sensors which henceforth will be named P 1 , P 2 , P 3 , and P 4 , and section B included the sensors denoted as P 5 , P 6 , P 7 , and P 8 .The location of the sensors relative to the rods is schematically displayed in Fig. 2. The sensors at each section were separated by 90 deg from their neighboring ones.In addition, the sensors with the same circumferential orientation were separated by 22.5 deg from the nearest rod.Such arrangement of the sensors not only allowed the decomposition of the signals into the rotating and plunging modes but also enabled investigation on the effects of the rod protrusion both upstream and downstream the rods.
To visualize the RVR, and to enable a qualitative measure of how the rods affect the RVR spatial structure, HSV was performed under cavitating conditions.The HSV recordings were performed using a Redlake MotionPro X3 High-speed camera at 100 fps.To increase the clarity of the vortex rope, small amounts of atmospheric air were introduced to the low-pressure vortex rope from the runner cone tip through the shaft.The low pressure in the vortex cavity created a suction for atmospheric air to travel through the shaft.The amount of introduced air was monitored by visualizing the vortex structure.The flow visualization was conducted under OP1 and OP2.
All the signals were recorded with a National Instruments (NI) compactRIO 9014 chassis at a sampling frequency of 2 kHz.The analog pressure signals were converted using two NI 9215 ADC  cards with a resolution of 16 bits and a voltage range of 610 V.In addition to the pressure data, the signals corresponding to the global flow parameters and turbine operation were converted using a 16-bit NI 9205 ADC card.The parameters recorded through NI 9205 are presented in Table 2 with their measurement uncertainties.
For each operating condition, the experiments were performed for the protrusion lengths varying from L Ã ¼ 0 to L Ã ¼ 0.79 in an increasing order with intervals of DL Ã ' 0.05.The pressure signals were continuously recorded during each operating condition, and the rods were in position at each step for a duration of t ¼ 180 s.Thus, enough data could be accumulated for the analysis of the RVR.All the cases were repeated at least three times to assure the repeatability of the results.

Data Analysis
The analysis of the data was performed using MATLAB scripts.After the performance of a sensitivity analysis, a duration between t ¼ 15 s and t ¼ 75 s after each change of protrusion length was chosen to study so that the mean flow parameters could stabilize after the change of rod position.The fluctuating part of the pressure signals was extracted based on the following equation [46]: where P i t ð Þ, P i , and Pi t ð Þ are the time-resolved, time-averaged, and fluctuating components of the pressure signals collected by the i-th sensor.The signals were then decomposed based on Eqs. ( 3) and ( 4) to obtain the synchronous and asynchronous modes' contribution to the pressure data where P p t ð Þ and P r,i t ð Þ are the plunging mode of the pressure fluctuations at the section of interest and rotating mode of the sensor P i , respectively.Discrete Fourier transform (DFT) using Welch's method was also performed on the fluctuating part of the raw and decomposed signals to enable a spectral analysis of the pressure oscillations [47].Thus, the general impact of rod protrusion on the RVR-related frequencies could be explored, and the mitigation trends could be extracted as a function of the protrusion length and operating condition.

Statistical Representation of the Rotating Vortex Rope
Amplitudes and Frequencies.To investigate the periodic behavior of the RVR, a statistical analysis introduced by Shiraghaee et al. [48] was applied to the pressure data.The utilized procedure is graphically represented in Fig. 3. Ideally, the RVR induces sinusoidal pressure oscillations.However, the real pressure data contains higher frequencies and noise as shown by the black line in Fig. 3(a).Therefore, the RVR frequency was extracted from the Fourier spectra for each case, and the corresponding pressure data was bandpass filtered around the RVR frequency (f BPF ' f RVR 60:5f RVR ).From the consequent signals, the peaks were identified as the initial points of the RVR revolutions.Thus, the data collected between every two consecutive peaks is attributed to one RVR revolution, and the difference between the maximum and minimum amplitude of every cycle corresponds to the peak-to-peak pressure amplitude (P2P) of the cycle.Also, the time from every peak to its subsequent one is the duration of the cycle which is the inverse of the respective frequency (Fig. 3(a)).
After the RVR P2P amplitudes and frequencies were extracted from the filtered data (Figs.3(b) and 3(c)), each pressure P2P amplitude point was plotted against its corresponding RVR revolution frequency.The distribution of the points in the consequent plot represents the periodicity and amplitude spread of the RVR pressure pulsations, as shown in Fig. 3(d).Finally, the average values for the frequencies and P2P amplitudes were calculated along with their respective standard deviations to quantify both the values as well as their spread (see Fig. 3(e)).
3.2 Phase-Averaged Pressure Data.The pressure data collected at each section were also phase-averaged to study the phase difference variation between the sensors.Thus, the RVR structure could be classified when subjected to rod protrusion.To this end, the raw pressure data (from P 1 ) was bandpass filtered around the RVR frequency over the same range mentioned above.Next, the peaks were identified for one sensor on the filtered semisinusoidal signals.
Then, the beginning point and the end point of each RVR period were specified.Thereafter, the sample length between every two consecutive peaks was divided into 360 bins of 1 degree each.The average of data in each bin -that corresponds to the data of a certain phase in all the recorded RVR rotations-was calculated.Thus, the phase average of the pressure data was calculated for the number of RVR revolutions that occur within one recording of a sensor.
The same procedure was then conducted for the other sensors at both sections, except that the RVR revolution beginning points and end points of the initial sensor were used as the references for the bin discretization of the data.Thus, the phase-averaged pressure data of the sensors were drawn with respect to one another.

Coherence
Calculations.The sensors P 1 and P 3 were used in this study to quantify the coherence of pressure oscillations in the draft tube.Magnitude-squared coherence was calculated according to the following equation [49]: where G P1P3 is the cross-power spectral density of the signals collected by P 1 and P 3 while G P1 and G P3 are the power-spectral density of the aforementioned sensors, respectively.The spectral amplitude of G P1P3 is a function of the linear relationship that exists between the phase-separated P 1 and P 3 signals at different frequencies.The value of C P1P3 varies between 0 and 1, which demonstrates how well the signals collected at P 1 correspond to those captured by P 3 through a normalized measure of system linearity.In a linear system with constant parameter, C P1P3 ¼ 1 while the corresponding value equals zero when the signals P 1 and P 3 are completely unrelated at the frequency of interest.

Image
Processing.The images captured through HSV had to be processed to properly analyze the RVR visualization results.The vortex core under certain conditions -such as OP1 or when the rods reached the RVR at OP2-could not maintain the introduced air.Thus, the cavitating vortex rope under these conditions appeared as a sparse chain of bubbles while the background reflections from the draft tube wall intensified.Therefore, a procedure graphically displayed in Fig. 4 was used to reduce the background noise and enable the analysis of the images.
First, a number of consecutive images corresponding to four RVR periods were chosen for each case.The structural similarity index measure (SSIM) was employed to compare the consecutive images [50].Then, the SSIM matrices were created based on the comparison of each image with its subsequent one.For each array within the consequent SSIM matrices, the similarity indexes above 0.6 were filtered out.After similarity filtering, the values above and below a certain intensity were filtered out again to reduce the background noises.Then, the remaining intensity values were replaced by ones in their respective arrays.Thus, the image matrices were binarized into arrays of zeros and ones.Finally, the corresponding arrays were averaged among the images, and the amplitudes below 0.2 were filtered out while the remaining arrays were binarized again.
For the cases where the vortex could retain the air suctioned through the shaft, similarity filtering was not performed.Thus, only the last two steps in Fig. 4 were used.

Results and Discussions
The results in this paper are organized as follows: first, the repeatability of the draft tube pressure oscillations under the different conditions is investigated.Then, the RVR-induced pressure oscillations are discussed without mitigation based on spectral analysis of the pressure data.Next, the results of the RVR mitigation are presented in the form of spectral and statistical analyses of the pressure data.Finally, the mechanisms leading to the RVR mitigation under each operating condition are discussed based on flow visualizations, phase-averaged pressure data, and vortex rope coherence.The pressure, frequency, and protrusion length are normalized by the net head (P Ã ¼ P= qgH ð Þ), runner frequency (f Ã ¼ f =f n ), and draft tube radius at the position of the rod axis L Ã ¼ L=R DT , respectively.
4.1 Repeatability of the Pressure Oscillations.The repeatability of the pressure oscillations was studied to investigate the uncertainty of the setup in acquiring the pressure oscillation range specific to every operating condition.For this purpose, the probability distributions corresponding to five repetitions were extracted for the fluctuating part of the raw pressure signals ( PÞ captured by the sensor P 1 under each operating point.Then, the distribution ranges were divided into 100 bins, and the average probability of the different repetitions was calculated within each bin along with the associated standard error (S.E).Thus, the average probability distribution and the S.E distribution were calculated for the pressure oscillations under every operating condition as shown in Fig. 5.  [49] attributed such distribution to narrow bandwidth noise.The sinusoidal pressure oscillations induced by the RVR under these conditions possess low amplitudes.Thus, the sinusoidal shape of the RVR Oscillations does not demonstrate a significant dominance over the short-period oscillations as shown in Fig. 3(a).Two peaks appear around zero in the distribution corresponding to OP3 (see Fig. 5(c)).Such distributions are representative of sine waves with random noise [49].The strong oscillations (as implied by the distribution range) under this condition lead to the RVR-induced sinusoidal waves appearing more pronounced in comparison to high-frequency pressure oscillations contained in the raw pressure data.
The uncertainties are generally smaller in the center of the distributions corresponding to the smaller amplitudes of pressure oscillations.The uncertainties increase close to the distribution tails that are attributed to the large-amplitude oscillations corresponding to the RVR under PL condition.The S.E is the lowest under OP2, being below 5% throughout the 95% confidence interval of the distribution range.The S.E increases under deeper PL operations due to the increased randomness of the oscillations as a result of the increased flow instability.The S.E is particularly high under OP4, both at the center and the tails of the distribution-reaching 15%-due to the stochastic nature of the RVR under this condition.

Unperturbed Rotating Vortex Rope Under the Different
Part-Loads.To identify the significant sources of pressure fluctuations and the RVR strength under each condition, a Fourier spectrum of the different sensors at the draft tube inlet (section A) is represented for the unperturbed measurements in Fig. 6.
The dominant frequencies of the pressure oscillations occur at f Ã ' 0.2 and f Ã ¼ 6 for all operating points.The low-frequency peak is associated with the RVR precession, while the high-frequency peak in the small subplot corresponds to the blade passage that is stronger than its adjacent runner frequency harmonics in Kaplan turbines [10].The RVR frequency increases with the decrease of the discharge from f Ã ¼ 0.15 at OP1 to f Ã ¼ 0.21 at OP4. Theoretically, the flow at BEP leaves the runner blades in the axial direction as demonstrated in Fig. 7(a).As the discharge decreases under PL, the absolute velocity of the flow (V) decreases while the runner velocity (U) and blade angle (b) remain constant in single-regulated turbines.Therefore, the flow enters the draft tube with some tangential component (v t ) as shown in Fig. 7(b).Draft tube velocity field measurements on the U9 model verify the existence of the tangential flow component under PL condition [8].Further decrease of the discharge causes further increase of v t along with the decrease of flow axial component that is represented by v a in Fig. 7(c).Thus, the RVR frequency increases as the turbine operates at deeper PLs.
The amplitude of the RVR-related pressure oscillations is at the same level as the blade passing frequency under the upper PL (OP1) at P Ã ' 0.01, which is around 1% of the net head (see the small subplot in Fig. 6).With the increase of the tangential flow that enters the draft tube, the RVR amplitude magnifies at lower discharge until it reaches 4.8% of the net head under OP3.However, with further decrease of the discharge, the peak corresponding to the RVR spreads at OP4, and its amplitude drops to P Ã ' 0.02 while the wideband pressure pulsations in the low-frequency region (f Ã <1) intensify.The trend for the amplitude and frequency of the RVR harmonic under the different operating conditions is similar to that of the fundamental mode.In addition, another peak also appears at a slightly higher frequency (f Ã ' 0.5) than the RVR harmonic under OP4, which will be discussed in the upcoming sections.Transactions of the ASME pressure oscillations is presented in this section.To this end, the amplitudes corresponding to the RVR peaks were extracted from the Fourier spectra and averaged among the different sensors at each section.The standard deviations were also calculated for the different cases to present the variation of the induced pressure amplitudes between the sensors of the same section.
The spectral amplitudes of the RVR-related pressure pulsations at section A are displayed in Fig. 8.The RVR amplitude variation for the different sensors -that is indicated by the error bars (representing 6r)decreases in general with the onset of rod protrusion except for OP4, where it doubles from L Ã ¼ 0 to L Ã ¼ 0.09.Likewise, the RVRinduced pressure amplitudes decrease initially with protrusion at every operating point, except for OP4, where a 95% aggravation is observed.
Under OP1, the RVR is almost completely mitigated with an 83% amplitude decrease at L Ã ¼ 0.35 (see Fig. 8(a)).Further insertion of the rods causes the reappearance of the peak associated with the RVR with the corresponding variation increasing by 100%.The reduction for OP2 remains below 30% until L Ã ¼ 0.44 along with a monotonic decrease in the variations from 2r ¼ 0.01 at L Ã ¼ 0 to 2r ¼ 0.005 at L Ã ¼ 0.44.A significant decrease occurs from L Ã ¼ 0.53 both in the amplitude and the variations.The highest mitigation under this operating point is achieved at L Ã ¼ 0.62 with 53%, and 2r drops to 34% of the average value compared to 72% of the unperturbed case.Further increase of the L Ã leads to a small magnification of the amplitudes at L Ã ¼ 0.70 whereas the variations exceed 100% of the average amplitude.The RVR under OP3 induces the highest amplitudes of pressure oscillations, as shown in Fig. 6.Opposite to the other cases, increased protrusion does not yield any significant mitigation under this condition (Fig. 8(c)).Only minor reductions below 20% are obtained at some lengths, and the least amplitude variation for OP3 is obtained at L Ã ¼ 0.53 with 2r ¼ 0.005.At OP4, the RVR amplitudes are first intensified with the protrusion, and their variation increases by nearly 100% at L Ã ¼ 0.09 as shown in Fig. 8(d).They then plummet until the RVR is almost completely damped at L Ã ¼ 0.44 (83% mitigation).
Figure 9 displays the spectral amplitudes of the RVR-related pressure pulsations in section B. The unperturbed RVR amplitude induced under OP1 is about 30% higher in section B than that induced in section A. In the case of OP2, the amplitudes observed at both sections are almost similar.However, the amplitudes significantly decrease at section B under OP3 and OP4 to P Ã ' 0.15 and P Ã ' 0.06, respectively.The substantial decrease at section B for OP3 and OP4 has been attributed to the RVR dissipation downstream in a previous study [48].
The protrusion of the rods has the same effect on the variation of amplitude at section B as observed upstream.The variations decrease at first and then increase at longer protrusion lengths.But unlike section A, protrusion does not lead to reduction for all operating conditions.Under OP1 and OP4, the trends are similar in sections B and A. However, at OP2, the RVR peak amplitude is magnified up to 75% at L Ã ¼ 0.35 while 2r steadily decreases from 0.012 to 0.008.The amplitudes then decrease -with a minimal variation of 2r ¼ 0.002 at L Ã ¼ 0.62-and increase again with further protrusion.At OP3, the amplitudes slightly increase with a small reduction of the variations, and the general impact on the RVR is insignificant similar to section A. Overall, one should consider that the RVR mitigation is of highest importance in the proximity of the turbine moving parts, which is located upstream of the rods.

4.3.2
The Analyses of the Decomposed Pressure Data.The variation of the pressure levels for the different sensors' rotating modes extracted in section A was below 5% after the decomposition.Also, given the global nature of the plunging mode, the trends obtained for the latter were similar in both sections.In other words, the amplitude of the RVR plunging mode decreased initially and increased afterward at section B for OP1, OP2, and OP3.For OP4, the mentioned amplitude at section B increased at L Ã ¼ 0.09, and then disappeared as in section A. Consequently, the difference in the spectral trends between the two sections is associated with the local differences caused by the rotating mode [7].Thus, only the rotating mode for the sensor P 1 (denoted as P r,1 ) and the plunging mode for section A (denoted as P p,A ) were chosen for the representation of the decomposed spectra in Fig. 10.
With an amplitude six times larger than the rotating component, the plunging mode dominates the latter at OP1 (Fig. 10(a)), indicating that the pressure oscillations occur mostly in the axial direction under the upper PL condition.The earlier appearance of the RVR plunging mode at upper PL operations as observed in Ref. [6] explains its dominance over the rotating mode under such conditions.The ratio of the rotating mode to plunging mode -for the unperturbed RVR-increases from 0.14 at OP1 to 9 at OP4, as the turbine operates at lower PL conditions.With the protrusion of the rods, the plunging mode initially decreases in all operating conditions at shorter lengths.However, the axial pressure oscillations increase at higher L Ã for OP1 and OP2 due to the effects on the RVR behavior, which will be scrutinized in the upcoming sections.
Under OP1, the plunging and rotating modes drop significantly until L Ã ¼ 0.35.The distinct peaks in both modes then almost disappear to scattered peaks in a wide-band frequency at L Ã ¼ 0.35 with 80% and 53% mitigations for the plunging and rotating modes, respectively.Further increase of the L Ã leads to the reappearance of distinct RVR peaks with the plunging mode reaching P Ã p,A ¼ 0.003 at L Ã ¼0.70.At OP2, small reductions (less than 15% for P Ã r,1 and less than 30% for P Ã p,A ) are obtained for the amplitude of both modes at shorter lengths before L Ã ¼ 0.53 (Fig. 10(b)).For L Ã ¼ 0.53, a mitigation of 77% is observed for the plunging mode while the rotating mode is reduced by 24%.Afterwards, both modes undergo significant amplitude mitigation at L Ã ¼ 0.62, where a 50% mitigation of P Ã r,1 and 80% mitigation of P Ã p,A is achieved at the corresponding RVR frequency.Further protrusion of the rods to L Ã ¼ 0.70 results in a 460% aggravation of the plunging mode (compared to L Ã ¼ 0.62) while the rotating modes remain unchanged.The decomposition of the signals demonstrates that most of the RVR peak aggravation in Fig. 8(b) that is observed between L Ã ¼ 0.62 and L Ã ¼ 0.70 occurs due to the increased oscillations in the axial direction (plunging mode).At OP3, the plunging mode decreases by 50% with rod protrusion until L Ã ¼ 0.44 (Fig. 10(c)).The rotating mode does not change significantly; only minor reductions can be observed at smaller lengths.The minimum rotating mode is obtained for OP3 at L Ã ¼ 0.26 with a 15% reduction.At OP4, the spectral magnitude of the RVR peak increases drastically for both modes at L Ã ¼ 0.09.The rotating mode increases by nearly 100% from P Ã r,1 ¼ 0.017 at L Ã ¼ 0 while the small amplitude of P Ã p,A under this condition increases from P Ã p,A ¼ 0.002 at L Ã ¼ 0 to P Ã p,A ¼ 0.004 at L Ã ¼ 0.09.With the increase of L Ã , the amplitude of the RVR-associated peaks decreases continuously.From L Ã ¼ 0.26, the plunging mode disappears completely, and the rotating mode of the RVR appears in the form of scattered peaks in a wide-band frequency range.The minimum peak amplitude of the rotating mode for OP4 is observed at L Ã ¼ 0.44 with an RVR mitigation of 80% (see Fig. 10(d)).Nevertheless, the pressure peak at f Ã ' 0.5 intensifies at longer protrusions in the rotating mode.The observations show that the changes in the amplitude of the RVRinduced oscillations under OP4 are in opposite trend compared to the mentioned frequency.The amplitude of the latter peak initially decreases by 57% from P Ã r,1 ' 0.0067 at L Ã ¼ 0, and then reappear with the increase of L Ã .Further increase of L Ã is accompanied by increased amplification of the aforementioned peak until it reaches P Ã r,1 ' 0.019 at L Ã ¼ 0.66.
To examine the relationship between the behavior of the RVR and the peak appearing at f Ã ' 0.5, the P2P amplitudes were obtained for P Ã r,1 signals filtered around f RVR and f Ã ' 0.5.Both filtered signals exhibited moments of notable fluctuations, when the P2P amplitude occasionally dropped to almost zero and then rose again as shown in Fig. 11(a).These random occasions of low amplitude and short period, correspond to the collapse of the investigated signal followed by its reappearance.To detect and quantify the occurrence of these nonperiodic collapses, a detection function (DF) was used as introduced in Ref. [51].A large averaging window equivalent to Dt¼ 50% of the time-series signal was defined to include the time scale of the random collapses.Then, the localized pressure amplitude variances were calculated as follows: For the negative values of Var, a threshold level of k ¼ 0.5 was applied to the DF as follows: where P rms is the root-mean-square of P Ã r,1 over the total recording time.Thus, the occurrence of collapses was detected for the RVR and the peak at f Ã ' 0.5 as shown in Fig. 11(b).The instances of collapse are noncoincident for f RVR and f Ã ' 0.5 implying that the collapse of one contributes to the aggravation of the other.A possible reason is that the flow under OP4 is in a bistable state that shifts between two topologies.The number of collapses was divided by the number total periods for the different protrusion length.Figure 11(c) displays the effect of rod protrusion on the collapse rate of the RVR and f Ã ' 0.5.The unperturbed RVR and the peak at f Ã ' 0.5 have collapse rates of 16% and 18%, respectively.For L Ã ¼ 0.09, the RVR collapse rate drops significantly to 3% where a spectral aggravation was observed in Fig. 10(d).The collapse rate f Ã ' 0.5, however, changes slightly (by less than 1%).Further insertion of the rods further facilitates the RVR collapsing, thereby contributing to the increase of the RVR collapse rate along with a significant reduction of the collapse rate for f Ã ' 0.5 until L Ã ¼ 0.62, where respective rates of 29% and 0.5% were obtained.For L Ã ¼0.7, the RVR collapse rate decreases to 26% while the oscillations at f Ã ' 0.5 do not collapse at all.The aggravation of the oscillations at f Ã ' 0.5 as the RVR is mitigated occurred specifically under OP4.A similar phenomenon has been observed at lower PL in Ref. [52] when investigating the draft tube guide vane system in a Francis turbine.Therein, increased mitigation levels of the RVR-related spectral peak were accompanied by the increase of an aggravated peak at a higher frequency.However, it was not examined whether the oscillations at these two frequencies were noncoincident or not.
To investigate the impact of rods on the RVR periodic characteristics, the results of the statistical analyses on P Ã r,1 signals are displayed in Fig. 12.The amplitude of P Ã r,1 for the unperturbed  RVR at OP1 is densely distributed around 0.3% of the total net head (Fig. 12(a)).The concentrated distribution pertains to the high periodicity and dense amplitude spread of P Ã r,1 pulsations for the vortex structure under this condition, which means that the vortex path remains relatively the same over different revolutions.The amplitude of the RVR-induced P Ã r,1 oscillations decreases by 25% at L Ã ¼ 0.26, and the frequency distribution range increases by nine times reaching 2r f ¼ 0.14.Consequently, the RVR becomes increasingly unstable, as is manifested by the frequency variations.Further protrusion (L Ã ¼ 0.70) causes the distribution to be denser again with the variations in the frequency and amplitude reaching 2r f ¼ 0.05 and 2r P ' 0.002, respectively.Also, the amplitude of the oscillation increases notably at L Ã ¼ 0.70 to 30% higher than that of the unperturbed state.However, the increased randomness of the RVR pressure oscillations at L Ã ¼ 0.70 compared to L Ã ¼ 0 causes the appearance of a 30% smaller peak for the former in the Fourier spectra as shown in Fig. 10(a).
The distribution of P Ã r,1 pressure oscillations for the unperturbed RVR at OP2 is located at f Ã ' 0.18 and P Ã r,1 ¼ 0.03 with respective variations of 2r f ¼ 0.004 and 2r P ' 0.002.By inserting the rods and increasing the protrusion length to L Ã ¼ 0.53, the pressure amplitudes and their variations do not change significantly.Only a decrease of 13% is observed at L Ã ¼ 0.35 which later drops to below 10% at L Ã ¼ 0.53 as shown in Fig. 12(b).Nevertheless, the RVR frequency increases by more than 40% at L Ã ¼ 0.53 and r f increases by 25%.The decreased periodicity of the RVR at L Ã ¼ 0.53 explains its smaller spectral amplitude (Fig. 10(b)) compared to L Ã ¼ 0.35 despite the actual pressure amplitudes of the latter being 15% smaller in Fig. 12(b).The distributions become significantly more scattered after L Ã ¼ 0.53 (particularly in terms of frequency), showcasing a high impact on the RVR periodicity and possible changes in its structure.In addition, the pressure amplitudes drop substantially, which explains the notable drop in the RVR-related peak in Fig. 10(b).The distribution parameters for L Ã ¼ 0.62 and L Ã ¼ 0.70 are presented in Table 3.
At a protrusion length of L Ã ¼ 0.70, a 50% mitigation of the rotating mode is achieved, which is the maximum mitigation in terms of the amplitude under OP2.However, the distribution for L Ã ¼ 0.70 is denser than that of L Ã ¼ 0.62, which implies its higher periodicity, leading to slightly higher amplitude in the spectra (Fig. 10(b)).
No significant reduction of the RVR rotating mode amplitude can be observed for OP3 in Fig. 12(c).The decrement in the amplitudes is below 10% (less than 1% of the net head) for all lengths of rod protrusion.However, the RVR frequency is influenced by the presence of the rods in a pattern similar to that of OP2.The distributions also become considerably denser with the increase of protrusion length (44% decrease in r P and 56% decrease in r f at L Ã ¼ 0.62), showing that the rods in fact improve the RVR periodicity under OP3.
The unperturbed RVR structure is highly nonperiodic under OP4, as illustrated by the wide distribution both in frequency and amplitude (Fig. 12(d)).The decrease of the guide vane opening angle leads to the expansion of the quasi-stagnant region leading to the RVR precession closer to the wall as discussed in the literature [48].This results in the erratic behavior of the RVR, which collapses and reappears subsequently, as shown in Fig. 11.At L Ã ¼ 0.09, the periodicity of the RVR increases by 75%.The periods of high frequency and low amplitude discussed earlier, do not occur at L Ã ¼ 0.09.Consequently, spectral analyses display almost a 100% aggravation from L Ã ¼ 0 to L Ã ¼ 0.09 as shown in Fig. 10(d).The increase of L Ã leads to further disruption of the RVR periodic characteristics, and a mitigation of 66% is obtained at L Ã ¼ 0.70.Moreover, the average amplitude of P Ã r,1 for L Ã ¼ 0.70 shows a difference equal to 1.6% of the net head compared to L Ã ¼ 0.26.In contrast, the changes in the spectral amplitude of the RVR were insignificant from L Ã ¼ 0.26 due to the high impact of the RVR periodicity on its spectral amplitude (see Fig. 10(d)).
To summarize the results of the RVR mitigation, the best percentages obtained from Fourier spectra and the results obtained for the same L Ã (as the best Fourier-based mitigation) based on the statistical method are presented in Table 4.
It has been briefly discussed so far that the insertion of the rods leads to changes in the frequency of pressure oscillations attributed to the RVR.To investigate the pattern of the RVR frequency function of the protrusion length, the frequency values corresponding to the RVR peak in the pressure spectra are extracted and displayed Fig. 13.It is worth mentioning that when the RVRinduced oscillations appeared as scattered peaks in a wideband range, the frequencies corresponding to the maximum spectral pressure amplitude in the RVR-expected range were chosen.
The RVR frequency under OP1 undergoes minor increment levels of below 10% up to L Ã ¼ 0.26.At L Ã ¼ 0.35, an abrupt increase is observed in the RVR frequency to f RVR ' 0.19, where the spectral RVR peak appears in a wideband frequency range in Fig. 10(a).The maximum RVR frequency under OP1 is obtained for L Ã ¼ 0.66 with a 31% increment from the unperturbed state.
The patterns for the other operating conditions are similar to one another.The RVR frequency increases initially with the protrusion of the rod, followed by a decrement from the maximum point.However, the protrusion length where the highest frequency is obtained (L Ã fmax ) decreases from OP2 to OP4.In addition, f RVR shows the highest change with protrusion under OP2 with a 47% increase compared to the unperturbed case.The least impact of protrusion on f RVR is observed for OP3 where the maximum RVR frequency is only 17% higher than that of the unperturbed case.

Mitigation Mechanisms
4.4.1 Flow Visualization. Figure 14 shows the averaged and instantaneous cavitating vortex visualizations for different protrusion lengths under OP2.
The precession of the unperturbed RVR takes place over a wide pitch as implied by Fig. 14(a).With the insertion of the rods by L Ã ¼ Transactions of the ASME 0.26, the RVR path becomes constricted similar to the observations made for the draft tube guide vane installations in Ref. [33].Consequently, the average vortex spread angle decreases from 12.5 deg to 10.5 deg (Fig. 14(b)).Short protrusions of the rods cause flow redirection to the draft tube center which leads to the stretch of the vortical structure and ultimately its constriction to the center as discussed in Ref. [53].Thus, the rotational frequency of the vortex rope increases in order to conserve the angular momentum.The increase of the RVR frequency continues until the rods reach the RVR path.Since the stagnant region is wider at deeper PL operations, the RVR pitch is wider too.Consequently, the maximum RVR frequency is reached at a shorter L Ã for these operations.At L Ã ¼ 0.70 under OP2, the rods have already reached the vortex structure as shown in Fig. 14(f).Vortex pitch is notably smaller while it is bent toward the rod number 4 (see Fig. 2).The small RVR frequency under this condition (compared to the maximum at L Ã ¼ 0.53) is, therefore, an outcome of the rods contacting the RVR structure.Under OP1, the pattern for the RVR frequency was different from other conditions.For the other conditions, a gradual and significant increase was observed in the RVR frequency followed by a notable decrease with the increment of protrusion.For OP1, however, two sections of slight changes were separated by an abrupt jump in the frequency at L Ã ¼ 0.35.Under OP1, the unperturbed vortex path is limited and is mostly concentrated in the draft tube center as shown in Fig. 15(a).The stagnant region at upper PL operations partially forms at the bottom of the draft tube [10].Therefore, the vortex pitch is small under such conditions, and longer protrusions are required to reach it.Rod protrusion does not lead to any significant change in the vortex frequency until L Ã ¼ 0.35.At L Ã ¼ 0.4, the vortex path is wider compared to L Ã ¼ 0 (Fig. 15(b)).Thus, flow redirection to the draft tube center causes the vortex form to shift from a relatively straight shape, into a spiral form which explains the jump in the RVR frequency.
4.4.2Phase-Averaged Pressure Data. Figure 16 displays the phase-averaged signals of the different sensors for the RVR under OP1.For L Ã ¼ 0 at section A, the RVR is almost symmetric and located in the draft tube center as the pressure signals are nearly in phase with similar phase-averaged amplitudes.The small pitch of the vortex in the upstream section results in a very small phase difference (Du ' 20 deg) and amplitude difference at section A (Fig. 16(a)).The in-phase nature of the oscillations at section A implies that they occur mostly in the form of plunging mode, which is in line with the results from the decomposed spectra (Fig. 10(a)).The phase difference between the sensors at section B increases to Du ' 40 deg.In addition, the amplitudes excited at different sensors located at section B vary substantially compared to section A, pointing out an increase in the RVR asymmetry close to the elbow (see Fig. 16(b)).The amplitude of the sensor P 7 is seven times that of P 5 due to the RVR asymmetry toward the inner elbow radius at section B. In other words, the vortex tail bends in the direction of the elbow bend.
At L Ã ¼ 0.40, the phase difference and the amplitude variation slightly increase at section A (Du ' 30 deg), as shown in Fig. 16(c).However, the minor differences indicate that the vortex is still relatively straight below the runner, which is in line with the visualization results in Fig. 15.On the other hand, the phase differences increase by more than 100% at section B, showing that the tail of the vortex is transforming into a wide spiral shape.Moreover, the reduced amplitude difference demonstrates that the vortex asymmetry has decreased (Fig. 16(d)).At L Ã ¼ 0.66, the vortex tail has completely shifted to a spiral form, indicated by the phase difference of Du ' 90 deg in Fig. 16(f).Also, the asymmetry of the vortex tail has further decreased, as implied by the decrease in the amplitude difference.The minimum amplitude induced at section B under the latter condition is 40% of the maximum amplitude.
Figure 17 represents the phase-averaged pressure data at sections A and B for OP2 where the unperturbed RVR circulates in a wide path as shown by its wide average zone of presence in Fig. 14.Thus, the signals at section A are well out of phase with some asymmetry, as implied by the 60% amplitude difference between the minimum and maximum values in Fig. 17 With the increase of L Ã , the difference between the amplitude of the different sensors decreases at both sections due to the RVR path constriction until L Ã ¼ 0.62 (Figs.17(c) and 17(d)).At L Ã ¼ 0.62, where the rods have reached the RVR, the pressure signals from the different sensors are equally separated in phase.The antiphase trait of the opposite signals at both sections results in the significant plunging mode mitigation that was observed in Fig. 10(b).The flow redirection to the draft tube center -as a result of the imposed blockage by the rods-causes the RVR to oscillate faster in the circumferential directions (shown by the RVR frequency increase and the opposite sensor signals being antiphase).Consequently, the blockage results in the decrease of the oscillations in the axial direction.Moreover, the asymmetry is at its minimum with the smallest amplitude being 70% of the highest (Fig. 17(c)).Further increase of protrusion to L Ã ¼ 0.70 leads to a decrease of phase difference (Du ' 50 deg) between the sensor signals in Fig. 17(e).Additionally, the smallest and highest amplitudes are captured at P 3 and P 1 , respectively.This indicates the asymmetry of the of the vortex toward P 1 , which is in line with the visualization result presented in Fig. 14(c).At L Ã ¼ 0.70 under OP2, the in-phase nature of the signals at section A leads to their cumulative effect (according to Eq. ( 3)) contributing to the increase of the plunging mode.

4.4.3
Rotating Vortex Rope Coherence.The impact of rod protrusion on the RVR coherence is presented in this section.For this purpose, the magnitude-squared coherence was calculated based on the sensors P 1 and P 3 that are installed opposite to one another in section A of the draft tube.The details of the method are presented in Sec. 3. Afterwards, the frequency containing the highest spectral amplitude in the range associated with the RVR was obtained, and the corresponding value of C P1P3 was extracted.Figure 18 demonstrates the values of C P1P3 for the RVR function of L Ã .The unperturbed vortex structure possesses a coherence of above 0.95 under all conditions except OP4.The loss of RVR coherence at lower PL operations has been reported in the literature which explains the drop in its spectral amplitude of pressure pulsations [54].
The highest coherence (C P1P3 ' 1) is observed under OP1, and the rod protrusion results in a slight decrease in its coherence until L Ã ¼ 0.26, where C P1P3 ¼ 0.98.Afterwards, C P1P3 of the vortex drops corresponding to the vortex rope plummets after L Ã ¼ 0.53 (along with the associated frequency as shown in Fig. 13), and the minimum coherence is observed at L Ã ¼ 0.70 where C P1P3 ¼ 0.60.Thus, as the rods reach the RVR path, significant changes in the RVR coherence are observed.
The RVR structure is highly coherent under OP3 (C P1P3 ' 0.98), and the protrusion of rods has little impact on the RVR coherence.The RVR coherence slightly increases with the insertion of rods until L Ã ¼ 0.35 and remains almost unchanged afterwards.The significant mitigations observed for the other operating conditions take place when the RVR coherence is significantly manipulated by the presence of the rods.However, the inability of the mitigation method to impose a significant effect on the RVR coherence at OP3 results in the lack of a notable mitigation.
A protrusion of L Ã ¼ 0.09 under OP4 leads to the increase of the RVR coherence from 0.93 to 0.94, while with further protrusion, C P1P3 first decreases to 0.92, and then, reaches C P1P3 ' 0.94 at L Ã ¼ 0.35.Afterwards, the RVR coherence drops substantially to 0.81 at L Ã ¼ 0.53.Further increase of L Ã leads to a recovery in the RVR coherence to C P1P3' 0.89 at L Ã ¼ 0.70.

4.5
The Impact of the Rotating Vortex Rope Mitigation on the Efficiency.The insertion of solid obstacles creates losses in the turbine draft tube that ultimately impact the efficiency of the machine.The hydraulic efficiency was calculated according to the following equation: where x is the runner angular frequency.The mechanical output of the turbine is directly determined by the torque extracted from the runner [55].Thus, the changes in the hydraulic torque and efficiency can display the effects of the rods on turbine hydraulic losses and performance.Figure 19 displays the hydraulic efficiencies and the corresponding mitigations and torque values against the protrusion ).The torque and hydraulic efficiency follow the same pattern function of L Ã .The efficiency drops first with the increase of L Ã and then increases again.Under OP1, OP2, and OP3, the maximum efficiency loss (between 2.5% to 3.5% of î BEP ) and the lowest torque is obtained at protrusion length where L Ã ¼ L Ã fmax .Under OP4, however, the minimum efficiency (75% of î BEP ) is observed for L Ã ¼ 0.44, whereas L Ã fmax ¼ 0.35 due to the increased randomness of the RVR behavior.The values of L Ã fmax for OP1, OP2, and OP3 are 0.66, 0.53, and 0.44, respectively.After L Ã fmax , both the efficiency and torque are alleviated with further protrusion of the rods as the RVR frequency decreases.It was discussed above that the RVR frequency increases initially when the rods protrude into the draft tube, and it subsequently decreases as the rods reach the vortex area.So, the rod insertion into tangential flow region causes an increase in the hydraulic losses whereas a direct perturbation of the vortex path causes a modification of the hydraulic losses.The similarity of the trends between the hydraulic torque and efficiency shows that the insertion of the rods impacts the efficiency mainly through the manipulation of the extracted torque.In addition, the pressure pulsation amplitude reduction levels displayed in Fig. 19 imply that the RVR mitigation with this method does not necessarily entail a modification of the hydraulic efficiency losses.However, the RVR angular frequency and its variations as a result of rod protrusion are closely related to the variations in the extracted torque (see Fig. 13).
The performance losses due to the RVR mitigation are vital considerations that should be taken into account.Similar to the previous investigation [42], an optimum length was found in the present study for every operating condition, where the highest reduction of the pressure oscillation was obtained.Nonetheless, the present study was performed through a parametric study for discrete values of protrusion length, and no controller was used to find any optimum.With the use of a controller, the optimum length can be identified more accurately and modified actively depending on the new flow conditions imposed by the insertion of the solid bodies.The protrusion length does not need to be identical for all rods as the presence of the elbow influences the RVR asymmetry.Also, the target criteria of mitigation can be optimized to achieve the highest mitigation within a defined threshold of performance losses.

Conclusion
Time-resolved pressure measurements and high-speed videography (HSV) were conducted to investigate the RVR mitigation with radial protrusion of four variable-length cylindrical rods into the draft tube of an axial turbine.The impact of the mitigation method on the RVR was studied under four PL conditions ranging from upper PL to deep PL.The wall pressure measurements were performed at 0.25 D n upstream and 0.75 D n downstream of the runner cone lower end in sections termed A and B, respectively.In addition, the effect of rod protrusion on the turbine performance was scrutinized for the different operating conditions.
The obtained mitigations were investigated using spectral analysis, and the RVR frequency and peak-to-peak pressure amplitude distributions.Short lengths of rod protrusion caused an increase in the RVR frequency due to flow redirection to the draft tube center.Afterwards, the RVR frequency decreased at higher L Ã as a result of the rods reaching the RVR path.The obtained L Ã fmax was smaller for the deeper PL operations because of wider RVR path at smaller guide vane angles.
Statistical analyses of the pressure oscillations showed that rod protrusion directly impacts the RVR periodicity, thereby affecting its spectral content.Significant rotating mode mitigations were obtained only when the RVR coherence was notably affected by the presence of the rods.The RVR coherence did not change under OP3, where no significant rotating mode mitigation was observed.On the other hand, the incoherent RVR at OP4 was easily perturbed and hence mitigated.However, with the RVR mitigation, new oscillations appeared at f Ã ¼ 0.5 under OP4.
Phase-averaged analyses showed that the mitigation of RVR plunging mode takes place when the pressure signals on the same level act in an opposite phase to their opposite sensor, thus minimizing their cumulative effect.This is particularly important since the plunging mode may contribute to resonance in the complete water circuit.Finally, the insertion of the rods reduced the hydraulic efficiency by a maximum of 3.5% of î BEP for all conditions until L Ã ¼ L Ã fmax .The efficiency was alleviated along with the torque afterward as the rods reached the RVR path.Consequently, to achieve a substantial mitigation with smaller losses in turbine efficiency, the rods should be inserted to disrupt the RVR path.
The impact of the mitigation technique proposed in this study can be actively optimized with the implementation of a feedback loop controller.Thus, an optimization objective can be defined depending on the RVR characteristics under a certain operating condition.Consequently, the proposed method can be directed toward the maximum mitigation of the rotating mode, plunging mode, or an optimized combination of pressure pulsation reduction with smaller efficiency losses.

2. 1
Model Setup.The experiments in the current study were performed on an IEC-certified test rig at the Vattenfall R&D facility in € Alvkarleby, Sweden.During the present measurements, a 1:3.875 scaled-down model of the U9 Kaplan turbine was mounted in the test rig.Both the model and the 10 MW prototype have been extensively investigated under different conditions and reported in the literature

Fig. 2
Fig. 2 Schematic view of the elbow draft tube with the location and orientation of the rods and pressure sensors.The sensors P 1 to P 8 are denoted as 1 to 8.

Fig. 3
Fig.3The procedure of obtaining the frequency versus P2P distributions for the RVR-related pressure oscillations

4. 3
Rotating Vortex Rope Mitigation 4.3.1 Rotating Vortex Rope Spectral Amplitudes.The impact of rod protrusion on the spectral amplitude of the RVR-induced

Fig. 5
Fig. 5 Probability distribution of the pressure oscillations for five repetitions under the investigated operating conditions with their corresponding average distribution and standard error.(a) OP1, (b) OP2, (c) OP3, (d) OP4.The dotted dashed lines represent the 95% confidence interval.

Fig. 6 Fig. 7
Fig.6Fourier spectra of the pressure data for the unperturbed flow under the different PL conditions.The small subplot is the spectra of the different operating conditions at a higher range of frequency.

Fig. 10
Fig. 10 Fourier spectra of the pressure oscillation rotating mode from P 1 and the plunging mode from section A: (a) OP1, (b) OP2, (c) OP3, and (d) OP4

Fig. 11 (
Fig. 11 (a) The amplitude of P Ã r,1 signals bandpass-filtered around the RVR frequency and the detected collapses under OP4.(b) DF values of P Ã r,1 bandpass filtered around f RVR and f Ã 5 0.5.(c) Effect of rod protrusion on the collapse rate.

Fig. 13
Fig.13The effect of protrusion length on the RVR frequency (a).At section B, the phase difference for the unperturbed RVR decreases to Du ' 45 deg, which indicates the increased contribution of the plunging mode due to the elbow proximity.

Fig. 16
Fig. 16 Phase-averaged signals of the RVR under OP1.Left: section A. Right: section B

Fig. 17
Fig. 17 Phase-averaged signals of the RVR under OP2.Left: section A. Right: section B

Fig. 19
Fig. 19 Turbine efficiency, shaft torque, and the obtained RVR mitigation rates at different protrusion lengths for the different operating conditions: (a) OP1, (b) OP2, (c) OP3, and (d) OP4 BEP ¼ 0.446 m 3 s À1 at a guide vane opening of a BEP ¼ 26 deg.

Table 3
Distribution parameters of P Ã r ,1 pressure oscillations for L Ã 5 0.62 and L Ã 5 0.70 under OP2

Table 4
The best mitigations in percent obtained at section A from Fourier spectra compared to the mitigation level for the same cases obtained from statistical analyses.