The Measured Flow at the Inlet of a Francis Turbine Runner Operating in Speed No-Load Condition

Hydropower stands out as a highly reliable, predictable, and flexible renewable energy source, surpassing other renewables like wind power and solar energy. It plays a crucial role in baseload generation and grid stabilization, compensating for fluctuations caused by the intermittent nature of other renewable sources or varying electricity demand. The Francis turbine, among various turbine types, holds the title of the most widely employed globally. Despite the possibility to design Francis turbines for different specific speeds, the high-efficiency zone of a given Francis turbine is inherently limited due to fixed blades and the original design optimized for peak efficiency. Recently, the increase of other renewable energy sources in electricity production has led to more frequent transient operations like startups and shutdowns, and prolonged use in off-design operating conditions [1], including part-load and speed-no-load (SNL). No-load condition is an operating condition where the runner generates no net torque. SNL is a particular no-load condition in which the runner rotates at the synchronous speed of the generator. It is a phase where the turbine is ready to be connected to the grid. SNL is an essential part of startups and is also independently utilized to enhance operational flexibility. This operating condition significantly contributes to turbine damage in both base load and grid stabilization operations [1].

At SNL, the swirl generated at the turbine inlet is not extracted by the runner and is transmitted to the draft tube. The high swirl of the flow gives rise to significant instabilities that in turn generate large stochastic pressure and velocity fluctuations [2,3]. The throughflow concentrates in the peripherical zone of the turbine and swirls significantly. The central portion of the draft tube is occupied by recirculating water [3], including a portion that flows back inside the runner [4]. The backflow is generated by the radial pressure gradient due to centrifugal forces [3]. As presented by the numerical analysis of Gagnon [5], the backflow impacts the runner blades on their suction side near the trailing edge, splitting into two. A portion of the flow bypasses the blade's trailing edge and enters the adjacent channel, while the other part moves upstream toward the runner entrance on the blade's suction side. Bourgeois [6] showed that the backflow presence in the Tr-Francis test case at SNL is not linked to the runner, as it persisted in a simulation conducted without the runner. This follows similar demonstrations for other turbines [7,8].

Vortices in the runner channels of the Francis turbine have previously been detected at SNL, but only in numerical studies. Nicolle et al. [9] identified vortices in the interblade channels of a low-head turbine during the SNL stage of a transient Reynolds-averaged Navier–Stokes (RANS) simulation employing the k-epsilon turbulence model. Gagnon [5] also numerically detected interblade vortices at the inlet of a Francis turbine under SNL conditions using unsteady Reynolds-averaged Navier–Stokes (URANS) with the scale-adaptive simulation-shear stress transport (SAS-SST) turbulence model. Laboratory visualizations by Gilis et al. [4] for the same turbine under SNL conditions did not show interblade vortices. However, as cavitation was employed as the visualization tracer, the possibility of noncavitating vortices cannot be ruled out. In the case of an axial turbine, the AxialT turbine, Houde et al. [8] studied SNL through experimental measurements and numerical simulations using URANS with the SAS-SST turbulence model. They found multiple vortices attached to the headcover at the point where the main flow encounters the backflow. Remarkably, similar vortices were observed even when they simulated the flow without runner blades, suggesting that these vortices originate from a shear-layer instability. Although the presence of blades influences the number of observed vortices, the flow instability persists even in their absence. A comparable study conducted by Bourgeois [6] on the Tr-Francis test case at SNL yielded similar conclusions. In the absence of the runner, three to five axial vortices formed in the runner channel instead of 13 interblade vortices. The author suggested that removing the runner facilitated the development of a shear layer instability at a different circumferential wave number, resulting in distinct configurations.

Since the swirl and its unextracted portion are similar between deep part-load and SNL, one can expect similar backflow and vortices in the runner channels in both cases. As in SNL, velocity measurements at the inlet of the Francis turbine runner at deep part-load are not available. However, more information can be found about the interblade vortices at this operating point than at SNL. Yamamoto et al. [10] found that interblade vortices, attached to the hub, are one of the flow phenomena inducing stochastic pressure fluctuations on the runner blades of a medium specific speed Francis turbine at deep part-load condition. Through visualizations using an instrumented guide vane, they observed cavitating interblade vortices by adjusting their Thoma number. In a separate study, Yamamoto et al. [11] simulated these vortices using an unsteady RANS multiphase flow simulation with the SAS-SST turbulence model. They attributed their inception to the collision of the inflow with a separated flow region close to the hub and blade trailing edge. The generation of these vortices is generally attributed to two distinct mechanisms: the high negative angle of attack at the leading edge causing localized separation [10–12] and the backflow coming from the draft tube [12,13]. The influence of turbine speed and discharge on the position of the interblade vortices was studied by Liu et al. [13] and Zhou et al. [14]. Decreasing the flowrate (by closing the guide vanes) at part-load operating conditions increases the swirl and the extent of backflow at the turbine center, which pushes the throughflow to the periphery of the runner and draft tube. As a result, the interblade vortices also move toward the blade leading edge.

Returning to the SNL condition under consideration in this paper, the aforementioned numerical studies must be complemented with comprehensive experimental results to validate the numerical findings and to provide a more detailed insight of the flow behavior. However, the experimental information at the inlet of Francis turbines is mostly limited to visualizations and pressure and strain measurements. Furthermore, all the visualizations that were performed used water vapor (cavitation) as the tracer material, limiting their capability to identify cavitating vortices exclusively. Consequently, these visualizations provide only qualitative insights into cavitating structures, lacking a comprehensive depiction of interblade vortices and quantitative velocity information. The sole PIV measurement covering a small segment of the interblade channels of a Francis turbine is a two-component, low sampling-rate measurement at part load [15]. As far as the authors are aware, there is currently no published study with velocity measurements in the interblade channels of a Francis turbine at SNL.

This paper presents high-speed, stereoscopic and endoscopic particle image velocimetry (PIV) measurements at the vaneless space and the first part of the interblade channel of the Tr-Francis turbine at SNL. Tr-Francis is an ongoing project at Heki Hydro-electricity Innovation Center in Université Laval. It aims to investigate fluid–structure interactions in a medium-head Francis turbine operating in speed-no-load and during startup at model and prototype scale. The objective is to identify the mechanisms associated with high strain levels in the runner under those two conditions. The paper discusses the flow topology at the runner entrance, backflow on the suction side of the blades, and the behavior of two types of vortices near the blade leading edge.

Experiments are conducted on the Tr-Francis turbine, which is a reduced scale medium-head Francis turbine with a scale ratio of 14.4:1. The homologous model is composed of a 13-blade Francis runner, and a 20 guide vanes distributor. Its $NQE$ at best efficiency point is 0.21. The model is installed on hydraulic turbine closed-loop test stand of Heki between a 10 m³ upstream and a 15 m³ downstream tanks (Fig. 1). Measurements are performed in cavitation-free conditions by adjusting the pressure in the downstream tank through the use of a free surface. More details on the test bench can be found in the work of Gilis et al. [4].

The measurements were performed with a Dantec Dynamics 2D3C TR-PIV system supplemented with additional optical devices, calibration system and synchronizing circuits. The imaging equipment includes two Phantom VEO-640 cameras with 2560 × 1600 pixel CMOS sensors, two camera endoscopes (KARL STORZ 10-mm rigid borescopes, 89370AF), and inhouse, custom-made lenses designed to widen the field of view. These lenses are machined in a window installed on the headcover. The lighting system uses a Litron LDY-304 dual cavity, high-frequency Nd:YLF laser with an energy capacity of 30 mJ per cavity. A specifically designed laser endoscope generates the light sheet within an instrumented guide vane and projects it through an acrylic window. The configuration of the PIV components is illustrated in Fig. 2. To better visualize the setup, the Tr-Francis runner is cut and only one guide vane is plotted.

Measurements were made in four radial-azimuthal planes, but the present paper focuses on the measurement plane shown in Fig. 3. Figure 3(a) displays a section cut of the runner at the axial position of the measurement plane. Figures 3(a) and 3(b) provide a schematic representation of the measurement plane's position and orientation. This radial-azimuthal plane is situated at 61.5% span (s) from the band of the runner inlet, covering the vaneless space and the first part of the interblade channels. It is the closest plane to the headcover that allows a satisfactory spatial coverage. The measurements are conducted at SNL condition with 24-m head (H). $NED$ and $QED$ relative to the values at the best efficiency point are 0.991 and 0.144, respectively. The measured operating point is presented in the normalized hill chart of Fig. 4. The operating points used in this figure are measured at 12 m head. The slight difference between the $NED*$ of the BEP and SNL is due to the slight change in the $NED$ of the best efficiency point between these two heads.

A three degrees-of-freedom mechanism is designed to perform the calibration of the cameras. This mechanism allows to align the target with the laser sheet at any desired axial position. A slit-based method is used to ensure precise alignment of the target with the laser sheet. To minimize reprojection error and increase calibration accuracy, the field of view is divided into four overlapping subzones after acquisition. In each of these subzones, a third-order polynomial serves as the calibration mapping function. Consequently, the images of the particles are also divided into the same subzones used for calibration, and velocity vectors are calculated separately in these subzones. The vector fields in these four subzones are then merged to create the full velocity field. Tracer particles consist of silver-coated hollow glass spheres with an average diameter of 13 μm. The full details of the measurement method and tools can be consulted in Ref. [16].

Particle image velocimetry is performed in single exposure, double-frame mode, with each laser pulse in a pair generated by a distinct laser cavity. To revalidate the superposition of the two laser sheets from each of the laser cavities just before measurements, a pair of PIV images is captured in still water. The cross-correlation of these two frames in the stationary water of the test bench yields zero displacement vectors across the cameras' field of view, indicating a perfect match of the particle positions between frames and a seamless overlap of the laser sheets.

Two acquisition modes, high sampling frequency and phase-locked, are utilized in this study. In the latter, six runner phases (positions) with 6 deg angular separation are chosen for these series, covering two interblade passages. The runner completes three rotations between two consecutive velocity fields. This reduces the number of fields required for statistical convergence by decreasing the correlation between snapshots. The acquisition frequency in this mode is 5.58 Hz and 4200 velocity fields are acquired. These data are used to compute the statistics and the mean flow features. In the high sampling frequency measurements, velocity fields are acquired at 1120 Hz, with approximately 5000 velocity fields recorded in each realization.

The PIV images are first dewarped and then analyzed using an adaptive image interrogation method with fast Fourier transform (FFT)-based cross-correlation in DynamicStudio 8.1. This method iteratively adapts the size, shape, and location of the interrogation areas based on the velocity gradients and particle densities. The process begins with a 64 × 64 pixel window size, progressively reducing to a minimum of 16 × 16 pixels at each grid point, with the condition that the particle density must be larger than or equal to 6 particles per interrogation area. The grid resolution of the velocity vector field is set at 10 × 10 pixels. Thus, the overlap between the neighboring interrogation areas is a function of the final size of the interrogation window at each grid point. At each iteration, peak height, peak height ratio, and signal-to-noise ratio are employed to identify spurious vectors with minimum accepted values of 0.25, 2.0, and 4.0, respectively. A universal outlier detection algorithm is applied at each iteration to detect and substitute false vectors based on a normalized median test. No substitution is applied in the final iterations.

To analyze the phase-locked images, static masks are used. Wall windowing function is employed during the analyses of these images to identify the location of the walls in the flow field, i.e., the blade surfaces, and mitigate wall bias. For the time-resolved images, on the other hand, a dynamic mask is applied to account for the change of the runner blade position in the images. No wall windowing is used in these series of measurements. However, since these images are not used in the statistical analyses, a small wall bias does not influence the results. Once the vector fields are calculated, the point-by-point N-sigma validation with a limit of 5 is applied to the in-plane velocity components. The vectors are then converted to the rotating reference frame before applying a universal outlier detection filter. This conversion helps to better detect spurious vectors. A universal outlier validation with a neighborhood size of 5 × 5 vectors is applied. Using the validated velocity fields from both cameras, three-component velocity fields are reconstructed. Another universal outlier validation with a 3 × 3 vector neighborhood size is applied to the three-component velocity fields. Finally, the last remaining erroneous velocity vectors are removed using range validation on the magnitude of the velocity vectors. Once the three-component vector fields are validated for each subzone, the merging process is done in Matlab. Details of the merging process can be found in Ref. [16]. The uncertainty of the in-plane velocity components is estimated to be 3% of the velocity magnitude based on the method proposed by Sciacchitano et al. [17]. The results of cross-correlation, rounded to the closest integer value, are used in this method to match image pairs by shifting the second frame toward the first frame. Then, the disparities of the individual particles within the inter-rogation area are statistically analyzed to give the estimation of the uncertainty.

where the index $i$ can be $r$⁠, $θ$⁠, and $z$ to indicate, respectively, the radial, tangential, and axial velocity components in a cylindrical reference frame. The axial direction is along the runner's axis of rotation and facing upward (opposite to throughflow direction in the draft tube).

Velocity spectra are calculated at fixed points in the spatial domain. These points are placed in the vaneless space to ensure valid velocity components at every time-step, thereby avoiding the omission of velocity information due to the blades and their shadows. Welch's method, employing nine blocks with a 75% overlap, is utilized for spectrum calculation. This approach enables good statistical convergence with a reasonable frequency resolution (⁠$Δf=0.55 Hz$⁠). Figure 5 shows the normalized power spectra of the radial $(GWrWr*)$ and tangential $(GWθWθ*)$ velocity components at the magenta probe point illustrated in Fig. 5(a). This point is located at a radial distance of $1.016rrunner,$near the leading edge of the runner blades and relatively far from the guide vanes, where $rrunner$ is the runner radius at the axial position of the measurement plane. The spectra are normalized with $U02 (2 rrunnergH)$⁠.

where $fa, fBP, and fs$ are aliased, blade passing, and sampling frequencies, respectively. Given the current sampling frequency, the second to fourth harmonics of BPF are aliased at 28, 15, and 2$fn$⁠, respectively. The blade-passing signature at BPF is more pronounced in the radial velocity component compared to the others. The peak at BPF has a lower amplitude in the axial velocity (not shown) and an even lower amplitude in the tangential velocity component (Fig. 5(c)). Figure 6 shows the spectra of the radial velocity component at different radial positions in the vaneless space. The parameter $δ$ represents the percentage of the distance of the probe from the runner with respect to the vaneless space width. As expected, the amplitude of the blade passing effect decreases with distance from the runner. Close to the guide vanes, the amplitude at BPF in the radial velocity spectra decreases drastically.

The amplitude at the first harmonic of BPF (⁠$f/fn=$ 26) is higher than that at BFP itself in the spectra presented in Figs. 5(b) and 5(c). However, in other spectra, the amplitudes of the harmonics decrease monotonically. Whether this phenomenon is solely related to the nonlinearities of the blade passing effect or is associated with another flow phenomenon at 26$fn$ needs further investigation, utilizing additional tools such as proper orthogonal decomposition and its variants. No other periodic phenomena are detected by the velocity spectra. Velocity spectra in the rotating frame, which cannot be obtained with the present PIV measurements, would have been necessary to clearly detect the flow phenomena discussed in Sec. 3.2. Figure 7 shows the low-frequency zone of the spectrum from Fig. 5(b) using a logarithmic scale. As evident in this figure, stochastic low-frequency phenomena do not exist in the velocity field at the inlet of the runner. While such phenomena were detected in the pressure signals at the draft tube of the Tr-Francis turbine by Gilis et al. [4], they were not observed in the pressure signals at the vaneless space.

This section discusses the backflow near the suction side of the blades, circulation zones, and vortices in the vaneless space and the first part of the interblade channels. In all the velocity-field figures presented in this paper, a schematic position of the blade is shown to help visualize the positioning of the plane. The guide vane is approximately aligned with the $X$-axis of the plots and is illustrated in Fig. 8(b). The presented velocity fields are subsampled from the measurements by a factor three (one vector out of three in each direction, unless otherwise indicated) to simplify the analyses. The velocity vectors represent in-plane velocities where the vectors with negative and positive radial velocities are colored in black and red to identify, respectively, regions of throughflow and backflow. Backflow in this paper refers to regions of positive radial velocity flow. In SNL, it is generally associated with the large backflow extending from the runner entrance to the draft tube bend. All reaction turbines at SNL have a large backflow region [3,8,18]. Throughflow refers to the zones with a negative axial velocity that actively contribute to the positive flowrate of the turbine and through which the flow is carried into the draft tube. Before calculating derivatives and plotting velocity fields, a 5 × 5 Gaussian filter is applied on the full-resolution grid to smooth the fields and enhance the detection of flow phenomena by reducing noise.

where $λ2$ [19,20] is the second eigenvalue of the pressure Hessian, and $σ(λ2)$ is the spatial standard deviation of $λ2$ in each snapshot. Connected zones where $λ2′$ falls below the threshold value of −3 are considered as vortices, and the centroid of these zones is assumed to be the center of the vortices in the following analyses. The spatial coordinates (⁠$X$ and $Y$⁠) and velocities are normalized with the span (s) and tip velocity (⁠$U0)$ of the runner, respectively.

Phase-averaged velocity fields are plotted in Fig. 8 for two runner phases, $ϕ=$ 342 deg and 330 deg, representing the pressure and suction sides of the same blade channel, respectively. The trajectory of the runner tip and its rotational direction are illustrated in Fig. 8(a) by a red dashed line and an arrow, respectively. One out of four calculated velocity vectors in each direction is included in the vector plots of this figure. As shown in Fig. 8(b), a strong backflow exists on the suction side of the blades at SNL. It even extends beyond the runner and reaches the vaneless space. The incident flow is oriented toward the suction side of the blades.

The interaction of the incident flow with the backflow likely creates the circulation zone on the suction side clearly depicted by the pseudo-streamlines in Fig. 8(d). A substantial portion of the throughflow is carried by this circulation zone (large negative $W¯z$ in Fig. 8(b)). Similar findings were reported in the numerical simulation results of Gilis et al. [4], where a similar circulation zone in the mean flow field was identified. It is important to distinguish between mean flow and phase-averaged behaviors, which depict global tendencies in the flow, and instantaneous behavior, which reflects the highly dynamic and stochastic nature of the flow phenomena at SNL. For instance, the backflow is intermittent, exhibiting variable strength and size at different times. This variability is illustrated in the instantaneous velocity fields in Figs. 9 and 10. In the field of Fig. 9(a), the backflow extends deeply into the vaneless space in a disorganized manner, while in the field of Fig. 10(a), it remains within the runner and in a smaller zone near the suction side of the blade.

The instantaneous flow field in Fig. 9(a) reveals that the large circulation comprises multiple vortices of various sizes. Similar vortices are formed on the side of the backflow that is proximate to the pressure side, called pressure-side vortices, hereafter. An example of such vortices is illustrated in Fig. 11. Figure 11(a) shows their position in the velocity field and Fig. 11(b) provides a more detailed view, in which the approximate convective velocity of the two vortices has been subtracted from the velocity vectors. These vortices are a manifestation of the instabilities due to the interaction between the main flow and the backflow. The suction-side vortices form within the strong shear layer at the interface of the countercurrent inflow and backflow. The pressure-side vortices appear in the shear layer between the backflow and a recirculation zone on the pressure side of the blade. The vortices display stochastic behavior, as expected by the intermittent nature of the backflow at this location. Depending on the size, orientation, and intensity of the backflow, the number and configuration of the vortices vary. The intensity and shape of the circulation zones also vary considerably over time, as depicted by the instantaneous pseudo-streamlines in Figs. 9(b), 10(b), and 12. Figure 12 indicates that, at times, several circulation zones are present near the blade suction side.

Going back to the phase-averaged depiction of the flow, two counter-rotating circulation zones are evident on the pressure side of the blade, as illustrated in Fig. 8(a) and designated as C1 and C2 in Fig. 8(c). However, these zones have weaker circulation compared to the circulation zone on the suction side. Circulation zone C1 does not significantly contribute to the throughflow. C1 appears to be formed by the interaction of the backflow coming from the suction side and the strongly swirling flow circulating in the vaneless space. Thus, inflow to the channel is blocked by various phenomena everywhere, except in the vicinity of the blade suction side, where it is carried downstream through the intense circulation zone (Figs. 8(b) and 8(d)). The central part of the interblade channel entrance has a negative radial velocity. However, this inflow to the channel is later decelerated to a zero radial velocity region due to the trailing edge blockage at this axial position that has been presented by Gilis et al. [4] and Gagnon et al. [5].

where $σ2$ (.) represents the variance of the parameter within the parentheses. This high kinetic energy zone may be related to the possible pressure fluctuations at the surfaces of the runner blades at SNL (not measured), and even the fluctuations observed by Yamamoto et al. [5] at deep part-load. The origin of these pressure fluctuations, whether stemming from the intermittency of the backflow, the stochastic nature of suction-side vortices, or a combination of both, remains unknown.

As mentioned previously, the key flow characteristics at SNL in the Tr-Francis runner include the backflow on the suction side of the blades, the large circulation zones, and the vortices emerging in the shear layers. The stochastic nature of these phenomena is evident. In this section, statistics are employed to enhance our understanding of the flow at the runner inlet at SNL. These statistics are computed using phase-locked velocity fields, and within three rectangular regions illustrated in Fig. 14. The turquoise region is utilized to evaluate statistics related to the suction-side vortices. This region encompasses, at most instants, the entirety of vortices found in both the shear layer and the circulation zone on the suction side. The magenta zone is designated for the pressure-side vortices, as they are predominantly encountered in that area. Finally, properties of the backflow within the vaneless space are computed within the region enclosed by the dashed purple line.

The pseudo-streamlines of the conditional mean velocity field are depicted in Figs. 15(a) and 15(b) for cases in which the backflow extends into the vaneless space and cases where it does not, respectively. The condition defining “backflow extending into the vaneless space” is met when the spatial average of the radial velocity component within the zone marked by the purple dashed line in Fig. 14 is positive. Figures 15(a) and 15(b) suggest that a stronger backflow results in more intense circulations, indicated by smaller spacing between pseudo-streamlines. It is also observed that a stronger backflow shifts the suction-side circulation toward the vaneless space. Despite these observations, the differences between the two fields of conditional pseudo-streamlines are small, and they are small even when compared to the phase-averaged pseudo-streamlines in Figs. 8(c) and 8(d). Thus, the backflow and the circulation zones are persistent features of the flow.

Let's shift our focus to the vortices. The probabilities of at least one and two suction-side vortices occurring in the turquoise rectangle are 85% and 36%, respectively. The likelihood of the presence of pressure-side vortices is approximately 52%. Notably, when pressure-side vortices are present, suction-side vortices are also observed 84% of the time. This suggests a consistent relationship between both types of vortices, likely associated with the backflow on the suction side of the blades. This association of the suction- and pressure-side vortices with the backflow is revealed in Fig. 16. The horizontal axis of this figure is a binning of all vector fields at $ϕ=336 deg$ into 8 equi-width $Wr̃/U0$ categories, where $Wr̃$ is the spatial average of the radial velocity component in the purple zone shown in Fig. 14. The vertical axis represents the number of instances with at least one vortex among the instances in a $Wr̃/U0$bin. The analysis is carried out separately for the suction- and pressure-side vortices. The numbers next to the markers indicate the count of velocity fields present in each $Wr̃/U0$ bin. This figure clearly demonstrates a correlation between the intensity of the backflow and the presence of vortices. The suction-side vortices have a direct relationship with the backflow strength. The pressure-side vortices appear more frequently as the backflow strength increases up to certain point, after which the frequency of appearance decreases. A stronger backflow shifts circulation zone C1 in Fig. 8 away from the blade pressure side, reducing the shear on the pressure side of the backflow. Lower shear strain generates fewer vortices.

Figure 17 presents the histogram of the velocity of the centroid of suction-side vortices normalized by $U0$⁠. As indicated by this histogram, most suction-side vortices have positive radial velocities, suggesting they are convected toward the vaneless space, where they appear to break down into smaller structures and eventually disappear. The histogram of the normalized radial velocity of the pressure-side vortices is shown in Fig. 18. The average normalized radial velocity is 0.018, approximately five times smaller than that of the suction-side vortices, which is calculated at 0.1. These vortices are not convected radially like their suction-side counterparts. In fact, they barely change position.

In this paper, velocity fields are measured for the first time at the inlet of a Francis turbine—specifically, the Tr-Francis turbine—under speed-no-load operating conditions. The measurements are conducted in a radial-azimuthal plane with high-speed stereoscopic and endoscopic particle image velocimetry. The measured plane covers the vaneless space and a significant region in the interblade channel of the runner. Analysis of the average and instantaneous velocity fields unveils intermittent flow phenomena associated with high velocity fluctuations on the suction side of the runner blades. The predominant phenomenon is a backflow along the suction side of the blades. As it interacts with the high-swirl inflow, it gives rise to the formation of large circulation zones and smaller vortices. Stronger backflow events tend to strengthen the circulation zones and shift the suction-side circulation zone toward the guide vanes. Smaller vortices form at the interface of the backflow and throughflow close to the suction and pressure sides of the blades. The suction-side vortices are convected into the vaneless space, while the pressure-side ones are almost stationary in the rotating reference frame. Further analysis is underway, incorporating additional measurement planes, to determine whether pressure fluctuations on the suction side of the runner blades result from the intermittency of the backflow, the stochastic nature of suction-side vortices, or a combination of both.

This work is done in Heki Hydro-electricity Innovation Center in Université Laval as part of the Tr-Francis project. We would like to thank Professor Jérôme Vétel for his logistical support that was invaluable to the success of this project.

• Natural Sciences and Engineering Research Council of Canada (Grant No. NSERC CRDPJ 507814-16; Funder ID: 10.13039/501100000038).

• The Consortium on Hydraulic Machines (Andritz Hydro, Électricité de France, GE Renewable Energy, Hydro-Québec, Vattenfall, Voith Hydro, Université Laval and École Polytechnique de Montréal).

• InnovÉÉ.

The authors attest that all data for this study are included in the paper.

D =

diameter, m

E =

specific hydraulic energy of machine, J/kg

f =

frequency, Hz

g =

gravitational acceleration, m/s²

$G*$ =

normalized power spectra

H =

head, m

K =

kinetic energy, $m2/s2$

N =

runner rotation speed, $s−1$

$NED$ =

speed factor, $NDE0.5$

$NED*$ =

normalized speed factor, $NED/NED, at BEP$

$NQE$ =

specific speed, $NQ0.5E0.75$

$Q$ =

measured discharge, $m3/s$

$QED$ =

discharge factor, $QD2E0.5$

$QED*$ =

normalized discharge factor, $QED/QED, at BEP$

r =

radius, m

s =

span, m

$U0$ =

linear velocity of the runner blade tip at the measurement plane, m/s

W =

velocity in the rotating reference frame, m/s

W′ =

velocity fluctuations, m/s

$W¯$ =

phase-averaged velocity, m/s

$W̃$ =

spatial average of the velocity in the region of interest, m/s

X =

X-component of the position, m

Y =

Y-component of the position, m

Greek Symbols

α =

guide vane opening angle, deg

$δ$ =

ratio of the distance from the runner with respect to the vaneless space width, %

$λ2$ =

second eigen value of the pressure Hessian

$λ′2$ =

normalized $λ2$

$σ$ =

standard deviation

$σ2$ =

variance

$ϕ$ =

runner phase angle, deg

Subscripts

a =

aliased

BP =

blade passing

n =

runner rotation

r =

radial component

runner =

runner blade tip at the axial position of the measurement plane

s =

sampling

z =

axial component

$θ$ =

tangential component

Abbreviations

BEP =

best efficiency point

BPF =

blade passing frequency

FFT =

fast Fourier transform

PIV =

particle image velocimetry

SAS =

scale-adaptive simulation

SNL =

speed no-load

SST =

shear stress transport

Tr-Francis =

transient Francis

URANS =

unsteady Raynolds-averaged Navier–Stokes