Design of a Simplified Experimental Test Case to Study Rotor–Stator Interactions in Hydraulic Machinery

Abstract

Because of the introduction of significant amounts of electricity from intermittent energy, such as solar and wind, on power grids, hydraulic turbines undergo more transient operation with varying rotation speeds. Start and stop sequences are known to induce significant mechanical stress in the runner, decreasing its lifespan. Complex fluid–structure interactions are responsible for those high-stress levels, but the precise mechanisms are still elusive, even if many experimental and numerical studies were devoted to the subject. One possible mechanism identified through limited measurements on large turbines operating in powerhouses is rotor–stator interactions. It is already known that rotor–stator interaction (RSI) in constant-speed operating conditions can lead to runner failure when the RSI frequency is close to the natural frequencies of specific structural modes. Start and stop sequence investigations show that RSI can induce a transient resonance while the runner is accelerating/decelerating, which generates a frequency sweep that excites the structure. Studying transient RSI-induced resonance of structural modes associated with hydraulic turbine runners is complex because of the geometry and the potential impacts from other flow-induced excitations. This paper presents the development and validation of an experimental setup specifically designed to reproduce RSI-induced resonances in a rotating circular structure with cyclic periodicity mimicking the structural behavior of a Francis runner. Such a setup does not exist in the literature and will be beneficial for studying RSI during speed variations, with the potential to provide valuable insights into the dynamic behavior of turbines during transient conditions. The paper outlines the different design steps and the construction and validation of the experiment and its simplified runner. It presents important results from preliminary analyses that outline the approach’s success in investigating transient RSI in hydraulic turbines.

1. Introduction

With the integration of renewable energy sources such as solar and wind, hydraulic turbines are going more often through transient operations such as speed no-load, total load rejection, start-up, and shutdown. Start-up and shutdown have been demonstrated to be highly damaging for the runner, both experimentally Coutu et al. [1] and through numerical structural simulation Huang et al. [2]. When turbines are used for grid stabilization, the damage is even more significant due to the increased number of start-ups, as reported by Unterluggauer [3].

The effect of the start-up sequence was studied for prototype turbines by several authors, including Seidel [4], Unterluggauer [5], and Gagnon et al. [6,7]. They demonstrated that severe stresses are induced in the runner blades when the guide vane opening sequence is not optimized. By varying the guide vane opening sequence, the acceleration rate of the turbine can be varied to achieve a slow or fast start sequence, which will result in different hydraulic excitations on the runner. All their results show that increasing the time to reach synchronous speed leads to lower dynamic stress induced in the runner blades. However, an overly long start-up sequence gives enough time for hydraulic instabilities to establish themselves and can cause significant mechanical stress, as highlighted by Gagnon et al. [8]. Recently, Schmid et al. [9] conducted a 1D transient simulation combined with multi-objective optimization of start-up sequences for a 5 MW reversible Francis pump turbine equipped with a full-size frequency converter. These optimized start-up sequences were experimentally investigated by Nicolet et al. [10] (part 1) and Biner et al. [11] (part 2). Biner et al. [11] showed that the start-up sequence optimized to minimize the runner fatigue significantly reduces the damage to the runner compared to the standard start-up sequence. The proposed approach is applicable only to variable-speed pump turbines with full-size frequency converter capable of controlling the rotational speed. No details are provided on excitation phenomena or how the optimized start-up sequence influences them.

When operating under off-design conditions, turbines are subjected to various fluid–structure interaction (FSI) phenomena that can induce significant mechanical stress. For instance, Seidel et al. [4] show that a vortex rope can occur in the draft tube at part-load conditions due to the remaining swirl at the outlet. This vortex rope generates most of the structural vibrations at this operating condition, as presented by Favrel et al. [12]. At deep part-load and no-load conditions, Lui et al. [13] show that the flow is generally turbulent, and inter-blade vortices happen, which generate dynamic loads in the turbine. During start-up and shutdown, the precise excitation mechanisms that induce high-stress levels remain elusive despite numerous experimental and numerical studies devoted to the subject.

Trivedi et al. [14] investigated the impact of guide vane opening sequences on flow behavior during the start-up and shutdown of a high-head Francis model turbine. Using pressure sensors positioned in the vaneless space and on the runner, they compared the effects of a slower and a faster opening sequence. Their results indicate that the opening and closing of the guide vanes have a significant effect on pressure fluctuations both in the vaneless space and on the runner, occurring at the blade and guide vanes’ passing frequency, respectively. These frequencies correspond to the rotor–stator interaction (RSI) in the stationary and rotating frames, respectively. Valentin et al. [15] studied the effect of the start-up sequence on the structural response of a prototype Francis turbine runner using strain gauges installed on the blades. Their results indicate that the first RSI harmonic induces a strong structural response as the runner accelerates from a standstill to synchronous speed. They also observed that the third RSI harmonic excites the runner with sufficient energy to be detected when passing through resonances. Similarly, Gagnon et al. [16] identified multiple resonances excited by RSI harmonics during the start-up and shutdown of a Francis prototype turbine. Their results show that some of the runner’s natural modes excited by RSI can be predicted using the classical theory proposed by Kubota et al. [17] and Tanaka [18], while several others cannot. Recently, Dollon et al. [19] introduced a more general mathematical model to describe how all rotational speed harmonics can excite the runner rather than focusing solely on RSI harmonics, which are the rotational speed harmonics corresponding to multiples of the number of guide vanes. Their model builds upon the works of Wildheim [20,21], who studied interactions in circumferentially periodic structures. The model considered a one-per-revolution excitation as the root generating a wide harmonic content that can induce resonances. Dollon et al. [19] termed this excitation mechanism “non-trivial runner–casing interactions” (NTRCI). Experimental measurements performed on a prototype Francis turbine runner with strain gauges provided support for their model. The physical phenomenon underlying this one-per-revolution excitation mechanism has not been identified, but they suggest that turbine casing asymmetries might be the cause.

Instrumenting prototypes and model runners is both time-consuming and costly, and observing specific phenomena remains challenging as it depends on the design of individual turbines. A carefully designed experimental setup, operated under well-controlled conditions and allowing at least two modes to be excited by RSI, would provide significant benefits for studying rotor–stator interactions (RSI) and, more broadly, NTRCI. Experimental setups using submerged disks in stationary fluid have been extensively studied in the literature to determine the effect of added mass, rotating speed, and confinement on natural frequencies. Many of these studies were conducted by Presas et al. [22,23,24,25,26,27] and Valentin et al. [28,29] from Universitat Politècnica de Catalunya. Their work provides valuable insight into the behavior of Francis turbine runners but was not intended to investigate RSI, as their setups operate in stationary water where RSI does not occur. Closer attempts to investigate the effect of RSI have been conducted in the last decade, focusing on the coupling effect generated by the fluid between a rotating disk and its casing. These experiments were conducted first by Presas et al. [24] and later by Weder et al. [30,31], both demonstrating that vibrations can be transmitted from one reference frame to the other through the fluid. In the literature, two studies better align with the objective of developing a simplified test case to study RSI. First, Kubota et al. [17] conducted an experiment using a rotating bladed disk in air, created by attaching small steel pieces evenly spaced along its circumference. The disk was excited by multiple water jets positioned on the fixed frame, generating a cyclic–symmetric excitation. Their results successfully validated the analytical predictions of RSI. Second, Presas et al. [22] investigated the compatibility between the mode shapes of a rotating disk and the RSI excitation pattern. The rotating disk was submerged in still water, and RSI was simulated by exciting the disk with piezoelectric actuators following different nodal diameter patterns. While both studies provide valuable insights into RSI, their experimental setups were not designed to allow the rotating structure to be excited by RSI induced by the water flow, as it would be in an actual Francis turbine.

This paper presents an experimental test case specifically designed to study transient RSI-induced resonances in a cyclic structure that responds with structural mode shapes similar to those of Francis runners. It consists of a simplified rotating cyclic–symmetric structure submerged in water and rotating inside an actual distributor. The test case is carefully designed to ensure that the dominant excitation is the RSI caused by the flowing water, while the rotational speed and angular acceleration can be precisely controlled to study the RSI effect during rotational speed variation. The paper is organized as follows. The fundamental theory describing rotor–stator interactions involving a cyclic–symmetric rotor and stator is first introduced in Section 2. Section 3 then describes the key elements related to the design of the test case, focusing on the expected structural behavior of the cyclic–symmetric rotating structure, the design of a flow straightener to prevent backflow, and the instrumentation and data acquisition system. In Section 4, the experimental measurement campaign conducted to identify the actual structural behavior of the runner is presented, along with some relevant results that support the suitability of the test case to investigate RSI.

2. Theoretical Framework

2.1. Rotor–Stator Interaction

Rotor–stator interaction (RSI) in hydraulic turbines refers to the excitation produced by the flow field generated by the guide vanes (a cyclic–symmetric fixed structure), acting on the runner blades (a cyclic–symmetric rotating structure) when the runner rotates. Figure 1a shows a sectional view of a Francis runner with 13 blades rotating at an angular velocity ω within a distributor containing 20 guide vanes. The blue arrows represent the water flow at the outlet of the guide vanes. As the runner rotates, this flow generates periodic fluctuating forces on each blade, producing excitation patterns characterized by specific frequencies and nodal diameters (ND).

Tanaka [18] derived equations to predict the excitation frequencies and nodal diameter (ND) shapes caused by RSI, which are summarized below. The excitation frequencies caused by RSI ($f_{RSI}$) can be expressed as follows:

$$f_{RSI}=m{Z}_g{f}_{rot}$$

(1)

where $m$ is a positive integer ($m>0$) representing the harmonic order, $Z_g$ is the number of guide vanes, and $f_{rot}$ is the runner’s rotating speed in Hz.

The excitation patterns are further characterized by the number of nodal diameters $k$, which is not unique for a given harmonic order $m$. The number of nodal diameters for a specific harmonic $m$ is defined as follows:

$$\pm k=p{Z}_r-m{Z}_g$$

(2)

where $Z_r$ is the number of runner blades, $p$ is a positive integer ($p>0$), and the $\pm$ sign indicates the direction of excitation: positive for forward wave excitation and negative for backward wave excitation.

According to Tanaka [18], the first harmonic $(m=1$) typically dominates the excitation, and excitation levels decrease with higher harmonics ($m=2, 3,\dots$). Additionally, for a given harmonic order $m$, the excitation with the lowest number of nodal diameters $k$ is generally the most significant.

RSI can excite the runner at resonance, potentially causing severe damage. For resonance to occur, both the excitation frequency and excitation pattern must match the runner’s natural frequency and mode shape. The natural modes most susceptible to excitation by RSI for Francis turbine runners are those in which the radial deformation of the band dominates over blade and crown deformation. These modes are known as band modes. Each of these modes is described by a natural frequency ($f_n$) and a mode shape with a specific number of nodal diameters ($n$). To avoid resonance, one must ensure the following:

$$f_n\ne f_{RSI}\hspace{1em}\mathrm{and}\hspace{1em}k\ne n$$

Conversely, in this study, we intentionally seek to satisfy the following to investigate the effects of RSI at resonance:

$$f_n=f_{RSI}\hspace{1em}\mathrm{and} k=n$$

2.2. Non-Trivial Runner–Casing Interaction

This excitation mechanism was initially modeled by Wildheim [20,21] in the 1980s and was recently revisited by Dollon et al. [19] with a modern formulation adapted to the field of hydraulic turbines. As shown in Figure 1b, their models consider a single point force in the fixed frame, acting sequentially on each rotating blade. This results in a one-per-revolution periodic excitation in the rotating frame, which can be decomposed into harmonic components.

Dollon et al. [19] showed that each harmonic component excites a specific nodal diameter, which can be predicted using the following equation:

$$h=q{Z}_r\pm v$$

(3)

with $v\in ⟦0,⎣Z_r/2⎦⟧$, where h is the harmonic component, $q$ is a positive integer ($q\ge 0$), $Z_r$ is the number of runner blades, and $v$ is the number of nodal diameters of the excitation shape ($v<0$ for forward excitation and $v>0$ for backward excitation). The function $⎣.⎦$ denotes the floor function.

If both the frequency of a harmonic component of the NTRCI ($f_{NTRCI}=h{f}_{rot}$) and the excited nodal diameter $v$ matches the natural frequency and the mode shape of a nodal-diameter mode, resonance occurs.

The key difference between NTRCI and RSI is that each harmonic of the rotating speed ($f_{RSI}=h{f}_{rot}$) has the potential to induce resonance for NTRCI, whereas only harmonics that are multiples of the number of guide vanes ($m{Z}_g{f}_{rot}$) can induce resonance for RSI.

3. Test Case Design

3.1. Objectives

The following criteria were used to design the test case:

• It must be integrated into an existing test rig and operated within the test rig’s capability.
• The hydrodynamics of rotor–stator interactions must be replicated to represent conditions similar to Francis turbines. Hence, it must involve a static and rotating structure with cyclic periodicity based on a given number of blades.
• The rotating structure must be designed to enable the excitation of at least two ND modes by rotor–stator interactions. Because of their lower natural frequencies, the 2ND and 3ND modes are preferred.
• Rotor–stator interaction must be the dominant excitation phenomenon.
• The rotational speed and the acceleration rate of the structure must be controllable.
• The rotating structure must be instrumented using strain gauges.

3.2. General Configuration

The experimental test case was designed to integrate with an existing test rig available at Université Laval. The test rig is a closed-loop hydraulic system using a centrifugal pump driven by a 22.4 kW (30 hp) variable-speed AC motor. The test rig was previously used to test 20 cm hydraulic turbine models. For this work, an existing spiral casing with a 16-guide-vane distributor was reused from a Kaplan model. The runner and draft tube were replaced by custom-designed components.

Figure 2 shows the test section of the test rig and highlights the inlet pipe, the spiral casing, and the outlet pipe. The rotating structure is located after the distributor (Figure 2b) and connected by a hollow shaft to a 2.2 kW variable-speed AC motor with a maximum rotational speed of 1800 RPM. A variable frequency drive (model TD20-2R2G-2-EU from Techtop Canada, Toronto, ON, Canada) is used to control the velocity and acceleration rate of the motor and, consequently, the rotating structure. A 15-channel slip ring (model GT38109-S30 from Moflon, Shenzhen, China) is installed on the shaft to transmit signals from the rotating frame to the stationary frame. An encoder (model HS35A from Avtron, Cleveland, OH, USA with 1024 pulses per revolution) is used to measure the rotation of the shaft, and a metal bellows coupling connects the AC motor to the shaft. The main parameters of the test rig are summarized in

Table 1. Main parameters of the test rig.

Parameters	Values
Nominal head	4 m
Nominal flowrate	0.102 m3/s
$\mathrm{Number} \mathrm{of} \mathrm{guide} \mathrm{vanes} (Z_r$)	16
Maximum rotating speed	1800 RPM
Maximum diameter for the rotating structure	200 mm

for reference.

Figure 2b shows a sectional view of the test section. The rotating structure, highlighted in yellow, consists of a band connected to a crown via seven rods that mimic the leading edges of Francis turbine runners. The hollow shaft is supported by two ball bearings. The slip ring is mounted between the AC motor and the right ball bearing. The stationary components include the head cover, spiral casing, distributor with 16 guide vanes, the bottom ring, flow straightener assembly, and draft tube. During operation, the water flows from the spiral casing (indicated by the red arrows), passes through the distributor, the rotating structure, and a flow straightener before exiting through the draft tube (indicated by the blue arrows).

The exploded view shown in Figure 2c shows that a groove was machined in the bottom ring, enclosing the band to prevent interaction with the main flow. A 3 mm radial gap and a 1 mm axial gap were chosen to minimize leakage flow and shear torque while also preventing mechanical contact.

Figure 2c also highlights that the flow straightener is assembled in a holder inserted between the left cover and the draft tube assembly. A 1 mm clearance was designed between the rotating structure and the non-rotating flow straightener to prevent any contact.

3.3. Hydrodynamic Design

The objectives of the test case and the availability of the Kaplan model with a spiral casing and a 16-guide-vanes distributor provided the starting point of the test case design. Hence, the design efforts were focused on manufacturing a runner with a structural response yielding RSI-induced resonance on specific rotation speeds and on specific modes that represented the typical band mode shapes of Francis turbines. Preliminary mechanical evaluation indicated that the ideal rotating structure should be composed of a circular band linked to a rigid crown by thin rods. Given the characteristics of the distributor and the rotation speeds achievable on the test stand, such runner configuration was necessary to achieve RSI-induced resonance of specific mode shapes within a given speed interval.

Other simplifications were introduced to minimize other potential sources of hydraulic excitations on the runner besides RSI. The guide vanes angle was fixed at a relatively high angle, ensuring a strong RSI response while minimizing the swirl number [32]. Since the rotating structure is composed of thin rods that will not extract the angular momentum generated by the distributor, the high swirl at the runner exit was expected to generate a large backflow associated with columnar vortices attached to the head cover [33].

Numerical flow simulations were thus used to evaluate and correct potential sources of hydrodynamic excitation that might dominate RSI. A mix of 2D and 3D Unsteady Reynolds Averaged Navier Stokes (URANS) simulations were performed using Star-CCM+ 17.02.008-R8 and the k-ω SST turbulence model with meshes of sufficient resolution to achieve Y+ of 1. The simulations, detailed in Bédard [34], were based on computational domains that included the distributor, runner, and draft tube with different types of interface between the rotating and stationary domains. The flow conditions in terms of discharge, head, and rotation speed were evaluated from the test stand characteristics curves. The simulations follow best practices and guidelines for flow simulation in hydraulic turbines. Since CFD, in the present context, is used as a design tool, the validation of the simulation results stems from the success of the experiment, as outlined in Section 4.

Two issues arose from the flow simulations: (1) Using truncated circular rods for the runner would lead to vortex shedding of significant amplitude at frequencies close to the targeted RSI frequencies; (2) Without a decrease in angular momentum in the runner, a large backflow would develop in the draft tube leading to the formation of columnar vortices that would induce pressure fluctuations on the runner larger than those associated with RSI.

Correcting the rod shape was achieved through an iterative process that involved flow simulations and structural simulations to assess the impact of the shape modification on natural frequencies. The original circular shape was designed with a flat surface oriented downstream of the flow to allow the installation of strain gages, as shown in Figure 3a. The modified shape was a truncated ellipsoid, as shown in Figure 3b. The vortex shedding in the wake of the rotating rods is illustrated using vorticity contours in Figure 4 for the final geometry. With the ellipsoid geometry of Figure 3b, the shedding frequency decreased by 30%, and the associated pressure fluctuation amplitudes by about 50%.

The flow simulations also confirmed the presence of a large backflow at the outlet of the rotating structure linked to the high swirl at the draft tube inlet. To prevent these undesired excitations, a flow straightener was designed and installed after the runner, as shown in Figure 2 and Figure 5. On top of the requirements of inhibiting the backflow and columnar vortices, the structure was also designed to have minimal impacts on the hydraulic losses. The resulting flow straightener is composed of fixed vanes and an enlarged hub (Figure 5). The hub’s purpose is to prevent the formation of the backflow close to the runner, while the geometry of the vanes was optimized to convert most of the angular momentum into static pressure.

Figure 6 compares the axial velocity field (along the Z-axis) inside the rotating structure and the draft tube (a) without and (b) with the designed flow straightener. In Figure 6a, without the flow straightener, positive axial velocities are observed beneath the crown of the rotating structure, indicating that the flow moves upward toward the crown and the rods. This upward flow has the potential to excite the rotating structure at frequencies other than those expected from RSI. In Figure 6b, the benefits of the flow straightener are evident, as the axial velocities are negative near the rotating structure, indicating that backflow is effectively mitigated. The flow straightener was manufactured by additive manufacturing using polylactide (PLA) material with 100% infill.

3.4. Design of the Rotating Structure

Figure 7a shows the designed rotating structure, which consists of a band, a crown, and rods that replace the blades found in a Francis turbine runner. The rotating structure has overall dimensions of 200 mm in diameter and 87 mm in height. The band features a rectangular cross-section with a radial thickness of 6.35 mm and a height of 18 mm. Figure 7b details the cross-section of the rods. Each rod has an elliptical cross-section with major and minor axes of 6.35 mm and 5 mm, respectively, and features a 4 mm wide flat surface on the inner side, designed for strain gauge installation.

3.4.1. Material Selection and Fabrication Method

A high-leaded tin bronze alloy (C93200 or SAE 660) was selected for its high density and low Young’s modulus. It also offers excellent machinability and good corrosion resistance, making it suitable for various water-related applications, including pump and valve components. The high density-to-Young’s modulus ratio of leaded bronze allows for lower natural frequencies compared to aluminum or steel alloys. This, in turn, allows for lower RSI frequencies and rotational speeds of the rotating structure. The density $\rho$ of the high-leaded tin bronze alloy is 8980 kg/m³, and its Young’s modulus E is 75.8 $\pm 15.8$ GPa (mean ± standard deviation). Both properties were measured experimentally. The Young’s modulus was measured in accordance with the ASTM E8-03 standard for tension testing of metallic materials using sixteen specimens.

The rotating structure was machined as a single piece using five-axis CNC milling machines. While different assembly methods have been studied, machining the rotating structure as a single piece was the easiest approach given the small dimensions of the rods and facilitating both experimental and simulation work.

3.4.2. Number of Rods

The number of rods and the number of guide vanes determine which ND modes are likely to be excited by RSI and by which harmonic (see Equation (2)). The test rig imposes the number of guide vanes ($Z_g=16$), while multiple options exist for the number of rods $Z_r$. In practice, lower ND modes are easier to excite, and excitation levels typically decrease with higher harmonics [18]. Hence, the optimal scenario would be a test case where the 2ND and 3ND modes are excited by the first and second RSI harmonics in either order. A unique solution exists to achieve this situation, which requires $Z_r=7$. In this configuration, it can be demonstrated that the 2ND mode is excited by the first RSI harmonic, and the 3ND mode by the second RSI harmonic.

3.4.3. Expected Natural Frequencies

The natural frequencies of the two modes of interest (2ND and 3ND) are influenced by both the rods and band dimensions. Through multiple design iterations with various rod and band dimension combinations, two key trends were identified. First, reducing the rod size lowers the natural frequencies. Second, increasing the band height decreases the natural frequencies significantly. These two trends guided the choice of the final dimensions of the rotating structure.

During the design phase, the natural frequencies of the 2ND and 3ND modes were calculated both in the air and submerged in still water. The simulations were performed with ANSYS Mechanical 2021 R2 using Modal simulations in air and Coupled-field modal simulations for the structure submerged in water, without rotation. The latter were used to predict the critical rotational speeds.

Figure 8a shows the boundary conditions used for these two types of simulations. A fixed boundary condition is applied to each hole, as indicated by the red circles on the crown of the structure. Figure 8b shows a section view of the acoustic volume used to perform the simulation for the structure submerged in water. The submerged structure is shown in red, and the water in blue. Particular attention was given to accounting for the effect of confinement on the dynamic behavior of the structure, as can be seen with the fluid gap around the structure, varying from 1 mm to 3 mm depending on the location. For simplification purposes, all holes and cavities in the crown are neglected in the Coupled-field modal simulation.

Table 2. Parameters used for Modal and Coupled-field modal simulations with ANSYS Mechanical 2021 R2.

Parameter	Value
FEM Modal simulation in air
Element size	3.5 mm
Element order	Quadratic
Number of elements	209,736
Coupled-field modal simulation in submerged water
Solid and acoustic element size	3 mm
Solid and acoustic element order	Quadratic
Number of solid elements	264,237
Number of acoustic elements	442,584
Water density	998.2 kg/m3
Water speed of sound	1482.1 m/s

summarizes the key parameters used for the two types of simulations.

Figure 9 shows the numerical prediction for (a) the 2ND and (b) the 3ND mode shapes of the rotating structure submerged in water. The natural frequencies in air and submerged in water are summarized in

Table 3. Simulated natural frequencies in air and submerged in water, and critical rotational speeds.

Mode Shape	Natural Frequency (Hz)		${\mathit{N}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}$ (RPM)
	Air	Submerged in Water
2 ND	291.7	221.7	831.4
3 ND	694.5	520.1	975.2

, as well as the corresponding critical rotational speed $N_{crit}$. The critical rotational speed (in RPM) is defined as follows:

$$N_{crit}={\displaystyle \frac{60f_{RSI}}{m{Z}_g}}$$

(4)

where $f_{RSI}$ is the RSI frequency that matches a predicted natural frequency of the rotating structure submerged in water, m is the harmonic order, and $Z_g=$ 16 guides vanes. With the proposed design, the predicted critical rotational speeds are 831.4 RPM for the 2ND modes and 975.2 RPM for the 3ND modes. These critical rotational speeds are within the operating range of the AC motor of the test rig.

3.5. Instrumentation of the Rotating Structure

Semiconductor strain gauges were installed on the band and rods of the rotating structure. They were selected over classical strain gauges due to their higher sensitivity and the anticipated low dynamic vibration amplitude of the rotating structure. The semiconductor strain gauge model used is SS-080-050-345-PB from Piezo Metrics (0.21 mm × 1.27 mm), Simi Valley, CA, USA, with a resistance of 345 ohms, a gauge factor of 140, and a backing (3.43 mm × 7.62 mm) that facilitates the installation. The main disadvantage of semiconductor strain gauges is the high-temperature dependence of their gauge factor. For instance, the thermal coefficient of the gauge factor for the selected strain gauge model is 23% per 100 °C. However, since the study focuses on identifying which ND mode and at what frequency the RSI induces resonance, the temperature dependence is not an issue, as identification will be performed using phase shift analysis between consecutive strain gauges and considering only the dynamic component of the signals. Moreover, since the water tank (270 m³) is part of the closed-loop hydraulic system, the water temperature remains nearly constant during the tests ($\pm 1 °\mathrm{C}$).

Figure 10a shows the locations of the strain gauges. First, a strain gauge is installed on the inner flat surface near the crown of each of the seven rods. Finite element simulations were conducted, and the anticipated high strain at these locations was confirmed. Second, seven strain gauges are evenly distributed along the inner side of the band at mid-height to measure the circumferential strain. This second set will be used, among other things, to identify the mode shapes. In total, 14 semiconductor strain gauges were installed.

Seven flexible printed circuit boards (flexible PCB) were used instead of wires, as shown in Figure 10b. Each flexible PCB is installed on a rod to connect a first strain gauge located there and extends onto the band to reach a second strain gauge. The flexible PCB was specifically designed to connect directly to the solder pads of each strain gauge, facilitating the installation process. Compared to small wires, using a flexible PCB is also more robust and makes it easier to install a protective layer. Additionally, grooves were machined into the crown for the insertion of flexible PCBs. Each flexible PCB terminates with a soldering pad positioned in a small cavity at the end of the groove on the crown, allowing for the connection of wires to the slip rings through the hollow shaft. Filling the grooves and cavities after inserting the flexible PCB ensures proper sealing and a uniform hydrodynamic profile.

A protective layer was applied over the strain gauges and the flexible PCBs, as depicted in Figure 10c. The protective layer used is Belzona 1321, which is an epoxy coating providing good wear resistance and is commonly used to protect surfaces of components such as propellers, centrifugal pumps, and turbine pumps. When properly applied, Belzona 1321 also provides the necessary waterproof protection. The protective layer on the rods and the band is 1 mm thick.

3.6. Wiring, Signal Conditioning, and Data Acquisition System

Figure 11 shows a schematic representation of the wiring, signal conditioning, and data acquisition system used for the measurements. In the rotating frame, RG-178 coaxial cables are connected to each soldering pad at the end of a flexible PCB. The coaxial cables pass through dedicated holes in the crown and then through the hollow shaft to reach the slip ring. In the stationary frame, each signal first passes through a custom analog high-pass filter that removes low-frequency content, including the DC component and its drift. Hence, any static component of strain, such as that potentially caused by the centripetal or temperature effect, is eliminated. The output signal of the high-pass filter thus contains only the dynamic component of the strain. It then enters a Wheatstone bridge, followed by an amplifier with a gain of 180. A quarter-bridge configuration is used for each strain gauge. The amplified signals are acquired using two synchronized eight-channel NI-9231 Sound and Vibration Modules installed in a cDAQ-9189 chassis. These modules have a maximum sampling frequency of 51.2 kHz.

A NI-9411 Digital Module and a NI-9263 Voltage Output Module are also installed in the cDAQ-9189 chassis. The NI-9411 Digital Module records the motor encoder signal, while the NI-9263 Voltage Output Module generates an analog signal corresponding to the desired rotation speed. This signal is fed to the variable frequency drive, which controls the AC motor’s rotation speed. A Labview program is used to record the measured signals and set the desired rotation speed.

4. Experimental Measurement Campaign

This section presents the results of an experimental measurement campaign conducted to validate the suitability of the designed test case.

4.1. Dynamic Behavior of the Structure in Air

An experimental modal analysis was conducted prior to the installation of the strain gauges to identify the natural frequencies and mode shapes of the structure in the air. The structure was suspended using bungee cords, and impact tests were performed with an instrumented impact hammer (model 086C01, 11.2 mV/N, PCB Piezotronics Depew, NY, USA) using the roving hammer method. Impacts were repeated three times at each of the 28 locations along the outer side of the band. An accelerometer (model 352C22, 10 mV/g, PCB Piezotronics Depew, NY, USA) was installed on the inner side of the band to measure the dynamic responses of the structure. A sampling frequency of 8192 Hz and a record length of 8 s were used. Average frequency response functions were obtained using spectrum averaging, and the natural frequencies and mode shapes were identified. Figure 12 illustrates the amplitude of a typical average frequency response function, highlights the natural frequencies for the ND modes, and depicts the associated mode shapes within the 235–1500 Hz frequency range. It can be observed that each ND mode appears in pairs at distinct frequencies. In a perfectly axisymmetric structure, ND modes occur in pairs, known as sine and cosine modes, at the same frequency. However, when the structure is not perfectly axisymmetric, these modes appear at distinct frequencies. This case was reported by Allaei et al. [35] and Fox et al. [36] for rings with non-axisymmetric mass and stiffness properties. Similar observations were reported by Rodriguez et al. [37] during an experimental modal analysis conducted on a model-scale Francis runner using the roving hammer method. In their case, they attributed this phenomenon to the impossibility of achieving a perfectly symmetric structure without specifying the source of the asymmetry. In our case, variability in Young’s modulus ($\pm 15.8$ GPa; see Section 3.4) is the most plausible explanation. However, the distinct frequencies of the sine and cosine modes do not present an issue for the proposed test case.

4.2. Identification of the Natural Modes Excited by RSI

To identify the natural modes that are excited by the first and second RSI harmonics, measurements were performed on the rotating structure using the strain gauges for 62 different constant rotational speeds, ranging from 258 to 1016 RPM. The guide vane opening angle was fixed at 30 deg, and the flow rate was adjusted to 0.1 m³/s for each test. A sampling frequency of 2133.33 Hz and a record length of 120 s were used. For each strain gauge signal, a fast Fourier transform (FFT) was performed with a block length of 2048 points (0.96 s), and the spectrum was obtained by performing 125 averages without overlapping and without a windowing function.

Figure 13 shows the amplitude spectrum of a strain gauge located on the band for rotational speeds of 704.4, 774.3, 836.4, 855.9, 910.3, and 949.2 RPM, arranged from left to right and top to bottom. In each graph, the amplitude at the first RSI harmonic (or at 16 times the rotational speed) is marked with a red dot. It can be observed that this amplitude increases with the rotational speed for lower rotational speed, reaches a maximum between 855.9 and 910.3 RPM, and decreases for higher rotational speed. This means that a natural mode is excited by the first RSI harmonic when the rotational speed is between 855.9 and 910.3 RPM. The same observation can be made for the second RSI harmonic, which is marked with a red cross in each graph and reaches a maximum amplitude near 910 RPM.

By tracking the amplitude at the first and second RSI harmonics for each of the 62 tests performed at different constant rotational speeds, the amplitude of the response corresponding to each of the two RSI harmonics can be plotted as a function of the frequency, as shown in Figure 14. The blue curve corresponds to the amplitude measured at the first RSI harmonic, and the red curve corresponds to the amplitude measured at the second RSI harmonic. The line connecting the data points was added solely to better illustrate the trend. For the proposed design (see Section 3.4), the theory predicts that the 2ND mode is excited by the first RSI harmonic, and the 3ND mode is excited by the second RSI harmonic. Hence, the natural frequency of both modes can be determined with the frequency corresponding to the peak value of each curve. The corresponding critical rotational speeds can be determined using Equation (4), yielding 878.9 RPM for the 2ND mode and 914.1 RPM for the 3ND mode. These rotational speeds are within the operational range of the AC motor.

Table 4. Natural frequencies and critical rotational speeds for the first and second RSI harmonics.

RSI Harmonic Order	Expected Excited Mode	Measured Natural Frequency (Hz)	Critical Rotational Speed (RPM)
1st	2ND	234.4	878.9
2nd	3ND	487.5	914.1

summarizes the natural frequencies and critical rotational speeds obtained experimentally.

To confirm which ND mode is excited by each RSI harmonic, a phase analysis can be performed using the strain gauge located on the band. The theoretical phase shift φ between two consecutive strain gauges is given by the following:

$$\phi =k·{\displaystyle \frac{2\pi }{Z_r}}$$

(5)

where $k$ is the nodal diameter, and $2\pi /Z_r$ is the angle between consecutive strain gauges. This equation assumes that the strain gauges are evenly distributed along the circumference and installed in the same orientation.

From the experimental measurements, the phase shift can be determined by calculating the phase of the averaged cross-spectrum between two consecutive strain gauges. FFT length of 2048 points (0.96 s) is used, with no overlapping and no windowing function, to calculate the cross-spectrum with 125 averages.

Table 5. Comparison of the theoretical and experimental phase shift between a pair of consecutive strain gauges installed on the band for the first and second RSI harmonics.

Pair of Strain Gauges	Phase Shift (Rad)		Error (%)
			1st RSI/2 ND	2nd RSI/3 ND
Theoretical	2ND	3ND
	1.79	2.69	-	-
Experimental	1st RSI harmonic	2nd RSI harmonic
1–2	1.71	2.53	4.74	6.00
2–3	1.82	2.76	1.15	2.41
3–4	1.80	2.71	0.12	0.59
4–5	1.78	2.74	0.74	1.70
5–6	1.91	2.82	6.36	4.55
Mean value	1.80	2.71	2.62	3.05

compares the theoretical phase shifts with the experimental values for each pair of consecutive strain gauges on the band. The theoretical phase shifts are 1.79 rad and 2.69 rad for the 2ND mode and the 3ND mode, respectively. By comparing these theoretical phase shifts with the experimental mean values calculated for the modes excited by the first and second RSI harmonics (1.80 rad and 2.71 rad), it can be confirmed with certainty that the first RSI harmonic excites the 2ND mode, while the second RSI harmonic excites the 3ND mode. Indeed, the mean error on phase shift is 2.62% and 2.71% for the 2ND and 3ND modes, respectively. The maximum error in the experimental phase shift is 6.36% and can be primarily attributed to positioning errors during installation, including those related to location and orientation. Strain gauge #7 was excluded from the analysis due to a malfunction.

The results at constant rotational speed confirm that the test case is suitable for studying RSI. The 1st RSI harmonic clearly excites the 2ND modes, while the 2nd RSI harmonic clearly excites the 3ND modes. The measured strain levels at both resonances are significant and can be easily captured using semiconductor strain gauges. The critical rotational speeds for both modes are within the operational range of the AC motor and are easily achievable.

4.3. Natural Modes Excited by RSI During Speed Variation

The primary goal of the proposed test case is to study RSI during speed variations. To verify the capability of the design to study RSI under speed variations, measurements were performed during a slow acceleration at a sampling rate of 4266.67 Hz. The AC motor was accelerated from 105 to 1090 RPM over 1200 s. A time–frequency analysis using short-time Fourier transforms (STFT) was performed to identify the natural modes excited by RSI. Figure 15 presents the spectrogram of a strain gauge installed on the band (top) and the rotational speed of the structure (bottom). The FFT length is set to 4096 points, a Hanning window is applied, and a 75% overlap is used. The resulting time and frequency resolutions are 0.24 s and 1.042 Hz, respectively. Two rectangles (labeled 2ND and 3ND) and six circles (numbered 1 to 6) are drawn with dashed lines to highlight key observations.

By examining the rectangle labeled 2ND, we conclude that a broadband noise excites both 2ND modes throughout the acceleration, as their natural frequencies remain visible at all times. Additionally, we observe that the natural frequencies of both 2ND modes diverge as the rotational speed increases. This behavior is expected for a structure rotating in a dense fluid, as described by Presas et al. [23]. Similarly, by examining the rectangle labeled 3ND, the same observation applies to the 3ND modes.

Each of the six circles identifies a resonance, numbered 1 to 6. For each resonance,

Table 6. Resonances identified during the acceleration from 105 to 1090 RPM in 1200 s.

Identification Number	Excited Mode *	Resonance Frequency (Hz)	$\mathbf{Strain} \mathbf{Level} (\mathbf{\mu }\mathbf{\epsilon }$)	Critical Rotational Speed (RPM)	Harmonic Order (h)	$\mathbf{Positive} \mathbf{Integer} (\mathit{q}$)	$\mathbf{Number} \mathbf{of} \mathbf{ND}$ (v)
1	2ND (bwd)	233.3	42.9	874.6	16
2	3ND (fwd)	488.5	9.46	916.1	32
3	2ND (bwd)	231.2	3.96	603.7	23	3	+2
4	2ND (fwd)	204.2	10.7	1023.8	12	2	−2
5	3ND (bwd)	520.8	2.72	822.0	38	5	+3
6	3ND (bwd)	524.0	6.81	1015.0	31	4	+3

* fwd: forward wave excitation; bwd: backward wave excitation.

provides information about the natural frequency, the excited mode, the strain level, the critical rotational speed, and the corresponding rotational speed harmonic.

The first two resonances are induced by RSI. Resonance 1 corresponds to the 2ND mode excited by the first RSI harmonic, i.e., the 16th harmonic of the rotational speed. Resonance 2 corresponds to the 3ND mode excited by the second RSI harmonic, i.e., the 32nd harmonic of the rotational speed. The resonance frequency and critical rotational speed are very close to those obtained in the tests performed at multiple constant speeds (see Table 4). The differences can be explained by the coarse discretization used for the tests performed at constant speeds.

The last four resonances (numbered 3 to 6) are induced by NTRCI. These resonances are induced by the 23rd, 12th, 38th, and 31st rotational speed harmonic and correspond to the 2ND mode (backward), 2ND mode (forward), 3ND mode (backward), and 3ND mode (backward), respectively. These results are corroborated by Equation (3), which is satisfied using the values of $h$, $q,$ and $v$ listed in the last three columns of Table 6. These results are similar to those obtained by Dollon et al. [19] during a coast-down with a prototype turbine, where five resonances were induced by NTRCI.

While the highest strain level observed corresponds to the 2ND mode excited by the first RSI harmonic (Resonance 1), the strain level at a resonance excited by NTRCI can also be significant. For instance, when comparing Resonance 4 with Resonance 2, we observe that the strain level for the 2ND mode excited by the 12th rotational speed harmonic (9.46 $\mathrm{\mu }\mathrm{\epsilon }$) is approximately the same as the strain level for the 3ND mode excited by the 2nd RSI harmonic (10.7 $\mathrm{\mu }\mathrm{\epsilon }$). This demonstrates that studying NRTCI is relevant and that the test case is also suitable for such investigations.

5. Conclusions

RSI is a complex FSI phenomenon that represents one of the main excitation sources acting in Francis turbines. RSI can generate significant mechanical stress in the runner, depending on the turbine’s operating regime. Transient operations, in particular, are known to be highly damaging due to the increased number of start-up and shutdown events, which are likely to excite the runner at resonance. The precise excitation mechanisms during the acceleration and deceleration of a turbine still need to be addressed to mitigate the high-stress levels induced. As a first step, this paper presented a simplified experimental test case specifically designed to study RSI phenomena occurring during speed variation.

A closed-loop hydraulic system previously used for a 20 cm diameter Kaplan model, including the spiral casing and distributor, was repurposed to develop the test case. The original runner was replaced with a simplified cyclic–symmetric rotating structure, designed to allow the 2ND modes to be excited by the first RSI harmonic and the 3ND modes by the second RSI harmonic within the system’s operating range. The rotating structure structural behavior is assimilable to the structural response of Francis runners. It features band modes that will be excited by RSI through seven rods located close to the guide vanes’ trailing edges. The rod radial location represents the typical position of the leading edges of the Francis runner’s blades. The rotating structure’s rotation speed and angular acceleration rate are controlled by an AC motor driven by a variable frequency drive. Additionally, a flow straightener was installed at the draft tube inlet to mitigate potential fluctuations associated with vortical structures in the draft tube.

Experimental modal analysis performed in air shows good agreement between the predicted and experimental natural frequencies for both the 2ND and 3ND modes of the rotating structure. However, the natural frequencies of the 2ND sine and cosine modes, as well as the 3ND modes, are distinct rather than identical as predicted in theory. This phenomenon is not unique to our structure. Indeed, Rodriguez et al. [37] observed the same behavior for a model Francis turbine runner. The variability in Young’s modulus is the most plausible explanation in our case. Nevertheless, this is not an issue, as RSI can still be effectively studied, and the modes inherently split during rotation.

Measurements were conducted on the structure in water under two conditions: constant rotational speeds and slow acceleration across a wide range of speeds. The measurements performed at constant rotational speeds validated that the 1st RSI harmonic excites the 2ND mode, and the 2nd RSI harmonic excites the 3ND mode, as expected during the design phase. A cross-spectrum phase analysis was performed using the signals from a pair of consecutive strain gauges to identify the number of nodal diameters. Additionally, the spectrogram of measurements conducted during a slow acceleration demonstrated that the test case is not only suitable for studying RSI but can also be used to investigate NTRCI, where a harmonic order of the rotational speed can excite a resonance, provided that the excitation matches both the natural frequency and mode shape. Future work will experimentally investigate RSI and NTRCI under different operating conditions, with a focus on the effect of the acceleration rate on transient RSI-induced resonances.