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Research Papers: Flows in Complex Systems

# Experimental and Numerical Investigations on the Origins of Rotating Stall in a Propeller Turbine Runner Operating in No-Load ConditionsPUBLIC ACCESS

[+] Author and Article Information
Sébastien Houde

Hydraulic Machines Laboratory,
Faculté des sciences et de génie,
Laval University,
1065 rue de la médecine,
e-mail: sebastien.houde@gmc.ulaval.ca

Guy Dumas

Laboratoire de Mécanique des
Fluides Numérique,
Faculté des sciences et de génie,
Laval University,
1065 rue de la médecine,
e-mail: guy.dumas@gmc.ulaval.ca

Claire Deschênes

Hydraulic Machines Laboratory,
Faculté des sciences et de génie,
Laval University,
1065 rue de la médecine,
e-mail: Claire.Deschenes@gmc.ulaval.ca

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received November 1, 2017; final manuscript received February 8, 2018; published online May 28, 2018. Assoc. Editor: Shawn Aram.

J. Fluids Eng 140(11), 111104 (May 28, 2018) (18 pages) Paper No: FE-17-1712; doi: 10.1115/1.4039713 History: Received November 01, 2017; Revised February 08, 2018

## Abstract

Hydraulic turbines are more frequently used for power regulation and thus spend more time providing spinning reserve for electrical grids. Spinning reserve requires the turbine to operate at its synchronous rotation speed, ready to be linked to the grid in what is termed the speed-no-load (SNL) condition. The turbine's runner flow in SNL is characterized by low discharge and high swirl leading to low-frequency high amplitude pressure fluctuations potentially leading to blade damage and more maintenance downtime. For low-head hydraulic turbines operating at SNL, the large pressure fluctuations in the runner are sometimes attributed to rotating stall. Using embedded pressure transducer measurements, mounted on runner blades of a model propeller turbine, and numerical flow simulations, this paper provides an insight into the inception mechanism associated with rotating stall in SNL conditions. The results offer evidence that the rotating stall is in fact associated with an unstable vorticity distribution not associated with the runner blades themselves.

## Introduction

In 2015, hydraulic turbines contributed to 70% of all the electricity production from renewable resources [1]. Using natural and artificial lakes/reservoirs, hydraulic energy, of all the major renewable energy sources (solar, wind, hydraulic), is the only one which can be stored prior to transformation, thus providing the possibility of long-term energy planning. Furthermore, hydraulic turbine technology offers the distinct advantage of being close to an “on-demand” production technology with little time lag required between demand and delivery. Henceforth, hydraulic turbines have become an important asset to provide power regulation on decentralized power grids with large contributions from wind and solar energy (e.g., the European market) or to provide short-time production to fulfill spot-market contracts.

In such conditions, hydraulic turbines undergo more frequent start/stop sequences or operate for longer time in speed-no-load (SNL) conditions, ready to be linked to the power grid. Those events, however, can have a significant impact on the residual life of the turbine [24] and can lead to premature cracking of turbine runners. Transient conditions and no-load regimes have thus become an important research subject in order to improve the reliability of hydraulic power-based systems [5,6]. The main difficulty in considering those conditions at the turbine design stages is the prediction of the flow-induced loading on the runner blades [4,710]. Besides the time-varying boundary conditions for start-up/shut-down, such operating regime also involves large backflow regions, separations, and a complex vorticity dynamics that lead to important vortical interactions inside the turbine. Specifically, for no-load conditions, some authors observed the presence of rotating stall in the runner, with a dominant impact on the pressure and strain signals [11,12].

This paper presents an original numerical and experimental contribution to the analysis of the flow dynamics in SNL conditions for a low-head propeller turbine. This turbine was the focus of a comprehensive experimental and numerical study, AxialT [13], of its flow dynamics in different operating conditions. Among many others, measurements based on 31 miniature pressure transducers mounted on two runner blades were performed in speed-no-load conditions [14,15]. Those measurements form the basis of the present study and serve to validate numerical flow simulations. The simulation's results based on an unsteady Reynolds-averaged Navier–Stokes (URANS) approach using the scale adaptive simulation (SAS) turbulence model reveal that the behavior inside the AxialT runner in speed-no-load is dominated by vortical structures stemming from instability of the flow in the vaneless space between the guide vanes and the runner. This paper provides evidence that those structures exist independently of the runner blades, but lead to what has been termed “rotating stall” within the runner interblade passages.

The first part of the paper presents an overview of the no-load problematic in hydraulic turbines and introduces the AxialT turbine. Analysis of the experimental measurements covering no-load conditions is then presented to outline the presence of rotating stall in the turbine runner. The details of the numerical setup, including an overview of the SAS turbulence treatment, follow along with the core of the paper consisting in the analysis of the numerical simulation results in that AxialT turbine. Finally, results of URANS simulations without the runner blades are used to demonstrate how the rotating stall affecting the runner of AxialT in no-load conditions stems fundamentally from an instability of the vorticity distribution within the vaneless space.

## No-Load Conditions in Hydraulic Turbines

Figure 1 illustrates the main components of the AxialT propeller hydraulic turbine. The water flows from the spiral casing and the distributor to the runner and exits through the draft tube. Within the distributor, a series of pivoting guide vanes regulate the discharge and the entrance conditions to the runner while converting static pressure into swirl. Angular momentum exchange takes place within the runner where swirl is converted into torque. The draft tube, a bended diverging duct, serves to lower the pressure at the runner exit and thus to increase the net head available for extraction by the turbine. A majority of hydraulic turbines are linked to a constant speed generator and thus operate at a predefined rotating speed referred to as synchronous speed (ns).

Constant speed reaction hydraulic turbines operate in no-load conditions during a normal start-up sequence. Speed-no-load is the final part of this sequence in which the turbine has reached the generator synchronous rotation speed (ns) while delivering no net torque (Tgen) under a given net head (H) and discharge (Q). On the efficiency η = Tgenn/ρEQ hill chart, where E = gH, of a particular turbine with a reference diameter D, using the dimensionless blade (nED = nD/E0.5) and discharge (QED = Q/DE0.5) parameters, the no-load line (also referred to as “runaway line”) corresponds to a line with η = 0 where Tgen = 0 (Fig. 2). For a given E, the speed-no-load conditions are defined at the point where nED yields the synchronous speed. Typically, SNL conditions are associated with low discharge and small guide vanes' opening angles (α). For a turbine operator, the speed-no-load conditions must be stable and must not generate unduly high stresses within the turbine [16].

Euler turbine equation states that the torque on a turbine blade is related to the change in angular momentum from inlet to outlet (T = ρQ(rVθin − rVθout)). Hence, in no-load conditions, there is no change in angular momentum of the fluid, and thus there is no extraction of the swirl injected into the turbine. In other words, there is no network exchanged between the fluid and the runner blades. This somehow implies that the runner blades have only a passive role. In no-load conditions, the ratio between the angular momentum and discharge at the runner exit is high with respect to normal operating conditions. No-load conditions are high swirl conditions for a hydraulic turbine.

Such swirl dominated flows are often unstable [16,17], leading to flows in the main parts of the turbine described as dominated by “stochastic” fluctuations [8,18]. However, some studies in no-load conditions have identified dominant coherent flow features. Two frequently recurring observations on flow dynamics in no-load regimes are large flow separations extending from the runner entrance to the draft tube and the presence of interblade channels vortices [12,1820]. For some turbines, speed-no-load conditions have also been linked to the onset of what is termed rotating stall [11,12,21,22].

The presence of a large recirculation zone is often referred to as “pumping” from the runner [12,18] since part of the runner apparently acts like a pump moving fluid upstream. There is, however, no proof that this pumping is actually induced by the runner itself. However, that recirculation in the center of the draft tube implies that most of the water discharge is moved toward the walls, leading to a strong shear layer within the draft tube and runner. Shear layers (vorticity layers in effect) are generally unstable and may give rise to vortical structures [17,23].

Interblade channel vortices are typically associated with low-load regimes [16] including no-load conditions. They are often attributed to steep, generally negative, angles of attack at the runner leading edges. Those vortices are typically attached to either hub or shroud and spans the interblade channels from leading edge toward the trailing edges of the blades. Interblade channel vortices are dominant flow features of low-load conditions [16] where they are often associated with cavitation.

Rotating stall is a flow phenomenon where a number of interblade channels in a rotor are “stalled” in the sense that large reverse flow regions prevent the effective contribution of those channels to the runner useful work. Those reverse flow cells rotate at their own angular speed relatively to the runner thus inducing periodical fluctuations in the interblade channels. Rotating stall is typically associated with pumps since the pressure rise through the rotor and the stator can trigger flow separation when nonoptimal angles of attack are present at the blades leading edges [24]. In hydraulic turbines, this phenomenon is only documented for operation near the no-load conditions.

Rotating stall may be considered a “wave-like” instability propagating circumferentially [25]. The wavelength of rotating stall is thus no longer than the circumference of the rotor and its lowest propagation speed is about 20% of the rotor speed. In fully established regimes, rotating stall involves reverse flow cells turning at a fraction of the runner speed. The inception mechanism of rotating stall in compressors and pumps can be divided into two categories: amplification of compressible long wavelength disturbances or local flow separation in the rotor (spike inception). The former being a compressible flow feature, it is unlikely to be present in hydraulic turbines. The description of rotating stall in hydraulic turbines often refers to flow separations associated with unfavorable angles of attack on the rotor blades leading to the formation of vortices and blockage of interblade channels [12,21,22]. This inception mechanism seems to have similarities with “spike inception” in pumps.

Spike inception is typically thought to originate from local nonuniformity of the compressor/fan blade geometries triggering a flow separation on one particular blade. This separation leads to blockage of an interblade channel and consequently, due to discharge redistribution, to increasing and decreasing angles of attack of the two adjacent blades [25]. The separation, and thus the blockage, moves to the upstream blade with the increasing angle of attack. This mechanism fundamentally requires the presence of the blade themselves. Measurements and analysis in axial compressor further suggest a complex mechanism involving a separation vortex originating near the blade tip and leading to significant blockage in the interblade channel and increased flow spillage at the tip [2628].

For axial hydraulic turbines, however, some preliminary evidence suggests that rotating stall inception might be of a very different nature. Pulpitel et al. [29] observed on a model turbine similar to AxialT, the presence of structured vortices in the vaneless space in no-load conditions. Those vortices, in precession with the runner, were generally stable and took different configurations depending on the operating parameters of the no-load conditions (head, runner speed, guide vane opening). They were associated with backflow regions extending into the draft tube. Their visualizations indicate stable configuration of either three or four vertical vortices attached to a fixed circumferential vortex in the vaneless space (Fig. 3). The number of vortices and their rotation speed are closely correlated with the dominant pressure fluctuations measured in the runner and the vaneless space. Their hypothesis on the origin of this flow configuration is linked to the presence of the shear layer arising from the backflow cells extending to the vaneless space in high swirl conditions. This mechanism thus appears entirely independent of the flow incidence at the runner blades leading edges.

## AxialT Project and Experimental Investigations of No-Load Conditions

AxialT was a research project aimed at studying the flow dynamics in a propeller turbine model. The project used a combination of experimental and numerical approaches to investigate primarily steady-state flow conditions in the runner and the draft tube. Extensive measurements using laser Doppler velocimetry, particle image velocimetry, laser induced fluorescence, pressure transducers, and strain gages were performed on the model turbine installed on the test stand of the Hydraulic Machines Laboratory (LAMH) in Laval University, Quebec City, QC, Canada. An overview of the project main results can be found in Deschênes et al. [13].

The AxialT turbine model has a semispiral casing, a distributor with 24 stay vanes and guide vanes, a six blades propeller runner with a 0.8 specific speed, and a single pier bended draft tube (Fig. 1). The scale model was set up on the LAMH turbine test rig. This test rig consists of a classical closed-loop hydraulic facility with a discharge up to 1 m3/s, a head up to 50 m, and a maximal net power output of 170 kW.

The speed-no-load conditions were studied as part of a larger investigation into transient regimes [15]. For those measurements, 31 piezo-resistive pressure transducers were mounted on two runner blades (referred to as blade I and blade II, see Fig. 4). Furthermore, one pressure sensor was installed in the vaneless space and one in the draft tube conical diffuser. To complement the pressure data, four strain gages, mounted in full bridge, were installed on the runner blade opposite to blade I. The strain gages measured the main bending loads at the blade root in the most probable strain hotspot. The pressure acquisition frequency was 5 kHz. The pressure sensors signal uncertainty, based on calibration in the operating condition useful range, is below 0.1% of the reference pressure obtained from DH Instruments pressure scale with a precision of 10 Pa. The main uncertainty comes from the zero drift during the measurements, evaluated as below 0.5% for the SNL tests. Further details about the experimental setup can be found in Houde et al. [14].

Pressure and operating conditions (Q, H, n) were recorded in two different transient scenarios linked to the SNL conditions: run 1 going from SNL to full load and run 2 going from full load to SNL. As illustrated in Fig. 5, the first run yielded 30 s of continuous operation in SNL conditions and the second run 15 s. The head and rotation speed were maintained constant throughout those measurements to obtain the nominal nED of about 0.65. The guide vane's opening for the SNL conditions represents 20% of the full load opening. The Thomas cavitation number (σ) was relatively high and the recorded pressure during the test was safely above the water vapor pressure, hence SNL conditions were cavitation free.

###### Overview of the Experimental Results.

The instantaneous pressure variations during runs 1 and 2 are illustrated for sensors S6 (pressure side) and S14 (suction side) in Fig. 6. The signal from those sensors is found to be representative of the signals from the other 31 sensors. The pressure fluctuations in SNL conditions dominate largely over the pressure fluctuations at full load. Furthermore, run 1 large pressure fluctuations are also present in the first phase of the transition from SNL to full load while for run 2, the same fluctuations appear before reaching the SNL conditions. The pressure signals in SNL regime for both tests were cross-correlated and showed high coherence, implying that the phenomena responsible for the large pressure fluctuations and their inception are independent of the initial conditions for the same operating parameters.

A spectral analysis of the pressure signals during both runs was made using wavelet routines described in Fortin [30]. Wavelet analysis is well suited for the study of low frequency content during transient events since it is not affected by the window size limitation of short time Fourier transform for example. As illustrated for sensors S6 and S14 in Fig. 7, the largest fluctuations occur in the speed-no-load regime and in the first instants preceding or following it. That energy is spread mainly in the subsynchronous range (log[f/n] < 0). One frequency stands out among the other, f/n = 0.88, which carries a higher energy level over the entire SNL range. Interestingly, that frequency also dominates in low-load regime just before and after the no-load conditions, and even gains in amplitude, indicating that the associated flow phenomena actually gain intensity in low-load regimes. As expected, the wavelet analysis reveals a rich stochastic energy content in the low frequency range for SNL operation but with a dominant frequency that appears linked with coherent and sustained flow structures.

In order to ascertain the origin of the f/n = 0.88 frequency, deeper spectral analysis, using fast Fourier transform-based methodologies, was carried out on the pressure signal of run 1 for the first 30 s corresponding to the permanent SNL regime. As illustrated by the energy spectra of pressure sensors S6 and S14 (Fig. 8), f/n = 0.88 is a dominant frequency in the subsynchronous range, although the energy content of other frequencies is still high. This observation holds true for most sensors except the ones located on the suction side near the leading edge (S1, S2, S22, S23). For the measured strains, the spectral analysis also reveals that the f/n = 0.88 component dominates the signal (Fig. 9(a)). Cross-spectral analysis between all sensors shows that the only frequency for which the coherence function is constantly above 0.9 is f/n = 0.88. These results indicate clearly that a coherent fluid phenomenon is responsible for the f/n = 0.88 pressure fluctuations while at other frequencies, the fluctuations are less structured and definitely of stochastic nature in most cases.

The phase angle of the pressure signal from sensor pairs located on both instrumented suction sides (e.g., S6 versus S27), which are physically at 60 deg from one another, is 180 deg for the f/n = 0.88 component. This implies that the flow structure associated with the f/n = 0.88 has an azimuthal period of 120 deg (2 deg × 60 deg). Considering the entire runner, this implies the existence of three flow structures in precession with the runner. This would be coherent with a three-cell rotating stall.

It is possible to evaluate the rotating stall angular speed (ωRS) in the fixed reference frame using the measured frequency of the three cells in the rotating frame of reference (0.88n) by considering that the relative velocity (n − ωRS) of N = 3 cells is linked with the measured excitation frequency by Display Formula

(1)$0.88n=N(n−ωRS)$

Hence, the angular speed of the rotating stall is ωRS = 0.7067n, which should yield a measured frequency of 2.12n for the three cells in the fixed reference frame. The spectral analysis of the pressure signal from the vaneless space is indeed dominated by the f/n = 2.12 component (Fig. 9(b)) providing further indication that a rotating stall is indeed present in the turbine in the SNL condition. Also, of interest in Fig. 9(b) is the presence of a peak at the runner blade frequency of f/n = 6 indicating that the rotor passing frequency felt by the stator in SNL conditions can induce, locally at least, pressure waves of similar amplitudes to rotating stall.

## Simulations of AxialT Speed-No-Load Conditions

The methodology selected for the simulations of the SNL flow dynamics of the AxialT turbine is based on the few published turbine simulations in no-load conditions. For general no-load operations, Nennemann et al. [31] compared results obtained with two types of URANS turbulence modeling strategies: kε and SAS, where the latter is a Boussinesq-based two-equation model but with eddy viscosity correction for large-scale turbulent flow structures. Their results indicate that stochastic torque fluctuations on the runner, and thus pressure fluctuations, are simulated more accurately with the SAS model. Although this result was expected, detailed validation of both simulations was lacking since they based their comparison on limited experimental data gathered on a prototype turbine. Their analysis indicates, however, that transient coherent vortical structures do exist within the runner at no-load operation. In a follow-up work using one-way fluid-structure simulations based on SAS, Morissette et al. [8] imputed underprediction of the simulated strain spectral content to lack of turbulence content due to the turbulence modeling used and to unaccounted for cavitation effects.

More recently, Mende et al. [18] compared simulation results from large eddy simulations (LES) and SAS for a Francis turbine in no-load conditions with limited validation data. They pointed out that, as performed, both modeling approaches gave similar results. Their simulations showed a complex flow pattern in the runner with dominant stochastic fluctuations near the band. The significant computational cost of running LES simulations in a hydraulic turbine is clearly exemplified by their limited duration. Mende et al. [18] simulated only 1.29 runner revolution, clearly not enough to resolve the high amplitude low frequency content observed on most Francis turbines operating at no-load conditions.

Nicole et al. [12] performed simulations of the entire start-up sequence of a low-head Francis turbine including the speed-no-load conditions using URANS along with the kε turbulence model. In SNL conditions, they managed to capture a rotating stall instability validated against measurements. Their conclusion was that the dominant coherent structures in SNL could be captured with a relatively light URANS approach while, in accordance with later studies, the stochastic fluctuations were damped out.

In the present study, simulations of AxialT speed-no-load conditions are performed using SAS turbulence treatment in order to confirm the presence of the rotating stall and to study its dynamics. The simulation strategy rests on the use of unsteady simulations in the distributor, runner, and draft tube. The semispiral casing is omitted to limit computing time. Sections 4.14.4 provide a description of the simulation strategy, a comparison with the experimental results and a vorticity dynamics-based analysis of the flow in the turbine in speed-no-load conditions. The present simulations indicate that the flow dynamics are similar to what was observed experimentally by Pulpitel et al. [29] on a similar turbine with the added insight of a quantitative flow description provided by the simulations.

###### Scale Adaptive Simulation Turbulence Model.

The most basic approach for flow simulations in speed-no-load conditions rests on the use of URANS simulations with an eddy viscosity turbulence closure model such as kε. Results by Nicolle et al. [12] clearly show that such simulations are able to capture rotating stall at speed-no-load at a reasonable computing cost. However, Reynolds-averaged Navier–Stokes (RANS) simulations lack the richness reflected by the measurements since they do not resolve the stochastic turbulent flow fluctuations. Based on the AxialT measurements, the low frequency stochastic fluctuations at SNL have non-negligible energy levels that may affect the turbine residual life according to Morissette et al. [8]. It thus becomes interesting to validate a numerical strategy that enables the simulations of at least part of the turbulent spectrum in the runner.

Scale adaptive simulation is considered a hybrid turbulence approach where a standard RANS model (in the present case the shear stress transport (SST) model) is modified to work as a scale-resolving simulation model [32]. Practically, the SAS modification consists of a source term (QSAS) added to the transport equation of the turbulence eddy frequency (ω) of the two-equation SST model. QSAS takes the form Display Formula

(2)$QSAS=maxρζ2κS2LLυK2−C2ρκσΦmax∇ω2ω2,∇κ2κ2,0$

where L represents a turbulence length scale and LυK the von Kármán length scale calculated from the second derivative of the velocity according to

$LυK=κU¯′U¯′′;U¯′′=∂2U¯i∂xk2∂2U¯i∂xj2;U¯′=S=2SijSij;Sij=12∂U¯i∂xj+∂U¯j∂xi$

In effect, Menter et al. [32] and Krappel et al. [33] explain that this procedure reduces the turbulent viscosity in free shear regions of the flow leading to the resolution of turbulent structures up to the grid size limits. A limiter, related to the LES Smagorinsky model, is applied to the eddy viscosity to ensure proper dissipation of turbulent structures at grid scale. In essence, the SAS model provides an improved RANS model to account for turbulent and unsteady flow structures in free shear flows. The SAS model has so far demonstrated surprisingly good convergence and robustness for off-design simulations of hydraulic turbine flows while delivering results similar to wall-modeled LES [34]. It presents significant advantages over the detached eddy simulation family of models since it appears less dependent on mesh quality and boundary conditions to deliver accurate results for hydraulic turbine flows.

###### Computing Domain and Grids.

The computing domains are illustrated in Fig. 10. The semispiral casing was omitted from the simulations but the stay vanes' channels include a converging section representing the midsection of the semispiral casing. The use of such geometrical simplifications was validated internally many times to ensure that it does not affect the dominant hydrodynamics of the turbine. The interface between the runner and the guide vanes is located on a cross-stream surface about midway between the guide vane's trailing edges and the runner blades leading edges. The runner geometry was built from a reverse engineered digital model [13]. The runner does not include the 0.1 mm tip gaps. A priori this may seem a significant omission, but simulations performed on the AxialT turbine [35] have shown that the impact of such gap on the velocity profiles at the runner exit is within the measured dispersion of laser Doppler velocimetry velocity profiles. The draft tube features a short extension at the exit to provide more controlled outlet conditions. The runner rotates around the Z-axis and the origin (0, 0, 0) is located at midheight of the distributor.

The computational grids (Figs. 11 and 12(a)) of all components are based on structured hexahedral elements. ANSYSicemhexa was used for the stay vanes' channel, runner, and draft tube while NUMECA AutoGRID was used for the distributor. All the grids respect basic quality criteria: minimal orthogonality of 20 deg, minimum 3 × 3 × 3 determinant of 0.5, and maximum aspect ratio of 300. The expansion ratio was kept below 1.2 in most regions. The wall resolution satisfies the requirements for the application of ANSYScfx scalable wall model, i.e., Y+ in the interval 20 < Y+ < 200. The final mesh sizes (Table 1) of all components were checked to deliver a grid influence of less than 5% on the individual component head losses and less than 1% on the stay vanes, guides vanes, and runner blades torque. The total number of elements for the entire simulation was 15.8 M.

Grid convergence tests were made using RANS simulations with the kε turbulence model. This choice is justified by three factors:

1. (1)The formulation of SAS is based on the SST, which is itself a blend of the kω and kε model.
2. (2)To limit the computational mesh size, a wall model is used; thus, near wall resolution is adjusted to keep Y+ > 20. The SAS model therefore acts, in attached regions near the wall, as a kε model with scalable wall functions.
3. (3)Scale adaptive simulation being a hybrid model with LES-like behavior in the core flow, mesh independence depends on the length and time-scales of interest. In the present case, for SNL operations, the dominant structures have time and length scales relatively large with respect to the runner (diameter and rotation speed); hence, the mesh is made to resolve turbulence scales one or two orders of magnitude beyond the structures of interest. This choice, although arbitrary, is based on published results with the SAS model.

Results of the runner grid convergence test are presented in Fig. 12(b) in terms of extracted torque since the main validation data in this study come from pressure measurements on the runner blades. The test shows that the selected mesh delivers results within the “mesh independent” range for torque prediction.

###### Boundary Conditions and Solver Parameters.

The boundary conditions used in the SNL simulations are presented in Table 2. The discharge (Q) and the rotation speed (n) are the averaged value coming from the measurements. A slip-wall condition is used on the frontiers of the draft tube extension.

The solver used is ANSYScfx v17.2, a finite element based finite volume unstructured solver [36]. CFX is a multigrid coupled solver using a modified Rhie and Chow algorithm for the pressure velocity coupling. The time integration scheme is based on second-order implicit Euler formulation. The time-step (Δt) used in the SNL simulations yielded 1 deg of runner rotation per iteration. Within each time-step, five linear solver iterations were used to yield maximum residuals below 5 × 10−5 on all solved variables. The so-called high resolution scheme was chosen for the advection part of the momentum equations. This scheme adjusts a blend factor (β) between a purely first-order upwind scheme (β = 0) and a centered second-order scheme (β = 1) depending on mesh quality and local flow dynamics. For the SNL simulations, the scheme delivered an averaged value of β = 0.9 over the entire simulation domain, with very few elements in the runner and distributor with a value below 0.75, which is deemed acceptable based on the authors' experience through internal tests and published results with the same solver. The turbulence transport equations were solved with a first-order upwind scheme. The SAS simulation ran for 72 runner revolutions and the last 55 revolutions were used to get statistically significant results for analysis purposes.

To assess how close to an LES or an RANS solution the SAS model is, CFX calculates a blending function (Blending Function for detached eddy simulation model) where a value of 1 indicates a RANS solution based on the SST model while a value of zero indicates a solution with LES-like behavior. Figure 13 illustrates the values of the blending function for one time-step representative of the behavior of the entire solution. As expected, the near wall region with attached flow is mostly treated as RANS while most of the vane less space, the runner, and the core of the draft tube flow are treated as LES. This result indicates that the computational meshes are indeed appropriate for the SNL conditions where most of the expected unsteady flow dynamics should occur in the vane less space, the runner, and the draft tube core flow.

###### Experimental–Numerical Comparison.

In order to validate the numerical methodology, comparison between the simulations and the measurements are based on statistical and spectral analysis of the pressure signal of sensors 1-31. Figure 14 compares the experimental and numerical average pressure levels (Pavg) of the 31 blade-mounted pressure sensors. The average difference between numerical and experimental values is 0.8% with a maximum of 3% for sensor S15. This result indicates that on average, the simulated pressure distribution is accurate; therefore, implying that the boundary conditions and the geometry used deliver relevant results albeit the geometrical simplifications. Indeed, even the sensors located close to the shroud, where the effect of the tip gap absence in the simulations should be strongest, show only differences of around 1%.

Figure 15 compares the pressure standard deviation (σ(p)) between measurements and simulations. The average difference is 10%. The largest differences (between 10% and 30%) are observed for sensors located close to the shroud. This is coherent with the absence of tip gap, changing locally the pressure fluctuations without affecting the average pressure distribution. However, with the exception of sensors S9, S10, S30, and S31, the measured σ(p) is consistently under the simulated values. This observation can stem from a number of possible causes such as the finite size of the pressure sensors (3 mm) in the laboratory in contrast to the point extraction process of the simulations.

In order to perform spectral analysis using similar frequency resolution (Δf) between simulations and measurements, the 30 s experimental signal was subsampled into ten 3 s samples. This procedure yielded a Δf/n ∼ 0.027 close to the Δf/n coming from the simulated signal. Figures 16 and 17 illustrate the numerical and experimental power spectra for sensors S6 and S14. Figure 16 shows the complete spectrum up to the cutoff frequency (f/n = 160) while Fig. 17 focuses on the subsynchronous range (f/n < 1).

The simulated power spectrum up to f/n = 160 is representative of the expected results associated with the use of the SAS model. The most energetic part of the power spectrum, up to f/n = 20, is well represented by the simulations considering the limited sampling time. Both the modal amplitude and the slope compare qualitatively well with the measurements. After f/n = 20, the simulations exhibit a steeper decrease in energy levels compared to measurements. This is coherent with the mesh-based filtering of the SAS model limiting the size of the resolved turbulent structures and increasing energy dissipation.

For the subsynchronous range, 0.44 < f/n < 1 (Fig. 17), the simulation results exhibit the same dominant frequency as the experiments around f/n = 0.88 and capture some of the minor side bands. The amplitude of the dominant frequency is overestimated for most sensors in the simulation. This could be the result of the limited sampling size in the simulations or, as mentioned earlier, could also stem from the numerical methodology itself with respect to pressure treatment and the finite size of the sensors. However, the simulations do capture the rotating stall affecting local pressure on the blade at f/n = 0.88.

The evolution of the individual generating blade torque (Fig. 18(a)) shows an alternating pattern around an overall mean value slightly positive and close to zero (Tavg = 0.78 N·m). The dominant frequency of those torque fluctuations is indeed f/n = 0.88. Interestingly, the torque alternates between positive and negative values in a 120 deg cyclic pattern where blades #1, 3, and 5 show quite often a torque of opposite sign with respect to blades #2, 4, and 6. This 120 deg phase shift is coherent with the measurements results presented in Sec. 3. The axial trust is also dominated by the 0.88n component.

Analysis of the torque distribution of blade 1 on both half-spans of the blade (Fig. 18(b)) reveals that the torque on the inner span (starting from the hub) does change sign in an alternate manner. Concurrently, the outer span is either not delivering any torque (T = 0) or operates in turbine brake mode (T < 0). The turbine brake mode operation of the outer part coincides generally with the inner part being in turbine mode. However, the zero torque indicates that at least part of the runner is working in its designed range for SNL operation based on Euler equation.

In terms of flow dynamics, this behavior of the torque is associated with complex interactions within the runner channel. To illustrate those time-dependent interactions, two instants (t1, t2) shown in Fig. 18(b) were selected for visualization purposes based on no torque condition of the outer span (t1) and the inverse peaks of both spans (t2).

In order to investigate the torque distribution on blade 1 at time t1 and t2, Fig. 19 illustrates streaklines on two constant span (30% and 70%) planes projected in Meridional-theta coordinates. Clearly, for the inner span (30%), the flow field for both times is far from design conditions, with large backflow regions at the pressure side and the suction side. Considering the tridimensional nature of the flow close to the hub, using standard two-dimensional analysis to describe the flow can be quite subjective. However, at 70% of the span, the flow is more amendable to such analysis, with no discernable separation at the blade surface. It appears that the angle of attack relative to the blade profile might be coherent with the torque values at both times: slightly negative at t1 yielding T = 0 and more negative at t2 yielding the observed negative torque.

The number of coherent vortices, based on iso-surfaces of λ2 criterion, in the vaneless space and the runner (Fig. 20) outlines the richness of the simulated flow field using SAS. The simulations do resolve a large range of chaotic flow structures making direct interpretation based on a singular vortex, as presented by Pulpitel et al. [29], difficult. However, careful observation reveals a clumping of vortices in three regions around the hub corresponding to three low-pressure zones.

The normalized pressure (Pn = P/E) distribution at the rotor stator interface (Fig. 21) clearly reflects the effects of the “bunched vortices.” With respect to the runner, the relative rotation speed of those three low-pressure regions corresponds to 0.88n. The pressure reflecting the complex vortex dynamics more coherently indicates that indeed the three “clumped vortices” regions are associated with the measured dominant pressure fluctuations in the runner.

Figure 22 presents backflow regions (in red) in three locations within the runner rotating reference frame. At the rotor–stator interface, backflow regions are concentrated around the hub in three distinct areas rotating at 0.88n. Backflow regions close to the shroud do not extend into the runner. Within the interblade channels, the backflow regions mostly evolve around the hub. For time t1, a backflow is present only at the suction side of blade 1 while at t2, backflow regions are present on both pressure and suction sides of the blade. This is coherent with observations at SNL by Pulpitel et al. [29] and agrees with the simulation results of Nicolle et al. [12]. Those rotating backflow regions within the runner are generally what are termed rotating stalls.

At the runner exit, backflow is present in all runner blade channels. Hence, near the hub, the turbine is locally in “reverse pump” mode. Furthermore, a large backflow region extends from the runner exit far into the draft tube trumpet after the bend (Fig. 23). Most of the discharge in the draft tube occurs on a narrow band close to the wall generating a “quasi-cylindrical” shear layer in the conical diffuser. This shear layer does not appear to undergo any coherent roll-up that would lead, for example, to a corkscrew vortex in the draft tube as was observed when the AxialT turbine goes into runaway from full load conditions (essentially a high discharge no-load conditions) [30]. The SNL draft tube flow is coherent with results from Mende et al. [18] or Nennemann et al. [31] in no-load conditions.

Hence, the SAS simulations indicate that the frequency f/n = 0.88 dominating the measured and simulated pressure signals is linked with three pairs of vortex/backflow in precession with the runner. Those vortices are attached to the head cover in the vaneless space and extend into the runner modulating the backflow regions almost to the runner exit. Experimental measurements and simulation results show no significant trace of their presence in the draft tube. The question now is whether those vortices/backflow pairs are a consequence of local flow separation on the runner blade or from an instability arising within the vaneless space as postulated by Pulpitel et al. [29].

## Numerical Analysis of the Rotating Stall Inception

The SNL simulation indicates the presence in the vaneless space of three regions with bunched vortices. Those bunched vortices are associated with regions of low pressure that would entrain air bubbles just as in the experiment of Pulpitel et al. [29]. Considering that, theoretically, an axial runner producing no torque should not change the flow direction from the inlet to the outlet, the runner blades should play a passive role in SNL conditions. So, in order to assess whether the observed rotating stall in the AxialT is related to a flow instability in the vaneless space or to local separation on the runner blades, simulations were performed on the AxialT turbine without the runner blades present. Those simulations used the same computational setup described in Sec. 4. The rotating domain is still the same with a rotating hub and fixed shroud.

Those simulations are based on the standard high-Re kε model. The use of kε is justified from the need to limit the number of flow structures for analysis purposes, simplifying the chaotic nature of turbulence, to outline the existence of the flow instability presented by Pulpitel et al. [29]. The use of kε in such a context is supported by the results of Nicolle et al. [12] who managed to reproduce rotating stall and its pressure fluctuations at speed-no-load in a low head Francis turbine.

Besides the absence of runner blades and the use of kε, those simulations use a new no-blade runner mesh with a circumferentially uniform distribution (Fig. 24) and a resolution similar to the original runner mesh. The absence of runner blades leads to a mesh size of 3M elements. The initial conditions for those simulations are based on a no-swirl initial guess. The same monitor points based on the position of the blade pressure sensors (Fig. 4) were used; thus, a direct frequency comparison can be performed.

The simulation was run for about 2 s of physical time using the same time-step as the SAS simulations with runner blades. The flow evolved over the simulated time span but reached a permanent state after 1.3 s. For analysis purpose, a 0.35 s segment was isolated from t = 1 s onward since it exhibited some quite revealing flow evolution. Figure 25 illustrates coherent structures based on iso-surfaces of λ2 criterion at the start of the 0.35 s sequence (ti) and at the end (tii). Figure 26 shows the backflow on the same planes as Fig. 22. Clearly, the simulation captures the evolution of an unstable vorticity distribution within the vaneless space quite similar to what was observed by Pulpitel et al. [29]. During the time span considered, discrete vortices and backflow regions in precession with the runner dominate the vaneless space. The number of vortices evolves from three to five, the three-vortex configuration having been the dominant configuration from t = 0.5 s until about t = 1.2 s. This evolution of the number of vortices is most probably linked with the growth of the most unstable modes of the vorticity distribution. That evolution could be compared to the evolution of geostrophic zero circulation vortex presented by Kloosterziel and coworkers [37]. For the remainder of the simulation, the five-vortex structure remains stable.

Interestingly, the backflow around the hub, often referred to as pumping, is present without the blades present (Fig. 27). This backflow spans from the hub too deep into the draft tube core. The observed backflow is most probably a consequence of two mutually interacting mechanisms:

1. (1)Balance between centrifugal forces associated with the high swirl content at the draft tube inlet and the adverse pressure gradient stemming from the draft tube divergence.
2. (2)Boundary layer separation on the head cover.

Analysis of pressure signals (Fig. 28) through spectral analysis reveals that two frequencies dominate f/n = 0.9 and f/n = 1.4. The frequency resolution of the spectra was improved by a factor 4 by signal replication. The f/n = 0.9 frequency corresponds to the three vortex evolution, which is relatively close to the f/n = 0.88 frequency associated with the runner SAS simulations. The f/n = 1.4 frequency is close to the excitation produced by the five vortex configuration considering that the rotating speed of an individual vortex is 0.88n/3 = 0.29n, and thus would be 1.47n for five vortices. This result indeed confirms that the observed vortical structures simulated with kε have the same origins as those observed in the simulations with runner blades.

Thus, without the runner blades, both discrete vortices and backflow regions are the dominant flow structures in the vaneless space and they move in precession with respect to the runner with an angular velocity close to that of the structures observed at rotating stall in AxialT turbine. This indeed confirms that vortical structures associated with an unstable vorticity distribution in the vaneless space is the primary mechanism leading to rotating stall in the AxialT turbine at SNL conditions. However, with respect to the chosen SNL condition, the blades do influence the evolution of the instability by exciting a specific mode. In the case of AxialT, the six-runner blades at the specified speed-no-load condition appear to favor the three-vortex configuration above the more stable five-vortex configuration.

The origin of the unstable vorticity distribution appears to be linked both with a boundary layer separation on the curved head cover surface and the backflow stemming from the draft tube (Fig. 29). Indeed, Bettocchi et al. [38], using time-averaged velocity measurements in the vaneless space of a propeller turbine with and without runner, showed that in high swirl conditions, similar to speed-no-load, a large separation develops on the head cover accompanied by significantly increased vibration levels. Although their measurements method was limited, they also measured a vorticity layer extending from the vaneless space to the draft tube entrance. Skotak [39] using axisymmetric RANS simulations of the distributor and runner channel of a Kaplan turbine also demonstrated the presence of the head cover separation without the presence of the runner blades. Both cited studies confirm that large separations extending from the head cover to the draft tube in low-load condition are not linked to the runner dynamics. Those studies also hypothesized the existence of a link between that separation and the presence of structured vortices in the runner. However, neither managed to measure nor simulate their dynamic to validate this hypothesis.

In our case, using three-dimensional simulations, it is possible to dissect the flow and confirm that the head cover separation is structured in three zones in precession with the vaneless space vortices. The origin of that separation cannot be easily resolved since it dominates the flow in the first few time steps of the kε simulations. It appears to stem both from the head cover separation moving downstream and a flow separation from the hub wake moving upstream merging somewhere in the runner channel.

## Analysis and Conclusion

Theories on rotating stall inception mostly hinge on flow separation at the leading edge of the runner blades for both pumps and turbines. However, without identifying rotating stall per se, Pulpitel et al. [29] identified vortical structures in the large vaneless space between guide vanes and runners of a Kaplan turbine operating with fixed blade angles, a configuration quite similar to AxialT. They studied no-load regimes with different rotational speed and observed changing flow patterns depending on Q, n and H. They concluded that the presence of those vortical structures stems from the presence of a shear layer associated with the pumping region located near the hub at SNL, much like what is observed on AxialT and on low head Francis turbine [12]. However, they could only speculate on the origin of the shear layer lacking either velocity measurements or simulations.

The results from AxialT SNL simulations, both with and without runner blades, and SNL measurements showed that:

• The inception mechanism of the rotating stall is the instability of a shear layer originating in the vaneless space and extending past the runner. The pumping action around the hub typically associated with the runner is independent of the blades themselves;

• The shear layer in the vaneless space seems linked to a boundary layer separation on the head cover and from a backflow region in the core of the draft tube flow;

• The blades influence the number of observed vortices but the flow instability nevertheless exists without them.

Hence, the rotating stall mechanism identified in the AxialT turbine is similar in nature to the suggestion by Pulpitel et al. [29]. The inception mechanism associated with the vortical instability is most probably only affecting low-head turbines with large vaneless space. Like many other vortical instabilities in complex geometries, the precise mechanism associated with that flow instability has yet to be identified. A better understanding of the inception mechanism could be gained by performing transient simulations at a constant guide vanes opening (α = constant) with slow variation of the dimensionless blade parameter (nED = nD/E0.5) starting from a point to the left of the no-load lines (Fig. 2). The present authors have indeed performed simulations of such a point close to the onset of the instability but without the presence of either flow separation, draft tube backflow and consequently without rotating stall. A numerical campaign of such transient simulations is underway in order to gain deeper understanding in the inception process.

In summary, based on the signal from 31 pressure transducers mounted on two runner blades of a propeller turbine, a URANS-SAS methodology has been validated to study the flow dynamics in speed-no-load conditions. Simulations and measurements indicated that the dominant pressure fluctuations in the runner for the SNL conditions are linked with the occurrence of rotating stall on top of some non-negligible stochastic large-scale flow phenomena. That rotating stall is linked with the presence of three vortices originating from the vaneless space. Simulation of the turbine without runner blades shows that those vortices stem from the presence of an unstable shear layer apparently linked with a flow separation on the turbine head cover and from a large backflow in the draft tube. This inception process has only been validated for one turbine in one particular speed-no-load condition. A deeper understanding of the vorticity dynamics at play is required to better understand the nature of this flow instability. Simulations are underway at the LAMH in order to provide more information on that particular aspect of the AxialT speed-no-load operation.

## Acknowledgements

The authors would like to thank the participants on the Consortium on Hydraulic Machines for their support and contribution to this research project: ANDRITZ Hydro Canada Inc., GE Renewable Energy, Hydro-Québec, Laval University, NRCan, Voith Hydro Inc. Our gratitude goes as well to the Canadian Natural Sciences and Engineering Research Council who participated in the funding for this research and to Compute Canada who provided the necessary computing allocation resources. The authors would also like to acknowledge the important role of Richard Fraser for the experimental measurements, the role of Mélissa Fortin who provided the wavelet analysis routines and the support of Professor Yvan Maciel through many fruitful discussions.

## Nomenclature

• D =

turbine reference diameter (m)

• E =

specific energy = gH (J/kg)

• Ek =

spectral energy

• f =

frequency (Hz)

• g =

gravity (m/s2)

• H =

net head available for extraction (m)

• k, TKE =

turbulence kinetic energy (m2/s2)

• L, LνK =

turbulence length scale (m), von Karman length scale (m)

• n =

turbine rotation speed (rev/s)

• N =

number of rotating stall cells

• nED =

• ns =

turbine synchronous rotation speed (rev/s)

• P =

pressure (kPa)

• Pn =

normalized pressure

• Q =

discharge (m3/s)

• QED =

discharge parameters

• QSAS =

source term of the SAS model

• (r θ z) =

cylindrical coordinates (m)

• S =

strain field (m/m/s)

• t =

time (s)

• T =

torque (N·m)

• Tgen =

net generated torque (N·m)

• T*, Q* =

normalized torque and discharge

• Uij =

velocity field in Einstein notation (m/s)

• U′, U″ =

operators in the definition of LνK in the SAS model

• Vθin/out =

angular flow velocity at the runner entrance (in) and exit (out) (m/s)

• (x y z) =

Cartesian coordinates (m)

• Y+ =

height of the first grid cells in wall coordinates (m)

• α =

guide vanes opening angle (deg)

• β =

• Δf =

frequency resolution for discrete Fourier transform (Hz)

• Δt =

simulation time step (s)

• ε =

turbulence dissipation rate (m2/s3)

• η =

efficiency

• κ =

constant

• λ2 =

second eigenvalue of the λ2 method to identify coherent vortex

• νt =

turbulence viscosity (m2/s)

• ρ =

fluid density (kg/m3)

• σ =

structural strain levels (microstrains)

• σ =

standard deviation

• ω =

turbulence eddy frequency (1/s)

• ωRS =

angular rotation speed of rotating stall cells (rev/s)

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View article in PDF format.

## References

REN21, 2016, “ Renewables 2016 Global Status Report,” Renewable Energy Policy Network for the 21st Century, Paris, France, Report.
Seidel, U. , Mende, C. , Hübner, B. , Weber, W. , and Otto, A. , 2014, “ Dynamic Loads in Francis Runners and Their Impact on Fatigue Life,” 27th IAHR Symposium on Hydraulics Machinery and Systems, Montreal, QC, Canada, Sept. 22–26.
Huang, X. , Chamberland-Lauzon, J. , Oram, C. , Klopfer, A. , and Ruchonnet, N. , 2014, “ Fatigue Analyses of the Prototype Francis Runners Based on Site Measurements and Simulation,” IOP Conf. Ser.: Earth Environ. Sci., 22, p. 012014.
Hübner, B. , Weber, W. , and Seidel, U. , 2016, “ The Role of Fluid-Structure Interaction for Safety and Lifetime Prediction in Hydraulic Machinery,” 28th IAHR Symposium on Hydraulics Machinery and Systems, Grenoble, France, July 4–8, pp. 473–481.
Liu, X. , Luo, X. , and Wang, Z. , 2016, “ A Review on Fatigue Damage Mechanism in Hydro Turbines,” Renewable Sustainable Energy Rev., 54, pp. 1–14.
Trivedi, C. , and Cervantes, M. , 2017, “ Fluid-Structure Interactions in Francis Turbines: A Perspective Review,” Renewable Sustainable Energy Rev., 68, pp. 87–101.
Monette, C. , Marmont, H. , Chamberland-Lauzon, J. , Skagerstrand, A. , Coutu, A. , and Carlevi, J. , 2016, “ Cost of Enlarged Operating Zone for an Existing Francis Runner,” 28th IAHR Symposium on Hydraulics Machinery and Systems, Grenoble, France, July 4–8, pp. 733–742.
Morissette, J.-F. , Chamberland-Lauzon, J. , Nennemann, B. , Monette, C. , Giroux, A.-M. , Coutu, A. , and Nicolle, J. , 2016, “ Stress Predictions in a Francis Turbine at No-Load Operating Regime,” 28th IAHR Symposium on Hydraulics Machinery and Systems, Grenoble, France, July 4–8, pp. 713–722.
Côté, P. , Dumas, G. , Moisan, É. , and Boutet-Blais, G. , 2014, “ Numerical Investigation of the Flow Behavior Into a Francis Runner During Load Rejection,” 27th IAHR Symposium on Hydraulics Machinery and Systems, Montreal, QC, Canada, Sept. 22–26.
Gauthier, J. P. , Giroux, A. M. , Etienne, S. , and Gosselin, F. P. , 2017, “ A Numerical Method for the Determination of Flow-Induced Damping in Hydroelectric Turbines,” J. Fluids Struct., 69, pp. 341–354.
Botero, F. , Hasmatuchi, V. , Roth, S. , and Farhat, M. , 2014, “ Non-Intrusive Detection of Rotating Stall in Pump-Turbines,” Mech. Syst. Signal Process., 48(1–2), pp. 162–173.
Nicolle, J. , Giroux, A.-M. , and Morissette, J.-F. , 2014, “ CFD Configurations for Hydraulic Turbine Startup,” IOP Conf. Ser.: Earth Environ. Sci., 22(3), p. 032021.
Deschênes, C. , Ciocan, G. D. , De Henau, V. , Flemming, F. , Huang, J. , Koller, M. , Arzola Naime, F. , Page, M. , Qian, R. , and Vu, T. , 2010, “ General Overview of the AxialT Project: A Partnership for Low Head Turbine Developments,” 25th IAHR Symposium on Hydraulic Machinery and Systems, Timisoara, Romania, Sept. 20–24.
Houde, S. , Fraser, R. , Ciocan, G. D. , and Deschênes, C. , 2012, “ Part 1: Experimental Study of the Pressure Fluctuations on Propeller Turbine Runner Blades During Steady-State Operation,” 26th IAHR Symposium on Hydraulic Machinery and Systems, Beijing, China, Aug. 19–23.
Houde, S. , Fraser, R. , Ciocan, G. D. , and Deschênes, C. , 2012, “ Experimental Study of the Pressure Fluctuations on Propeller Turbine Runner Blades—Part 2: Transient Conditions,” 26th IAHR Symposium on Hydraulic Machinery and Systems, Beijing, China, Aug. 19–23.
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## Figures

Fig. 1

AxialT turbine model

Fig. 2

Schematic representation of a hydraulic turbine efficiency hill chart in terms of nED and QED. The dark shaded area would be the useful operating range. The runaway line correspond to the line where T = 0. The darker dot would be the location of the SNL condition.

Fig. 3

Vortical structures observed experimentally by Pulpitel et al. [29] in two different no load conditions: (a) three vertical vortices configuration and (b) four vertical vortices configuration with attached vortex ring on the head cover (Reproduced with permission from Pulpitel et al. [29]. Copyright 1996 by Springer.)

Fig. 4

Positions of the miniature pressure transducers on AxialT runner blades

Fig. 5

Measured normalized discharge (Q*) and torque (T*) of AxialT: (a) run 1 from SNL to full load and (b) run 2 from full load to SNL

Fig. 6

Pressure signal for sensors S6 and S14 for the two test runs

Fig. 7

Wavelet analysis of sensors S6, S14 for run 1 and run 2

Fig. 8

Power spectra of sensors S6 and S14: (a) 0 < f/n < 10 and (b) 0.8 < f/n < 1

Fig. 9

(a) Power spectra of the blade mounted strain gage and (b) power spectra of the pressure transducer in the vaneless space

Fig. 10

Computational domains, boundary conditions locations and axis orientations

Fig. 11

Illustration of the computational meshes: (a) distributor and wicket gates and (b) draft tube

Fig. 12

(a) Illustration of the runner hexahedral mesh and (b) results of the grid convergence test. T* refers to the torque normalized by the torque value with the coarser mesh. The selected final grid size is circled.

Fig. 13

Contours of the detached eddy simulation blending function for a selected time-step: (a) ZX plane at y = 0 and (b) XY plane at midrunner height

Fig. 14

Comparison of the averaged pressure from SAS simulations and measurements

Fig. 15

Comparison of the pressure standard deviation (σ) from SAS simulations and measurements

Fig. 16

Numerical and measured power spectra for sensors S6 and S14 (full-frequency range)

Fig. 17

Numerical and measured power spectra for sensors S6 and S14 (subsynchronous range)

Fig. 18

(a) Simulated torque for each blade over 0.2 s and (b) inner span (black) and outer span (gray) torque on blade 1 for 0.2 s

Fig. 19

Streaklines on two constant span surfaces projected in Meridional-theta coordinates of blade 1: (a) 30% span close to the hub and (b) 70% span close to the shroud

Fig. 20

(a) Iso-surfaces of λ2 criterion in the vaneless space and runner. (b) Iso-surfaces of λ2 criterion with contours of Pn = P/E showing the clumping of vortices. Blade 1 is in black.

Fig. 21

Normalized pressure (Pn) contours at the rotor–stator interface for times t1, t2

Fig. 22

Contours of axial velocity (wz) on three planes within the runner for time t1, t2. Darker regions are backflow regions.

Fig. 23

Contours of axial velocity (wz) on ZY plane within the runner and draft tube for time t1, t2. Darker regions are backflow regions.

Fig. 24

Fig. 25

Iso-surfaces of λ2 criterion for time ti and tii for the runner without blades

Fig. 26

Backflow regions (in darker color) for time ti and tii at the rotor–stator interface and midrunner XY plane

Fig. 27

Backflow regions (in darker color) on an YZ plane for time ti and tii

Fig. 28

(a) Pressure signal of the kε simulations without runner blades at monitor points S6 and S14 for a period of 0.35 s and (b) fast Fourier transform of the pressure signal of S6 and S14

Fig. 29

Contours of vorticity and two-dimensional streaklines on a YZ plane for time ti, tii

## Tables

Table 1 Number of elements per domain
Table 2 Boundary conditions of the SAS simulations

## Errata

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