Abstract

Hydraulic turbines are operated more frequently at no-load conditions, also known as speed-no-load (SNL), to provide a spinning reserve that can rapidly connect to the electrical grid. As intermittent energy sources gain popularity, turbines will be required to provide spinning reserves more frequently. Previous studies show vortical flow structures in the vaneless space and the draft tube and rotating stall between the runner blades of certain axial turbines operating at SNL conditions. These flow phenomena are associated with pressure pulsations and torque fluctuations which put high stress on the turbine. The origin of the instabilities is not fully understood and not extensively studied. Moreover, mitigation techniques for SNL must be designed and explored to ensure the safe operation of the turbines at off-design conditions. This study presents a mitigation technique with independent control of each guide vane. The idea is to open some of the guide vanes to the best efficiency point (BEP) angle while keeping the remaining ones closed, aiming to reduce the swirl and thus avoid the instability to develop. The restriction is to have zero net torque on the shaft. Results show that the flow structures in the vaneless space can be broken down, which decreases pressure and velocity fluctuations. Furthermore, the rotating stall between the runner blades is reduced. The time-averaged flow upstream of the runner is changed while the flow below the runner remains mainly unchanged.

Introduction

In an attempt to reduce the climate change pace, member countries of the United Nations have agreed to push for carbon emission neutrality by the second half of the 21st century [1]. In some cases, more ambitious goals are proposed, for instance, Sweden has set a net zero greenhouse emission target for 2045 and a negative emission target beyond that [2]. A necessary step is to aim for electricity production to be completely renewable by 2040. Wind power plays a vital role in this scenario [3] and has gained popularity with worldwide year-over-year growth of 53% in recent years [4]. However, wind power is an intermittent energy source that is challenging to predict over time. Hydropower has become increasingly important with the growth of wind power, as it is utilized to regulate intermittency and ensure a balance on the electrical grid. A recent study shows that hydropower can be coordinated with wind power to meet electricity requirements [3].

Hydropower plants are designed to be operated at the best efficiency point (BEP). However, the operation of hydropower plants at off-design operating conditions will increase as intermittent energy resources gain popularity. Operation at off-design conditions is often described as chaotic, with largely separated regions, stagnant regions, and recirculation zones. Shear layers between flow regions can be a source for vortical flow structures which can cause stresses on critical parts of the turbine and unfavorable pressure pulsations dangerous for the turbine. Damage on the runner blades, guide vanes, parts of the draft tube, and bolts are documented and force the shutdown of the turbine. The downtime for repair is usually costly as the powerplant is out of operation [5].

Some turbines are expected to provide a spinning reserve to respond rapidly to power shortages on the electrical grid [6,7]. The spinning reserve operating condition in hydraulic turbines is often referred to as speed-no-load (SNL). As the name suggests, the runner is rotating without producing any power. More specifically, it is rotating at the synchronous speed, which is the operational angular velocity of the turbine when the generator is connected to the grid. The flowrate is relatively low, with a high swirl, as the guide vanes are only slightly opened to restrict the flowrate. The runner is not extracting any power, meaning that the energy of the fluid must be dissipated somehow through the flow. The flow from the guide vane trailing edge to the draft tube elbow in the axial direction has a high rotation and is mainly located near the wall in a thin region. The rest of the flow is stagnant or recirculating upstream. The flow field is often characterized as unstable with time-varying flow structures extending from the vaneless space to the draft tube. The flow structures eventually break down into smaller structures until they dissipate entirely. They are associated with large pumping regions and fluctuating pressure- and velocity fields. The peak-to-peak pressure and strain fluctuations on the runner are important factors affecting the lifespan [8].

Experimental studies have been conducted to investigate the flow at SNL on Francis and Kaplan turbines, but they are in space. A study of the Francis 99 turbine model shows that the pressure pulsations in the vaneless space and on the runner blades are twice as high during SNL operation compared to BEP operation. The pressure loading is similar to transient operating conditions, such as load variation, start-stop, and emergency shutdown [9]. Other studies show that transient operation is harmful to hydraulic turbines, as it is associated with detrimental pressure pulsations [1012]. Jonsson et al. present an experimental study of a refurbished prototype and model of a Kaplan runner, which shows that flow disturbances develop in the vaneless space due to small guide vane openings and high swirl [13]. The flow disturbances induce severe rotor vibrations. The study also concluded that the frequency of the flow disturbance depends on the runner blade angle, and the frequency of the measured bending moment on the runner could vary between 2.6·fr and 3.3·fr, where fr is the runner rotational speed. Another experimental study by Půlpitel et al. visualizes the flow structures in the vaneless space of a Kaplan turbine operated off-cam [14]. The configuration of the structures varies between a three-flow structure attached to the hub and a four-flow structure with a vortex ring attached to the head cover. The number of flow structures depends on the guide vane opening.

Several numerical studies aim to analyze the SNL and low-load operating conditions. Nennemann et al. [8] and Seidel et al. [15] study Francis runners and report recirculating flow from the draft tube extending past the runner, also referred to as pumping. The pressure pulsations from the chaotic flow field seem stochastic. The same studies also mention that numerical simulation of such operating conditions generally requires a fine computational mesh combined with a small-time-step to resolve the flow field, which is computationally expensive [8,15]. For instance, Seidel et al. [15] mention a numerical study deploying a large eddy simulation (LES) turbulence model aiming to resolve the stochastic fluctuations at SNL using a computational mesh consisting of 100 × 106 elements. Yamamoto et al. [16] study the flow close to the blade of a Francis runner. The study shows the presence of interblade vortices, which are related to the recirculating flow close to the runner hub. They can sometimes be associated with steep angles of attack on the runner blades and typically span the interblade channel from the leading to the trailing edge while being attached to either the hub or shroud [17]. Trivedi [18] presents a detailed LES study of a Francis runner showing that the flow is recirculating at the runner blade's suction side while accelerating on the pressure side. The recirculating flow has a source in the outer radius of the draft tube elbow. The flow is also separated from the blade's leading edge, leading to vortical flow structures that break up into four parts and travel downstream the runner [18]. The flow dynamics on the blade's suction side, especially at the trailing edge, and the flow topology seem to depend on the turbulence model chosen [19].

Axial turbines are not as frequently studied using numerical models. Nevertheless, there are some valuable and significant findings. Iovănel et al. [20] present a study of the Porjus U9 prototype (Porjus, Sweden), which compares numerical simulations to experimental pressure data on the runner blades. Similar to other studies at SNL, it reports a pumping region from the draft tube extending up to the runner. Initially, two flow structures are identified in the draft tube. Eventually, the flow evolves into four structures that join in pairs below the runner. The resulting pressure fluctuations on the runner blades are analyzed and agree well with the experimental data. The primary frequency found on both sides of the blade is 0.92·fr. The experimental measurements show a frequency peak at 3.2·fr, close to what is considered a shaft torsional natural frequency at 3.16·fr. This frequency is captured at 2.78·fr during the simulations. The study also concludes that the predicted amplitude of the pressure pulsations is more accurate on the suction side of the blade, while the mean pressure value is more accurate on the pressure side. The amplitude of the shaft torque is challenging to capture and is underestimated [20].

A study of a different axial turbine operating at SNL, by Houde et al. [21], shows that the rotating stall in the runner interblade passages is linked to vortical flow structures in the vaneless space. Rotating stall refers to the blockage of flow passages between the runner blades due to recirculating flow, which prevents the runner from working effectively. The flow instabilities are independent of the runner blades; they exist when the blades are excluded from the model but can change in quantity. Configurations consisting of three or five structures are found, with the three-structure configuration being the most stable. Moreover, the runner blades are subject to pressure fluctuations with a frequency of 0.88·fr in combination with a 2.12·fr frequency in the vaneless space, which corresponds to a three-structure flow configuration. The mechanism responsible for the flow structures and rotating stall is linked to an unstable vorticity distribution in the vaneless space and a shear layer formation caused by boundary layer separation from the head cover and pumping from the draft tube [21].

A similar study of an axial propeller turbine is conducted at SNL by Yang et al. [22], which shows that vortexes dominate the flow between the runner blades. The runner blades are subjected to the highest pressure pulsations, and it is recommended to operate the turbine at SNL for as limited time as possible [22].

In terms of modeling, the scale adaptive simulation-shear stress transport (SAS-SST) turbulence model is widely used for numerical simulation of hydraulic turbines at SNL operation [8,18,20,21]. The model provides LES-like behavior in detached flow regions without having the same mesh- and time-step requirements, thus being more cost-effective. It is also more effective for computing vortex behavior [23] than other Reynolds-averaged Navier–Stokes (RANS) models, as it can resolve the turbulence rather than modeling it [8]. Extensive detached flow regions are expected during SNL operation, making the SAS-SST turbulence model a good choice. The SAS solution can be used as an initial condition for a more advanced scale-resolving model like LES [18]. Nevertheless, the SAS-SST model operates under the unsteady Reynolds-averaged Navier–Stokes (URANS) methodology and is an extension of the two-equation eddy viscosity SST model, which has some known limitations. For instance, mean quantities of the flow field are calculated, which dampens instantaneous fluctuations, unlike more advanced methods such as LES. Moreover, eddy viscosity models are inherently insensitive to streamline curvature and anisotropic flows, as no term accounts for anisotropic effects in the turbulence kinetic energy term [24].

Several mitigation techniques aiming to neutralize pressure fluctuations at off-design operating conditions are studied experimentally and numerically. A numerical study shows that axial water injection can mitigate the rotating and plunging components of the pressure fluctuations caused by a rotating-vortex-rope (RVR) during part-load (PL) operation. The axial momentum below the runner cone increases by injecting water, lowering the swirl number, and breaking down the flow structure [24]. A different study summarizes some mitigation techniques used to mitigate the RVR [25]. Besides the injection of axial momentum, it mentions the injection of water in the tangential direction opposite to the swirl. One of the main drawbacks of injecting water is that the water bypasses the turbine, which is a loss of production. Fins in the draft tube can also redirect the flow and improve flow stability. However, this can lead to a reduction in efficiency. Sometimes, air is injected from different locations to change the core of the flow and reduce pressure pulsations [25]. There do not seem to be any mitigation techniques designed to mitigate the flow structures during SNL operation of axial turbines. Unlike PL operation, where one single flow structure usually is found (RVR), the flow is more complicated at SNL with multiple flow structures.

Hydraulic turbines are operated at SNL more frequently and will increasingly be as intermittent energy resources gain popularity. There is a need to study the harmful flow at SNL operation, especially for axial turbines with large vaneless spaces where flow instabilities develop. The origin of the instabilities is not fully understood and not extensively studied. Moreover, mitigation techniques for SNL must be designed and explored to ensure the safe operation of the turbines at off-design conditions. There exist mitigation techniques designed to mitigate flow structures at off-design operation, such as PL; however, no publications exist on mitigation techniques for SNL. At SNL, the flow field is more complicated, and the flow structures can extend from the vaneless space to the draft tube, meaning they pass by the runner. As a result, the runner is subject to detrimental pressure pulsations leading to torque fluctuations on the runner shaft. Different publications show that flow structures at off-design operating conditions, especially at low- or no-load, are present because of the high swirl. The guide vanes setup the swirl and have a narrow or nearly closed angle to restrict the flowrate at this operating condition. No power is produced at SNL, meaning the shaft's net torque is zero. The objective of this paper is to develop a numerical tool that can be used to investigate if the flow structures found during SNL operation of an axial turbine can be mitigated by individually controlling some of the guide vanes. The idea behind the mitigation technique is to reduce the swirl generated by the guide vanes, and thereby mitigate the flow structures and pressure fluctuations, by opening some of the guide vanes at a large opening angle while keeping the rest of them closed. The number of opened guide vanes and their angle is restricted by the flowrate corresponding to zero net torque on the runner shaft. A flow with a lower swirl should give more stable flow conditions with less or no flow structures, fewer pressure and velocity fluctuations, and reduced rotating stall, stagnant regions, and shear layer formation. This will help extend the turbine's lifespan and widen the operating range, i.e., safer operation with fewer detrimental pressure pulsations for a wider range of guide vane openings. The development of the numerical model is still in an early stage, meaning that the outcome and learnings from this study will be implemented progressively.

Method

Numerical Domain.

The 10 MW prototype Porjus U9 Kaplan turbine is located along the Luleå River, close to Porjus in northern Sweden. A 1:3.875 scale model for experimental analysis exists at Vattenfall R&D in Älvkarleby, Sweden. The model analyzed during this study has a 400 mm runner diameter operating under a 7.5 m head with a synchronous rotational speed n = 870.39 rpm. There are 20 guide vanes, 18 stay vanes, and the runner is six-bladed. The spiral casing, distributor, runner, and draft tube are all included in the numerical model shown in Fig. 1.

Fig. 1
The numerical setup containing spiral casing, distributor, runner, and draft tube
Fig. 1
The numerical setup containing spiral casing, distributor, runner, and draft tube
Close modal

The commercial solver ansyscfx 2020 R2 [24] is used to perform the numerical simulations. cfx is an algebraic multigrid solver and the code discretizes the governing mass and momentum equations by a finite volume approximation. Furthermore, a modified Rhie and Chow algorithm is implemented for coupling pressure and velocity.

The time-averaged continuity and momentum equations, for an incompressible flow without body forces, are shown in the following equations:
(ρui¯)xi=0
(1)
(ρui¯)t+xj(ρui¯uj¯+ρuiuj¯)=p¯xi+τij¯xj
(2)
where u is the velocity, ρ the density and p the pressure. The variables, here denoted as ϕ, are decomposed into a mean and fluctuating part by Reynolds decomposition: ϕ(xi,t)=ϕ¯(xi)+ϕ(xi,t). The mean viscous stress tensor components are expressed in the following equation, where μ is the dynamic viscosity:
τij¯=μ(ui¯xj+uj¯xi)
(3)
The set of equations is not closed due to the presence of the Reynold stress, i.e., uiuj¯, in Eq. (2). The Reynolds stress in eddy viscosity turbulence models is modeled as
ρuiuj¯=μt(ui¯xj+uj¯xi)23ρδijk
(4)
where the turbulent kinetic energy, k, is expressed in the following equation:
k=12uiui¯
(5)

The eddy viscosity, μt, from Eq. (4), which closes the set of equations needs to be solved. Two equation models model the eddy viscosity through two characteristic turbulent flow parameters to describe time and space. In the SST turbulence model, for instance, it is modeled through the turbulent kinetic energy and turbulent frequency. In turn, these are solved through transport equations; see Ref. [26] for more details. The SAS-SST turbulence model is an extension of the SST turbulence model and was selected for this study as it can predict vortex dynamics and is reasonably cost-efficient compared to other hybrid scale-resolving models such as detached eddy simulation (DES). This is accomplished by introducing the von Kármán length scale through a source term into the transport equation of the turbulence frequency in the SST two-equation eddy viscosity model [24].

A total pressure inlet boundary condition is used for all simulations. The boundary condition is from a simulation with a known flowrate. The pressure at the outlet is set to 0 Pa relative pressure. Transient rotor–stator general grid interfaces are utilized between the domains, which interchange the fluxes between the sides of the interfaces. This is performed at each time-step because of the updated relative position between the domains. Moreover, this type of interface can handle imperfectly matching meshes.

The studied turbine, being a Kaplan, has adjustable blades, meaning that there is a clearance at the hub beside the one present at the shroud side. A previous study of the same turbine showed the importance of including the correct blade clearances as they affect the torque and velocity prediction. Therefore, varying clearances are included on both sides of the blade. The blade axle is also included and modeled as a rotating wall.

The second-order backward Euler transient scheme is used for the temporal derivatives. To approximate the solution of some variable, ϕ, over the control volume, V, after a time-step, Δt, the integral in the following equation is evaluated:
tVϕdV=VΔt(32ϕn2ϕn1+12ϕn2)
(6)

where n denotes the current time-step. This implicit scheme is second-order accurate in time [24]. Furthermore, a high-resolution advection scheme is used. The advection scheme utilizes a blending function changing between a first-order upwind scheme and a centered second-order scheme. During the current simulations, more than 95% of the flow field is solved with a second-order scheme by selecting the high-resolution setting.

All transient simulations are run for 50 runner revolutions, and the last 40 are used for analysis. The flow structures extending from the vaneless space to the draft tube take approximately 10 runner rotations to develop and introduce significant pressure pulsations, hence the choice to analyze the 40 last runner rotations. The same initial condition of 25 runner revolutions is used for each simulation, aiming to exclude initial flow behavior from the results. The flow never settles in a stable configuration over time at this operating condition because of its unsteady and chaotic nature, which makes it challenging to determine the required simulation length. The time-step during the simulations is set to 10 deg of the runner rotation for two reasons: First, it should be sufficient to capture the expected frequencies from previous axial turbine studies [13,20,21]. Second, this study aims to try out a new concept with different configurations, meaning that the simulations must be cost-efficient. Detailed analyses are not feasible at this stage. The resulting RMS courant number from the chosen time-step is approximately 220.

The convergence criteria are set to 1 × 10−4 RMS for mass and momentum, which is reached within a maximum of six inner loop iterations. This is considered sufficient at this stage, with the same cost-efficiency reasoning as for the time-step selection. Point data is also considered for convergence by ensuring that unrealistic variations between the different time steps do not exist.

A computational mesh with hexahedral elements is used for the spiral casing, runner, and draft tube, while the distributor mesh consists of tetrahedral elements with prism layers. During this study, the guide vane angle is small, making it very challenging to use structured hexahedral elements. The distributor's meshing method is automated to ease the meshing of the different guide vane angle configurations. Special attention is given to the distributor and runner domain meshes as the highest pressure pulsations are found here and the flow instabilities seem to originate in the vaneless space [21]. Moreover, the flow close to the runner blade and hub is also of interest. A mesh study of the current runner was performed in a previous study, using a similar methodology as the mesh study of the distributor domain. It showed that ∼5.7 M elements are sufficient; the mesh used follows its recommendation.

A mesh study for the distributor domain is conducted using Richardson's extrapolation as described in Ref. [27]. Steady-state simulations are run until the total pressure drop and mass flowrate converge. These variables are suitable to determine the grid convergence as they are global measures and represent the entire computational domain. However, one drawback is that they give a value related to the complete domain. Therefore, they can be misleading as local deviations can cancel each other, not being reflected in global variables. The spiral, runner, and draft tube are also included with fixed meshes, according to Table 1. The runner blade angle was β = –15.23 deg, and the guide vane angle α = 3.08 deg. Table 2 shows the results of the mesh study.

Table 1

The mesh statistics showing a number of elements, minimum angle, maximum volume change, aspect ratio, and the mean y+ value

PartElements (106)Minimum angle (deg)Maximum volume change Aspect ratio Mean y+
Spiral1.8218.6212124∼3.31
Distributor∼6.06∼25.7∼20∼87∼30.1
Runner∼5.69∼39.4∼17∼7621∼4.09
Draft tube1.3829.5175667∼0.65
Total14.95
PartElements (106)Minimum angle (deg)Maximum volume change Aspect ratio Mean y+
Spiral1.8218.6212124∼3.31
Distributor∼6.06∼25.7∼20∼87∼30.1
Runner∼5.69∼39.4∼17∼7621∼4.09
Draft tube1.3829.5175667∼0.65
Total14.95
Table 2

Mesh study of the distributor domain showing the two studied variables: the total pressure drop and mass flowrate

Total pressure drop distributor (104 Pa)Mass flow (kg s−1)
Number of elements N1, N2, N3 (106)14.76.062.3314.76.062.33
Refinement ratio r21, r321.341.371.341.37
Computed variable η1, η2, η31.711.691.5068.169.777.9
Apparent order P8.025.03
Extrapolated value ηext211.7167.6
Approximate relative error ea210.86%2.37%
Extrapolated relative error eext210.09%0.70%
Grid convergence index GCIfine210.11%0.86%
Total pressure drop distributor (104 Pa)Mass flow (kg s−1)
Number of elements N1, N2, N3 (106)14.76.062.3314.76.062.33
Refinement ratio r21, r321.341.371.341.37
Computed variable η1, η2, η31.711.691.5068.169.777.9
Apparent order P8.025.03
Extrapolated value ηext211.7167.6
Approximate relative error ea210.86%2.37%
Extrapolated relative error eext210.09%0.70%
Grid convergence index GCIfine210.11%0.86%

The indexes indicate the three different meshes, 1 being the most refined.

The study shows that ∼6 × 106 elements in the distributor domain are sufficient. The mesh density in the spiral- and draft tube domain balances between accuracy and computational time. No flow structures and fully attached flow are expected in the spiral casing; therefore, a coarser mesh is acceptable. The draft tube mesh is relatively coarse compared to the distributor and runner mesh. Nevertheless, it is sufficiently fine for the scale adaptive model to resolve the flow. The final mesh statistics are shown in Table 1. The distributor and runner domain mesh statistics vary slightly depending on the runner blade angle and guide vane angle. The value of y+ differs somewhat with the operating condition. A similar number of global elements was used in previous studies [20,21]. Figure 2 shows the final mesh and the blade clearances.

Fig. 2
(a) Figure shows the full mesh, (b) shows the distributor mesh, (c) shows the mesh on the runner hub and blades, (d) displays the mesh at the shroud clearance, (e) the mesh at the hub clearance, (f) a cross section of the mesh in the runner domain perpendicular to the runner axis, and (g) shows the clearances on both sides of the blade at blade angle β = 0.8 deg highlighted in pink
Fig. 2
(a) Figure shows the full mesh, (b) shows the distributor mesh, (c) shows the mesh on the runner hub and blades, (d) displays the mesh at the shroud clearance, (e) the mesh at the hub clearance, (f) a cross section of the mesh in the runner domain perpendicular to the runner axis, and (g) shows the clearances on both sides of the blade at blade angle β = 0.8 deg highlighted in pink
Close modal

Figure 3 shows the blending function for hybrid scale-adaptive models taken from the last time-step of the simulation with β = 0.8 deg and α = 6.8 deg. It displays the resolved and modeled part of the flow field, where the blue part is resolved and the red part is modeled. The modeled part is given by a RANS solution while the resolved part resembles a more detailed LES solution. The degree of resolution depends on the mesh density and time-step. A large part of the flow is resolved using the SAS formulation. The flow in the spiral and along the walls is modeled using RANS formulation, which is expected. Note that the flow field in the draft tube is resolved even though the mesh is coarse. From this, the mesh density and chosen time-step seem acceptable as the flow field is resolved in regions of interest.

Fig. 3
Blending function showing resolved regions in blue and modeled regions in red
Fig. 3
Blending function showing resolved regions in blue and modeled regions in red
Close modal

Simulation Plan.

The simulation plan for the study is presented below. All simulations are performed in model scale. First, three simulations with different runner blade angles, β, and guide vane angles, α, are run to match experimental conditions. Two simulations correspond to on-cam operation, meaning that the runner blade angle is optimized to the guide vane angle, and one simulation corresponds to off-cam operation, meaning that the runner blade angle is fixed at the BEP angle, i.e., the turbine is operated as a propeller.

  • On-cam M: SNL with β = –16.8 deg and α = 2.54 deg. Results are compared with experimental pressure measurements in the vaneless space and in the draft tube and torque on the shaft from the model.

  • On-cam P: SNL with β = –15.23 deg and α = 3.08 deg. Results are compared with experimental pressure measurements performed on the pressure and suction side of a runner blade from the prototype. The rotational speed is slightly different compared to the model simulation to match experimental conditions, see Ref. [28]. It is calculated from n=n11HD and is n = 857 rpm.

  • Off-cam M: SNL with β = 0.8 deg and α = 6.8 deg. Results are compared with experimental pressure measurements in the vaneless space and in the draft tube and torque on the shaft from the model.

The experimental campaign was carried out at the Vattenfall Research and Development Center in Älvkarleby, Sweden, as part of the active flow control system for improving hydraulic turbine performances at off-design operation project, also known as AFC4Hydro.1 The experimental rig fulfills the requirements of IEC60193 test code, which is for hydraulic turbines, storage pumps, and pump-turbines model acceptance tests. More details on the experimental setup can be found in Ref. [29].

The mitigation technique is tested on the three cases above. For the on-cam simulations, two different scenarios are tested: opening two guide vanes to the BEP angle, α = 26.5 deg, while keeping all other guide vanes closed and opening four guide vanes to α = 12.7 deg while keeping all other guide vanes closed which both give a net-torque close to zero. Four guide vanes are opened to the BEP angle for the off-cam simulation while keeping all the other guide vanes closed. Figure 4 shows the distributor setup for the different mitigation scenarios.

Fig. 4
(a) Figure shows the scenario where two guide vanes are opened to α = 26.5 deg, (b) shows the scenario where four guide vanes are opened to α = 26.5 deg, and (c) shows the scenario where four guide vanes are opened to α = 12.7 deg
Fig. 4
(a) Figure shows the scenario where two guide vanes are opened to α = 26.5 deg, (b) shows the scenario where four guide vanes are opened to α = 26.5 deg, and (c) shows the scenario where four guide vanes are opened to α = 12.7 deg
Close modal

Analysis of Results.

The results are analyzed by first investigating the FFT of the pressure and torque signals. Four locations are considered for the signals on the head cover in the vaneless space, six on the pressure side of the runner blade, six on the suction side, and four on the draft tube wall 90 deg apart with the first point 22.5 deg clockwise off the elbow axis. Figure 5 shows the different locations. The pressure signals are normalized by the reference pressure.

Fig. 5
(a) Pressure measurement on the head cover in the vaneless space ∼0.25 m above the center of the runner, (b) on the runner blade pressure and suction side, and (c) the draft tube wall where the line shows the height ∼0.52 m below the center of the runner
Fig. 5
(a) Pressure measurement on the head cover in the vaneless space ∼0.25 m above the center of the runner, (b) on the runner blade pressure and suction side, and (c) the draft tube wall where the line shows the height ∼0.52 m below the center of the runner
Close modal
The number of flow disturbances is calculated according to Ref. [30] by the following equation:
Zd=fs+fbfr
(7)
where fs is the frequency of the pressure pulsation in the stationary frame, fb is the frequency of pressure pulsation on the blade and fr is the runner frequency. The expected frequency visible in the torque signal can be calculated from the following equation:
fT=kZrfbZd
(8)

where k is 1, 2, 3…, Zr is the number of blades and Zd is calculated from Eq. (1).

The flow field is visualized by means of axial velocity contours to identify recirculating and stagnant regions, vorticity contours, and Q-criterion and pressure isosurfaces to identify vortex regions. The Q-criterion is formulated in the following equations:
Q=Cq(Ω2S2)
(9)
S=2SijSij
(10)
Ω=2ωijωij
(11)
Sij=12(Uixj+Ujxi)
(12)
ωij=12(UixjUjxi)
(13)

where S is the absolute value of the strain rate, Ω is the absolute value of the vorticity, and Cq = 0.25 [18]. The eddy viscosity colors the isosurfaces of the Q-criterion to visualize the turbulence intensity. A fully developed turbulence has ∼1000 times larger eddy viscosity compared to the molecular dynamic viscosity [24]. Time-averaged velocity profiles are created to investigate how the flow field changes after the mitigation. Three different sections are considered; see Fig. 6. The velocities are normalized by the bulk flow velocity defined from the mass flow and runner diameter.

Fig. 6
Velocity profiles at three different sections. AR is above the runner, RC is below the runner at the runner cone and DT is in the draft tube.
Fig. 6
Velocity profiles at three different sections. AR is above the runner, RC is below the runner at the runner cone and DT is in the draft tube.
Close modal
The swirl number is calculated on three different planes perpendicular to the runner axis to investigate how the flow changes with the mitigation technique: in the middle of the distributor, on a plane 0.06 m above the runner, and a plane 0.06 m below the runner. The swirl number is the ratio of the axial flux of the angular momentum to the axial momentum flux times the equivalent radius [31]. The swirl number is calculated from the following equation:
Sw=RoRiUaUθr2dr(RiRo)RoRiUa2rdr
(14)

where Ua is the absolute value of the axial velocity, Uθ is the absolute value of the tangential velocity and R corresponds to radii.

Results

The results are presented below; Table 3 displays the flowrate from the different operating conditions. The mass flowrates vary slightly throughout the simulations because of the total pressure inlet boundary condition.

Table 3

Flowrates for the different simulations

SimulationMass flow (kg s−1)
On-cam M∼61
On-cam P∼69
Off-cam M∼162
SimulationMass flow (kg s−1)
On-cam M∼61
On-cam P∼69
Off-cam M∼162

FFT.

The resulting pressure pulsations and fluctuations in torque from the vortical flow structures and rotating stall are interesting to analyze as experimental measurements are available to validate the numerical simulations. The sensors located in the same domain roughly capture the same frequencies but have slightly different amplitudes. This is true for both the experimental and numerical results. Therefore, the result of one sensor in each domain is presented. In the vaneless space, the sensor on the positive side of the y-axis is used, in the draft tube the one closest to the outlet 22.5 deg off the elbow axis, and on the runner blade pressure and suction side the ones closest to the trailing edge on the shroud side (see Fig. 5). Note that the experimental and numerical results are from the model for on-cam M and off-cam M. For the on-cam P on the other hand, the experimental results are from the prototype while the numerical results are from the model.

On-Cam M.

In the vaneless space, two numerical peaks are captured close to the experimental results shown in Fig. 7. The first and highest numerical peak is at 1.07·fr and the corresponding experimental one is at the runner frequency. However, they do not necessarily correspond to the same phenomena. The second numerical peak is captured at 1.50·fr and an experimental one at 1.52·fr. The simulation also captures peaks at lower frequencies not captured with the experiment. The draft tube signal is mostly stochastic or chaotic because of the highly turbulent flow, both from the simulation and experiment. This means that the signal is random without a clear pattern, in contrast to the signal from the vaneless space. The only clear peak from the experimental data is at the runner frequency, while the highest peak from the simulations is toward the lower side of the spectrum at 0.05·fr.

Fig. 7
Pressure signal in the vaneless space and the draft tube comparing numerical and experimental results for on-cam M with β = –16.8 deg and α = 2.54 deg
Fig. 7
Pressure signal in the vaneless space and the draft tube comparing numerical and experimental results for on-cam M with β = –16.8 deg and α = 2.54 deg
Close modal

The signals change when the mitigation scenarios are deployed. Figure 8 shows that mitigation with two opened guide vanes reduces the spike at 1.07·fr while it increases when four guide vanes are opened. Opening two guide vanes slightly changes the frequency of the spike at 1.50·fr and opening four slightly decreases the amplitude. Moreover, frequencies in the lower range increase in amplitude when two guide vanes are opened, while they decrease when four guide vanes are opened. The spectrum from the draft tube is still stochastic but different; the amplitudes increase by opening two guide vanes while the amplitudes are similar to the regular case when four guide vanes are opened, but new frequencies arise between 1.5·fr and 2.5·fr.

Fig. 8
Pressure signal for the mitigation scenarios in the vaneless space and in the draft tube for on-cam M with β = –16.8 deg and α = 2.54 deg
Fig. 8
Pressure signal for the mitigation scenarios in the vaneless space and in the draft tube for on-cam M with β = –16.8 deg and α = 2.54 deg
Close modal

The FFT of the torque is presented in Fig. 9. The experimental results present a high peak of around 3.6·fr which could be related to a shaft torsional natural frequency like the one found in Ref. [20]. Another frequency peak is captured at 2.94·fr both from the experiment and simulation. All other peaks captured from the experiment are multiples of the runner frequency. The numerical result also captures some low-frequency spikes, one at 0.17·fr, which could be related to the 0.05·fr in the stationary frame.

Fig. 9
FFT of the torque signal on the shaft for on-cam M with β = –16.8 deg and α = 2.54 deg; a zoomed-in version is shown on the right-hand side
Fig. 9
FFT of the torque signal on the shaft for on-cam M with β = –16.8 deg and α = 2.54 deg; a zoomed-in version is shown on the right-hand side
Close modal

Opening two of the guide vanes increases the amplitudes significantly, which is shown in Fig. 10. Opening four guide vanes gives a similar spectrum to the regular case except for higher amplitudes in the lower frequency range and a small peak at 1.55·fr.

Fig. 10
Torque signal on the shaft for the mitigation scenarios for on-cam M with β = –16.8 deg and α = 2.54 deg
Fig. 10
Torque signal on the shaft for the mitigation scenarios for on-cam M with β = –16.8 deg and α = 2.54 deg
Close modal

On-Cam P.

For the second on-cam case, numerical results are compared to experimental pressure measurements on the prototype blade's pressure- and suction sides. Figure 11 presents the results from the FFT. A peak is captured at 0.92·fr both from the experiment and simulation. The experimental results also show a peak in the runner frequency, which is not captured clearly by the simulation. Peaks at some other frequencies are visible in the numerical spectrum. The numerical simulation underestimates the amplitudes by a factor of 10. This deviation is most likely related to the difference in pressure between the model simulation and the prototype experiment which are performed under different heads; 7.5 versus 55.5 m, respectively. However, the accuracy of the numerical model might also contribute to the deviation as the difference is not exactly proportional to the difference in heads. Reference [20] shows that even when the prototype is simulated, the amplitude is still underestimated by a comparable factor.

Fig. 11
Pressure signal on the pressure side of the blade comparing numerical and experimental results for on-cam P with β = –15.23 deg and α = 3.08 deg. The experimental results are from the prototype, while the simulation results are from the model. A zoomed-in version is shown on the right-hand side. Note the different amplitude scales on the right and left sides of the plots.
Fig. 11
Pressure signal on the pressure side of the blade comparing numerical and experimental results for on-cam P with β = –15.23 deg and α = 3.08 deg. The experimental results are from the prototype, while the simulation results are from the model. A zoomed-in version is shown on the right-hand side. Note the different amplitude scales on the right and left sides of the plots.
Close modal

The experimental amplitude on the suction side of the blade is significantly lower and more stochastic, as shown in Fig. 12. The same frequencies are captured compared to the pressure side of the blade. Interestingly, the runner frequency is also captured from the simulation on this side of the blade. The amplitudes from the experiment and simulation are in the same range. This is in line with results from Ref. [20], where the amplitudes are predicted better on the blade's suction side.

Fig. 12
Pressure signal on the suction side of the blade comparing numerical and experimental results for on-cam P with β = –15.23 deg and α = 3.08 deg. Note that the experimental results are from the prototype, while the simulation results are from the model. A zoomed-in version is shown on the right-hand side.
Fig. 12
Pressure signal on the suction side of the blade comparing numerical and experimental results for on-cam P with β = –15.23 deg and α = 3.08 deg. Note that the experimental results are from the prototype, while the simulation results are from the model. A zoomed-in version is shown on the right-hand side.
Close modal

Deploying the mitigation scenarios shows that the peak amplitude at 0.92·fr on both sides of the blade decreases, as shown in Fig. 13. Peaks at some other frequencies increased slightly by opening two guide vanes. On the pressure side of the blade, a large peak at twice the runner frequency is visible when opening two of the guide vanes. A peak at four times the runner frequency is also visible for both mitigation scenarios.

Fig. 13
Pressure signal on the pressure- and suction side of the blade from the mitigation scenarios for on-cam P with β = –15.23 deg and α = 3.08 deg
Fig. 13
Pressure signal on the pressure- and suction side of the blade from the mitigation scenarios for on-cam P with β = –15.23 deg and α = 3.08 deg
Close modal

Off-Cam M.

The off-cam operation is analyzed and shown in Fig. 14. The amplitudes are much higher and more frequencies stand out compared to on-cam operation. The highest peak-to-peak numerical pressure pulsation in the vaneless space recorded by the monitoring point was 22,000 Pa (76,000 Pa to 54,000 Pa). In the vaneless space, one peak stands out from the experiment at 1.92·fr. The simulation captures a frequency corresponding to three rotating vortexes at 1.85·fr, which has the highest amplitude of the peaks nearby the experimental one. The calculation of the flow structures confirms that the numerical frequency corresponds to a three-vortex configuration. The simulation also captures a peak with a higher amplitude at 1.35·fr, corresponding to a two-vortex configuration, and an additional peak with a lower amplitude at 1.40·fr. The experiment captures a frequency at 1.43·fr, but with a significantly lower amplitude which could be related to the same phenomena. In addition, the experiment captures a peak at 2.15·fr, simultaneously as the simulation captures one at 2.25·fr. The runner frequency is captured by both the experiment and the simulation. In the draft tube, on the other hand, both signals are stochastic without any peak standing out. The amplitude from the simulation is significantly higher.

Fig. 14
Pressure signal in the vaneless space and the draft tube comparing numerical and experimental results for off-cam M with β = 0.8 deg and α = 6.8 deg
Fig. 14
Pressure signal in the vaneless space and the draft tube comparing numerical and experimental results for off-cam M with β = 0.8 deg and α = 6.8 deg
Close modal

The effect of the mitigation is clearly visible in the vaneless space, as shown in Fig. 15. The high amplitude peaks are mitigated and the remaining peaks have shifted in frequency. The peaks toward the lower frequencies in the spectra remain relatively unchanged. The highest peak-to-peak pressure pulsation is reduced by 68% to 7000 Pa (69,000 Pa to 62,000 Pa). On the other hand, the signal from the draft tube remains relatively unchanged. The scenario with two opened guide vanes is not tested for off-cam M as the opening angle would exceed the BEP one.

Fig. 15
Pressure signal in the vaneless space and the draft tube comparing numerical results for off-cam M with mitigation, β = 0.8 deg and α = 6.8 deg
Fig. 15
Pressure signal in the vaneless space and the draft tube comparing numerical results for off-cam M with mitigation, β = 0.8 deg and α = 6.8 deg
Close modal

Similarly to the on-cam operation, the experiment captures a frequency in the torque signal close to 3.6·fr, which might be related to a shaft torsional natural frequency, as shown in Fig. 16. Besides that, there is a peak at 2.17·fr from the experiment, which likely corresponds to the experimental peak at 1.92·fr in the vaneless space. A frequency peak is captured by the simulation at 2.3·fr, which corresponds to a configuration of three rotating vortices confirmed by calculating the flow structures. The experiment and simulation both have higher amplitude peaks on the lower range of the spectrum. However, the amplitude is significantly higher from the simulation, with a peak at 0.1·fr standing out. The torque fluctuations on the shaft from the simulations are up to 10 times as high during off-cam operation compared to on-cam.

Fig. 16
Torque signal on the shaft for off-cam M with β = 0.8 deg and α = 6.8 deg; a zoomed-in version is shown on the right-hand side
Fig. 16
Torque signal on the shaft for off-cam M with β = 0.8 deg and α = 6.8 deg; a zoomed-in version is shown on the right-hand side
Close modal

The mitigation effect is clearly visible in the torque signal in Fig. 17. The amplitudes are lower throughout the spectrum without any clear peak except in the low-frequency range close to 0.1·fr. Some peaks have shifted in frequency.

Fig. 17
Torque signal on the shaft for off-cam M with mitigation, β = 0.8 deg and α = 6.8 deg; a zoomed-in version is shown on the right-hand side
Fig. 17
Torque signal on the shaft for off-cam M with mitigation, β = 0.8 deg and α = 6.8 deg; a zoomed-in version is shown on the right-hand side
Close modal

Flow Structures.

The number of flow structures is calculated from the numerical results. The frequency in the stationary frame is taken from the vaneless space and the frequency in the rotating frame is from the runner blades. Table 4 shows the results; note that not all frequencies are presented in the FFT section. Possible combinations for both two and three flow disturbances can be found for both on- and off-cam operation. The amplitude of the frequency in the torque signal corresponding to three flow structures is dominant over the two flow structures one for all simulations. Those frequencies are captured relatively close to the experiments. When the mitigation scenarios are deployed, there is no possible combination of frequencies that represent an even number of flow disturbances indicating that the flow structures in the vaneless space are mitigated.

Table 4

Calculation results for the number of flow disturbances and expected frequency in the torque signal

Simulationfs/frfb/frZdfTcalculated/frfTnumerical/fr
On-cam M1.070.9222.772.69
On-cam M1.501.5033.002.94
On-cam P1.070.9222.782.75
On-cam P1.471.5233.053.07
Off-cam M1.350.6722.032.02
Off-cam M1.851.1532.302.30
Simulationfs/frfb/frZdfTcalculated/frfTnumerical/fr
On-cam M1.070.9222.772.69
On-cam M1.501.5033.002.94
On-cam P1.070.9222.782.75
On-cam P1.471.5233.053.07
Off-cam M1.350.6722.032.02
Off-cam M1.851.1532.302.30

Flow Field Comparison On-Cam M and Off-Cam M.

The flow field for on-cam M and on-cam P from the simulations are very similar; therefore, only on-cam M is visualized and compared to off-cam M. The instantaneous axial velocity at the end of the simulation is shown in Fig. 18, where blue shows recirculation regions. The recirculating action is also referred to as pumping because the runner works as a pump close to the hub. Large pumping regions are present in the central region of the draft tube for both cases; however, more fluctuations are present at off-cam. The flow recirculates along the hub past the blades, which can be the onset of rotating stall and interblade vortices due to shear layer interaction. There is also some recirculation at the shroud close to the blade's leading edge, which could depend on a nonoptimal angle of attack on the blade. Flow separation from the blade's leading edge can also give rise to interblade vortices. In addition, it can generate vortical flow structures that travel downstream of the runner. The angle of attack on the runner blade is less optimal during off-cam operation because of its fixed position. Recirculation regions are also observed in the vaneless space.

Fig. 18
Instantaneous axial velocity on a central plane with (a) on-cam M and (b) off-cam M. A zoomed-in version of the flow close to the hub is included for the off-cam case where the recirculating flow past the runner blade is visible. Note that a small region with low velocity exists above the blade close to the hub.
Fig. 18
Instantaneous axial velocity on a central plane with (a) on-cam M and (b) off-cam M. A zoomed-in version of the flow close to the hub is included for the off-cam case where the recirculating flow past the runner blade is visible. Note that a small region with low velocity exists above the blade close to the hub.
Close modal

Instantaneous vorticity contours are shown in Fig. 19. The distribution in the vaneless space is uneven for both operating conditions. This has been linked to flow separation from the head cover and, combined with backflow from the draft tube, can lead to vortical flow structures in the vaneless space and rotating stall between the runner blades [21]. Moreover, a high vorticity region is attached to the suction side of the blade near the hub (where the flow recirculates) and travels downstream along the draft tube wall. The flow is more chaotic on the blade's suction side, which is in line with Ref. [18]. All effects are magnified during the off-cam operation.

Fig. 19
Instantaneous vorticity on a central plane with (a) on-cam M and (b) off-cam M. A zoomed-in version of the flow at the suction side of the blade is visible for the off-cam case. A region of high vorticity attached to the blade's suction side and a high vorticity wake from the previous blade are visible.
Fig. 19
Instantaneous vorticity on a central plane with (a) on-cam M and (b) off-cam M. A zoomed-in version of the flow at the suction side of the blade is visible for the off-cam case. A region of high vorticity attached to the blade's suction side and a high vorticity wake from the previous blade are visible.
Close modal

Flow structures visualized by the Q-criterion are shown in Fig. 20. The Q-criterion is presented on planes at different locations where regions colored differently than blue represent vortex regions. In addition, isosurfaces of Q-criterion at 10,000 s−2 are shown in gray. The figure on the left-hand side shows the on-cam operation and the right side shows the off-cam. It is challenging to identify individual flow structures during on-cam operation other than a region with a high value of the Q-criterion in the vicinity of the hub in the vaneless space and the draft tube wall. On the other hand, two regions of flow structures are visible during off-cam, which extend from the vaneless space to the draft tube. Moreover, flow structures exist between the runner blades. This is in line with a pumping action at the hub and high vorticity close to the suction side of the blade and in the vaneless space.

Fig. 20
Isosurface of Q-criterion at 10,000 s–2 shown in gray and Q-criterion on different planes for (a) on-cam M and (b) off-cam M. Two regions with flow structures extending from the vaneless space to the draft tube are visible for the off-cam operation. Meanwhile, no flow structures are recognized with the same means of visualization for the on-cam case.
Fig. 20
Isosurface of Q-criterion at 10,000 s–2 shown in gray and Q-criterion on different planes for (a) on-cam M and (b) off-cam M. Two regions with flow structures extending from the vaneless space to the draft tube are visible for the off-cam operation. Meanwhile, no flow structures are recognized with the same means of visualization for the on-cam case.
Close modal

The evolution of the flow structures contributes to understanding the phenomena. It is studied for off-cam M in Fig. 21 where the top figures show the Q-criterion on a plane in the center of the distributor and the bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity. The number of flow structures differs between the time steps, for instance, four exist in the draft tube after 28 runner rotations, three after 37, and two after 40. The vortex regions in the vaneless space, which are connected to the hub in the upper figures, are attached to the ones in the draft tube in the bottom figures. They pass through the runner channel and can contribute to the rotating stall. However, they do not rotate at the same frequency as the runner, which means that the structures are cut by the runner blades at some point and recreated later, which might be a source of pressure pulsations. The flow structures in the vaneless space, which are not attached to the hub, only exist locally and are not connected to anything below the runner. After 34 runner rotations, there are three flow structures in the draft tube but only two attached to the hub in the vaneless space; some of the structures connect below the runner, which is in line with results from Ref. [20]. The number of flow structures in the vaneless space seems to change between two and three, which confirms the results presented in Table 4.

Fig. 21
Flow evolution for off-cam M at four different time steps, the last one at the end of the simulation. The upper figures show the Q-criterion on a plane in the center of the distributor and the bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity.
Fig. 21
Flow evolution for off-cam M at four different time steps, the last one at the end of the simulation. The upper figures show the Q-criterion on a plane in the center of the distributor and the bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity.
Close modal

Flow Field Mitigation Off-Cam M.

Visualization of the flow field for off-cam M with the mitigation technique with four opened guide vanes is presented below, as it provides the most visible and easily understandable results of the different mitigation attempts. Instantaneous axial velocity and vorticity contours are presented in Fig. 22. In contrast to the regular case without mitigation, the recirculation in the draft tube is more stable and concentrated in the central zone. Furthermore, the recirculating flow between the runner blades is decreased, which should reduce the rotating stall. Interestingly, the vorticity in the vaneless space is lower and more evenly distributed and separated regions are smaller, which is favorable.

Fig. 22
Instantaneous contours for off-cam M with mitigation of (a) axial velocity and (b) vorticity
Fig. 22
Instantaneous contours for off-cam M with mitigation of (a) axial velocity and (b) vorticity
Close modal

The evolution of flow structures is also studied with the mitigation technique. Figure 23 presents the Q-criterion on a plane in the center of the distributor in the top figures. The bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity. The significant flow disturbances attached to the hub at the top of Fig. 21 have disappeared. In addition, there no longer exists a connection between the flow structures in the vaneless space and the draft tube. There also are fewer flow structures between the runner blades. Some smaller flow disturbances appear in the vicinity of the opened guide vanes. The bottom figures clarify that the flow still is chaotic in the draft tube with large flow structures. In other words, the effect of the opened guide vanes is most visible in the vaneless space and between the runner blades.

Fig. 23
Flow evolution at four different time steps for off-cam M with mitigation, the last one at the end of the simulation. The upper figures show the Q-criterion on a plane in the center of the distributor and the bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity.
Fig. 23
Flow evolution at four different time steps for off-cam M with mitigation, the last one at the end of the simulation. The upper figures show the Q-criterion on a plane in the center of the distributor and the bottom figures show a pressure isosurface at 90,000 Pa colored by the eddy viscosity.
Close modal

Flow Field Distributor Mitigation on-Cam M and Off-Cam M.

The flow field in the distributor is unavoidably affected by only opening some of the guide vanes. Figure 24 shows the time-averaged absolute pressure and circumferential velocity on a central plane in the distributor for the mitigation scenario with two opened guide vanes for on-cam M. A distinct pressure distribution emerges with two slightly lower pressure tails reaching from the hub to the opened guide vanes. This explains the peak at twice the runner frequency in Fig. 13. In addition, low-pressure regions arise at the leading edge of the guide vane behind the opened one, where the absolute pressure is as low as 1000 Pa. This is closely connected with a high velocity in the same location, which is around 13 m/s. This low pressure may indicate the occurrence of cavitation which is not positive. The flow recirculates close to the guide vanes behind the opened one and between some of the stay vanes.

Fig. 24
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for on-cam M with two opened guide vanes
Fig. 24
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for on-cam M with two opened guide vanes
Close modal

The pressure distribution and recirculation zones are also visualized for the mitigation scenario with four opened guide vanes for on-cam M in Fig. 25. The pressure distribution is not as distinct and has four tails, which explains the peak at four times the runner frequency in Fig. 13. The low-pressure region at the leading edge of the guide vane behind the opened one has a minimum value of 81,700 Pa, much higher than opening only two guide vanes. With such pressure, cavitation is not expected. The recirculating zones are smaller.

Fig. 25
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for on-cam M with four opened guide vanes
Fig. 25
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for on-cam M with four opened guide vanes
Close modal

Finally, the same contours are shown in Fig. 26 for the mitigation scenario with four opened guide vanes for off-cam M. In contrast to the previous figure, four low-pressure tails are now clearly visible. The lowest pressure is around 1000 Pa. This low pressure indicates the possibility of cavitation. Recirculating regions are present, especially between some of the stay vanes.

Fig. 26
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for off-cam M with four opened guide vanes
Fig. 26
(a) Absolute pressure distribution and (b) recirculating fluid on a central plane in the distributor for off-cam M with four opened guide vanes
Close modal

Velocity Profiles On-Cam M and Off-Cam M.

Time-averaged velocity profiles are compared at three different locations before and after the mitigation to analyze how the flow is affected. Figure 27 shows tangential and axial velocity profiles for on-cam M. The tangential velocity is much larger than the axial velocity above the runner. Below the runner, the tangential and axial velocities are high close to the outer wall and lower in the central region where the flow is recirculating upstream. This is explained by a combination of centrifugal forces and recirculating fluid pushing upstream from the central draft tube region. Tip vortices originating from the blade clearance at the shroud side can potentially increase the momentum close to the outer wall and contribute to the higher velocity in that region. When the mitigation scenario with two opened guide vanes is deployed, the time-averaged velocity upstream of the runner changes. Especially the tangential velocity is lower, which is expected as the swirl should be lower. The velocities are not changed much downstream of the runner, except that the axial recirculating velocity is slightly smaller. The flow is similar to the regular case when opening four of the guide vanes.

Fig. 27
Time-averaged axial and tangential velocities at AR, RC, and DT before and after the mitigation for on-cam M
Fig. 27
Time-averaged axial and tangential velocities at AR, RC, and DT before and after the mitigation for on-cam M
Close modal

Time-averaged velocity profiles are also analyzed for off-cam operation before and after mitigation in Fig. 28. The tangential velocity above the runner and the velocities along the outer wall are lower than during on-cam operation. Interestingly, when employing the mitigation scenario, the tangential velocity above the runner increases close to the hub and decreases toward the shroud while the axial velocity increases. A distinct jump in the velocity profiles appears nearby r* = 0.92 above the runner, which partly can be explained by the mitigation of the flow structures, which no longer disturb the flow allowing a higher axial flowrate close to the shroud. In addition, a combination of centrifugal forces pushing the flow outward and recirculation fluid pushing upstream from the draft tube also contribute to the high mass flux close to the shroud. This phenomenon is however not observed for the on-cam case, which might be related to the difference in mass flow between the two operating modes; the flow characteristics intensify during off-cam operation. This velocity distribution might be more optimal as the flow does not separate as much in the vaneless space. The time-averaged velocities remain relatively unchanged below the runner.

Fig. 28
Time-averaged axial and tangential velocities at AR, RC, and DT before and after the mitigation for off-cam M
Fig. 28
Time-averaged axial and tangential velocities at AR, RC, and DT before and after the mitigation for off-cam M
Close modal

Swirl Number.

The main idea behind the mitigation technique is to reduce the swirl to avoid instability from developing in the vaneless space. The swirl number is calculated and averaged over time on three different planes. After that, the percent change between the regular and mitigation case is calculated and presented in Fig. 29. By employing the mitigation technique, the swirl number above the runner decreases by as much as 30%. This is true in all cases except with four opened guide vanes for on-cam P. On the other hand, the swirl downstream the runner remains relatively unchanged, which confirms that the flow downstream the runner does not change much.

Fig. 29
The percent change in swirl on a central plane in the distributor, a plane 0.06 m upstream the runner and a plane 0.06 m downstream of the runner. Percentages are calculated in relation to the regular cases without mitigation.
Fig. 29
The percent change in swirl on a central plane in the distributor, a plane 0.06 m upstream the runner and a plane 0.06 m downstream of the runner. Percentages are calculated in relation to the regular cases without mitigation.
Close modal

Discussion

The flow at off-cam operation is clearly more chaotic than on-cam as larger fluctuations in pressure and velocity are observed. At this operating condition, flow structures extend from the vaneless space past the runner blades to the draft tube. It is difficult to visualize the flow structures during on-cam. Nevertheless, the flow is disrupted, which can be seen from the FFT of the pressure signal in the vaneless space and the torque on the runner shaft. It was also shown that the frequencies correspond to a configuration of two and three flow structures. Although, the amplitudes are lower for the on-cam case, which suggests that the flow structures are less energetic and could explain why it is more difficult to visualize them. This indicates that the SNL likely is more harmful to propeller turbines as the blade angle is constant and cannot be adapted to the high swirl. Interestingly, the swirl is lower at off-cam, meaning that it cannot explain the existence and intensity of the flow disturbances alone. The mass flow during off-cam is significantly higher compared to on-cam. This means that more energy must be dissipated, hence the more chaotic nature of the flow. Moreover, the angle of attack on the runner blade is less optimal at off-cam, which also might contribute to the increased chaos of the flow.

It was previously shown that the instabilities exist regardless of the runner blades, according to Ref. [21]. This observation is consistent with the results presented in this paper, as the flow structures are present regardless of the runner blade angle. On the other hand, the blades influence the flow field even though the net torque is zero and they only have a passive role. For instance, the swirl is much lower below the runner, and regions with high vorticity are attached to the blade's suction side and travel downstream. Flow separation on the blade's leading edge combined with recirculating flow on the hub generates interblade vortices. This can contribute to rotating stall, meaning that part of the runner blade channel is blocked, obstructing the flow and preventing the runner from working effectively. The flow structures extending from the vaneless space to the draft tube also contribute to the rotating stall. The number of flow structures is not constant and does not find a stable configuration during the simulation time. A longer simulation time might be required to find a stable configuration. This is an additional motivation for using a 10 deg time-step to increase the feasibility of the simulations.

The presented mitigation technique can potentially mitigate the flow disturbances in the vaneless space and decrease the rotating stall. The structures disappear and are no longer connected to the draft tube structures. This means that they do not pass the runner and are not cut by the blade, which otherwise likely is an origin for pressure pulsations. The structures break down into a more stochastic flow visible in the frequency analysis. The decreased swirl can explain this; the flow does not separate as much and the vorticity is lower and more evenly distributed, which counteracts the evolution of the flow structures in the vaneless space and the rotating stall between the runner blades. The resulting pressure and velocity fluctuations are smaller and less harmful to the turbine.

Nevertheless, some aspects of this method require more extensive studies. For instance, the low-pressure regions inside the distributor can introduce cavitation problems, such as erosion on the guide vanes. There is a possibility that the collapse of the cavitation vapor bubbles induces high-frequency pressure pulsations. The present simulation cannot capture this as the time-step is too large to capture high-frequency pulsations and no cavitation model is deployed. Even though a cavitation model is not included, the low-pressure regions can still indicate where to expect cavitation problems. The pressure field inside the distributor is felt by the runner blades at both two and four times the runner frequency. It should be investigated whether this is harmful to the turbine.

The numerical model roughly captures the same pressure pulsation and torque frequencies as the experiments. However, the simulations show some additional frequencies not captured by the experiments and the amplitudes are considerably different, especially during off-cam operation for the model. This indicates that there are some shortcomings in the simulations. Moreover, some deviation can be observed in the low-frequency range for both on- and off-cam operation. A longer simulation time will help to better resolve the low-frequency range. As mentioned before, it is challenging to decide the required simulation time because the flow never settles in a stable configuration at this operating condition. This means that full periods of vortex motion are not always completed which affects the FFT of the pressure and torque signals, giving uncertainties in the frequency and amplitude prediction. However, the unsteady flow will produce stable time-averages of the variables, assuming the simulation time is sufficiently long. In addition, the resolution of the FFT increases with a longer simulation time. This presents a constant balance between resources, time efficiency, and obtaining sufficiently reliable results.

The chaotic nature of the flow at off-cam operation is captured less accurately than the on-cam operation with the chosen time-step. This will be considered carefully further developing the numerical model for future simulations; a smaller time-step might be preferable and should better capture the rapid flow changes between the time steps. There also exists a significant deviation for the amplitude prediction of the pressure on the runner blade's pressure side between the numerical and experimental results for the on-cam operation. This also suggests that a smaller time-step must be considered to capture the pressure fluctuations in the vicinity of the blade. Also, the shaft is not modeled during the simulations. Therefore, the response of the system might not be correct in the vicinity of the natural frequencies, which could cause some deviations. For instance, the frequency in the torque signal close to 3.6·fr is not captured by the simulations.

Nevertheless, the current simulation results are reasonably accurate considering the relatively large time-step and short simulation time, as all critical frequencies from the experiments are captured. The FFT also confirms that the off-cam operation is more challenging to model numerically as a wider band of frequencies is captured with considerably higher amplitudes. By deploying the mitigation technique, the flow structures are broken down into stochastic flow with lower amplitude. The amplitude of pressure pulsation on the runner blade is decreased, both on the pressure and suction side, which suggests that the flow is less harmful to the turbine. The decrease of the maximal pressure fluctuation in the vaneless space is 68%, which strengthens this argument.

One interesting observation is that the pressure pulsations are reduced more by opening four guide vanes at on-cam instead of only two, even though the swirl number is higher. A new type of instability might be introduced if the flow is forced through only two openings. The mitigation works best during off-cam operation. One of the reasons might be that the flow is very chaotic, which makes the mitigation effects easily visible. The decreased swirl likely improves the angle of attack on the runner blade, which is most noticeable in this operating condition because of the fixed blade angle. Moreover, the BEP angle of the opened guide vanes matches the runner blade angle better at off-cam compared to on-cam; at BEP β = 0.8 deg and α = 26.5 deg.

The velocity profiles and the swirl number confirm that the instantaneous and time-averaged flow upstream of the runner is changed when the mitigation technique is deployed because the swirl is setup differently. The effect is most visible for on-cam M when only two guide vanes are opened; the flow is very similar to the regular case when four are opened because of the smaller guide vane angle. The time-averaged flow downstream the runner does not change noticeably. However, the fluctuations might be dampened, which does not affect the time-averaged flow field. For a fixed operating condition, i.e., constant β and α, lowering the swirl mitigates a lot of the pressure pulsations and torque fluctuations.

The exact implementation of the mitigation technique requires more extensive studies. There exist many possible combinations of number of opened guide vanes, guide angle, their placement relative to each other, and relative the distributor inlet; it is a system with many degrees-of-freedom. The only constraints are the zero net torque on the shaft and minimum pressure pulsations. The layout will probably have to be designed for each specific turbine. For instance, by opening four guide vanes for on-cam P, the swirl actually increases compared to the regular case, which is undesirable. A decreased axial velocity might explain this. On the other hand, opening four guide vanes significantly reduces the cavitation risk inside the distributor compared to opening two. One limitation of this study is that the opened guide vanes are distributed symmetrically. The pressure pulsations might be reduced even more by opening them unsymmetrically or with an uneven number. In this way, the excitation of frequencies related to the number of runner blades or runner frequency might be avoided. The setup can even be changed to avoid the excitation of other harmful frequencies, such as those related to the shaft. Opening the guide vanes at an angle close to the BEP one might be useful to reduce the swirl and limit the required number of opened guide vanes. A smaller number of opened guide vanes will be easier to implement on existing power plants as only a few will require individual control. The studied mitigation technique generally shows a decrease in pressure fluctuations on the runner blades and in the vaneless space, which is promising. The distributor setup is an optimization problem that likely will have different solutions depending on the turbine and operation requirements.

Conclusions

The flow field at SNL operation of the present axial turbine is chaotic and dominated by flow structures extending from the vaneless space, past the runner blades, to the draft tube. The number of flow structures is unstable and changes between a two- and three-structure configuration, with the three-structured one being the most dominant. This study shows that the pressure pulsations and fluctuations in torque can be mitigated by individually controlling some of the guide vanes. Decreasing the swirl upstream of the runner for a fixed operating condition breaks down the flow structures in the vaneless space into stochastic behaving flow, reduces the rotating stall between the runner blades, and decreases fluctuations in pressure and velocity. The time-averaged flow upstream of the runner is changed while the flow downstream of the runner remains relatively unchanged. The flow at SNL is more chaotic during off-cam operation than on-cam, which might be one of the reasons why the mitigation technique is more successful at off-cam. There exist many possible distributor configurations: number of opened guide vanes, guide angle, their placement relative to each other, and the distributor inlet; it is a system with many degrees-of-freedom. The configuration must be studied more extensively to avoid problems with recirculating flow and low-pressure regions. Ultimately, it will be an optimization problem specific to each turbine.

Acknowledgment

The research presented was carried out as a part of the “Swedish Hydropower Centre”—SVC has been established by the Swedish Energy Agency, Elforsk, and Svenska Kraftnät, together with the Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology, and Uppsala University.

Part of this research received funding from the European Union's Horizon H2020 research and innovation program under grant agreement No 814958.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

D =

diameter

ea =

approximate relative error

eext =

extrapolated relative error

fb =

frequency on blade

fr =

runner frequency

fs =

frequency stationary frame

fT =

frequency torque

H =

head

k =

turbulent kinetic energy

n =

rotational speed

N =

number of elements

n11 =

unit speed

p =

pressure

P =

apparent order

Q =

Q-criterion

r =

refinement ratio

R =

radius

S =

strain rate

Sw =

swirl number

t =

time

u¯ =

mean velocity

u =

fluctuating velocity

U =

velocity

V =

volume

y+ =

nondimensional wall distance

Zd =

number of flow disturbances

Zr =

number of blades

Greek Symbols
α =

guide vane angle

β =

runner blade angle

δij =

Kronecker delta

Δt =

time step

η =

mesh study variable

ηext =

extrapolated variable

μ =

dynamic viscosity

μt =

eddy viscosity (turbulent viscosity)

ρ =

density

τij =

viscous stress tensor

ϕ =

variable

Ω =

vorticity

Abbreviations
BEP =

best efficiency point

DES =

detached eddy simulation

GCI =

grid convergence index

LES =

large eddy simulation

PL =

part-load

RANS =

Reynolds-averaged Navier–Stokes

RVR =

rotating-vortex-rope

SAS =

scale adaptive simulation

SNL =

speed-no-load

URANS =

unsteady Reynolds-averaged Navier–Stokes

Footnotes

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