608532 - hyperbole - deliverable d4 documents/deliverables... · 2020. 7. 22. ·...

Deliverable Project number 608532

Project title HYPERBOLE ‐ HYdropower plants PERformance and flexiBle Operation towards Lean integration of new renewable Energies

Call (part) identifier FP7‐ENERGY‐2013‐1

Funding scheme Collaborative project

Date February 23, 2017

Partner Author P1 – Ecole Polytechnique Fédérale de Lausanne P6 – Power Vision Engineering Sàrl

Name e‐mail

Arthur Favrel [email protected]

Joao Gomes Pereira Junior [email protected]

Christian Landry Christian.landry@powervision‐eng.ch

Sébastien Alligné Sebastien.alligne@powervision‐eng.ch

Christophe Nicolet Christophe.nicolet@powervision‐eng.ch

François Avellan [email protected]

Deliverable Number Deliverable 4.4

FP7‐ENERGY‐2013‐1 – N° 608532 – HYPERBOLE Deliverable 4.4

02.23.2017 page 2/43

This page is intentionally left blank


02.23.2017 page 3/43

Report describing the hydro‐acoustic model of power plant unit and validation

of the hydro‐acoustic parameters transposition by comparison with experimental results from WP5

Content

1 Introduction ....................................................................................................................................... 5

2 One‐dimensional modelling of the hydropower plant unit at off‐design conditions ....................... 6

2.1 One‐dimensional SIMSEN model of the hydropower plant unit ............................................... 6

2.2 Modelling of the cavitation vortex rope in the draft tube ........................................................ 6

2.3 Validation of 1D SIMSEN model of the hydropower plant unit by transient simulations ......... 7

3 Description of hydropower plant on‐site measurements and main results ................................... 10

3.1 Investigated operating points .................................................................................................. 10

3.2 Precession and natural frequencies at part load conditions ................................................... 10

3.3 Resonance at part load conditions .......................................................................................... 11

3.4 Instability at full load conditions ............................................................................................. 12

4 Transposition of the hydro‐acoustics parameters and resonance prediction at part load conditions

13

4.1 Part load resonance prediction based on the swirl number at the model scale..................... 13

4.1.1 Determination of the vortex precession frequency frope and the natural frequency f0 ... 13

4.1.2 Resonance conditions ...................................................................................................... 13

4.1.3 Swirl number ................................................................................................................... 15

4.1.4 Empirical laws for prediction of frope and f0 at the model scale ...................................... 15

4.1.5 Determination of the wave speed and bulk viscosity at the model scale ....................... 16

4.2 Part load resonance prediction at the prototype scale ........................................................... 20

4.2.1 Transposition of the wave speed and the bulk viscosity ................................................. 20

4.2.2 Natural frequency of the prototype ................................................................................ 21

4.2.3 Precession frequency of the vortex on the prototype .................................................... 21


02.23.2017 page 4/43

4.2.4 Prediction of the resonance conditions on the prototype and comparison with on‐site

measurements ................................................................................................................................. 22

4.3 Part load resonance prediction based on similitude of local cavitation factor ....................... 24

4.3.1 Discrepancy of local cavitation factor between model and prototype ........................... 24

4.3.2 Wave speed as a function of the swirl number and the local cavitation coefficient ...... 25

4.3.3 Dimensionless wave speed as a function of the swirl number and the local cavitation

coefficient ........................................................................................................................................ 26

4.3.4 Determination of the wave speed corresponding to the conditions during on‐site tests

27

4.3.5 Update of the predicted natural frequency and resonance conditions at the prototype

scale 28

4.4 Prediction of pressure fluctuations amplitude at part load conditions on the prototype ...... 29

4.4.1 Determination of the pressure excitation source at the model scale ............................. 29

4.4.2 Simulation in the time domain at the prototype scale .................................................... 31

5 Transposition of the hydro‐acoustics parameters and prediction of the instability at full load ..... 35

5.1 Limits of stability at full load at the model scale and comparison with on‐site measurements

35

5.2 Empirical law for prediction of finst. at the model scale ........................................................... 36

5.3 Determination of the hydro‐acoustic parameters at the model scale .................................... 37

5.4 Transposition of the hydro‐acoustic parameters at the prototype scale................................ 38

5.5 Prediction of instability frequency at full load and comparison with on‐site measurements 39

6 Conclusions ...................................................................................................................................... 40

6.1 Potential impacts of the presented results ............................................................................. 41

7 References ....................................................................................................................................... 43


02.23.2017 page 5/43

1 Introduction Hydraulic machines operating at off‐design operation are subject to the development of hydraulic

instabilities involving the presence of cavitation flow in the draft tube and the propagation of pressure

pulsations through the hydraulic circuit. At part load conditions, the cavitation vortex rope is described

as an excitation source for the hydraulic system and the interaction between this excitation source and

the eigenfrequencies of the system may result in resonance phenomena and induce draft tube surge

and electrical power swings. At full load conditions, the axisymmetric cavitation vortex rope can enter

self‐oscillation and is described as an internal source of energy, resulting in the propagation of pressure

fluctuations of high amplitude.

Experimental tests are commonly performed on reduced scale physical models enabling the perfect

prediction of the hydraulic behaviour of the machine in terms of efficiency and cavitation. Although both

geometric and kinematic similarities between model and prototype scales are fulfilled according to IEC

Standards, the amplitude of the pressure fluctuations experienced by the hydraulic circuit in off‐design

conditions cannot be directly transposed to the prototype. Indeed, the eigenfrequencies of the system

depend on both the cavitation volume within the vortex core and the characteristics of the hydraulic

circuit. Therefore, the interaction on the prototype between the excitation source and the hydraulic

circuit is different to the one observed during model testing.

The methodology developed in the framework of the HYPERBOLE project for the prediction of pressure

fluctuations is based on the one‐dimensional modelling of the hydraulic circuit and the identification of

hydro‐acoustic parameters describing the cavitation flow at the model scale. The transposition of these

hydro‐acoustic parameters from the model scale to the prototype scale is the last step for the prediction

of the pressure fluctuations on the real machine.

This deliverable reports the transposition of the hydro‐acoustic parameters modelling the cavitation

draft tube flow at both part load and full load conditions and its validation by comparisons with on‐site

measurements performed on the hydropower plant Francis turbine unit.

To identify the hydro‐acoustic parameters on the complete part load operating range, empirical laws

linking both precession and natural frequencies with the swirl degree of the flow exiting the runner are,

first, established on the reduced scale model. The corresponding hydro‐acoustic parameters are, then,

identified by using the dimensionless wave speed and bulk viscosity introduced by Landry et al. [1‐ 2],

see Deliverable D4.2, and the 1D SIMSEN model of the test rig. The corresponding excitation sources

induced by the vortex rope are also identified. The transposed hydro‐acoustic parameters are finally

injected in the 1D SIMSEN model of the hydropower plant unit, enabling the computation of its

eigenfrequencies as a function of the discharge factor, as well as the prediction of the amplitude of the

pressure fluctuations.

The same procedure is used at full load conditions. The frequency of the instability, identified as the first

eigenfrequency of the hydraulic system, is identified on the reduced scale model and expressed as a

function of the degree of swirl of the flow exiting the runner. The corresponding hydro‐acoustic

parameters are, then, identified and transposed to the prototype scale. This enables the prediction of

the frequency of the instability at the prototype scale.

The results are compared with the results of on‐site measurements performed in November 2016 in the

hydropower plant unit.


02.23.2017 page 6/43

2 One‐dimensional modelling of the hydropower plant unit at off‐design conditions

2.1 One‐dimensional SIMSEN model of the hydropower plant unit

The objectives are to predict the pressure pulsations induced by the cavitation vortex rope occurring in

off‐design conditions and to assess the power plant stability on its complete operating range. For this

purpose, a 1D SIMSEN model of the main electrical, hydraulic and mechanical components of the

hydropower plant unit is implemented [4]. A detailed presentation of this model can be found in the

Deliverable D4.3.

As the simulations performed and presented in the Deliverable D4.3 have shown, in case the unit

operates with a synchronous generator, there is no need to model the electrical components and the

related power network to which the machine is connected if only the stability of the unit is investigated.

Indeed, their effects on the amplitude and the frequencies of the pressure pulsations experienced by

the hydraulic circuit are negligible.

Since the hydropower plant unit has a synchronous generator and this Deliverable focuses on the

pressure pulsations observed on its hydraulic circuit, only the hydro‐mechanical components of the

machine are modelled. Additionally, a few improvements on the original 1D model presented in D4.3

are made to increase the accuracy of the model. Based on drawings of the water passages, wave speeds

in the pipes have been computed taking into account the material and the wall thickness. Moreover an

extra pipe named “BS” (see Figure 1) is also added to model the effect of the spiral casing.

Figure 1 1D SIMSEN model of the hydropower plant unit

2.2 Modelling of the cavitation vortex rope in the draft tube

The hydro‐acoustic model of the cavitation draft tube flow is presented in details in the Deliverable D4.2.

It is represented in Figure 1 by the element DTUBE21 and its electrical analogy scheme is shown in

Figure 2.

The main hydro‐acoustic parameters modelling the cavitation draft tube flow, which will be transposed

from the model scale to the prototype scale, are the following:

‐ The wave speed a in the draft tube, which is linked to the cavitation compliance Cc by the

relation Cc = g AL / a with g being the gravity, A the cone section and L the cone length;

‐ The bulk viscosity μ’’ representing the internal processes breaking the thermodynamic

equilibrium between the cavitation volume and the liquid phase;

‐ The pressure excitation source Sh induced by the precession of the vortex rope at part load.

More details about the modelling of the cavitation draft tube flow and the corresponding hydro‐acoustic

parameters can be found in the references [1, 2, 3, 4, 10, 11].


02.23.2017 page 7/43

Figure 2 Representation of the draft tube featuring a cavitation vortex rope by an equivalent electrical scheme

A method for coupling experimental measurements on a reduced scale model with an external

excitation system and the 1D SIMSEN model of the test rig has been developed by Landry et al. [1,2] and

presented in the Deliverable D4.2. The corresponding measurements on the reduced scale physical

model were presented in the Deliverables D1.3 and D1.5 in the framework of WP1.

2.3 Validation of 1D SIMSEN model of the hydropower plant unit by transient simulations

To validate the 1D SIMSEN model of the hydropower plant, the simulation of an emergency shutdown

is compared to measurements performed during the on‐site tests. The operating condition of the Unit

2 before the emergency shutdown is summarized in Table 1.

Table 1 Operating condition of Unit 2 before emergency shutdown

Output power 434 MW Wicket gate opening 85 %

Guide vane angle 22.4° ZHeadwater 752.8 m asl ZTailwater 572.4 m asl

The non‐linearity of the guide vane angle closing law, featuring two slopes, has been measured and

reproduced in the SIMSEN model for the simulation. The kinematic relationship between the wicket

gate opening and the guide vane angle shown in Figure 3 has been used as input for the SIMSEN model.

Figure 3 Kinematic relationship between wicket gate opening and guide vane angle

In Figure 4, Figure 5 and Figure 6, the time history during the emergency shutdown of the guide vane

angle, the rotational speed, the pressure in the spiral case and the pressure in the draft tube are

compared to the measurements, respectively. The simulated quantities are indicated by the suffix “_s”.


02.23.2017 page 8/43

Figure 4 Time history of guide vane angle and rotational speed

Figure 5 Time history of guide vane angle and pressure in spiral case pSC10

Figure 6 Time history of guide vane angle and pressure in the draft tube pDT10

Regarding the steady state before the emergency shutdown, the measured guide vane angle of 70% is

imposed to the SIMSEN model. The resulting steady pressure in the spiral case pSC10 is similar to the

measurements, which means that the modelled energy losses and the upstream hydraulic layout are

valid. However, the steady pressure in the draft tube exhibits an offset of 0.4 bar. This difference could

be due to the swirling flow field in the draft tube that is not taken into account in the energy losses

modelling of the draft tube.


02.23.2017 page 9/43

The simulated overspeed of the unit and the induced maximum pressure in the spiral case of the

emergency shutdown are similar to the measurements. Moreover, one can observe that the period of

the water hammer is well reproduced, which validates the values of the wave speed in the water

passages.

However, the measurements of the draft tube pressure during the emergency shutdown show that the

unsteady flow field of the cavitation vortex rope in the draft tube induces pressure fluctuations, see

Figure 6. These fluctuations are not reproduced by the SIMSEN model since no momentum excitation,

modelling the cavitation vortex rope, is set. Time history of three pressure sensors located in the draft

tube are shown In Figure 7. The corresponding pressure fluctuations are synchronous in time, meaning

that plane waves are generated in the draft tube by the cavitation vortex rope.

Figure 7 Time history of pressure in the draft tube for three different sensors

Based on this comparison with measurements for emergency shutdown, the SIMSEN model is

considered as valid and calibrated for frequency analysis and prediction of stability of the power plant.


02.23.2017 page 10/43

3 Description of hydropower plant on‐site measurements and main results

3.1 Investigated operating points

On‐site tests were performed on the hydropower plant unit in November 2016. A detailed description

of the tests is already provided in Deliverable D5.2. In the following, only results from pressure sensors

located in the draft tube cone, the spiral casing and the penstock are used.

The output power of the hydropower plant unit was varied from its maximum P = 475 MW value to its

minimum value P = 78 MW, with stabilization at given operating points during several minutes. The list

of the operating points investigated during the tests is given in Table 2, together with the corresponding

operating parameters determined by using an interpolation of the test case hill chart.

Table 2 Operating parameters of the investigated operating points on the prototype

OP

P (MW)

nED (‐)

QED (‐)

σ (‐)

Fr (‐)

S (‐)

1a 474 0.2765 0.2450 0.1005 5.77 ‐0.054 Full load

1b 477 0.2787 0.2533 0.1176 5.72 ‐0.090 Full load

2 448 0.2773 0.2321 0.107 5.75 0.030

3 399 0.2770 0.2054 0.1055 5.76 0.220

4 347 0.2766 0.1800 0.1055 5.76 0.450

5 312 0.2763 0.1632 0.1021 5.77 0.644 Part load

6 295 0.2763 0.1556 0.1024 5.77 0.746 Part load

7 270 0.2764 0.1439 0.1038 5.76 0.924 Part load

8 262 0.2765 0.1406 0.1045 5.76 0.981 Part load

9 253 0.2764 0.1361 0.1036 5.76 1.060 Part load

10 240 0.2763 0.1302 0.1033 5.76 1.173 Part load

11 225 0.2763 0.1237 0.1036 5.76 1.310 Part load

12 214 0.2759 0.1188 0.1008 5.77 1.420 Part load

13 171 0.2757 0.0986 0.0994 5.77 ‐ Deep part load

14 127 0.2756 0.0767 0.0994 5.77 ‐ Deep part load

15 78 0.2756 0.0496 0.1002 5.77 ‐ Deep part load

3.2 Precession and natural frequencies at part load conditions

For each operating point corresponding to part load conditions, the cross‐spectrum between two

pressure signals measured in the same cross‐section of the draft tube cone is computed. An example is

given for OP10 in Figure 8, which includes the amplitude (a) and the phase of the auto‐spectrum (b).

The pressure fluctuations at the frequency f0 are of synchronous nature and this frequency can be

identified as the natural frequency of the system, similar to the reduced scale model [5]. Therefore, it is


02.23.2017 page 11/43

possible to determine both precession and natural frequency values of the prototype by a cross‐spectral

analysis between pressure signals measured in the same cross‐section of the draft tube cone.

(a) Amplitude (b) Phase

Figure 8 Cross‐spectrum between two pressure signals measured in the same cross‐section of the draft tube cone at OP10

The influence of the discharge factor on the precession frequency and the natural frequency is given in

Figure 9. The discharge factor is made dimensionless by its value at the Best Efficiency Point (BEP)

QED* = 0.20. Different flow regimes similar to those observed on the model [6] are observed, see

Deliverable 4.1. The precession frequency value increases linearly as the discharge factor is decreased.

The transition from the regime 2 to the regime 3 is comprised between QED / QED* = 0.59 and

QED / QED* = 0.48. However, no measurement was performed in this operating range and, therefore, it is

not possible to determine precisely the limits of the flow regimes. This range of QED values is however

in agreement with the observations made on the reduced scale model, see Deliverable D4.1.

Figure 9 Influence of the discharge factor on the precession frequency and the natural frequency of the prototype.

3.3 Resonance at part load conditions

The influence of the discharge factor on the amplitude and the phase of the cross‐spectrum between

two pressure signals measured in the cone is given in Figure 10.

The resonance conditions occurs for a discharge factor comprises between QED = 0.1406 (OP8) and

QED = 0.1361 (OP9). The amplitude of the cross‐spectrum at frope reaches its maximum at these points

and the phase shift at frope tends towards zero, which indicates that the synchronous pressure pulsation

is strongly amplified and is dominant compared with the convective pressure fluctuations in the draft

tube cone.

0 0.2 0.4 0.6 0.80

0.5

1.0

1.5

2.0

2.5

1.0

|Gxy

|(Hz-1)

(-)

f / n

f0

frope

×10-2

0 0.2 0.4 0.6 0.8 1.0

0

�

-�

�2

-�2

θxy

(rad)

(-)

f / n

f0

frope

0.08 0.1 0.12 0.14 0.160.2

0.6

1

1.4

0.18(-)

Tra

nsi

tion

regim

es 2

-3

f(Hz)

QED

frope

f0


02.23.2017 page 12/43

(a) Amplitude (b) Phase

Figure 10 Influence of the discharge factor on the cross‐spectrum amplitude and phase on the prototype. The pressure sensors are located in the same cross‐section of the cone.

3.4 Instability at full load conditions

Operating points at full load conditions were explored to investigate the onset of full load instability and

its impact on the mechanical behaviour of the unit. The opening angle of the guide vane was set to reach

the maximum output power of the unit, i.e. P = 475 MW. The output power was limited to this value

due to the limitation of the generator.

For this maximum output power, the system remained stable for the initial head value, i.e. H = 179.9 m

(OP1a). To increase the distance from the swirl‐free zone, the tailwater reservoir level has been

increased by operating simultaneously Unit 1 and Unit 3. As a result, the head value was decreased,

allowing an increase of the discharge for the same maximum output power P = 475 MW (OP1b). In such

conditions, the system enters self‐excitation, as illustrated in Figure 11. In this figure, the coefficient Cp

corresponds to the pressure pulsations made dimensionless by the turbine head. The frequency of the

instability in these conditions is equal to 0.79 Hz. It induces output power fluctuations of high amplitude,

whose peak‐to‐peak value reaches 40 MW, i.e. 9 % of the rated output power of the machine.

(a) Stable configuration (b) Unstable configuration

Figure 11 Time history of pressure signals measured in the draft tube cone in stable and unstable configurations. The pressure signals are made dimensionless by the net head

Tra

nsi

tion

regim

es 2

-3

0.4 0.5 0.6 0.7 0.80

1.0

2.0

3.0

4.0

5.0

|Gxy

|(Hz-1)

×10-2

(-)

QED / QED*

Res

onan

ce

0.4 0.5 0.6 0.7 0.8

0

�

-�

�2

-�2

(rad) θxy

Tra

nsi

tion

regim

es 2

-3

(-)

QED / QED*R

esonan

ce

0 5 10 15 20-0.05

0

0.05

0.1

Cp(-)

(-)

n × t

draft tube cone - left

draft tube cone - right

0 5 10 15 20-0.05

0

0.05

0.1

Cp(-)

(-)

n × t


02.23.2017 page 13/43

4 Transposition of the hydro‐acoustics parameters and resonance prediction at part load conditions

4.1 Part load resonance prediction based on the swirl number at the model scale

4.1.1 Determination of the vortex precession frequency frope and the natural frequency f0

Pressure fluctuations measurements are carried out on the reduced scale physical model of the Francis

turbine test case for values of the speed factor covering the head range of the prototype. For each nED

value, the discharge factor is varied in a wide range of value, from about 90 % to 50 % of the value at

the Best Efficiency Point (BEP). Moreover, the Thoma number σ is set at the rated value of the prototype,

which corresponds to the average level of the tailwater reservoir. The investigated operating points are

given in Table 3.

Table 3 Operating parameters of the operating points investigated on the model

nED (‐)

QED (‐)

σ (‐)

Fr (‐)

Comments

0.268 variation 0.095 8.73 ‐ No Froude similitude ‐ σ‐value corresponding to the average tailwater level

0.277 variation 0.102 8.73

0.288 variation 0.110 8.73

0.300 variation 0.119 8.73

0.317 variation 0.133 8.73

For each operating point, the precession frequency of the vortex frope and the first eigenfrequency f0 of

the test rig are identified by cross‐spectral analysis of two pressure signals measured in the same cross‐

section of the draft tube cone. An example is given in Figure 12.

Figure 12 Cross‐spectral analysis of two pressure signals measured in the same cross‐section of the cone in cavitation conditions

In addition to the precession frequency, a frequency f0 with a coherence level higher than 0.95 is

identified. The phase shift equal to zero at this frequency highlights the synchronous nature of the

pressure fluctuations propagating at this frequency. Favrel et al. [5] identified this frequency as the first

natural frequency of the hydraulic circuit. It is therefore possible for this particular test case to identify

the natural frequency without external excitation system, as it was used by Landry et al. [1‐2], see

Deliverable D4.2. This feature is exploited to determine the natural frequency for all the investigated

operating points.

4.1.2 Resonance conditions

The influence of the discharge factor on both the precession frequency and the natural frequency is

given in Figure 13 for a speed factor equal to nED = 0.288. Its influence on the hydro‐acoustic response

0 0.2 0.4 0.6 0.80.7

0.8

0.9

1

(-)

(-)

Cxy

f / n

f0 frope

0 0.2 0.4 0.6 0.80

1

2

3

4x 10

−4

|Gxy

|

(-)

(Hz -1)

f / n

f0

frope

0 0.2 0.4 0.6 0.8

0

(rad)

(-)

θxy

f / n

frope

f0


02.23.2017 page 14/43

of the system, represented by the amplitude of the auto‐spectrum of a pressure signal measured in the

upstream pipe, is also given. The discharge factor is made dimensionless by its value at the Best

Efficiency Point (BEP), QED* = 0.20. Different flow regimes are observed depending on the value of the

discharge factor, similarly to those observed in cavitation‐free conditions [6].

(a) Frequency (b) Auto‐spectrum amplitude

Figure 13 Influence of the discharge factor on the frequencies frope and f0 (a) and on the system response (b) for nED = 0.288 and σ = 0.11

The natural frequency rapidly decreases when the discharge factor is decreased within the regime 1.

Within the regime 2, the values f0 slowly decreases and remains quasi‐constant for low values of

discharge. Beyond the transition between the regimes 2 and 3, the natural frequency starts increasing,

which is the consequence of the incoherent fluctuations of the cavitation volume within the vortex core

at these operating conditions. When the value of the natural frequency approaches the value of the

precession frequency, the pressure pulsations are amplified as shown in Figure 13. Their amplitude

reaches its maximum for a discharge factor of QED / QED* = 0.78, when the natural frequency matches

with the precession frequency. That corresponds to the resonance conditions.

The resonance conditions are identified for all the investigated values of speed factor. The results are

presented in Figure 14 as a function of nED and QED. The operating points corresponding to the swirl‐free

conditions are also reported in this figure.

Figure 14 Resonance conditions identified by pressure measurements on the reduced scale physical model

The value QEDres of the discharge factor in resonance conditions linearly increases when the speed factor

is increased, in a similar way than the value QED0 in swirl‐free conditions. This suggests that the resonance

conditions occur for a given value of swirl degree of the flow exiting the runner. Based on this

observation, both precession and natural frequencies will be expressed as a function of the swirl number

of the flow exiting the runner, whose analytical expression as a function of the operating parameters of

the machine is derived in the following.

0.5 0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

QED

/ QED

*

(-) f / n

(-)

frope

f0

Regime 3 Regime 2 Regime 1

0.5 0.6 0.7 0.8 0.90

2

4

6

QED

/ QED

*

(-)

(Hz -1) Gxx

(frope

)

Resonance

×10 3

Regime 3 Regime 2 Regime 1

0.05 0.1 0.15 0.2 0.25 0.30.24

0.26

0.28

0.3

0.32

0.34

Resonance conditions

Swirl-free conditionsnED

QED

(-)

(-)


02.23.2017 page 15/43

4.1.3 Swirl number

The swirl degree of the flow exiting the runner can be characterized by the swirl number [7‐8], which is

defined by the ratio between the axial flux of angular momentum and the axial flux moment of axial

momentum as follows:

2

0

2

0

R

R

Cm Cu r dr

S

R Cm r dr

(1)

where Cm and Cu are the time‐averaged axial and tangential velocity components, respectively, and R

is the radius of the section at the runner outlet. By assuming that the axial flow velocity is independent

of the radial position, i.e. Cm = 4 Q / π D2, a simple analytical expression of the swirl number can be

derived and it can be finally expressed as a function of the operating parameters of the machine, i.e. the

speed and discharge factors as follows:

2

0

1 1( )

8ED

ED ED

S nQ Q

(2)

where QED0 is the discharge factor corresponding to the swirl‐free conditions for a given nED‐value. The

complete demonstration can be found in Favrel et al. [9], which is in press for publication in the Journal

of Hydraulic Research.

4.1.4 Empirical laws for prediction of frope and f0 at the model scale

The results presented in the section 3.1.2 suggest that, for a given σ‐value, the precession frequency

and the natural frequency of the test rig are mainly driven by the swirl degree of the flow exiting the

runner. That imply that simple dimensionless laws linking both frequencies with the swirl number may

represent the influence of both the speed and discharge factors on these frequencies. The Strouhal

number Strope and St0 are introduced:

3

St rope

rope

f D

Q (3)

3

00St

f D

Q (4)

The theoretical swirl number is computed for all the investigated operating points by using Equation 2.

The Strouhal numbers are, then, plotted as a function of the swirl number in Figure 15. The data collapse

to one linear function for the precession frequency and they collapse to one power function for the

natural frequency. Linear and power regression laws fitting the experimental data are given in Figure 15.

This result suggests that pressure measurements on the reduced scale model for a limited number of

operating points would enable the prediction at the model scale of both frequencies on the complete

part load operating range for which the vortex rope is coherent (Regime 1 and Regime 2).

For these conditions of σ and nED, the swirl number for which resonance conditions are expected to

occur can be determined by using Figure 15 and is equal to S = 0.756. Based on this value, an expression

between the speed factor and the discharge factor in resonance conditions can be derived:

2

8 11 / ( )

0.7252 0.0355res resED

ED ED

SQ

n n (5)


02.23.2017 page 16/43

In this expression, the value of QED0 is approximated by the linear relation QED

0 = 0.7252 nED + 0.0355.

The resonance conditions are identified at QED = 0.17 for a speed factor of 0.317 whereas the computed

value is equal to 0.1695, which confirm the ability of Equation 5 to fairly predict the resonance

conditions on the complete part load operating range at the model scale.

(a) Precession frequency (b) Natural frequency

Figure 15 Strouhal numbers as a function of the swirl number computed analytically

The measurements presented above were not performed in Froude similitude with the prototype.

Therefore, a second set of measurements were carried out in Froude and σ similitude with the prototype

for a given nED value. The corresponding parameters are nED = 0.288, Fr = 5.5 and σ = 0.11. The values of

the natural frequency as a function of the swirl number are reported in Figure 16 and compared with

the previous results and those obtained for two other Froude numbers (Fr = 6.56 and 7.65).

Figure 16 Natural frequency as a function of the swirl number for three different values of the Froude number

For a given value of swirl number, a decrease in the value of the Froude number results in a decrease of

the natural frequency of the test rig. Therefore, the Froude number influences the value of the hydro‐

acoustic parameters modelling the cavitation flow in the draft tube. However, the influence of the swirl

on the natural frequency is similar at both Froude numbers, i.e. Fr = 8.7 and Fr = 5.47, suggesting that

the same power regression law can be used for all the Froude numbers with however a shift.

In the following, the hydro‐acoustic parameters of the cavitation draft tube flow, i.e. the wave speed

and the bulk viscosity, are determined for all the investigated operating points and are, then, transposed

to the prototype scale to predict the resonance conditions on the real machine.

4.1.5 Determination of the wave speed and bulk viscosity at the model scale

For each operating point, the wave speed a and the bulk viscosity μ’’ in the draft tube are identified by

using the 1D SIMSEN model of EPFL test rig PF3. A detailed description of this model is already given in

Deliverable D4.2. Based on the value of the natural frequency identified experimentally, both

parameters can be determined to obtain the same natural frequency with the 1D model. The

nED

= 0.268

nED

= 0.277

nED

= 0.288

nED

= 0.300

nED

= 0.317

fitting law

0.50

0.5

1

1.5

(-) frope D

3 / Q

1 1.5 2(-)

S0.5 1 1.5 20

0.5

1

1.5

f0 D3 / Q(-) n

ED = 0.268

nED

= 0.277

nED

= 0.288

nED

= 0.300

nED

= 0.317

fitting law

S

(-)

0

2

4

6

8

0.5 1 1.5 2

f0

S

(Hz) Fr = 8.7

Fr = 6.56

Fr = 5.47

(-)

Fr = 7.65


02.23.2017 page 17/43

corresponding algorithms were developed and used by Landry et al. [1‐2] (see Deliverable D4.2). In this

case, an external excitation system was used and the procedure has to be adapted for the present case.

The determination of the parameters a and μ’’ is based on the empirical laws established by Landry et

al. [1‐2]. The dimensionless wave speed and bulk viscosity are defined as:

2w

out v

a

p p (6)

2 20 1 c

out v w

fM

p p

(7)

where is the dimensionless wave speed, M’’ is the dimensionless bulk viscosity, ρw is the water

density, ρc is the vapour density, pout is the pressure at the turbine outlet, pv is the vapour pressure, f0 is

the system’s natural frequency and β is the void fraction.

The relation between these dimensionless numbers and the void fraction β were defined in D4.2. The

results are presented once again in Figure 17.

(a) (b)

Figure 17 Dimensionless wave speed (a) and dimensionless bulk viscosity M’’ (b) as a function of

the void fraction

Initially, the bulk viscosity is set to zero and the wave speed is determined in order to obtain a value of

f0 with the 1D model equal to the value identified experimentally. The dimensionless wave speed is,

then, computed and the void fraction β is determined by using the law = f(β) presented in Figure 17.

The corresponding bulk viscosity is determined by using the law M’’ = f(β) and is, then, injected in the

1D model of the test rig. The wave speed is re‐calculated by considering the new value of the bulk

viscosity. From this new value of a, the corresponding void fraction is, then, determined. The new void

fraction leads to a new value of bulk viscosity that is again injected in the 1D model of the test rig. The

algorithm ends when the value of f0 computed by the 1D model tends towards the experimental value.

The algorithm is illustrated in Figure 18.


02.23.2017 page 18/43

Figure 18 Algorithm for the determination of the wave speed and the bulk viscosity at the model scale

The wave speed obtained by this procedure is given in Figure 19a as a function of the swirl number. For

a given Froude number, a power regression law can represent the influence of the swirl number on the

wave speed. It is also observed that the wave speed remains almost constant beyond S > 1. However,

for a given swirl number, a decrease of the Froude number results in a decrease of the wave speed. A

linear relation is observed between the swirl number and the minimum value of wave speed (for S > 1),

as illustrated in Figure 19b.

(a) am vs swirl number (b) Minimum am vs Froude number

Figure 19 Influence of the Froude number on the wave speed

Experimental value of f0

Iden�fica�on of a with 1D SIMSEN model (μ’’ = 0)

Determina�on of β with the law Π = f(β)

Determina�on of μ’’ withthe law M’’ = f(β)

| f0, num - f0 | < Δf

Iden�fica�on of a withthe new value of μ’’

Final values (a , μ’’)

No

Yes

mm

m

m

m

m

m

0

20

40

60

80

1.5 2

am

S

(m.s-1) Fr = 8.7

Fr = 6.56

Fr = 5.47

Fr = 7.65

0.5 1 (-)10

15

20

25

am

Fr

(m.s-1)

(-)5 6 7 8 95.76

13.3


02.23.2017 page 19/43

(a) Fr = 5.47 and Fr = 8.7 (b) Fr = 5.76

Figure 20 Power regression law with standard deviation for the wave speed as a function of the swirl number

(a) μ’’ vs swirl number (b) Minimum μ’’ vs Froude number

Figure 21 Influence of the Froude number on the bulk viscosity

The power regression fitting law with the corresponding standard deviation is defined for the influence

of the swirl number on the wave speed at Fr = 8.7. The same law can fairly represent the influence of

the swirl number on the wave speed at Fr = 5.47, as shown in Figure 20a. The interpolation presented

in Figure 19b is finally used to shift the power regression law and represent the influence of the swirl

number on the wave speed for Fr = 5.76 for which no experimental data are available. The resulting

curve is given in Figure 20b. This Froude number corresponds to the value of the prototype during the

on‐site measurements. However, in case of a different head value on the prototype, the curve a = f(S)

should be shifted according to Figure 19b.

(a) Fr = 5.47 and Fr = 8.7 (b) Fr = 5.76

Figure 22 Power regression law with standard deviation for the bulk viscosity as a function of the swirl number

The same procedure is used for the bulk viscosity. As shown in Figure 21 and Figure 22, the dispersion

of the results is higher for the bulk viscosity. However, it does not affect the accuracy of the results since

0

20

40

60

80

1.5 20.5 1 (-)

am

S

(m.s-1) Fr = 8.7 (exp.)

Fr = 5.47 (exp.)

Fr = 8.7 (fitting law with

standard deviation)


standard deviation)

0

20

40

60

80

1.5 20.5 1

am

S

(m.s-1)

(-)

0

2

4

6

8×104

1.5 2

μm’’

S

(Pa.s) Fr = 8.7

Fr = 6.56

Fr = 5.47

Fr = 7.65

0.5 1 (-)1

3

4

Fr

(-)5 6 7 8 95.76

1.96

×104

μm’’(Pa.s)

0

2

4

6

8×104

μm’’(Pa.s)

1.5 20.5 1 (-)

S

Fr = 8.7 (exp.)

Fr = 5.47 (exp.)


standard deviation)


standard deviation)

0

2

4

6

8

1.5 20.5 1

S(-)

×104

μm’’

(Pa.s)


02.23.2017 page 20/43

the influence of the bulk viscosity on the value of the natural frequency is negligible. The influence of

the swirl number on the bulk viscosity for Fr = 5.76 is finally given in Figure 22b under the form of a

power regression fitting law with the corresponding deviation.

4.2 Part load resonance prediction at the prototype scale

4.2.1 Transposition of the wave speed and the bulk viscosity

The hydro‐acoustic parameters of the draft tube cavitation flow are transposed from the model scale to

the prototype scale. The hydro‐acoustic parameters corresponding to the conditions in Froude

similitude with the operating conditions tested on the prototype (Fr = 5.76) are used.

For the transposition of the wave speed in Froude similarity, Equation (8) can be directly used. For the

bulk viscosity, the transposition of its value from the model scale to the prototype scale requires the

values of the natural frequency on both the model and the prototype as described in Equation (9) (see

Landry [2]).

prot prot

prot model

model model

D na a

D n (8)

2 2

0

0

prot prot modelprot model

model model prot

D n f

D n f (9)

The value of the natural frequency on the model at a Froude number equal to Fr = 5.76 is computed by

interpolating the results obtained at Fr = 5.52 and Fr = 5.76. For the transposition, the algorithm

presented in Figure 23 is used. For a given swirl number, only the mean value of the bulk viscosity is

considered as its influence on the natural frequency is negligible whereas the standard deviation of the

wave speed is taken into account.

Figure 23 Algorithm for the transposition of the wave speed and the bulk viscosity

The values of the wave speed and bulk viscosity on the prototype are given in Figure 24 as a function of

the swirl number.

Transposi�on of a

Computa�on of f0 on the prototype (μ’’ = 0)

Transposi�on of μ’’

| fi+1 - fi | < Δf

Computa�on of f0 withthe new value of μ’’

Final values (a , μ’’, f0 )

No

Yes

Transposi�on of μ’’ withthe new value of f0

m

p

m

p

p

pp

m

p pp


02.23.2017 page 21/43

(a) Wave speed (b) Bulk viscosity

Figure 24 Hydro‐acoustic parameters as a function of the swirl number at the prototype scale

4.2.2 Natural frequency of the prototype

The value of the natural frequency of the hydropower plant unit is given in Figure 25 as a function of the

swirl number and the discharge factor. The speed factor is equal to nED = 0.2764.

(a) (b)

Figure 25 Natural frequency of the prototype as a function of the swirl number (a) and the discharge factor (b) for a speed factor equal to nED = 0.2764

4.2.3 Precession frequency of the vortex on the prototype

To predict the resonance conditions on the prototype, the value of the precession frequency is also

required. It is expected that the precession frequency can be directly transposed since it depends only

on the hydraulic conditions of the machine, which are conserved from the model to the prototype for

given values of discharge factor and speed factor. The relation between the Strouhal number and the

swirl number established by model tests and presented previously is given once again in Figure 26a.

Additional results obtained with a different Froude number are also added. It is shown that the Froude

number does not affect the relation Str = f(S) for the precession frequency. A linear best‐fit curve, with

the corresponding standard deviation, can finally model the influence of the swirl number on the

Strouhal number for the precession frequency, as shown in Figure 26b.

Based on this relation, the influence of the discharge factor on the precession frequency at the

prototype scale for a speed factor of nED = 0.2764 is finally derived and presented in Figure 27.

Nevertheless, this linear relation between Strouhal and swirl number cannot reproduce the constant

precession frequency observed for high values of discharge factor (flow regime 1). However, considering

the low values of the precession frequency on the prototype, the absolute error can be neglected.

0

50

100

150

200

1.5 20.5 1

ap

S

(m.s-1)

(-)0

1

2

3

1.5 20.5 1

S(-)

×106

μp’’

(Pa.s)

0.5 1 1.50

0.5

1

1.5

2

S

(-)

f (Hz)

0

0.5

1

1.5

0.1 0.12 0.14 0.16 0.18

QED

(-)

f (Hz)


02.23.2017 page 22/43

(a) Raw data (model) (b) Linear fitting law

Figure 26 Linear regression law with standard deviation for the Strouhal number of the precession frequency as a function of the swirl number

Figure 27 Precession frequency on the prototype with the corresponding standard deviation as a function of the discharge factor for the condition nED = 0.2764

4.2.4 Prediction of the resonance conditions on the prototype and comparison with on‐site

measurements

The influence of the discharge factor on both the precession frequency and the prototype natural

frequency is given in Figure 28. That enables the determination of the range of discharge factors for

which resonance are expected to occur according to the transposition methodology developed in the

framework of the HYPERBOLE project.

Figure 28 Precession frequency and natural frequency of the prototype as a function of the discharge factor for the condition nED = 0.2764

0.5 1 1.50

0.5

1

1.5

2

(-) frope D

3 / Q

(-)

S = f(nED, QED)

nED

= 0.268

nED

= 0.277

nED

= 0.288

nED

= 0.300

nED

= 0.317

fitting law

nED

= 0.288

(Froude sim.)

0.50

0.5

1

1.5

(-) frope D

3 / Q

1 1.5 2(-)

S = f(nED, QED)

0.1 0.12 0.14 0.16 0.180

0.5

1

1.5

QED

(-)

frope (Hz)

0

0.5

1

1.5

0.1 0.12 0.14 0.16 0.18

QED

(-)

f (Hz)frope f0

Res

onan

ce p

red.

QED =[0.147-0.154] pred.


02.23.2017 page 23/43

For a speed factor of nED = 0.2764, the resonance conditions are predicted for a discharge factor

comprised between QED = 0.147 and 0.154. That corresponds to an output power generated by the

prototype unit comprised between 279 and 295 MW and an opening guide vanes angles between 13.7

and 14.4 deg. The frequency of the pressure pulsations in resonance conditions predicted by the model

is comprised between 0.52 and 0.59 Hz.

The predicted frequencies are now compared with those identified experimentally on the prototype

unit. The results are given in Figure 29. A perfect agreement is observed between the predicted

precession frequency and the one identified by on‐site measurements. It can be concluded that the

empirical law Str = f(S) established by model tests enables a very good prediction of the precession

frequency on the complete part load operating range of the prototype. As expected, the precession

frequency can be directly transposed from the model to the prototype scale, as both geometric

homology and kinematic similarity between model and prototype scales are fulfilled according to IEC

Standards (1999). As the law linking the Strouhal number with the swirl number is linear, pressure

measurements on the model for a limited number of operating points would enable the prediction of

the precession frequency on the complete part load operating range on the prototype.

Figure 29 Comparison between the predicted frequencies and the ones identified by on‐site measurements.

The results for the natural frequency of the prototype show a slight discrepancy between experimental

results and those based on the 1D SIMSEN model. The difference in frequency is comprised between

0.15 and 0.25 Hz. That results in a slight shift between the resonance conditions observed on the

prototype and those predicted by the methodology of transposition.

The resonance are observed on the prototype for a discharge factor between 0.1361 and 0.1406

whereas the 1D SIMSEN model predict the occurrence of the resonance conditions at a discharge factor

between 0.147 and 0.154. The results are reported on the complete hill chart of the machine in Figure 30

These differences correspond to a slight shift in terms of output power of the machine comprised

between 17 and 42 MW, i.e. 3.8% and 9.5% of the rated output power of the machine (Prated = 444 MW).

If the interpolation of the wave speed on the model as a function of the Froude number is performed

for all the Froude numbers covering the head range of the prototype, the resonance conditions can be

predicted on the complete part load operating range of the hydropower plant unit by using the

methodology presented previously. The predicted resonance conditions are reported in Figure 31 as a

function of the raw head and the output power of the generating unit. The raw head and the output

power are made dimensionless by their rated value, i.e. Hrated = 171 m and Prated = 444 MW.

Res

onan

ce p

red.

0

0.5

1

1.5

0.1 0.12 0.14 0.16 0.18

QED

(-)

f (Hz)

frope, pred.f0, pred.

Res

onan

ce e

xp.

frope, exp.

f0, exp.


02.23.2017 page 24/43

Figure 30 Test case hillchart with the predicted and observed resonance conditions.

Figure 31 Predicted resonance on the complete operating range of the generating unit

4.3 Part load resonance prediction based on similitude of local cavitation factor

4.3.1 Discrepancy of local cavitation factor between model and prototype

The model and prototype conditions investigated in the previous sections were fulfilling similitude

conditions in terms of Thoma number. However, the values of wave speed identified at the model scale

and transposed to the prototype scale are too low compared with the expected ones, which leads finally

to a discrepancy in terms of natural frequencies between prediction and experiment.

This discrepancy might be explained by different pressure conditions in the draft tube cone between

model and prototype, even if the similitude conditions are fulfilled, i.e. same values for the Thoma

number, Froude number, speed factor and discharge factor.

Observed resonance: P = 253 - 262 MW

Predicted resonance: P = 287 ± 8 MW fres = 0.555 ± 0.035 Hz pred.

QED

(-)

nED

(-)

0.26

0.27

0.28

0.29

0.30

0.31

0.32

0.05 0.10 0.15 0.20 0.25

Swirl-free

zone

Lower limit

part load

Interblade

vortices

limit

Prototype measurements

fres = 0.62 Hzexp.


02.23.2017 page 25/43

To compare the pressure conditions between model and prototype, the local cavitation coefficient is

introduced:

2 2

vxnD

p p

n D

(10)

where x

p is the mean pressure in a reference located in the cone.

The local cavitation coefficient is computed for all the operating points measured on the reduced scale

model with a Thoma number equal to σ‐rated. Its value is plotted in Figure 32 as a function of the swirl

number. The values of the local cavitation coefficient measured on the prototype are also given.

Figure 32 Local cavitation coefficient as a function of the swirl number at the model and prototype scales.

An important discrepancy in terms of local cavitation coefficient is observed between the model and

the prototype is observed whereas the value of the Thoma number are very similar. This difference in

terms of pressure conditions might be induced by different specific energy losses between model and

prototype and might explain the final discrepancy observed for the values of the natural frequency

between prediction and experiments.

4.3.2 Wave speed as a function of the swirl number and the local cavitation coefficient

To overcome this problem and identify the wave speed on the model scale corresponding to the

pressure conditions on the prototype, a 3‐D mapping of the wave speed identified on the model scale

as a function of the swirl number and the local cavitation coefficient is realized. Measurements at two

additional values of Thoma number, corresponding to the maximum and minimum levels of the

tailwater reservoir on the prototype, are added for the analysis. The 3‐D mapping of the wave speed for

a Froude number equal to Fr = 8.76 is given in Figure 33.

For a given Froude number, the relation between the wave speed, the swirl number and the local

cavitation coefficient can be represented by a single regression surface that is composed by the sum of

Hermite polynomials. The standard deviation of the experimental data regarding the regression surface

is also computed and included in Figure 33.


02.23.2017 page 26/43

Figure 33 Wave speed at the model scale as a function of the swirl number and the local cavitation coefficient, together with regression surface and corresponding deviation.

To obtain the surface corresponding to different values of Froude number, a linear shift is used. Its value

is identified by using the values of the wave speed identified at the model scale in Froude similitude, as

illustrated in Figure 34.

Figure 34 Wave speed at the model scale as a function of the swirl number and the local cavitation coefficient, together with regression surface and corresponding calculated wave speed

4.3.3 Dimensionless wave speed as a function of the swirl number and the local cavitation

coefficient

The value of the Froude number has an influence on the value of the wave speed, as shown in the

previous section. Therefore, a linear shift of the regression surface is used to obtain the surface

corresponding to different values of Froude number.

To obtain a single surface including the influence of the Froude number, the dimensionless wave speed

Π defined in Section 4.1.5 is used. The 3‐D mapping of Π as a function of the swirl number and the local

cavitation coefficient is given in Figure 35, including two values of the Froude number, Fr = 8.76 and Fr

= 5.52, respectively. Once again, all the data collapse in one single surface. Therefore, the influence of

the swirl number, the local cavitation coefficient and the Froude number on the value of the

dimensionless wave speed at the model scale can be represented by a single regression surface.


02.23.2017 page 27/43

Figure 35 Dimensionless wave speed at the model scale as a function of the swirl number and the local cavitation coefficient, together with regression surface.

4.3.4 Determination of the wave speed corresponding to the conditions during on‐site tests

The values of wave speed at the model scale corresponding to the prototype conditions during on‐site

measurements in terms of swirl number and local cavitation coefficient are, then, identified by using

first the surface a = f(S, χnD) corresponding to the Froude number during on‐site measurements. The

procedure is illustrated in Figure 36.

Figure 36 Identification of the wave speed at the model scale corresponding to prototype conditions.

The dimensionless wave speed at the model scale corresponding to the prototype conditions is also

identified by using the surface Π = f(S, χnD), which is illustrated in Figure 37.

The value of the wave speed at the model scale identified for each operating conditions investigated

during on‐site measurements is given in Figure 38 as a function of the swirl number. Both methods are

used, i.e. the identification based on the surface a = f(S, χnD) and the one based on the surface Π = f(S,

χnD). The results obtained by both methods are very similar.


02.23.2017 page 28/43

The experimental values of the wave speed corresponding to the conditions during on‐site tests are also

given. These values have been identified based on the 1D SIMSEN model of the hydropower plant unit

and the values of the prototype natural frequency identified experimentally.

The calculated values of the wave speed are in very good agreement with the values identified by

experimental tests on the prototype, except for the point with the lowest value of swirl number

Figure 37 Identification of the dimensionless wave speed at the model scale corresponding to prototype conditions.

Based on a = f(S, χnD) (b) Based on Π = f(S, χnD)

Figure 38 Values of the wave speed identified on the model scale, together with the values identified by measurements on the prototype.

4.3.5 Update of the predicted natural frequency and resonance conditions at the prototype

scale

The calculated values of the wave speed are then transposed from the model to the prototype scale by

using the equation given in Section 4.2.1. By injecting the values of the transposed wave speed in the

1D SIMSEN model of the hydropower plant unit, the natural frequency of the prototype is computed for

each operating conditions investigated during model tests. Its value is given as a function of the output

power of the machine in Figure 39 and is compared with the previous prediction that is based on Thoma

number similitude. An increase of the value of the predicted natural frequency is obtained for all the

operating conditions.

Finally, the results are compared with on‐site results in Figure 40.


02.23.2017 page 29/43

Figure 39 Precession frequency and natural frequency of the prototype as a function of the discharge factor for the condition nED = 0.2764

Figure 40 Comparison between the predicted frequencies and the ones identified by on‐site measurements.

A very good agreement is now achieved between the predicted and experimental values of the natural

frequency at the prototype scale. The experimental values are included in the range of uncertainty of

the predicted values. Therefore, the predicted resonance conditions are in very good agreement with

the conditions identified during on‐site tests, which validates the methodology based on the similitude

in terms of local cavitation coefficient.

4.4 Prediction of pressure fluctuations amplitude at part load conditions on the prototype

4.4.1 Determination of the pressure excitation source at the model scale

The excitation source induced by the precession of the vortex rope at part load conditions is modelled

as a periodical pressure source Sh in the 1D SIMSEN model of the test rig, characterized by its amplitude

A (in meters) and its frequency frope (in Hz) as follows:

2 ropej f t

Sh Ae (11)

The pressure source was previously modelled as a distributed source by representing its amplitude along

the draft tube as a Gaussian distribution defined by an average location L and a standard deviation e [2]:

2

2

( )( )

22 rope

x Lj f teSh Ae e

(12)


02.23.2017 page 30/43

In the following, the pressure source is modelled as a lumped source and is applied on one element in

the SIMSEN model of the test rig. Moreover, the location of the pressure source is considered as

independent of the operating conditions whereas it was adapted for each operating point in [2], see

Deliverable D4.2.

For a given operating point, the amplitude A is adapted in order to obtain a forced harmonic response

at the precession frequency simulated by the 1D SIMSEN model [10] similar to the experimental

response measured in the test rig by the pressure sensors located in the upstream pipes of the machine.

Regarding the precession frequency, its value is fixed at the experimental value.

The amplitude of the pressure excitation source is determined for all the operating points investigated

in the previous section. They include the measurements at Fr = 5.47 (Froude similitude with the

prototype) and Fr = 8.7. For each operating point, the wave speed and the bulk viscosity are fixed at the

value determined in Section 3 and the amplitude of the pressure source is, then, identified. The raw

amplitude A and its value made dimensionless by the net head of the turbine are plotted as a function

of the swirl number in Erreur ! Source du renvoi introuvable..

(a) Amplitude vs swirl (b) Dimensionless amplitude vs swirl

Figure 41 Amplitude of the pressure source and its value made dimensionless by the net head as a function of the swirl number for two different Froude numbers

Similar to the hydro‐acoustic parameters a and μ’’, the data for the amplitude of the pressure source at

Fr = 8.7 collapse in one single curve, except the operating points near the resonance conditions for which

the dispersion of the results is very high. The amplitude at Fr = 5.47 is slightly lower, which highlights

the influence of the Froude number on the pressure excitation source. However, if the amplitude of the

pressure source is made dimensionless by the net head of the turbine, the experimental data collapses

in one single curve as shown in Figure 41b, except once again the operating points near the resonance

conditions. This result will be exploited to determine the pressure excitation source at the Froude

conditions corresponding to the conditions during on‐site measurements, for which no data are

available on the reduced scale model.

It can be noticed that the shape of the curves in n Figure 41 is unexpected, as the amplitude of the

pressure source reaches its maximum when the system enters into resonance. On the contrary, it was

expected that the amplitude of the pressure source increases when the discharge factor decreases and

reaches its maximum for high values of swirl number [6]. There is no reason why the amplitude of the

pressure excitation source is affected by the occurrence of resonances, as its value depends on the

dynamics of the vortex rope and not on the interaction with the system.

It can be suggested that the occurrence of resonances, which induce strong non‐linearity in the system

due to the fluctuations of the cavitation volume, introduces an additional excitation. This excitation was

notably introduced by Alligné [11] as a mass source term. However, it cannot be directly transposed to

the prototype, contrary to the pressure excitation source induced by the precession of the vortex.

0.5 1 1.50

0.5

1

1.5

2

2

A

S

(m)Fr = 8.7

Fr = 5.47

(-)

Resonance

0.5 1 1.50

0.5

1

1.5

2

2

A / H

S

(-)Fr = 8.7

Fr = 5.47

(-)

Resonance


02.23.2017 page 31/43

Therefore, the 1D model and the methodology proposed in this work are not able to identify properly

the pressure excitation source when the system enter into resonance. For this reason, only the results

for the operating points far from the resonance, typically with S > 1.25, will be transposed in the

following.

As shown in n Figure 41b, the amplitude of the pressure source made dimensionless by the value of Hm

remains almost constant for S > 1.25, with however a slight dispersion of the results. The average

dimensionless amplitude and its corresponding standard deviation are equal to:

0.014 0.002 (‐)m

m

A

H (13)

4.4.2 Simulation in the time domain at the prototype scale

This section is focused on the operating point OP12 investigated during the on‐site measurements. This

operating point is characterized by a swirl number S = 1.420 and a Froude number of Fr = 5.77. The

amplitude of the pressure source corresponding to this Froude number at the model scale is equal to:

(0.014 0.002) 0.166 m 0.024 mm mA H (14)

The amplitude of the pressure source is transposed to the prototype scale as follows [2]:

p

p m

m

DA A

D (15)

where Dp and Dm are the diameter of the runner at the prototype scale and model scale, respectively.

The amplitude for the investigated operating point is finally equal to Ap = 2.57 m ± 0.37 m.

The hydro‐acoustic parameters corresponding to the investigated operating point on the prototype are

identified according to the results obtained in Section 3:

‐ a = 51.87 m.s‐1 ± 3.42 m.s‐1

‐ μ’’ = 117350 Pa.s ± 35000 Pa.s

‐ frope = 0.675 Hz ± 0.021 Hz

These parameters are injected in the 1D SIMSEN model of the hydropower plant unit and a simulation

in the time domain is performed. A total of 6 simulation is performed, including:

‐ Simulations with the mean value of a and μ’’ and the minimum, mean and maximum values of

A to highlight the effect of the uncertainty on the excitation amplitude (3 simulations);

‐ Simulations with the mean value of A and the minimum, average and maximum values of a and

μ’’ to highlight the effect of the uncertainty on the hydro‐acoustic parameters (3 simulations).

The time history of the pressure fluctuations in the draft tube cone simulated with the 3 different values

of the excitation amplitude is compared in Figure 42a with the experimental results measured on the

prototype. The experimental results are filtered with a low‐pass filter at 0.9 Hz. The auto‐spectrum of

the pressure fluctuations simulated with the mean value of A is also compared in Figure 42b with the

experimental results (without filtering). A similar comparison is provided in Figure 43 for the pressure

fluctuations in the penstock.


02.23.2017 page 32/43

(a) Time domain (b) Frequency domain

Figure 42 Comparison between numerical and experimental results for pressure fluctuations in the draft tube cone. The minimum, average and maximum values of the amplitude are

used for the simulation.

(a) Time domain (b) Frequency domain

Figure 43 Comparison between numerical and experimental results for pressure fluctuations in the penstock with the minimum, average and maximum values of the excitation amplitude

The amplitude of the pressure fluctuations simulated by the 1D model is lower than the experimental

one. It is confirmed by the auto‐spectra given in Figure 42b and Figure 43b. The peak at the precession

frequency for the experimental results is more than twice larger than that for the numerical results.

A wide‐band peak at a frequency of 0.51 Hz is observed in the auto‐spectra of the experimental results.

This frequency corresponds to the natural frequency f0. A similar peak is observed in the auto‐spectra

of the numerical results at a frequency of 0.37 Hz, with however a lower amplitude. This frequency

corresponds to the natural frequency predicted by the 1D SIMSEN model with the transposed hydro‐

acoustic parameters corresponding to this operating point

The values of the RMS and peak‐to‐peak indicators are computed for the pressure signals obtained by

1D simulation and are compared with those of the experimental signals. The comparison is given in

Table 4. For the experimental results, the RMS and peak‐to‐peak indicators are estimated by splitting

the complete signals into several windows since the pressure fluctuations have a stochastic component.

0 2 4 6 8 10-6

-4

-2

0

2

4

6∆p / ρg(m)

t(s)

Exp.Sim. (A

max)

Sim. (Amin

)Sim. (A

mean)

0 0.5 1 1.5 20

1

2

3∆p / ρg(m)

f

(Hz)

Exp.Sim. (A

mean)

frope

f0

-6

-4

-2

0

2

4

6

0 2 4 6 8 10

∆p / ρg(m)

t(s)

Exp.Sim. (A

max)

Sim. (Amin

)Sim. (A

mean)

0 0.5 1 1.5 20

1

2

3∆p / ρg(m)

f

(Hz)

Exp.Sim. (A

mean)frope

f0


02.23.2017 page 33/43

Table 4 Comparison between experimental and numerical values of the RMS and peak‐to‐peak indicators of pressure fluctuations with different values of source amplitude

DT cone (sim.)

DT cone (exp.)

Relative Error

Penstock (sim.)

Penstock (exp.)

Relative Error

RMS 1.12 m ± 0.06 m

2.32 m 52 % ± 3% 0.82 m ± 0.12 m

1.82 m 55 % ± 7 %

Peak‐to‐peak

3.32 m ± 0.47 m

7.16 m 54 % ± 7% 2.41 m ± 0.35 m

5.68 m 58 % ± 7 %

The relative error between experiments and simulation is comprised between 40 and 54% for the RMS

value of the pressure signals in the draft tube depending on the value of the amplitude of the predicted

excitation source which is used. Therefore, the pressure fluctuations are underestimated by the

methodology presented in this report.

(a) Draft tube cone (b) Penstock

Figure 44 Comparison between numerical and experimental prototype results for pressure fluctuations in the draft tube and penstock with the minimum, average and maximum

values of the wave speed

For the results presented above, the mean value of both the wave speed and the bulk viscosity was

used. However, as explained in Section 4.2, the value of these parameters transposed to the prototype

scale consists of a mean value and a standard deviation, which is induced by the dispersion of the results

obtained at the model scale. It is expected that a change in their value may affect the amplitude of the

simulated pressure signals since that modifies the natural frequency of the system and potentially the

interaction between the system and the pressure excitation source.

To highlight the effect of the uncertainties on the wave speed and the bulk viscosity, a comparison

between the numerical results obtained with the maximum, average and minimum values of this

parameters is presented in Figure 44. The average amplitude of the excitation source is used in this case.

The experimental results are also included.

Table 5 Comparison between experimental and numerical values of the RMS and peak‐to‐peak indicators of pressure fluctuations with different values of wave speed

DT cone (sim.)

DT cone (exp.)

Error Penstock (sim.)

Penstock (exp.)

Error

RMS 1.12 m ± 0.06 m

2.32 m 52 % ± 3% 0.82 m ± 0.05 m

1.82 m 55 % ± 3 %

Peak‐to‐peak

3.32 m ± 0.05 m

7.16 m 54 % ± 1% 2.41 m ± 0.1 m

5.68 m 58 % ± 2 %

-6

-4

-2

0

2

4

6

0 2 4 6 8 10

∆p / ρg(m)

t

Exp.Sim. (a

max)

Sim. (amin

)Sim. (a

mean)

(s)-6

-4

-2

0

2

4

6

0 2 4 6 8 10

∆p / ρg(m)

t

Exp.Sim. (a

max)

Sim. (amin

)Sim. (a

mean)

(s)


02.23.2017 page 34/43

The effect of a change of the wave speed and the bulk viscosity within their range of uncertainty almost

does not affect the amplitude of the simulated pressure signals. It is confirmed in Table 5, which provides

the value of the RMS and peak‐to‐peak indicators.

In the draft tube cone, the uncertainty on the RMS value of the simulated pressure signal is ± 0.06 m,

which corresponds to ± 5% of the mean value. Consequently, the relative error with the experimental

results remains high and is comprised between 49% and 55% of the experimental results.


02.23.2017 page 35/43

5 Transposition of the hydro‐acoustics parameters and prediction of the instability at full load

5.1 Limits of stability at full load at the model scale and comparison with on‐site measurements

Pressure fluctuations were performed on the reduced scale model for 5 different values of the speed

factor, see Table 3. For each speed factor, the discharge factor is varied from about 116% to 150% of

the value at the BEP. The Thoma number is set at the rated value of the prototype whereas the Froude

number is equal to Fr = 8.7, which does not corresponds to the Froude conditions on the prototype.

For each speed factor, the limits of stability are defined by analysing the pressure fluctuations measured

in the draft tube cone. When the vortex rope becomes unstable, pressure fluctuations propagates into

the entire hydraulic circuit at a given frequency, which corresponds to the first eigenfrequency of the

test rig. The pressure fluctuations in the draft tube at this frequency are highly coherent, with a

coherence level equal to Cxy ≈ 1. An example of pressure fluctuations measured in the draft tube cone

in stable and unstable conditions is given in Figure 45.

Figure 45 Time history of pressure signals measured in the draft tube cone at full load in stable and unstable conditions.

The limits of stability at full load determined on the reduced scale physical model are given in Figure 46.

For each nED value, the black point corresponds to the highest QED value for which the system is still

stable whereas the red point corresponds to the first unstable point.

Figure 46 Limits of stability of the reduced scale model at full load conditions

0 5 10 15 20 25 30-0.05

0

0.05

0.1

Cp(-)

(-)

n × t

Stable

Unstable

0.22 0.24 0.26 0.28 0.30.24

0.26

0.28

0.3

0.32

0.34

1 2 3

1 Stable operation

2 Transition

3 Unstable operation

Swirl-freenED

QED

(-)

(-)


02.23.2017 page 36/43

As described previously, two operating points were tested on the prototype. One was stable whereas

the second one, which features a lower head value, was unstable. These operating points are reported

in Figure 47 together with the limits of stability of the machine defined previously.

Figure 47 Limits of stability of the reduced scale model at full load conditions and comparison with stable and unstable operating conditions identified on the prototype

The stable operating point OP1a measured on the prototype is located in the stable zone identified on

the reduced scale model whereas the unstable operating point OP1b is located in the unstable operation

zone. Therefore, the results observed on the prototype seems to be in agreement with the stability zone

defined on the reduced scale model. That suggests that the stability zone at full load can be defined

during model tests and directly transposed to the prototype machine.

However, this result must be validated. Indeed, the measurements on the reduced scale model were

performed at the rated σ‐value, which corresponds to the average value of the TWEL over one year.

During prototype tests, the σ‐value for OP1a was slightly lower whereas the one at OP1b was higher

than the rated value. The instability may therefore occur for a lower value of discharge factor on the

prototype in the case the Thoma number corresponds to its average value. Moreover, the Froude

number was also lower during prototype tests.

5.2 Empirical law for prediction of finst. at the model scale

For all the operating points in unstable conditions, the frequency of the instability is determined by

spectral analysis of the pressure fluctuations. The corresponding Strouhal number are plotted as a

function of the swirl number in Figure 48.

Similar to the results obtained at part load conditions, the experimental data collapses in one single

curve. It suggests that the frequency of the instability can be predicted on the complete full load

operating range based on these empirical laws.

In the following, the hydro‐acoustic parameters of the full load vortex rope are determined for all the

investigated operating points by applying the procedure already used at part load conditions. It has been

demonstrated that the empirical laws linking the dimensionless wave speed and bulk viscosity with the

void fraction are valid for all the operating points, including full load conditions.

0.22 0.24 0.26 0.28 0.30.24

0.26

0.28

0.3

0.32

0.34

3

1 Stable operation (model)

2 Transition (model)3 Unstable operation (model)

Swirl-freenED

QED

(-)

(-)

1 2

Stable point (proto)

Unstable point (proto)


02.23.2017 page 37/43

(a) Frequency (b) Strouhal number

Figure 48 Frequency of full load instability and the corresponding Strouhal number as a function for the swirl number

5.3 Determination of the hydro‐acoustic parameters at the model scale

The wave speed and the bulk viscosity are given in Figure 49 as a function of the swirl number.


Figure 49 Wave speed and bulk viscosity on the reduced scale model as a function of the swirl number

The influence of the Froude influence on the hydro‐acoustic parameters at full load cannot be clearly

identified as the value of the hydro‐acoustic parameters in Froude similitude (Fr = 5.52) is comprised in

the range of uncertainties of the results at the higher Froude number. Therefore, the results at Fr = 8.7

will be directly transposed and compared with on‐site measurements, although this Froude number

does not corresponds to the prototype value.

-0.2 -0.15 -0.1 -0.05 00

2

4

6

8

10

(Hz) finst

(-)

S = f(nED, QED)

nED

= 0.268

nED

= 0.277

nED

= 0.288

nED

= 0.300

nED

= 0.317

fitting law

nED

= 0.273

(Froude sim.)

-0.2 -0.15 -0.1 -0.05 00

0.2

0.4

0.6

0.8

1

(-) finstD3/Q

(-)

S = f(nED, QED)

nED

= 0.268

nED

= 0.277

nED

= 0.288

nED

= 0.300

nED

= 0.317

fitting law

nED

= 0.273

(Froude sim.)

-0.2 -0.15 -0.1 -0.05 00

20

40

60

80Fr = 8.7 (exp.)

Fr = 5.52 (exp.)


standard deviation)

am(m.s-1)

(-)

S

-0.2 -0.15 -0.1 -0.05 00

2

4

6

8

10×104

Fr = 8.7 (exp.)

Fr = 5.52 (exp.)


standard deviation)

(Pa.s)

(-)

S

μm’’


02.23.2017 page 38/43

5.4 Transposition of the hydro‐acoustic parameters at the prototype scale

The wave speed and bulk viscosity transposed to the prototype scale are plotted as a function of the

swirl number in Figure 50. A power regression fitting law is included together with the corresponding

standard deviation.

The natural frequency of the system and the corresponding Strouhal number are finally plotted as a

function of the swirl number in Figure 51.

It is expected to observe full load instability at a frequency comprised between 0.32 Hz and 1.23 Hz

depending the operating conditions. The uncertainties is however higher for the highest values of swirl

number and corresponds to 8‐15 % of the mean value.


Figure 50 Hydro‐acoustic parameters as a function of the swirl number at the prototype scale at full load conditions

(a) Frequency (b) Strouhal number

Figure 51 Frequency of full load instability and the corresponding Strouhal number as a function for the swirl number at the prototype scale

-0.2 -0.15 -0.1 -0.05 00

50

100

150

200

250Transposition of model tests results

Fitting law with standard

deviation

ap(m.s-1)

(-)

S

-0.2 -0.15 -0.1 -0.05 00

1

2

3

4

5Transposition of model tests results

Fitting law with standard

deviation

(Pa.s)

(-)

S

μp’’

×106

-0.2 -0.15 -0.1 -0.05 00

0.5

1

1.5

2

(Hz) finst

(-)

S

p Results from transposed HA parameters

Fitting law with standard deviation

-0.2 -0.15 -0.1 -0.05 00

0.2

0.4

0.6

0.8

1

(-) finst

D3/Q

(-)

S

p Results from transposed HA parameters

Fitting law with standard deviation


02.23.2017 page 39/43

5.5 Prediction of instability frequency at full load and comparison with on‐site measurements

The value of the frequency of the full load instability is given in Figure 52 as a function of the swirl

number and the discharge factor under the form of power regression fitting laws with the corresponding

standard deviation. The speed factor is equal to nED = 0.279. The value of the frequency of the instability

identified during on‐site measurements is also provided. It is included in the range of uncertainty of the

values obtained by the transposition of the model test results. A deviation of 11 % from the mean value

of frequency obtained with 1D SIMSEN model is observed.

(a) Frequency vs swirl number (b) Frequency vs discharge factor (nED=0.279)

Figure 52 Comparison between the predicted frequencies and the one identified by on‐site measurements on the prototype at full load conditions.

-0.2 -0.15 -0.1 -0.05 00

0.5

1

1.5

2

S(-)

f (Hz)f0, pred. f0, exp. (unstable) f0, exp. (stable)

0.24 0.25 0.26 0.27 0.280

0.5

1

1.5

2

QED

(-)

f (Hz)

f0, pred. f0, exp. (unstable) f0, exp. (stable)


02.23.2017 page 40/43

6 Conclusions This report presents the transposition of the hydro‐acoustic parameters modelling the cavitation draft

tube flow at part load and full load conditions from the model scale to the prototype scale. The

cavitation vortex rope is described as an excitation source for the hydraulic system at part load

conditions whereas it can enter self‐excitation at full load conditions. It may result in resonance

phenomena, inducing pressure surge and electrical power swings.

For an accurate prediction of pressure pulsations and resonance conditions at part load and full load

conditions, a proper modelling of the cavitation draft tube flow is required. It has already been

presented in Deliverable D4.2. The hydro‐acoustic parameters modelling the cavitation draft tube flow,

such as the cavitation compliance and the bulk viscosity, and the pressure excitation source representing

the excitation induced by the vortex rope were identified on a reduced scale model. Measurements with

an external excitation source and numerical simulations with the 1D SIMSEN model of the test rig were

used for this purpose. The corresponding results were also presented in Deliverable D4.2.

The transposition of the hydro‐acoustic parameters and the pressure excitation source is the last step

of the methodology developed in the framework of the HYPERBOLE project for the assessment of

hydropower plants stability. To validate the complete methodology, the results of the simulations were

finally compared to the on‐site measurements performed on the hydropower plant unit.

However, the operating conditions during prototype tests did not correspond to the investigated

operating points during model tests in terms of speed factor, discharge factors and Froude number. To

overcome this problem, it is demonstrated that the influence of both the speed and discharge factors

on the vortex precession frequency and the test rig natural frequency can be reduced to single curves if

these frequencies are expressed as a function of the swirl number.

Based on the value of the natural frequency identified at the model scale, the value of the corresponding

hydro‐acoustic parameters is determined by using the dimensionless parameters derived by Landry et

al. [1, 2]. For a given Froude number, the influence of the speed and discharge factors on each hydro‐

acoustic parameters can be reduced to one single curve by expressing this parameter as a function of

the swirl number. Power regression laws linking the hydro‐acoustic parameters and the swirl number

were finally derived and transposed to the prototype scale. This enables the prediction of the hydro‐

acoustic parameters on the complete part load operating range of the prototype and finally the

prediction of the resonance conditions and the simulation of the resulting pressure fluctuations in the

time domain. The same procedure was used at full load conditions for the prediction of the instability

frequency.

The results were compared with the measurements performed on the hydropower plant unit and the

following remarks can be drawn regarding the accuracy of the predictions and simulations performed

with the 1D SIMSEN model:

Part load:

The curve linking the vortex rope frequency and the swirl number is very similar between model and

prototype, which enables a very good estimation of the vortex precession frequency on the prototype.

However, the natural frequency of the prototype hydraulic circuit is underestimated by about 0.15‐0.25

Hz, which corresponds to a relative error of 25%‐32%, if only similitude in terms of Thoma number

between model and prototype is considered. Indeed, the value of the wave speed in the draft tube is

underestimated.


02.23.2017 page 41/43

Different pressure conditions between prototype and model are observed even if similitude conditions

are fulfilled in terms of speed factor, discharge factor, Froude number and Thoma number, which may

explain the underestimation of the predicted values for the natural frequency.

To overcome this problem, a 3‐D mapping of the wave speed at the model scale as a function of the

swirl number and the local cavitation coefficient (defined in the draft tube cone) is realized, leading to

a new prediction of the natural frequency on the prototype. The experimental results and the new

predicted values are in very good agreement.

Regarding the amplitude of the pressure source, its value seems to be underestimated by the presented

method. The modelling of the pressure excitation source in the 1D SIMSEN model might be too simple

and it might be necessary to take into account the harmonics of the precession frequency. Further

investigation will focus on this aspect.

Full load:

Even though the model tests were done at a different value of the Froude number, the stable and

unstable zones on the model are in good agreement with the ones observed on the prototype (inside

the uncertainty margins). Moreover, the frequency of the instability observed during on‐site tests is

comprised in the uncertainty margin of the predicted values. Time domain simulations at full load

conditions were not performed since they require the value of the mass flow gain factor, which is

neglected in this work. Moreover, the non‐linearity of the wave speed must be included [11] to obtain

finite fluctuations called “limit cycle”.

Important findings:

The introduction of the swirl number in the analysis is an important new insight for hydropower

stability assessment. Indeed, this enables the prediction of several parameters, including hydro‐

acoustic parameters and precession frequency, on the complete operating range of the machine by

limiting the number of measurements at the model scale.

Moreover, it is necessary to use similitude in terms of local cavitation coefficient between model and

prototype to ensure similar cavitation volumes between both scales. Indeed, the classic approach, i.e.

similitude in terms of Thoma number, does not necessary ensure similar pressure conditions between

prototype and model.

To conclude, the methodology developed in the framework of the HYPERBOLE project is a decisive step

towards the complete assessment of hydropower plants stability. Although different aspects of the

modelling must be improved, the global methodology and the results are promising. Further

investigations must focused on the uncertainties for the identification of the pressure source and the

similitude between prototype and model in terms of pressure level in the draft tube. The project has

provided a large amount of data that will allow these developments and improvements.

6.1 Potential impacts of the presented results

The developed methodology might lead to the revision of IEC 60193 standards for model tests in an

industrial context. It will enable the prediction of the dynamic behaviour of the full‐scale prototype unit

on its complete operating range. Therefore, the numerical results provided by the developed

methodology will allow the operators to extend safely the operating range of hydropower units from 0

to 100 % of the rated power by avoiding operating conditions putting at risk the stability of the hydro‐

mechanical system and reducing the lifetime of the mechanical components.

The extension of the operating range of both conventional and reversible pumped storage hydropower

plants will enable hydropower to play a key role in the integration of new renewable energy sources


02.23.2017 page 42/43

into the electrical grid by providing additional services to the grid and by participating actively to its

stabilization.

Furthermore, for both greenfield projects at their design stage and refurbishment projects, the

prediction of hydro‐acoustic resonance frequencies will enable the hydraulic equipment manufacturer

to avoid possible matchings between these frequencies and the eigenfrequencies of the hydraulic,

mechanical, electrical and control components, which might lead to case of resonance and dramatic

consequences for the hydropower plant.

“This project has received funding from the European Union’s Seventh Programme for research, technological development and demonstration under grant agreement No ERC/FP7‐ ENERGY‐2013‐1‐Grant 608532”.


02.23.2017 page 43/43

7 References

[1] Landry C., Favrel A., Müller A., Yamamoto K. and Avellan F. (2016), Local wave speed and bulk flow viscosity in Francis turbines at part load operation, Journal of Hydraulic Research, vol. 54(2)

[2] Landry C. (2016), Hydro‐acoustic modelling of a cavitation vortex rope for a Francis turbine, PhD thesis n°6547, EPFL, Switzerland

[3] Alligné S., Nicolet C., Tsujimoto Y. and Avellan F. (2014), Cavitation surge modelling in Francis turbine draft tube, Journal of Hydraulic Research, vol. 52(3)

[4] Alligné S., Nicolet C., Béguin A., Landry C., Gomes J. and Avellan F. (2016), Hydroelectric system response to part load vortex rope, IOP Conference Series: Earth and Environmental Science, vol. 49 (28th IAHR Symposium, Grenoble, France)

[5] Favrel A., Müller A., Landry C., Yamamoto K. and Avellan F. (2016), LDV survey of cavitation and resonance effect on the precessing vortex rope dynamics in the draft tube of Francis turbines, Experiments in Fluids, vol. 57(11)

[6] Favrel A., Müller A., Landry C., Yamamoto K. and Avellan F. (2015), Study of the vortex‐induced pressure excitation source in a Francis turbine draft tube by particle image velocimetry, Experiments in Fluids, vol. 56(12)

[7] Gupta A.K., Lilley D.G. and Syred N. (1984), Swirl flows, vol. 1, Abacus Press, UK

[8] Müller A., Favrel A., Landry C. and Avellan F. (2017), Fluid‐structure interaction mechanism leading to dangerous power swings in Francis turbines at full load, Journal of Fluids and Structures, vol. 69, p. 56‐71

[9] Favrel A., Gomes J., Landry C., Nicolet C. and Avellan F. (2017), New insight in Francis turbine cavitation vortex rope: role of the runner outlet flow swirl number, Journal of Hydraulic Research, article in press

[10] Alligné S., Silva P.C.O., Béguin A., Kawkabani B., Allenbach P., Nicolet C. and Avellan F. (2014), Forced response analysis of hydroelectric systems, IOP Conference Series: Earth and Environmental Science, vol. 22 (27th IAHR Symposium, Montreal, Canada)

[11] Alligné S. (2011), Forced and Self Oscillations of Hydraulic Systems induced by Cavitation Vortex Rope of Francis Turbines, PhD thesis n°5117, EPFL, Switzerland

608532 - hyperbole - deliverable d4 documents/deliverables... · 2020. 7. 22. ·...

Documents