Experimental investigation of swirl number influence on spiral vortex structure dynamics

Abstract

The hydraulic turbines are recently forced to operate far away from the optimal conditions in order to balance fluctuations in electricity generation. In case of Francis, pump and propeller turbines, using only single control component of guide vanes, it means that in regions where the high residual swirl enters the draft tube, the flow is decelerated and convenient conditions for the vortex rope development are created. Such flow conditions are considered to be the triggering mechanism for occurrence of different forms of vortex structures in the Francis turbine draft tube, e.g. spiral or straight vortex rope at part load or full load respectively. Independently on the vortex rope shape the unsteady pressure fields develop producing periodic stress on turbine components and possibly resulting in noise, blade cracks, runner lift, power swing, etc. To study and mimic such flow conditions, a simplified device of vortex generator apparatus is employed. Thanks to its design, the vortex generator enables to change the ratio between fluxes of axial momentum and tangential moment of momentum of generated swirl. Then, the behavior of vortex structure changes in a similar way as the flow rate variation in the draft tube of Francis turbine. For above mentioned reasons the unsteady cavitating spiral vortex is experimentally studied using both high speed video record and particle image velocimetry (PIV). The main focus is on change of vortex dynamics regarding to the swirl number variation. The proper orthogonal decomposition (POD) together with the classical fast Fourier transformation (FFT) are employed to extract dominant modes and frequencies from experimental data.

Full text

IOP Conference Series: Earth and Environmental Science
PAPER • OPEN ACCESS
Experimental investigation of swirl number
influence on spiral vortex structure dynamics
To cite this article: D Štefan et al 2021 IOP Conf. Ser.: Earth Environ. Sci. 774 012085
View the article online for updates and enhancements.
You may also like
The numerical research of runner
cavitation effects on spiral vortex rope in
draft tube of Francis turbine
J Yang, L J Zhou and Z W Wang
-
Investigation of the Plunging Pressure
Pulsation in a Swirling Flow with
Precessing Vortex Rope in a Straight
Diffuser
S Muntean, C Tnas, A I Bosioc et al.
-
Effect of spiral vortices on the stability of
vortex structures in the diffusor
F Pochylý, S Fialová and D Štefan
-
This content was downloaded from IP address 209.242.215.37 on 09/01/2022 at 19:37

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
1
Experimental investigation of swirl number influence
on spiral vortex structure dynamics
Dˇ
Stefan1, M Hudec1, V Uruba2, P Proch´azka2, O Urban1, P Rudolf1
1V. Kaplan Department of Fluid Engineering, Faculty of Mechanical Engineering, Brno
University of Technology, Technick´a 2896/2, 61699 Brno, Czech Republic
2Institute of Thermomechanics, The Czech Academy of Science, Dolejˇskova 1402/5, Prague,
Czech Republic
E-mail: stefan@fme.vutbr.cz
Abstract. The hydraulic turbines are recently forced to operate far away from the optimal
conditions in order to balance fluctuations in electricity generation. In case of Francis, pump
and propeller turbines, using only single control component of guide vanes, it means that in
regions where the high residual swirl enters the draft tube, the flow is decelerated and convenient
conditions for the vortex rope development are created. Such flow conditions are considered to
be the triggering mechanism for occurrence of different forms of vortex structures in the Francis
turbine draft tube, e.g. spiral or straight vortex rope at part load or full load respectively.
Independently on the vortex rope shape the unsteady pressure fields develop producing periodic
stress on turbine components and possibly resulting in noise, blade cracks, runner lift, power
swing, etc. To study and mimic such flow conditions, a simplified device of vortex generator
apparatus is employed. Thanks to its design, the vortex generator enables to change the ratio
between fluxes of axial momentum and tangential moment of momentum of generated swirl.
Then, the behavior of vortex structure changes in a similar way as the flow rate variation
in the draft tube of Francis turbine. For above mentioned reasons the unsteady cavitating
spiral vortex is experimentally studied using both high speed video record and particle image
velocimetry (PIV). The main focus is on change of vortex dynamics regarding to the swirl
number variation. The proper orthogonal decomposition (POD) together with the classical fast
Fourier transformation (FFT) are employed to extract dominant modes and frequencies from
experimental data.
1. Introduction
The spiral vortex (special form of so called ”vortex rope” in hydro-turbine terminology) is one of
the main flow structures found in the draft tube (outlet diffuser) of Francis turbine operated at
part load. At part load the flow rate Qis lower than one at the best efficiency point QBEP and
the high residual swirl exits the turbine outlet and enters the draft tube where the remaining
kinetic energy is transformed to the static pressure. This decelerated swirling flow tends to be
unstable, thus the highly unsteady pressure field with the spiral vortex rotating around 25% of
the runner speed is formed. The resulting high pressure amplitudes strain the mechanical parts
of the turbine (e.g. blades, bearings) [1], cause power swing in electricity generation [2, 3, 4]
and produce significant mechanical noise. For this reason it is important to understand the
main source of dominant frequencies which could also cause resonance of some parts in hydraulic
system [5]. In order to study dynamical behavior of such vortex structure the simplified approach

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
2
consisting of swirling apparatus was employed in several researchers’ studies [6, 7, 8, 9, 10, 11]
and various spiral vortex behaviors were reported: transition from single to twin spiral vortex
[12, 13], self-oscillating vortex rope dynamic of synchronous character [14, 15] and finally the
vortex re-connection with formation of vortex rings [16, 17, 18, 19].
It is important to understand all these mentioned instabilities since the hydraulic turbines
are forced to operate far away from the optimal conditions in order to balance fluctuations in
electricity generation. For this reason the simplified device of swirl generator was used to study
changes in vortex dynamics regarding to the swirl number variation.
2. Swirl generator
The vortex generator presented in this study was developed with aim to change the ratio
between fluxes of axial momentum and tangential moment of momentum of generated swirl.
Such a geometry enables to adjust the swirl parameters while keeping the simple design for
manufacturing and operation (e.g. exclude any rotating parts). This was ensured by the inflow
separated to axial and tangential entrance. The tangential inflow Qtis mixed with axial one Qa
using the spiral geometry without any blade cascade. The swirl generator geometry is shown in
figure 1 where the red plane marks longitudinal cross-section in axial direction and blue plane
the mid-cross-section in the spiral. This swirl generator was already used in several numerical
[20, 21] and experimental studies [19].
Figure 1. Geometry of swirl generator. Figure 2. Test rig overview.
3. Measurements
Measurements were performed using the closed loop hydraulic circuit where the water is supplied
by the pump situated in the basement of laboratory. For both axial Qaand tangential Qtinflow
the static pressure level and flow rate were measured. The photo of test-rig with installed swirl
generator is shown in figure 2.
This paper presents results of two experimental campaigns. While the first one was
concentrated on visual observation of cavitating vortex using the high speed camera (HSC),
the second one employed stereoscopic particle image velocimetry (S-PIV) to obtain information
about velocity distribution in longitudinal planes situated within transparent diffuser. Assuming
the diffuser throat diameter Dt= 0.05 m as a reference length the Reynolds number for nominal
flow rates Qn= 10 l/s and Qn= 5 l/s is Re = 254647 and Re = 127323, respectively.
3.1. High speed camera recording
For the image recording of cavitating vortex the high speed camera Ximea CB120MG-CM-X8G3
was used. This camera, consisting of 12 MPx monochrome CMOSIS sensor, was equipped with
Canon EF 50 mm f/1.4 USM lens characterized by a very good aperture and focal length without
image distortion. For the scene illumination the 238 x 190 mm large LED lightening panel

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
3
Aputure Amaran HR consisting of 672 LED bulbs was used. In order to record vortex structure
filled with the water vapors, the so called back-light method was employed as suitable and
previously proved technique. According to the layout shown in figure 3 the camera, light source
and recorded object are situated inline. While the camera is heading towards the light source,
the recorded object is in the middle of the scene. In our case the diffuser was manufactured
from acrylic glass, thus it is fully transparent. The LED panel was placed just aside the diffuser
wall and camera was placed approximately in distance of 0.5 m. The final record was done with
aperture F2.5 and frame rate 1487.5 fps.
Figure 3. Layout of HSC recording. Figure 4. S-PIV calibration arrange-
ment for longitudinal target.
3.2. PIV measurements
The stereoscopic PIV (S-PIV) measurements were done employing the Pegasus PIV laser
Nd:YLF from New Wave Research and two CMOS NanoSense Mk III cameras from Dantec
Dynamic A/S. The laser has double head emitting coherent light with wavelength 527 nm by
maximal frequency equal to 10 kHz. The pulse energy is optimized for 1 kHz when its value
is 10 mJ (corresponding power is 10 W per head). Both cameras with resolution 1280 x 1024
pixels were equipped with Nikon lenses. As shown in figure 4 the multi-level calibration target
was designed and 3D printed exclusively for this application. The velocity vector maps with
129 x 110 vectors corresponds to space resolution approx. 1 mm. The water was saturated
by a borosilicate silver coated glass particles from Dantec. Diameter of particle is 10 µm with
spherical shape and smooth surface. For each measured flow ratio totally 1636 snapshots were
acquired with record length t= 3.27 s.
3.3. Measurements uncertainties
Several uncertainties were accounted for both PIV and high speed camera recording. Regarding
the PIV the positioning uncertainty of calibration target was lowered to minimum by using
the calibration target with outer dimensions equal to inner dimensions of diffuser cone. The
verticality of target placed in diffuser was checked by micro spirit level. The Scheinpflug lens
mounting was used for PIV cameras, however, the angle between lenses axes and the diffuser
outer surface was far from to be perpendicular, as recommended in similar situation [22]. This
configuration has been dictated by spatial restrictions. The uncertainty of flow rate plays
significant role since the ratio between axial and tangential inflow was set accordingly. The
flow rate uncertainty was quantified as combination of flow meter accuracy and actual measured
flow rate. For nominal flow rate Qn= 10 l/s the maximal uncertainty 0.05 l/s was estimated.
The pressure sensor behind test section was used to measure pressure level and consequently
evaluate Thoma cavitation number σas further described in equation (2). The measuring range
of this sensor is 0 −6 bars with uncertainty 0.00625 bars.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
4
4. Proper orthogonal decomposition
The proper orthogonal decomposition (POD) was introduced to the field of fluid mechanics in
1967 by Lumley [23] as a method for identification of coherent structures. POD can be used to
analyze experimental and numerical data that are applied to scalar or vector functions.
Let u=f(x, tk) be a variable dependent on spatial coordinates x= (x, y, z ) and discrete time
tk. Given a set of data can be expressed using POD as a set of orthogonal spatial basis functions
φj
i(x) (spatial modes) and temporal functions aj(tk) (temporal modes), where i= 1,2, ..., N (N
is number of grid points), k= 1,2, ..., M (Mis number of snapshots) and j= 1,2, ..., L (Lis
number of summed modes and L≤M). Accordingly, the approximation of the data set in the
first Lmodes can be written in terms of the spatial and temporal functions as follows:
ui(x, tk) =
L
X
i=1
φj
i(x)aj(tk) (1)
where L≤Mhas the largest mean square projection. In the method of snapshots, the general
N×Neigenvalue problem is reduced to an M×Meigenvalue problem. This solution enables
a substantial reduction in computational effort when the number of grid points Nsignificantly
exceeds the number of dataset snapshots M. The inner products of all pairs of the sampled
fields, i.e., the snapshots, are the temporal correlation matrix C(t′
k, tk) used as the kernel and
defined in discrete form as C(t′
k, tk) = M−1U(x, t′
k)U(x, tk), where U(x, tk) is the data matrix
where the columns are snapshots. On solving the eigenvalue problem of the form CA =λAwe
obtain the matrix of eigenvectors A, where ajis the jth column, and corresponding eigenvalues
λj. If uis velocity (obtained by PIV for example), we are minimizing the residual kinetic energy.
For this reason, POD is often referred to as the energetically optimal method. More details on
the computation of POD modes from discrete data might be found in [24, 25].
In this work POD was applied on measured flow fields by PIV to identify the main dynamics
features of generated swirling flow.
5. Results
Using the high speed camera the spatial shape of cavitating vortex was recorded for different flow
ratios. For each flow ratio four distinct snapshots are shown in figures 5 - 10. It is obvious that
the vortex shape is inconsistent in time and the spatial shape changes significantly with various
flow ratios. While at flow ratios with low tangential inflow (figures 5 - 7) the well developed
spiral vortex appears, at flow ratios with high tangential inflow (figures 9 - 10) the upper part
of vortex is formed by large amount of cavitation. Moreover in regimes close to flow ratio 50:50
(axial to tangential) it was observed that the vortex spiral grows and collapses periodically. Such
behavior is accompanied by creation of vortex ring and is documented in previous study [19].
In order to quantitatively describe the flow conditions with cavitating vortex, Thoma
cavitation number σwas calculated as:
σ=p−pv
1
2ρv2
0
(2)
where pis static pressure measured behind the diffuser section, pvis pressure of saturated
vapour, ρis water density and v0is mean velocity in diffuser throat area. Consequently, for
visualized cavitating vortex depicted in figures 5 - 10 average value of σ= 7.9.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
5
Figure 5. Cavitating vortex at 55:45 of axial
to tangential flow ratio, σ= 7.82, S= 0.21.
Figure 6. Cavitating vortex at 50:50 of axial
to tangential flow ratio, σ= 7.85, S= 0.26.
Figure 7. Cavitating vortex at 45:55 of axial
to tangential flow ratio, σ= 7.88, S= 0.32.
Figure 8. Cavitating vortex at 40:60 of axial
to tangential flow ratio, σ= 7.91, S= 0.38.
Figure 9. Cavitating vortex at 35:65 of axial
to tangential flow ratio, σ= 7.96, S= 0.45.
Figure 10. Cavitating vortex at 30:70 of
axial to tangential flow ratio, σ= 8.00, S=
0.52.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
6
The swirling flow magnitude as a ratio between axial to tangential inflow is determined by
the swirl number S. It is a dimensionless parameter commonly used to characterize a swirling
flow at high Reynolds numbers. When the critical swirl number is reached the vortex breakdown
occurs. The widely used swirl number is defined as S=´uwrdS
R´u2dS where uis the time-averaged
axial velocity, wis the time-averaged tangential velocity, ris the radial coordinate and Ris
reference radius. In our case the swirl number is evaluated on single line from planar PIV grid
(as shown in figure 11), thus the equation might be modified as
S=´R
0uwrdr
R´R
0u2dr (3)
This evaluation line is situated in throat of diffuser, close to the border of PIV grid and contains
only 39 points with valid PIV vectors. Together with relatively short record length of t= 3.27 s
the exact value of swirl number might not be precisely estimated. This was particularly case of
flow ratio 30:70 with nominal flow rate Q= 10 l/s. This regime was at the limit of cavitation
occurrence which could distort PIV record. That was reason why the larger portion of tangential
inflow was measured for Q= 5 l/s. As shown in figure 12 the swirl number gradually increases as
the portion of tangential inflow increases. The black dotted line denotes the trend line estimated
as a combination of both data sets.
Figure 11. Line for evaluation of swirl
number. Figure 12. Swirl number evolution.
Figure 13 shows contours of axial velocity vax for four different flow ratios. The red line
marks the border of backflow region where vax <0. It is obvious that the size of backflow region
increases and changes shape with variation of flow ratio. Comparing these results with figures
5 - 10 it is possible to conclude the influence on spatial shape of cavitating vortex. In order to
identify the region of largest velocity fluctuations the root mean square of velocity magnitude
was calculated as follows
RMSv=v
u
u
t
1
N
N
X
i=1
(vi−vavg )2(4)
where viis recorded sample of velocity magnitude, vavg is average from whole record. Results
are depicted in figure 14.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
7
Figure 13. Contours of axial velocity (red lines separate the backflow region).
While at flow regimes 60:40 and 50:50 (axial to tangential) the regions of largest velocity
fluctuations form the conical-like shape leading from the end of hub, at regimes with higher
portion of tangential inflow the largest velocity fluctuations are found in the outer periphery of
diffuser throat. This redistribution might point to the significant change in vortex spatial shape.
Figure 14. Contours of velocity magnitude root means square RM Sv.
5.1. Modal decomposition of PIV data
POD was primarily applied to the field of radial velocity vrad. It is known that even small
instability is well recognizable in a field of vrad and thus is suitable for such analysis. Figure
16 shows variation of radial velocity modes for four different flow ratios. Spatial shape in form
of contours plot, relevant modal frequency fand percentage contribution to the overall energy
spectra are presented for the first four dominant modes #1 −#4.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
8
The change in flow ratio brings the variation in spatial modes as well as change of modal
frequency and percentage contribution. While at flow ratio 60:40 (axial to tangential) the mode
#1 contributes to the overall energy spectra by 8.7 %, at flow ratio 30:70 (axial to tangential) it
is already more than three times higher (31.4%). As shown in figure 18 the frequency of mode
#1 increases linearly with increase of tangential inflow and denotes the dominant frequency of
generated vortex. One can see that frequency of mode #1 and #2 is identical and percentage
contribution is very similar. If we plot temporal mode a1against a2we will get so called phase
portrait as depicted in figure 15. Circular shape denotes π/2 phase shift between these two
modes #1 and #2, which means that those two POD modes create one Fourier mode in form
of traveling wave .
Figure 15. Phase portraits of temporal modes a1and a2
Flow ratio mode #1 mode #2 mode #3 mode #4
60ax:40tan
S = 0.16
f = 18.1 Hz f = 18.1 Hz f = 46.5 Hz f = 54.2 Hz
8.7 % 7.6 % 4.5 % 4.3 %
mode #1 mode #2 mode #3 mode #4
50ax:50tan
S = 0.26
f = 29.7 Hz f = 29.7 Hz f = 93 Hz f = 71 Hz
13 % 12.3 % 5.2 % 5 %
mode #1 mode #2 mode #3 mode #4
40ax:60tan
S = 0.38
f = 39.5 Hz f = 39.5 Hz f = 77.5 Hz f = 77.5 Hz
22.8 % 18.2 % 3.7 % 3.7 %
mode #1 mode #2 mode #3 mode #4
30ax:70tan
S = 0.52
f = 51 Hz f = 51 Hz f = 102 Hz f = 102 Hz
31.4 % 20 % 4.1 % 3.6 %
Figure 16. Variation of radial velocity modes

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
9
From spatial shapes, identical modal frequency, similar percentage contribution and phase
portraits we can conclude, that modes #1 and #2 create a complex Fourier mode [26]. These
modes appear especially in case that POD is applied on flow in axi-symmetric domain, e.g.
vortex rope in draft tube of hydraulic turbine [27, 28].
Different situation applies for modes #3 and #4. Their frequency sometimes varies and it is
not always dominant in frequency spectra as for modes #1 and #2. Percentage contribution is
also much smaller as shown in figure 17 and remains around same magnitude. It is important to
highlight that the spatial shape of modes #3 and #4 at flow ratios 40:60 and 30:70 is different
to all others pointing on some longitudinal wave propagation.
Finally, the instability of vortex breakdown starts to appear somewhere between flow ratio
80:20 and 70:30 (axial to tangential inflow). This might be well identified from figure 18 where
the frequency of mode #1 is plotted with lowest value at Qt/Q = 0.3.
Figure 17. Mode’s percentage contribution. Figure 18. Frequency of mode #1.
6. Conclusions
The unsteady spiral vortex structure was studied using simple device of swirl generator
apparatus. The spatial shape of generated vortex structure changes according to variation
of swirl number S. From video record of cavitating vortex it might be concluded, that type of
vortex breakdown is dependent on magnitude of swirl number S. While for interval 0.2< S < 0.4
the spiral form of vortex breakdown develops, for S > 0.4 the ”wasp-nest-like” shape with large
amount of cavitation region just under the hub and short spiral tail is formed. This fact is
reflected in modal decomposition of radial velocity field. At flow ratios 40:60 and 30:70 the
spatial shape of modes #3 and #4 reveal longitudinal wave with frequency twice higher than
for modes #1 and #2.
By swirl number estimation and POD analysis it was possible to identify the combination
of axial to tangential inflow when the vortex breakdown starts to develop. Such a ratio
is approximately 75:25 (axial to tangential inflow) with estimated swirl number S= 0.04.
Approximately at this flow ratio the frequency of mode 1 is becoming nonzero, thus the vortex
breakdown instability emerges.
In future work we will concentrate on CFD simulation employing scale resolving turbulence
model and presented experimental results will be used for verification.

30th IAHR Symposium on Hydraulic Machinery and Systems
IOP Conf. Series: Earth and Environmental Science 774 (2021) 012085
IOP Publishing
doi:10.1088/1755-1315/774/1/012085
10
Acknowledgments
The research has been supported by project “Computer Simulations for Effective Low-Emission
Energy” funded as project No. CZ.02.1.01/0.0/0.0/16 026/0008392 by Operational Programme
Research, Development and Education, Priority axis 1: Strengthening capacity for high-quality
research and by specific research project No. FSI-S-20-6235 of Brno University of Technology,
Faculty of Mechanical Engineering.
References
[1] Frunzˇaverde D, Muntean S, Mˇarginean G, Cˆampian V, Mar¸savina L, Terzi R and S¸erban V 2010 IOP
Conference Series: Earth and Environmental Science 12 1–10
[2] Rheingans W J 1940 Transactions of the ASME 62 1–14
[3] Silva P C O, Nicolet C, Grillot P, Drommi J L and Kawkabani B 2017 IEEE Transactions on Industry
Applications 53 3345–3354
[4] Valent´ın D, Presas A, Egusquiza E, Valero C, Egusquiza M and Bossio M 2017 Energies 10
[5] D¨orfler P, Sick M and Coutu A 2013 Flow-Induced Pulsation and Vibration in Hydroelectric Machinery:
Engineer’s Guidebook for Planning, Design and Troubleshooting (London: Springer London) pp 1–31
[6] Susan-Resiga R, Muntean S, Tanasa C and Bosioc A 2008 4th German—Romanian Workshop on
Turbomachinery Hydrodynamics (Stuttgart, Germany) pp 1 – 16
[7] Gramlich M 2012 Numerical Investigations of the Unsteady Flow in the Stuttgart Swirl Generator with
OpenFOAM Master’s thesis Chalmers University of Technology Gothenburg, Sweden
[8] Bosioc A I, Susan-Resiga R, Muntean S and Tanasa C 2012 J. Fluids Eng. 134 11
[9] ˇ
Stefan D, Rudolf P, Muntean S and Susan-Resiga R 2013 Engineering MECHANICS 20 339–353
[10] Javadi A and Nilsson H 2017 Engineering Applications of Computational Fluid Mechanics 11 30–41
[11] Litvinov I, Shtork S, Gorelikov E, Mitryakov A and Hanjalic K 2018 Experimental Thermal and Fluid Science
91 410 – 422
[12] Tsoy M A, Skripkin S G and Shtork S I 2015 2015 5th International Youth Conference on Energy (IYCE)
pp 1–5
[13] Skripkin S, Tsoy M, Shtork S and Hanjali´c K 2016 Journal of Hydraulic Research 54 450–460
[14] M¨uller A, Yamamoto K, Allign´e S, Yonezawa K, Tsujimoto Y and F F A 2015 J. Fluids Eng. 138 8
[15] Stuparu A and Susan-Resiga R 2015 6th IAHR International Meeting of the Workgroup on Cavitation and
Dynamic Problems in Hydraulic Machinery and Systems (Ljubljana, Slovenia) p 8
[16] Alekseenko S V, Kuibin P A, Shtork S I, Skripkin S G, Sonin V I, Tsoy M A and Ustimenko A S 2016 IOP
Conference Series: Earth and Environmental Science 49 1–7
[17] Skripkin S G, Tsoy M A, Kuibin P A and Shtork S I 2017 Journal of Fluids Engineering 139
[18] Skripkin S, Tsoy M, Kuibin P and Shtork S 2019 Experimental Thermal and Fluid Science 100 349 – 359
[19] ˇ
Stefan D, Rudolf P, Hudec M, Uruba V, Proch´azka P and Urban O 2019 IOP Conference Series: Earth and
Environmental Science 405 012033
[20] Pochyl´y F, Urban O and Fialov´a S 2019 IOP Conference Series: Earth and Environmental Science 240
072026
[21] Urban O and Rudolf P 2019 IOP Conference Series: Earth and Environmental Science 405 012034
[22] Raffel M, Willert C E, Scarano F, K¨ahler C, Wereley S T and Kompenhans J 2018 Particle Image Velocimetry,
A Practical Guide (Springer International Publishing) p 669
[23] Lumley J L 1967 Atmospheric Turbulence and Wave Propagation ed Yaglom A M and Tatarski V I (Moscow:
Nauka) pp 166–178
[24] Berkooz G, Holmes P and Lumley J L 1993 Annual Review of Fluid Mechanics 25 539–575
[25] ˇ
Stefan D, Rudolf P, Muntean S and Susan-Resiga R 2017 Journal of Fluids Engineering 139 ISSN 0098-2202
[26] Citriniti J H and George W K 2000 Journal of Fluid Mechanics 418 137–166
[27] Rudolf P and ˇ
Stefan D 2012 IOP Conference Series: Earth and Environmental Science 15 062008
[28] Stefan D and Rudolf P 2015 Journal of Physics: Conference Series 579 012002