The mitigation of the precessing vortex core developing in the draft tube of Francis turbines operating under part load conditions is crucial to increase the operation flexibility of these hydraulic machines to balance the massive power production of intermittent energy sources. A systematic approach following the optimal control theory is, therefore, presented to control this vortical flow structure. Modal analysis characterizes the part load vortex rope as a self-sustained instability associated with an unstable eigenmode. Based on this physical characteristic, an objective function targeting a zero value of the unstable eigenvalue growth rate is defined and subsequently minimized using an adjoint-based optimization algorithm. We determine an optimal force distribution that successfully quenches the part load vortex rope and sketches the design of a realistic control appendage.

References

References
1.
Trivedi
,
C.
,
Gandhi
,
B.
, and
Cervantes
,
J. M.
,
2013
, “
Effect of Transients on Francis Turbine Runner Life: A Review
,”
J. Hydraul. Res.
,
51
(
2
), pp.
121
132
.
2.
Alonso
,
M.
, and
Amaris
,
H.
,
2011
, “
Impact of Wind Farms in Power Networks
,”
Wind Farm
, IntechOpen, Rijeka, Croatia.
3.
Brush
,
M.
,
Münch
,
V.
,
Aichinger
,
M.
,
Khuhn
,
M.
,
Weymann
,
M.
, and
Schmid
,
G.
,
2011
, “
Power Blackout Risks
,” Emerging Risk Initiative, Position Paper, CRO Forum, Nov., pp. 1–32.
4.
Kim
,
J.
, and
Bewley
,
T. R.
,
2007
, “
A Linear Systems Approach to Flow Control
,”
Annu. Rev. Fluid Mech.
,
39
(
1
), pp.
383
417
.
5.
Nishi
,
M.
,
Wang
,
X. M.
,
Yoshida
,
K.
,
Takahashi
,
T.
, and
Tsukamoto
,
T.
,
1996
,
An Experimental Study on Fins, Their Role in Control of the Draft Tube Surging
,
Springer
,
Dordrecht, The Netherlands
, pp.
905
914
.
6.
Kurokawa
,
J.
,
Junichi
,
I.
, and
Young-Do
,
C.
,
2010
, “
Effect of j-Groove on the Suppression of Swirl Flow in a Conical Diffuser
,”
ASME J. Fluids Eng.
,
132
(
7
), p.
071101
.
7.
Brekke
,
H.
,
2003
, “
A Review of Some Dynamic Problems in Hydropower Plants
,”
11th International Meeting of IAHR Work Group on the Behaviours of Hydraulic Machinery Under Steady Oscillatory Conditions
, Stuttgart, Germany, Oct. 8–10.
8.
Qian
,
Z. D.
,
Li
,
W.
,
Huai
,
W. X.
, and
Wu
,
Y. L.
,
2012
, “
The Effect of Runner Cone Design on Pressure Oscillation Characteristics in a Francis Hydraulic Turbine
,”
Proc. Inst. Mech. Eng., Part A
,
226
(
1
), pp.
137
150
.
9.
Blommaert
,
G.
,
Prenat
,
J. E.
,
Avellan
,
F.
, and
Boyer
,
A.
, 1999, “
Active Control of Francis Turbine Operation Stability
,”
ASME
Paper No. FEDSM99-72103.https://www.researchgate.net/publication/37424262_Active_control_of_Francis_turbine_operation_stability
10.
Qian
,
Z.
,
Yang
,
J.
, and
Huai
,
W.
,
2007
, “
Numerical Simulation and Analysis of Pressure Pulsation in Francis Hydraulic Turbine With Air Admission
,”
J. Hydrodyn., Ser. B
,
19
(
4
), pp.
467
472
.
11.
Luo
,
X.
,
Yu
,
A.
,
Ji
,
B.
,
Wu
,
Y.
, and
Tsujimoto
,
Y.
,
2017
, “
Unsteady Vortical Flow Simulation in a Francis Turbine With Special Emphasis on Vortex Rope Behavior and Pressure Fluctuation Alleviation
,”
Proc. Inst. Mech. Eng., Part A
,
231
(
3
), pp.
215
226
.
12.
Susan-Resiga
,
R.
,
Vu
,
T. C.
,
Muntean
,
S.
,
Ciocan
,
G. D.
, and
Nennemann
,
B.
,
2006
, “
Jet Control of the Draft Tube Vortex Rope in Francis Turbines at Partial Discharge
,” 23rd IAHR Symposium on Hydraulic Machinery and Systems (IAHR), Yokohama, Japan, Oct. 17–21, Paper No.
F192
.https://www.researchgate.net/publication/228370287_Jet_control_of_the_draft_tube_vortex_rope_in_Francis_turbines_at_partial_discharge
13.
Goyal
,
R.
,
Gandhi
,
B.
, and
Cervantes
,
M.
,
2017
, “
Experimental Study of Mitigation of a Spiral Vortex Breakdown at High Reynolds Number Under an Adverse Pressure Gradient
,”
Phys. Fluids
,
29
(
10
), p.
104104
.
14.
Zhang
,
R.
,
Mao
,
F.
,
Wu
,
J.-Z.
,
Chen
,
S.-Y.
,
Wu
,
Y.-L.
, and
Liu
,
S.-H.
,
2009
, “
Characteristics and Control of the Draft-Tube Flow in Part-Load Francis Turbine
,”
ASME J. Fluids Eng.
,
131
(
2
), p.
021101
.
15.
Bosioc
,
A. I.
,
Susan-Resiga
,
R.
,
Muntean
,
S.
, and
Tǎnasǎ
,
C.
,
2012
, “
Unsteady Pressure Analysis of a Swirling Flow With Vortex Rope and Axial Water Injection in a Discharge Cone
,”
ASME J. Fluids Eng.
,
134
(
8
), p.
081104
.
16.
Tǎnasǎ
,
C.
,
Susan-Resiga
,
R.
,
Sebastian
,
M.
, and
Bosioc
,
A. I.
,
2013
, “
Flow-Feedback Method for Mitigating the Vortex Rope in Decelerated Swirling Flows
,”
ASME J. Appl. Mech.
,
135
(
6
), p.
061304
.
17.
Foroutan
,
H.
, and
Yavuzkurt
,
S.
,
2014
, “
Flow in the Simplified Draft Tube of a Francis Turbine Operating at Partial Load—Part I: Simulation of the Vortex Rope
,”
ASME J. Appl. Mech.
,
81
(
6
), p.
061010
.
18.
Joslin
,
R.
,
Gunzburger
,
M.
,
Nicolaides
,
R.
,
Erlebacher
,
G.
, and
Hussaini
,
M.
,
1997
, “
Self-Contained Automated Methodology for Optimal Flow Control
,”
AIAA J.
,
35
(
5
), pp.
816
824
.
19.
Gunzburger
,
M. D.
,
1999
, “
Sensitivities, Adjoints and Flow Optimization
,”
Int. J. Numer. Methods Fluids
,
31
(
1
), pp.
53
78
.
20.
Bewley
,
T. R.
,
Moin
,
P.
, and
Temam
,
R.
,
2001
, “
DNS-Based Predictive Control of Turbulence: An Optimal Benchmark for Feedback Algorithms
,”
J. Fluid Mech.
,
447
, pp.
179
225
.
21.
Passaggia
,
P.-Y.
, and
Ehrenstein
,
U.
,
2013
, “
Adjoint Based Optimization and Control of a Separated Boundary-Layer Flow
,”
Eur. J. Mech.–B/Fluids
,
41
, pp.
169
177
.
22.
Goit
,
J. P.
, and
Meyers
,
J.
,
2015
, “
Optimal Control of Energy Extraction in Wind-Farm Boundary Layers
,”
J. Fluid Mech.
,
768
, pp.
5
50
.
23.
Giannetti
,
F.
, and
Luchini
,
P.
,
2007
, “
Structural Sensitivity of the First Instability of the Cylinder Wake
,”
J. Fluid Mech.
,
581
, pp.
167
197
.
24.
Marquet
,
O.
,
Sipp
,
D.
, and
Jacquin
,
L.
,
2008
, “
Sensitivity Analysis and Passive Control of Cylinder Flow
,”
J. Fluid Mech.
,
615
, pp.
221
252
.
25.
Qadri
,
U. A.
,
Mistry
,
D.
, and
Juniper
,
M. P.
,
2013
, “
Structural Sensitivity of Spiral Vortex Breakdown
,”
J. Fluid Mech.
,
720
, pp.
558
581
.
26.
Camarri
,
S.
, and
Iollo
,
A.
,
2010
, “
Feedback Control of the Vortex-Shedding Instability Based on Sensitivity Analysis
,”
Phys. Fluids
,
22
(
9
), p.
094102
.
27.
Pasche
,
S.
,
Gallaire
,
F.
, and
Avellan
,
F.
,
2018
, “
Predictive Control of Spiral Vortex Breakdown
,”
J. Fluid Mech.
,
842
, pp.
58
86
.
28.
Barkley
,
D.
,
2006
, “
Linear Analysis of the Cylinder Wake Mean Flow
,”
Europhys. Lett.
,
75
(
5
), p.
750
.
29.
Pasche
,
S.
,
Avellan
,
F.
, and
Gallaire
,
F.
,
2017
, “
Part Load Vortex Rope as a Global Unstable Mode
,”
ASME J. Fluids Eng.
,
139
(
5
), p.
051102
.
30.
Avellan
,
F.
,
2000
, “
Flow Investigation in a Francis Draft Tube: The FLINDT Project
,” 20th IAHR Symposium on Hydraulic Machinery and Systems (IAHR), Charlotte, NC, Aug. 6–9, Paper No.
DY-03
.https://www.researchgate.net/publication/37424288_Flow_Investigation_in_a_Francis_Draft_Tube_the_Flindt_Project
31.
Arpe
,
J.
, and
Avellan
,
F.
,
2002
, “
Pressure Wall Measurements in the Whole Draft Tube: Steady and Unsteady Analysis
,” Proceedings of the 21st IAHR Symposium on Hydraulic Machinery and Systems (
IAHR
), Lausanne, Switzerland, Sept. 9–12, pp. 593–602.https://www.researchgate.net/publication/37424322_Pressure_Wall_Measurements_in_the_whole_Draft_Tube_Steady_and_Unsteady_Analysis
32.
Moody
L. F.
,
1932
, “Moody,” U.S. Patent No.
1,848,738
.https://www.google.com/patents/US1848738
33.
Arpe
,
J.
,
Nicolet
,
C.
, and
Avellan
,
F.
,
2009
, “
Experimental Evidence of Hydroacoustic Pressure Waves in a Francis Turbine Elbow Draft Tube for Low Discharge Conditions
,”
ASME J. Fluids Eng.
,
131
(
8
), p.
081102
.
34.
Viola
,
F.
,
Iungo
,
G. V.
,
Camarri
,
S.
,
Porté-Agel
,
F.
, and
Gallaire
,
F.
,
2014
, “
Prediction of the Hub Vortex Instability in a Wind Turbine Wake: Stability Analysis With Eddy-Viscosity Models Calibrated on Wind Tunnel Data
,”
J. Fluid Mech.
,
750
, R1.
35.
Sipp
,
D.
, and
Lebedev
,
A.
,
2007
, “
Global Stability of Base and Mean Flows: A General Approach and Its Applications to Cylinder and Open Cavity Flows
,”
J. Fluid Mech.
,
593
, pp.
333
358
.
36.
Mantič-Lugo
,
V.
,
Arratia
,
C.
, and
Gallaire
,
F.
,
2014
, “
Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake
,”
Phys. Rev. Lett.
,
113
(
8
), p.
084501
.
37.
Mauri
,
S.
,
2002
,
Numerical Simulation and Flow Analysis of an Elbow Diffuser
, EPFL/LMH, Lausanne, Switzerland, Thèse EPFL No. 2527 3.
38.
Ciocan
,
G.
,
Iliescu
,
M.
,
Vu
,
T.
,
Nenneman
,
B.
, and
Avellan
,
F.
,
2007
, “
Experimental Study and Numerical Simulation of the FLINDT Draft Tube Rotating Vortex
,”
ASME J. Fluids Eng.
,
129
(
2
), pp.
146
158
.
39.
Trivedi
,
C.
,
Cervantes
,
M. J.
, and
Dahlhaug
,
O. G.
,
2016
, “
Numerical Techniques Applied to Hydraulic Turbines: A Perspective Review
,”
ASME Appl. Mech. Rev.
,
68
(
1
), p.
010802
.
40.
ANSYS
,
2015
, “
Ansys 16.0 Release Documentation, Theory and Modelling Guide
,” ANSYS, Inc., Canonsburg, PA.
41.
Barth
,
T. J.
, and
Jespersen
,
D. C.
,
1989
, “
The Design and Application of Upwind Schemes on Unstructured Meshes
,”
AIAA
Paper No. 1989-366.
42.
Polak
,
E.
,
1997
,
Optimization Algorithms and Consistent Approximations
,
Springer-Verlag
,
New York
.
43.
Hecht
,
F.
,
2012
, “
New Development in Freefem FreeFem++
,”
J. Numer. Math.
,
20
(
3–4
), pp.
251
265
.https://pdfs.semanticscholar.org/549b/bc057248bfbc67b15d65d8a7f22432ed7c43.pdf
44.
Lehoucq
,
R. B.
,
Sorensen
,
D. C.
, and
Yang
,
C.
,
1998
,
ARPACK Users' Guide
(Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods),
SIAM
, Philadelphia, PA, pp. 1–7.
45.
Turton
,
S. E.
,
Tuckerman
,
L.
, and
Barkley
,
D.
,
2015
, “
Prediction of Frequencies in Thermosolutal Convection From Mean Flows
,”
Phys. Rev. E
,
91
(
4
), p.
043009
.
46.
Susan-Resiga
,
R.
,
Ciocan
,
G.
,
Anton
,
I.
, and
Avellan
,
F.
,
2006
, “
Analysis of the Swirling Flow Downstream a Francis Turbine Runner
,”
ASME J. Fluids Eng.
,
128
(
1
), pp.
177
189
.
47.
Nishi
,
M.
,
Kubota
,
T.
,
Matsunaga
,
S.
, and
Senoo
,
Y.
,
1982
, “
Flow Regimes in Elbow-Type Draft Tube
,” 11th IAHR Symposium on Hydraulic Machinery and Cavitation, Amsterdam, The Netherlands, Sept. 13–17.
48.
Mohammadi
,
B.
, and
Pironneau
,
O.
,
2004
, “
Shape Optimization in Fluid Mechanics
,”
Annu. Rev. Fluid Mech.
,
36
(
1
), pp.
255
279
.
49.
Tǎnasǎ
,
C.
,
Ciocan
,
T.
, and
Muntean
,
S.
,
2017
, “
Numerical Assessment of Pulsating Water Jet in the Conical Diffusers
,”
AIP Conf. Proc.
,
1906
, p.
050002
.
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