We present a novel method for the determination of N-factors in cross-flow transition scenarios. The method considers numerical simulations, in which a turbulent model is applied downstream from a predetermined point and solves a laminar flow upstream from this point. The solution is postprocessed using higher order dynamic mode decomposition to extract the leading spatial mode in several small sections along the streamwise direction. The spatial evolution of the amplitude of this mode will determine the N-factor. The results presented are compared with experimental measurements and linear stability theory, showing the good performance of this novel method, which does not assume parallel flow assumptions, is automatic and computationally efficient.

1.
D.
Arnal
and
G.
Casalis
, “
Laminar-turbulent transition prediction in three-dimensional flows
,”
Prog. Aerosp. Sci.
36
,
173
191
(
2000
).
2.
H.
Bippes
, “
Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability
,”
Prog. Aerosp. Sci.
35
,
363
412
(
1999
).
3.
V. I.
Borodulin
,
A. V.
Ivanov
, and
Y. S.
Kachanova
, “
Swept-wing boundary-layer transition at various external perturbations: Scenarios, criteria, and problems of prediction
,”
Phys. Fluids
29
(
9
),
094101
(
2017
).
4.
R.
Bose
and
P. A.
Durbin
, “
Transition to turbulence by interaction of free-stream and discrete mode perturbations
,”
Phys. Fluids
28
(
11
),
114105
(
2016
).
5.
L. N.
Cattafesta
 III
and
M.
Sheplak
, “
Actuators for active flow control
,”
Annu. Rev. Fluid Mech.
43
(
1
),
247
272
(
2011
).
6.
F. W.
Boltz
,
G. C.
Kenyon
, and
C. Q.
Allen
, “
Boundary-layer stability characteristics of an untapered wing at low speeds
,” Report No. NASA-TN-D-338,
NASA Ames Research Center
,
Moffett Field, California
,
1960
.
7.
L.
De Lathawer
,
B.
De Moor
, and
J.
Vandewalle
, “
A multilinear singular value decomposition
,”
SIAM J. Matrix Anal. Appl.
21
,
1253
1278
(
2000
).
8.
U.
Ehrenstein
and
F.
Gallaire
, “
On two-dimensional temporal modes in spatially evolving open flows: The flat-plate boundary layer
,”
J. Fluid Mech.
536
,
209
218
(
2005
).
9.
A.
Fedorov
, “
Transition and stability of high-speed boundary layers
,”
Annu. Rev. Fluid Mech.
43
(
1
),
79
95
(
2011
).
10.
E.
Ferrer
,
J.
DeVicente
, and
E.
Valero
, “
Low cost three-dimensional global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools
,”
Int. J. Numer. Methods Fluids
76
(
3
),
169
184
(
2014
).
11.
L. M.
Gonzalez
,
J. M.
Gimenez
, and
E.
Ferrer
, “
Instability onset for a submerged cylinder
,”
Phys. Fluids
31
(
1
),
014106
(
2019
).
12.
L. M.
Gonzalez
,
E.
Ferrer
, and
H. R.
Diaz-Ojeda
, “
Onset of three dimensional flow instabilities in lid-driven circular cavities
,”
Phys. Fluids
29
,
064102
(
2017
).
13.
Z. H.
Han
,
J.
Chen
,
K. S.
Zhang
,
Z. M.
Xu
,
Z.
Zhu
, and
W. P.
Song
, “
Aerodynamic shape optimization of natural-laminar-flow wing using surrogate-based approach
,”
AIAA J.
56
(
7
),
2579
2593
(
2018
).
14.
Z. H.
Han
,
F.
He
,
W. P.
Song
, and
Z. D.
Qiao
, “
A preconditioned multigrid method for efficient simulation of three dimensional compressible and incompressible flows
,”
Chin. J. Aeronaut.
20
(
4
),
289
296
(
2007
).
15.
Z. H.
Han
,
C. Z.
Xu
,
L.
Zhang
,
Y.
Zhang
,
K. S.
Zhang
, and
W. P.
Song
, “
Efficient aerodynamic shape optimization using variable-fidelity surrogate models and multilevel computational grids
,”
Chin. J. Aeronaut.
(
in press
).
16.
T.
Herbert
and
N.
Lin
, “
Studies on boundary-layer receptivity with parabolized stability equations
,” AIAA Paper No. 93–3053,
1993
.
17.
T.
Herbert
, “
Parabolized stability equations
,”
Annu. Rev. Fluid. Mech.
29
,
245
283
(
1997
).
18.
W.
Koch
, “
On a degeneracy of temporal secondary instability modes in Blasius boundary-layer flow
,”
J. Fluid Mech.
243
,
319
351
(
1992
).
19.
Y.
Kohama
,
T.
Onodera
, and
Y.
Egami
, “
A high-frequency, secondary instability of crossflow vortices that leads to transition
,” in
Proceedings of Royal Aerospace Society Conference Boundary-Layer Transition Control Conference
(
Peterhouse College
,
Cambridge, UK
,
1991
), pp.
4.1
4.13
.
20.
T. G.
Kolda
and
B. W.
Bader
, “
Tensor decompositions and applications
,”
SIAM Rev.
51
,
455
500
(
2009
).
21.
S.
Le Clainche
and
E.
Ferrer
, “
Reduced order model to predict transient flows around straight bladed vertical axis wind turbines
,”
Energies
11
(
3
),
566
(
2018
).
22.
S.
Le Clainche
,
J. M.
Pérez
, and
J. M.
Vega
, “
Spatio-temporal flow structures in the three-dimensional wake of a circular cylinder
,”
Fluid Dyn. Res.
50
,
051406
(
2018
).
23.
S.
Le Clainche
,
D.
Rodríguez
,
V.
Theofilis
, and
J.
Soria
, “
Flow around a hemisphere-cylinder at high angle of attack and low Reynolds number. Part II: POD and DMD applied to reduced domains
,”
Aerosp. Sci. Technol.
44
,
88
100
(
2015
).
24.
S.
Le Clainche
,
F.
Sastre
,
J. M.
Vega
, and
A.
Velazquez
, “
Higher order dynamic mode decomposition applied to study flow structures in noisy PIV experimental data
,” AIAA Paper 2017-3304,
2017
.
25.
S.
Le Clainche
and
J. M.
Vega
, “
Higher order dynamic mode decomposition
,”
SIAM J. Appl. Dyn. Syst.
16
(
2
),
882
925
(
2017
).
26.
S.
Le Clainche
and
J. M.
Vega
, “
Higher order dynamic mode decomposition to identify and extrapolate flow patterns
,”
Phys. Fluids
29
(
8
),
084102
(
2017
).
27.
S.
Le Clainche
,
J. M.
Vega
, and
J.
Soria
, “
Higher order dynamic mode decomposition of noisy experimental data: The flow structure of a zero-net-mass-flux jet
,”
Exp. Therm. Fluid Sci.
88
,
336
353
(
2017
).
28.
S.
Le Clainche
,
M. M.
Wu
,
Z. H.
Han
, and
E.
Ferrer
, “
An alternative method to calculate cross-flow instabilities
,” AIAA Paper 2018-3700,
2018
.
29.
M. R.
Malik
and
F.
Li
, “
Secondary instability of Gortler and crossflow vortices
,” in
Proceedings of International Symposium Aerospace & Fluid Science
(
Tohoku University
,
Sendai, Japan
,
1993
), pp.
460
477
.
30.
M. R.
Malik
,
F.
Li
,
M. M.
Choudhari
, and
C. L.
Chang
, “
Secondary instability of crossflow vortices and swept-wing boundary layer transition
,”
J. Fluid Mech.
399
,
85
115
(
1999
).
31.
D. I. A.
Poll
, “
Some observations of the transition process on the windward face of a long yawed cylinder
,”
J. Fluid Mech.
150
,
329
356
(
1985
).
32.
H.
Raposo
,
S.
Mughal
, and
R.
Ashworth
, “
Acoustic receptivity and transition modeling of Tollmien-Schlichting disturbances induced by distributed surface roughness
,”
Phys. Fluids
30
(
4
),
044105
(
2018
).
33.
H. L.
Reed
,
E.
Perez
,
J. J.
Kuehl
,
T.
Kocian
, and
N.
Oliviero
, “
Verification and validation issues in hypersonic stability and transition prediction
,”
J. Spacecr. Rockets
52
(
1
),
29
37
(
2015
).
34.
W. S.
Saric
,
H. L.
Reed
, and
E. B.
White
, “
Stability and transition of three-dimensional boundary layers
,”
Annu. Rev. Fluid. Mech.
35
,
413
440
(
2003
).
35.
W. S.
Saric
,
A. L.
Carpenter
, and
H. L.
Reed
, “
Passive control of transition in three-dimensional boundary layers, with emphasis on discrete roughness elements
,”
Philos. Trans. R. Soc., A
369
,
1352
1364
(
2011
).
36.
H.
Schlichting
and
K.
Gersten
, “
Fundamentals of turbulent flows
,” in
Boundary-Layer Theory
, 8th revised and enlarged edition (
Springer
,
Berlin
,
2000
), pp.
499
516
.
37.
L.
Sirovich
, “
Turbulence and the dynamics of coherent structures. Parts I–III
,”
Q. Appl. Math.
45
(
3
),
561
571
(
1987
).
38.
H.
Schlichting
, “
Zur entstehung der turbulenz bei der plattenstromung
,” in
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen
, Mathematisch-Physikalische Klasse (
Niedersäechsische Staats- und Universitäetsbibliothek
,
Gottingen
,
1933
), pp.
181
208
.
39.
P. J.
Schmid
, “
Dynamic mode decomposition of numerical and experimental data
,”
J. Fluid Mech.
656
,
5
28
(
2010
).
40.
L.
Schrader
,
S.
Amin
, and
L.
Brandt
, “
Transition to turbulence in the boundary layer over a smooth and rough swept plate exposed to free-stream turbulence
,”
J. Fluid Mech.
646
,
297
325
(
2010
).
41.
L.
Schrader
,
L.
Brandt
, and
D.
Henningson
, “
Receptivity mechanisms in three-dimensional boundary-layer flows
,”
J. Fluid Mech.
618
,
209
241
(
2009
).
42.
P. K.
Sharma
and
T. K.
Sengupta
, “
Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation
,”
Phys. Fluids
31
(
6
),
064107
(
2019
).
43.
P.
Sundaram
,
T. K.
Sengupta
, and
S.
Sengupta
, “
Is Tollmien-Schlichting wave necessary for transition of zero pressure gradient boundary layer flow
,”
Phys. Fluids
31
(
3
),
031701
(
2019
).
44.
D.
Tempelmann
,
L. U.
Schrader
,
A.
Hanifi
,
L.
Brandt
, and
D. S.
Henningson
, “
Swept wing boundary-layer receptivity to localised surface roughness
,”
J. Fluid Mech.
711
,
516
544
(
2012
).
45.
W.
Tollmien
, “
Uber die entstehung der turbulenz
,” in
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen
, Mathematisch-Physikalische Klasse (
Niedersäechsische Staats- und Universitäetsbibliothek
,
Gottingen
,
1929
), pp.
21
44
.
46.
L. R.
Tucker
, “
Some mathematical notes on three-mode factor analysis
,”
Psikometrica
31
,
279
311
(
1996
).
47.
M.
Wang
,
T.
Herbert
, and
G. K.
Stuckert
, “
PSE analysis of receptivity and stability in swept wing flows
,” AIAA Paper No. 94-0180,
1994
.
48.
M.
Wu
,
Z.
Han
,
H.
Nie
,
W.
Song
,
S.
Le Clainche
, and
E.
Ferrer
, “
A transition prediction method for flows over airfoils based on dynamic mode decomposition
,”
Chin. J. Aeronaut.
(
in press
).
49.
D.
Xiao
and
G.
Papadakis
, “
Nonlinear optimal control of bypass transition in a boundary layer flow method
,”
Phys. Fluids
29
(
5
),
054103
(
2017
).
50.
F.
Xie
,
W. P.
Song
, and
Z. H.
Han
, “
Numerical study of high-resolution scheme based on preconditioning method
,”
J. Aircraft
46
(
2
),
520
525
(
2009
).
51.
K. S.
Zhang
,
Z. H.
Han
,
Z. J.
Gao
, and
Y.
Wang
, “
Constraint aggregation for large number of constraints in wing surrogate-based optimization
,”
Struct. Multidiscip. Optim.
59
(
2
),
421
438
(
2019
).
You do not currently have access to this content.