A combinatorial technique merging image segmentation via K-means clustering and colormap of the barycentric triangle is used to investigate the Reynolds stress anisotropy tensor. The clustering aids in extracting the identical features from the spatial distribution of the anisotropy colormap images by minimizing the sum of squared error between the cluster center and all data points. The dataset used to investigate the applicability of the clustering technique consists of the flow in a large wind farm for different thermal stratification representatives of a characteristic diurnal cycle. Based on the attribute values defining the colormap of the Reynolds anisotropy stress tensor, the images are converted into color space and then the K-means algorithm assesses the similarities and dissimilarities via a distance metric. In unsupervised learning problems, the K-means algorithm runs independently for different numbers of clusters. The elbow criterion is used to determine the best trade-off between the cluster number and the total variance to select the optimal number of clusters. The clustering technique improves pattern visualization and allows us to identify characteristic regions of the flow based on the structure of the Reynolds stress anisotropy. The dominant patterns reveal that there are major perturbations that control the operation of the wind farm during the diurnal cycle, including the formation and growth of the convective boundary layer and the strong stratification among the flow layers during the stably-stratified period. These parameters attempt to redistribute energy into the velocity deficit region and contribute to the energy balance in the flow domain through the distributions of the momentum flux. As a result of the weak mixing and negligible buoyancy effect, the neutral wind farm displays gradual changes from a prolate turbulence state near the rotor to an oblate turbulence state at the top of the domain.

1.
N.
Ali
,
N.
Hamilton
,
G.
Cortina
,
M.
Calaf
, and
R. B.
Cal
, “
Anisotropy stress invariants of thermally stratified wind turbine array boundary layers using large eddy simulations
,”
J. Renewable Sustainable Energy
10
(
1
),
013301
(
2018
).
2.
R. A.
Antonia
,
J.
Kim
, and
L.
Browne
, “
Some characteristics of small-scale turbulence in a turbulent duct flow
,”
J. Fluid Mech.
233
,
369
388
(
1991
).
3.
P.
Krogstad
and
L. E.
Torbergsen
, “
Invariant analysis of turbulent pipe flow
,”
Flow, Turbul. Combust.
64
,
161
181
(
2000
).
4.
R.
Gómez-Elvira
,
A.
Crespo
,
E.
Migoya
,
F.
Manuel
, and
J.
Hernández
, “
Anisotropy of turbulence in wind turbine wakes
,”
J. Wind Eng. Ind. Aerodyn.
93
,
797
814
(
2005
).
5.
C.
Klipp
, “
Near-surface anisotropic turbulence
,”
Proc. SPIE
7685
,
768505
(
2010
).
6.
A.
Jimenez
,
A.
Crespo
,
E.
Migoya
, and
J.
Garcia
, “
Advances in large-eddy simulation of a wind turbine wake
,”
J. Phys.: Conf. Ser.
75
,
012041
(
2007
).
7.
W. D.
Smyth
and
J. N.
Moum
, “
Anisotropy of turbulence in stably stratified mixing layers
,”
Phys. Fluids
12
,
1343
1362
(
2000
).
8.
N.
Ali
,
N.
Hamilton
,
D.
DeLucia
, and
R.
Bayoán Cal
, “
Assessing spacing impact on coherent features in a wind turbine array boundary layer
,”
Wind Energy Sci.
3
(
1
),
43
56
(
2018
).
9.
J.
Rotta
, “
Statistical theory of nonhomogeneous turbulence
,”
Z. Phys.
131
,
51
77
(1951).
10.
K.
Choi
and
J. L.
Lumley
, “
The return to isotropy of homogeneous turbulence
,”
J. Fluid Mech.
436
,
59
84
(
2001
).
11.
K.
Liu
and
R. H.
Pletcher
, “
Anisotropy of a turbulent boundary layer
,”
J. Turbul.
9
,
N18
(
2008
).
12.
N.
Hamilton
,
M.
Tutkun
, and
R. B.
Cal
, “
Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition
,”
Phys. Rev. Fluids
2
(
1
),
014601
(
2017
).
13.
Z.-T.
Xie
,
O.
Coceal
, and
I. P.
Castro
, “
Large-eddy simulation of flows over random urban-like obstacles
,”
Boundary-Layer Meteorol.
129
(
1
),
1
23
(
2008
).
14.
D. A.
Philips
, “
Modeling scalar dispersion in urban environments
,” Ph.D. thesis (
Stanford University
,
2012
).
15.
E.
Bou-Zeid
,
X.
Gao
,
C.
Ansorge
, and
G. G.
Katul
, “
On the role of return to isotropy in wall-bounded turbulent flows with buoyancy
,”
J. Fluid Mech.
856
,
61
78
(
2018
).
16.
P.
Brugger
,
G. G.
Katul
,
F.
De Roo
,
K.
Kröniger
,
E.
Rotenberg
,
S.
Rohatyn
, and
M.
Mauder
, “
Scalewise invariant analysis of the anisotropic Reynolds stress tensor for atmospheric surface layer and canopy sublayer turbulent flows
,”
Phys. Rev. Fluids
3
(
5
),
054608
(
2018
).
17.
J.
Dougherty
, “
The anisotropy of turbulence at the meteor level
,”
J. Atmos. Terr. Phys.
21
(
2–3
),
210
213
(
1961
).
18.
H.
Liu
,
R.
Yuan
,
J.
Mei
,
J.
Sun
,
Q.
Liu
, and
Y.
Wang
, “
Scale properties of anisotropic and isotropic turbulence in the urban surface layer
,”
Boundary-Layer Meteorol.
165
(
2
),
277
294
(
2017
).
19.
S.
Banerjee
,
R.
Krahl
,
F.
Durst
, and
C.
Zenger
, “
Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches
,”
J. Turbul.
8
,
N32
(
2007
).
20.
M.
Emory
and
G.
Iaccarino
, “
Visualizing turbulence anisotropy in the spatial domain with componentality contours
,” in Annual Brief, Center for Turbulence Research,
2014
.
21.
J. L.
Lumley
and
G. R.
Newman
, “
The return to isotropy of homogeneous turbulence
,”
J. Fluid Mech.
82
,
161
178
(
1977
).
22.
A. J. M.
Spencer
, “
Part III. Theory of invariants
,”
Continuum Phys.
1
,
239
353
(
1971
).
23.
A. K.
Jain
, “
Data clustering: 50 years beyond K-means
,”
Pattern Recognit. Lett.
31
(
8
),
651
666
(
2010
).
24.
P. M.
Patel
,
B. N.
Shah
, and
V.
Shah
, “
Image segmentation using K-mean clustering for finding tumor in medical application
,”
Int. J. Comput. Trends Technol. (IJCTT)
4
(5),
1239
1242
(
2013
).
25.
N.
Ali
,
M.
Calaf
, and
R. B.
Cal
, “
Reduced-order modeling of the wake behind a single wind turbine
,” in
Progress in Turbulence VIII
(
Springer International Publishing
,
2019
), pp.
285
290
.
26.
J.
Han
,
J.
Pei
, and
M.
Kamber
,
Data Mining: Concepts and Techniques
(
Elsevier
,
2011
).
27.
S.
Yadav
and
M.
Biswas
, “
Improved color-based K-mean algorithm for clustering of satellite image
,” in
2017 4th International Conference on Signal Processing and Integrated Networks (SPIN)
(IEEE,
2017
), pp.
468
472
.
28.
E.
Kaiser
,
B. R.
Noack
,
L.
Cordier
,
A.
Spohn
,
M.
Segond
,
M.
Abel
,
G.
Daviller
,
J.
Östh
,
S.
Krajnović
, and
R. K.
Niven
, “
Cluster-based reduced-order modelling of a mixing layer
,”
J. Fluid Mech.
754
,
365
414
(
2014
).
29.
A.
Schmidt
,
C.
Hanson
,
B. E.
Kathilankal
, and
J.
Law
, “
Classification and assessment of turbulent fluxes above ecosystems in North-America with self-organizing feature map networks
,”
Agric. For. Meteorol.
151
(
4
),
508
520
(
2011
).
30.
A.
Clifton
and
J. K.
Lundquist
, “
Data clustering reveals climate impacts on local wind phenomena
,”
J. Appl. Meteorol. Climatol.
51
(
8
),
1547
1557
(
2012
).
31.
P.
Drineas
,
A.
Frieze
,
R.
Kannan
,
S.
Vempala
, and
V.
Vinay
, “
Clustering large graphs via the singular value decomposition
,”
Mach. Learn.
56
(
1
),
9
33
(
2004
).
32.
M.
Meilă
, “
The uniqueness of a good optimum for k-means
,” in
Proceedings of the 23rd International Conference on Machine Learning
(ACM,
2006
), pp.
625
632
.
33.
D.
Arthur
and
S.
Vassilvitskii
, “
k-means++: The advantages of careful seeding
,” in:
Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms
(SIAM,
2007
), pp.
1027
1035
.
34.
J. A.
Hartigan
,
Clustering Algorithms
(
Wiley
,
New York
,
1975
), Vol.
209
.
35.
E.
Gokcay
and
J. C.
Principe
, “
Information theoretic clustering
,”
IEEE Trans. Pattern Anal. Mach. Intell.
24
(
2
),
158
171
(
2002
).
36.
R. W. G.
Hunt
and
M. R.
Pointer
,
Measuring Colour
(
John Wiley & Sons
,
2011
).
37.
I. C.
Consortium
,
Specification ICC. 1: 2004-10
,
Image Technology Colour Management-Architecture, Profile Format, and Data Structure
(Specification ICC, 2004).
38.
L.
Lucchese
and
S. K.
Mitra
, “
Colour image segmentation: A state-of-the-art survey
,”
Proc.-Indian Natl. Sci. Acad. Part A
67
(
2
),
207
222
(
2001
).
39.
R. W. G.
Hunt
,
The Reproduction of Colour
, 6th ed. (
John Wiley & Sons
,
2004
).
40.
R. B.
Stull
,
An Introduction to Boundary Layer Meteorology
,
Atmospheric Science Library
(
Springer
,
1988
).
41.
E.
Bou-Zeid
,
C.
Meneveau
, and
M. B.
Parlange
, “
A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows
,”
Phys. Fluids
17
,
1
18
(
2005
).
42.
M.
Calaf
,
M. B.
Parlange
, and
C.
Meneveau
, “
Large eddy simulation study of scalar transport in fully developed wind-turbine array boundary layers
,”
Phys. Fluids
23
,
126603
(
2011
).
43.
V.
Sharma
,
M.
Calaf
,
M.
Lehning
, and
M. B.
Parlange
, “
Time-adaptive wind turbine model for an LES framework
,”
Wind Energy
19
,
939
952
(
2016
).
44.
G.
Cortina
,
V.
Sharma
, and
M.
Calaf
, “
Investigation of the incoming wind vector for improved wind turbine yaw-adjustment under different atmospheric and wind farm conditions
,”
Renewable Energy
101
,
376
386
(
2017
).
45.
C.
Moeng
, “
A large-eddy-simulation model for the study of planetary boundary-layer turbulence
,”
J. Atmos. Sci.
41
,
2052
2062
(
1984
).
46.
J. D.
Albertson
and
M. B.
Parlange
, “
Natural integration of scalar fluxes from complex terrain
,”
Adv. Water Resour.
23
,
239
252
(
1999
).
47.
C.
Canuto
,
M.
Hussainii
, and
A.
Quarteroni
,
Spectral Methods in Fluid Dynamics
(
Springer-Verlag
,
Berlin
,
1988
).
48.
G. S.
Poulos
,
W.
Blumen
,
D. C.
Fritts
,
J. K.
Lundquist
 et al, “
CASES-99: A comprehensive investigation of the stable nocturnal boundary layer
,”
Bull. Am. Meteorol. Soc.
83
(
4
),
555
(
2002
).
49.
V.
Sharma
,
M. B.
Parlange
, and
M.
Calaf
, “
Perturbations to the spatial and temporal characteristics of the diurnally-varying atmospheric boundary layer due to an extensive wind farm
,”
Boundary-Layer Meteorol.
162
(
2
),
255
282
(
2017
).
50.
N.
Ali
,
G.
Cortina
,
N.
Hamilton
,
M.
Calaf
, and
R. B.
Cal
, “
Turbulence characteristics of a thermally stratified wind turbine array boundary layer via proper orthogonal decomposition
,”
J. Fluid Mech.
828
,
175
195
(
2017
).
51.
N.
Ali
,
N.
Hamilton
,
M.
Calaf
, and
R. B.
Cal
, “
Turbulence kinetic energy budget and conditional sampling of momentum, scalar, and intermittency fluxes in thermally stratified wind farms
,”
J. Turbul.
20
(
1
),
32
63
(
2019
).
52.
G.
Cortina
,
R. B.
Cal
, and
M.
Calaf
, “
Distribution of mean kinetic energy around an isolated wind turbine and a characteristic wind turbine of a very large wind farm
,”
Phys. Rev. Fluids
074402
,
1
18
(
2016
).
53.
R. B.
Cal
,
J.
Lebrón
,
L.
Castillo
,
H. S.
Kang
, and
C.
Meneveau
, “
Experimental study of the horizontally averaged flow structure in a model wind-turbine array boundary layer
,”
J. Renewable Sustainable Energy
2
(
1
),
013106
(
2010
).
54.
M.
Calaf
,
C.
Meneveau
, and
J.
Meyers
, “
Large eddy simulation study of fully developed wind-turbine array boundary layers
,”
Phys. Fluids
22
(
1
),
015110
(
2010
).
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