It is well known that low- and high-speed velocity streaks are statistically asymmetric. However, it is unclear how different the low- and high-temperature structures (T-structures) are even though they are strongly coupled with the streamwise velocity. Therefore, this paper identifies three-dimensional wall-attached temperature structures in supersonic turbulent boundary layers over cooled and heated walls (coming from direct numerical simulations) and separates them into positive and negative families. Wall-attached T-structures are self-similar; especially, the length and width of the positive family are linear functions of the height. The superposed temperature variance in both positive and negative families exhibits a logarithmic decay with the wall distance, while the superposed intensity of the wall-normal heat flux in the negative family shows a logarithmic growth. The modified strong Reynolds analogy proposed by Huang, Coleman, and Bradshaw [“Compressible turbulent channel flows: DNS results and modelling,” J. Fluid Mech. 305, 185–218 (1995)] is still valid in the negative family. The relative position between T-structures of opposite signs depends on the wall temperature and that in the cooled-wall case differs significantly from the relative position between low- and high-speed streaks, especially those tall ones. In the cooled-wall case, although positive temperature fluctuations below and above the maximum of the mean temperature can cluster to large-scale wall-attached structures, they are very likely dynamically unrelated.

1.
F.
Moisy
and
J.
Jimenez
, “
Geometry and clustering of intense structures in isotropic turbulence
,”
J. Fluid Mech.
513
,
111
133
(
2004
).
2.
J.
Jiménez
, “
Coherent structures in wall-bounded turbulence
,”
J. Fluid Mech.
842
,
1
100
(
2018
).
3.
A. A.
Townsend
,
The Structure of Turbulent Shear Flow
, 2nd ed. (
Cambridge University Press
,
1976
).
4.
C.
Cheng
,
W.
Li
,
A.
Lozano-Durán
, and
H.
Liu
, “
Uncovering Townsend's wall-attached eddies in low-Reynolds-number wall turbulence
,”
J. Fluid Mech.
889
,
A29
(
2020
).
5.
S. K.
Robinson
,
S. J.
Kline
, and
P. R.
Spalart
, “
A review of quasi-coherent structures in a numerically simulated turbulent boundary layer
,”
NASA Technical Memorandum Report No. 102191
,
1989
.
6.
J. H.
Lee
and
H. J.
Sung
, “
Very-large-scale motions in a turbulent boundary layer
,”
J. Fluid Mech.
673
,
80
120
(
2011
).
7.
J. H.
Lee
and
H. J.
Sung
, “
Comparison of very-large-scale motions of turbulent pipe and boundary layer simulations
,”
Phys. Fluids
25
,
045103
(
2013
).
8.
J. A.
Sillero
,
J.
Jimenez
, and
R. D.
Moser
, “
Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to
δ+2000,”
Phys. Fluids
26
,
105109
(
2014
).
9.
J.
Sillero
, “
High Reynolds numbers turbulent boundary layers
,” Ph.D. thesis (
University Politécnica Madrid
,
2014
).
10.
J.
Hwang
and
H. J.
Sung
, “
Wall-attached clusters for the logarithmic velocity law in turbulent pipe flow
,”
Phys. Fluids
31
,
055109
(
2019
).
11.
J.
Kim
and
P.
Moin
, “
Transport of passive scalars in a turbulent channel flow
,” in
Turbulent Shear Flows
6
(
Springer-Verlag
,
Berlin
,
1989
), pp.
85
96
.
12.
R. A.
Antonia
,
H.
Abe
, and
H.
Kawamura
, “
Analogy between velocity and scalar fields in a turbulent channel flow
,”
J. Fluid Mech.
628
,
241
268
(
2009
).
13.
H.
Abe
and
R. A.
Antonia
, “
Near-wall similarity between velocity and scalar fluctuations in a turbulent channel flow
,”
Phys. Fluids
21
,
025109
(
2009
).
14.
S.
Pirozzoli
,
M.
Bernardini
, and
P.
Orlandi
, “
Passive scalars in turbulent channel flow at high Reynolds number
,”
J. Fluid Mech.
788
,
614
639
(
2016
).
15.
S.
Pirozzoli
,
J.
Romero
,
M.
Fatica
,
R.
Verzicco
, and
P.
Orlandi
, “
DNS of passive scalars in turbulent pipe flow
,”
J. Fluid Mech.
940
,
A45
(
2022
).
16.
S.
Pirozzoli
and
M.
Bernardini
, “
Turbulence in supersonic boundary layers at moderate Reynolds number
,”
J. Fluid Mech.
688
,
120
168
(
2011
).
17.
S.
Dong
,
F.
Tong
,
M.
Yu
,
J.
Chen
,
X.
Yuan
, and
Q.
Wang
, “
Effects of wall temperature on two-point statistics of the fluctuating wall shear stress and heat flux in supersonic turbulent boundary layers
,”
Phys. Fluids
34
,
065114
(
2022
).
18.
L.
Duan
,
I.
Beekman
, and
M. P.
Martín
, “
Direct numerical simulation of hypersonic turbulent boundary layers. II. Effect of wall temperature
,”
J. Fluid Mech.
655
,
419
445
(
2010
).
19.
M. S.
Shadloo
,
A.
Hadjadj
, and
F.
Hussian
, “
Statistical behavior of supersonic turbulent boundary layers with heat transfer at
M=2,”
Int. J. Heat Fluid Flow
53
,
113
134
(
2015
).
20.
A.
Hadjadj
,
O.
Ben-Nasr
,
M. S.
Shadloo
, and
A.
Chaudhuri
, “
Effect of wall temperature in supersonic turbulent boundary layers: A numerical study
,”
Int. J. Heat Mass Transfer
81
,
426
438
(
2015
).
21.
S.
Sharma
,
M. S.
Shadloo
, and
A.
Hadjadj
, “
Turbulent flow topology in supersonic boundary layer with wall heat transfer
,”
Int. J. Heat Fluid Flow
78
,
108430
(
2019
).
22.
M.
Yu
,
C.-X.
Xu
, and
S.
Pirrozoli
, “
Genuine compressibility effects in wall-bounded turbulence
,”
Phys. Rev. Fluids
4
,
123402
(
2019
).
23.
M.
Yu
and
C.-X.
Xu
, “
Compressibility effects on hypersonic turbulent channel flow with cold walls
,”
Phys. Fluids
33
,
075106
(
2021
).
24.
D.
Sun
,
Q.
Guo
,
X.
Yuan
,
H.
Zhang
,
C.
Li
, and
P.
Liu
, “
A decomposition formula for the wall heat flux of a compressible boundary layer
,”
Adv. Aerodyn.
3
,
33
(
2021
).
25.
D.
Xu
,
W.
Wang
, and
S.
Chen
, “
Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers
,”
J. Fluid Mech.
941
,
A4
(
2022
).
26.
M. V.
Morkovin
, “
Effects of compressibility on turbulent flows
,” in
Mécanique de la Turbulence
, edited by
A. J.
Favre
(
CNRS
,
1962
), pp.
367
380
.
27.
P. G.
Huang
,
G. N.
Coleman
, and
P.
Bradshaw
, “
Compressible turbulent channel flows: DNS results and modelling
,”
J. Fluid Mech.
305
,
185
218
(
1995
).
28.
Y.
Zhang
,
W.
Bi
,
F.
Hussain
, and
Z.
She
, “
A generalized Reynolds analogy for compressible wall-bounded turbulent flows
,”
J. Fluid Mech.
739
,
392
420
(
2014
).
29.
W.
Zhu
,
D.
Gu
,
W.
Si
,
S.
Chen
,
Y.
Zhu
, and
C.
Lee
, “
Reduced aerodynamic heating in a hypersonic boundary layer by a wavy wall
,”
Sci. Bull.
67
,
988
990
(
2022
).
30.
F.
Tong
,
S.
Dong
,
J.
Lai
,
X.
Yuan
, and
X.
Li
, “
Wall shear-stress and wall heat-flux in a supersonic turbulent boundary layers
,”
Phys. Fluids
34
,
015127
(
2022
).
31.
S.
Dong
,
F.
Tong
,
M.
Yu
,
J.
Chen
,
X.
Yuan
, and
Q.
Wang
, “
Positive and negative pairs of fluctuating wall shear stress and heat flux in supersonic turbulent boundary layers
,”
Phys. Fluids
34
,
085115
(
2022
).
32.
X.
Li
,
D.
Fu
, and
Y.
Ma
, “
Direct numerical simulation of hypersonic boundary-layer transition over a blunt cone
,”
AIAA J.
46
,
2899
2913
(
2008
).
33.
X.
Li
,
D.
Fu
, and
Y.
Ma
, “
Direct numerical simulation of hypersonic boundary-layer transition over a blunt cone with a small angle of attack
,”
Phys. Fluids
22
,
025105
(
2010
).
34.
J.
Hwang
and
H. J.
Sung
, “
Wall-attached structures of velocity fluctuations in a turbulent boundary layer
,”
J. Fluid Mech.
856
,
958
983
(
2018
).
35.
M.
Yoon
,
J.
Hwang
,
J.
Yang
, and
H. J.
Sung
, “
Wall-attached structures of streamwise velocity fluctuations in an adverse-pressure-gradient turbulent boundary layer
,”
J. Fluid Mech.
885
,
A12
(
2020
).
36.
Y. S.
Kwon
,
N.
Hutchins
, and
J. P.
Monty
, “
On the use of the Reynolds decomposition in the intermittent region of turbulent boundary layers
,”
J. Fluid Mech.
794
,
5
16
(
2016
).
37.
G.
Borrell
and
J.
Jiménez
, “
Properties of the turbulent/non-turbulent interface in boundary layers
,”
J. Fluid Mech.
801
,
554
596
(
2016
).
38.
J. C.
Del Álamo
,
J.
Jiménez
,
P.
Zandonade
, and
R. D.
Moser
, “
Self-similar vortex clusters in the turbulent logarithmic region
,”
J. Fluid Mech.
516
,
329
358
(
2006
).
39.
O.
Flores
,
J.
Jimenez
, and
J. C.
del Alamo
, “
Vorticity organization in the outer layer of turbulent channels with disturbed walls
,”
J. Fluid Mech.
591
,
145
154
(
2007
).
40.
A.
Lozano-Durán
,
O.
Flores
, and
J.
Jiménez
, “
The three-dimensional structure of momentum transfer in turbulent channels
,”
J. Fluid Mech.
694
,
100
130
(
2012
).
41.
A.
Lozano-Durán
and
J.
Jiménez
, “
Time-resolved evolution of coherent structures in turbulent channels: Characterization of eddies and cascade
,”
J. Fluid Mech.
759
,
432
471
(
2014
).
42.
S.
Dong
,
A.
Lozano-Durán
,
A.
Sekimoto
, and
J.
Jiménez
, “
Coherent structures in statistically stationary homogeneous shear turbulence
,”
J. Fluid Mech.
816
,
167
208
(
2017
).
43.
J. I.
Cardesa
,
A.
Vela-Martín
, and
J.
Jiménez
, “
The turbulent cascade in five dimensions
,”
Science
357
,
782
784
(
2017
).
44.
K.
Osawa
and
J.
Jimenez
, “
Intense structures of different momentum fluxes in turbulent channels
,”
Phys. Rev. Fluids
3
,
084603
(
2018
).
45.
C.
Cheng
,
W.
Li
,
A.
Lozano-Durán
, and
H.
Liu
, “
On the structure of streamwise wall-shear stress fluctuations in turbulent channel flows
,”
J. Fluid Mech.
903
,
A29
(
2020
).
46.
S.
Dong
,
Y.
Huang
,
X.
Yuan
, and
A.
Lozano-Durán
, “
Wall-layer models for large-eddy simulations
,”
J. Fluid Mech.
892
,
A22
(
2020
).
47.
A. E.
Perry
and
M. S.
Chong
, “
On the mechanism of wall turbulence
,”
J. Fluid Mech.
119
,
173
217
(
1982
).
48.
J. C.
del Álamo
and
J.
Jiménez
, “
Linear energy amplification in turbulent channels
,”
J. Fluid Mech.
559
,
205
213
(
2006
).
49.
S. E.
Guarini
,
R. D.
Moser
,
K.
Shariff
, and
A.
Wray
, “
Direct numerical simulation of a supersonic turbulent boundary layer
,”
J. Fluid Mech.
414
,
1
33
(
2000
).
50.
S.
Dong
, “
Coherent structures in statistically-stationary homogeneous shear turbulence
,” Ph.D. thesis (
University Politécnica Madrid
,
2016
).
51.
C.
Pan
and
Y.
Kwon
, “
Extremely high wall-shear stress events in a turbulent boundary layer
,”
J. Phys.: Conf. Ser.
1001
,
012004
(
2018
).
52.
B.
Guerrero
,
M. F.
Lambert
, and
R. C.
Chin
, “
Extreme wall shear stress events in turbulent pipe flows: Spatial characteristics of coherent motions
,”
J. Fluid Mech.
904
,
A18
(
2020
).
53.
R. J.
Adrian
, “
Hairpin vortex organization in wall turbulence
,”
Phys. Fluids
19
,
041301
(
2007
).
54.
R. J.
Adrian
,
C. D.
Meinhart
, and
C.
Tomkins
, “
Vortex organization in the outer region of the turbulent boundary layer
,”
J. Fluid Mech.
422
,
1
54
(
2000
).
55.
M.
Yu
and
C. X.
Xu
, “
Predictive models for near-wall velocity and temperature fluctuations in supersonic wall-bounded turbulence
,”
J. Fluid Mech.
937
,
A32
(
2022
).
You do not currently have access to this content.