The ability of the low-rank approximation of hypersonic turbulent boundary layers with/without wall cooling is examined with the linear resolvent operator in a compressible form. The freestream Mach number of the base flow is 5.86, and the friction Reynolds number is 420. The wall-to-recovery temperature ratio is set as 1.0 and 0.25, respectively, corresponding to an adiabatic wall condition and a cold-wall condition. Different from the resolvent analysis of incompressible turbulent boundary layers, the optimal response mode in the wave-parameter space exhibits a relatively subsonic and a relatively supersonic region [Bae et al., “Resolvent-based study of compressibility effects on supersonic turbulent boundary layers,” J. Fluid Mech. 883, A29 (2020)], divided by the freestream relative Mach number of unity. The features of energy distribution of the optimal response mode in space and scales are examined, and the energy spectra of streamwise velocity and temperature fluctuations, carried by the optimal response mode, are discussed with typical subsets of streamwise and spanwise wavelengths. This reveals the dynamics of the near-wall small-scale and outer larger-scale motions and the distinction in the relatively subsonic/supersonic region. Moreover, the coherent structures, including the velocity and temperature streaks, quasi-streamwise vortices, and large-scale/very-large-scale motions, are identified in the optimal response mode. Special attention is paid to the effects of wall cooling.
REFERENCES
1.
L.
Duan
,
I.
Beekman
, and
M. P.
Martin
, “
Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature
,” J. Fluid Mech.
655
, 419
–445
(2010
).2.
X.
Liang
and
X.-L.
Li
, “
Direct numerical simulation on mach number and wall temperature effects in the turbulent flows of flat-plate boundary layer
,” Commun. Comput. Phys.
17
(1
), 189
–212
(2015
).3.
C.
Zhang
,
L.
Duan
, and
M. M.
Choudhari
, “
Effect of wall cooling on boundary-layer-induced pressure fluctuations at Mach 6
,” J. Fluid Mech.
822
, 5
–30
(2017
).4.
C.
Zhang
,
L.
Duan
, and
M. M.
Choudhari
, “
Direct numerical simulation database for supersonic and hypersonic turbulent boundary layers
,” AIAA J.
56
(11
), 4297
–4311
(2018
).5.
M.
Yu
and
C.-X.
Xu
, “
Compressibility effects on hypersonic turbulent channel flow with cold walls
,” Phys. Fluids
33
(7
), 075106
(2021
).6.
D.-H.
Xu
,
J.-C.
Wang
,
M.-P.
Wan
,
C.-P.
Yu
,
X.-L.
Li
, and
S.-Y.
Chen
, “
Effect of wall temperature on the kinetic energy transfer in a hypersonic turbulent boundary layer
,” J. Fluid Mech.
929
, A33
(2021
).7.
D.-H.
Xu
,
J.-C.
Wang
,
M.-P.
Wan
,
C.-P.
Yu
,
X.-L.
Li
, and
S.-Y.
Chen
, “
Compressibility effect in hypersonic boundary layer with isothermal wall condition
,” Phys. Rev. Fluids
6
, 054609
(2021
).8.
M. V.
Morkovin
, “
Effects of compressibility on turbulent flows
,” Méc. Turbul.
367
, 380
(1962
).9.
Y.-T.
Fan
,
W.-P.
Li
, and
S.
Pirozzoli
, “
Energy exchanges in hypersonic flows
,” Phys. Rev. Fluids
7
, L092601
(2022
).10.
C.
Wenzel
,
T.
Gibis
, and
M.
Kloker
, “
About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers
,” J. Fluid Mech.
930
, A1
(2022
).11.
Y.-S.
Zhang
,
W.-T.
Bi
,
F.
Hussain
,
X.-L.
Li
, and
Z.-S.
She
, “
Mach-number-invariant mean-velocity profile of compressible turbulent boundary layers
,” Phys. Rev. Lett.
109
, 054502
(2012
).12.
Y.-S.
Zhang
,
W.-T.
Bi
,
F.
Hussain
, and
Z.-S.
She
, “
A generalized reynolds analogy for compressible wall-bounded turbulent flows
,” J. Fluid Mech.
739
, 392
–420
(2014
).13.
A.
Hadjadj
,
O.
Ben-Nasr
,
M. S.
Shadloo
, and
A.
Chaudhuri
, “
Effect of wall temperature in supersonic turbulent boundary layers: A numerical study
,” Intl. J. Heat Mass Transfer
81
, 426
–438
(2015
).14.
D.
Modesti
and
S.
Pirozzoli
, “
Reynolds and Mach number effects in compressible turbulent channel flow
,” Int. J. Heat Fluid Flow
59
, 33
–49
(2016
).15.
A.
Trettel
and
J.
Larsson
, “
Mean velocity scaling for compressible wall turbulence with heat transfer
,” Phys. Fluids
28
(2
), 026102
(2016
).16.
M. S.
Shadloo
,
A.
Hadjadj
, and
F.
Hussain
, “
Temperature-invariant scaling for compressible turbulent boundary layers with wall heat transfer
,” Heat Transfer Eng.
39
(11
), 923
–932
(2018
).17.
K. P.
Griffin
,
L.
Fu
, and
P.
Moin
, “
Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer
,” Proc. Natl. Acad. Sci. U. S. A.
118
(34
), e2111144118
(2021
).18.
B. J.
McKeon
and
A. S.
Sharma
, “
A critical-layer framework for turbulent pipe flow
,” J. Fluid Mech.
658
, 336
–382
(2010
).19.
L. N.
Trefethen
,
A. E.
Trefethen
,
S. C.
Reddy
, and
T. A.
Driscoll
, “
Hydrodynamic stability without the eigenvalues
,” Science
261
, 578
–584
(1993
).20.
K. M.
Butler
and
B. F.
Farrell
, “
Three-dimensional optimal perturbations in viscous shear flows
,” Phys. Fluids A
4
, 1637
–1650
(1992
).21.
B. J.
McKeon
, “
The engine behind (wall) turbulence: Perspectives on scale interactions
,” J. Fluid Mech
817
, P1
(2017
).22.
A. S.
Sharma
and
B. J.
McKeon
, “
On coherent structure in wall turbulence
,” J. Fluid Mech.
728
, 196
–238
(2013
).23.
R.
Moarref
,
A. S.
Sharma
,
J. A.
Tropp
, and
B. J.
McKeon
, “
Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels
,” J. Fluid Mech.
734
, 275
–316
(2013
).24.
S.
Symon
,
S. J.
Illingworth
, and
I.
Marusic
, “
Energy transfer in turbulent channel flows and implications for resolvent modelling
,” J. Fluid Mech.
911
, A3
(2021
).25.
L. I.
Abreu
,
A. V. G.
Cavalieri
,
P.
Schlatter
,
R.
Vinuesa
, and
D. S.
Henningson
, “
Resolvent modelling of near-wall coherent structures in turbulent channel flow
,” Int. J. Heat Fluid Flow
85
, 108662
(2020
).26.
S. T. M.
Dawson
,
B. J.
McKeon
, and
T.
Saxton-Fox
, “
Modeling passive scalar dynamics in wall-bounded turbulence using resolvent analysis
,” in AIAA AVIATION Forum, Fluid Dynamics Conference
, Atlanta, 2018
.27.
H. J.
Bae
,
S. T. M.
Dawson
, and
B. J.
McKeon
, “
Resolvent-based study of compressibility effects on supersonic turbulent boundary layers
,” J. Fluid Mech.
883
, A29
(2020
).28.
S. T. M.
Dawson
and
B. J.
McKeon
, “
Studying the effects of compressibility in planar Couette flow using resolvent analysis
,” in AIAA SciTech Forum
, San Diego, 2019
.29.
H. J.
Bae
,
S. T.
MDawson
, and
B. J.
McKeon
, “
Studying the effect of wall cooling in supersonic boundary layer flow using resolvent analysis
,” in AIAA SciTech Forum
, 2020
.30.
A.
Madhusudanan
and
B. J.
McKeon
, “
Subsonic and supersonic mechanisms in compressible turbulent boundary layers: A perspective from resolvent analysis
,” arXiv:2209.14223 (2022
).31.
A.
Towne
,
A.
Lozano-Durán
, and
X.
Yang
, “
Resolvent-based estimation of space-time flow statistics
,” J. Fluid Mech.
883
, A17
(2020
).32.
P.
Morra
,
O.
Semeraro
,
D. S.
Henningson
, and
C.
Cossu
, “
On the relevance of reynolds stresses in resolvent analyses of turbulent wall-bounded flows
,” J. Fluid Mech.
867
, 969
–984
(2019
).33.
F. R.
Amaral
,
A. V. G.
Cavalieri
,
E.
Martini
,
P.
Jordan
, and
A.
Towne
, “
Resolvent-based estimation of turbulent channel flow using wall measurements
,” J. Fluid Mech.
927
, A17
(2021
).34.
P.
Morra
,
P. A. S.
Nogueira
,
A. V. G.
Cavalieri
, and
D. S.
Henningson
, “
The colour of forcing statistics in resolvent analyses of turbulent channel flows
,” J. Fluid Mech.
907
, A24
(2021
).35.
B. J.
McKeon
,
A. S.
Sharma
, and
I.
Jacobi
, “
Experimental manipulation of wall turbulence: A systems approach
,” Phys. Fluids
25
(3
), 031301
(2013
).36.
S. J.
Illingworth
,
J. P.
Monty
, and
I.
Marusic
, “
Estimating large-scale structures in wall turbulence using linear models
,” J. Fluid Mech.
842
, 146
–162
(2018
).37.
L. M.
Mack
, “
Boundary-layer linear stability theory
,” Technical Report No. 709
(
NASA Jet Propulsion Laboratory
, 1984
).38.
B.-T.
Chu
, “
On the energy transfer to small disturbances in fluid flow (Part I)
,” Acta Mech.
1
(3
), 215
–234
(1965
).39.
L. N.
Trefethen
, Spectral Methods in MATLAB (SIAM, 2000
), pp. 125
–134
.40.
M. S.
Shadloo
,
A.
Hadjadj
, and
F.
Hussain
, “
Statistical behavior of supersonic turbulent boundary layers with heat transfer at
=2
,” Int. J. Heat Fluid Flow
53
, 113
–134
(2015
).41.
R.
Moarref
,
A. S.
Sharma
,
J. A.
Tropp
, and
B. J.
McKeon
, “
On effectiveness of a rank-1 model of turbulent channels for representing the velocity spectra
,” in 43rd Fluid Dynamics Conference
, 2013
.42.
J. P.
Monty
and
M. S.
Chong
, “
Turbulent channel flow: Comparison of streamwise velocity data from experiments and direct numerical simulation
,” J. Fluid Mech.
633
, 461
–474
(2009
).43.
A. A.
Townsend
, The Structure of Turbulent Shear Flow
(
Cambridge University Press
, 1976
).44.
S.
Pirozzoli
and
M.
Bernardini
, “
Turbulence in supersonic boundary layers at moderate Reynolds number
,” J. Fluid Mech.
688
, 120
–168
(2011
).45.
M.
Bross
,
S.
Scharnowski
, and
C. J.
Kähler
, “
Large-scale coherent structures in compressible turbulent boundary layers
,” J. Fluid Mech.
911
, A2
(2021
).46.
S.
Pirozzoli
,
F.
Grasso
, and
T. B.
Gatski
, “
Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at
.
,” Phys. Fluids
16
(3
), 530
–545
(2004
).47.
L.
Duan
,
I.
Beekman
, and
M. P.
Martin
, “
Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number
,” J. Fluid Mech.
672
, 245
–267
(2011
).48.
J.
Zhou
,
R. J.
Adrian
,
S.
Balachandar
, and
T. M.
Kendall
, “
Mechanisms for generating coherent packets of hairpin vortices in channel flow
,” J. Fluid Mech.
387
, 353
–396
(1999
).49.
V. K.
Natrajan
,
Y.
Wu
, and
K. T.
Christensen
, “
Spatial signatures of retrograde spanwise vortices in wall turbulence
,” J. Fluid Mech.
574
, 155
–167
(2007
).50.
Y.
Wu
and
K. T.
Christensen
, “
Population trends of spanwise vortices in wall turbulence
,” J. Fluid Mech.
568
, 55
–76
(2006
).51.
D.
Chung
and
B. J.
McKeon
, “
Large-eddy simulation of large-scale structures in long channel flow
,” J. Fluid Mech.
661
, 341
–364
(2010
).52.
A. J.
Smits
,
B. J.
McKeon
, and
I.
Marusic
, “
High-Reynolds number wall turbulence
,” Annu. Rev. Fluid Mech.
43
(1
), 353
–375
(2011
).53.
R. J.
Adrian
,
C. D.
Meinhart
, and
C. D.
Tomkins
, “
Vortex organization in the outer region of the turbulent boundary layer
,” J. Fluid Mech.
422
, 1
–54
(2000
).54.
R. J.
Adrian
, “
Hairpin vortex organization in wall turbulence
,” Phys. Fluids
19
(4
), 041301
(2007
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.