This paper demonstrates the maintenance of self-sustaining turbulence in a restricted nonlinear (RNL) model of plane Couette flow. The RNL system is derived directly from the Navier-Stokes equations and permits higher resolution studies of the dynamical system associated with the stochastic structural stability theory (S3T) model, which is a second order approximation of the statistical state dynamics of the flow. The RNL model shares the dynamical restrictions of the S3T model but can be easily implemented by reducing a DNS code so that it retains only the RNL dynamics. Comparisons of turbulence arising from DNS and RNL simulations demonstrate that the RNL system supports self-sustaining turbulence with a mean flow as well as structural and dynamical features that are consistent with DNS. These results demonstrate that the simplified RNL system captures fundamental aspects of fully developed turbulence in wall-bounded shear flows and motivate use of the RNL/S3T framework for further study of wall-turbulence.
REFERENCES
1.
J.
Kim
, P.
Moin
, and R.
Moser
, “Turbulence statistics in fully developed channel flow at low Reynolds number
,” J. Fluid Mech.
177
, 133
–166
(1987
).2.
M.
Simens
, J.
Jimenez
, S.
Hoyas
, and Y.
Mizuno
, “A high-resolution code for turbulent boundary layers
,” J. Comput. Phys.
228
, 4218
–4231
(2009
).3.
J. C.
del Álamo
, J.
Jiménez
, P.
Zandonade
, and R. D.
Moser
, “Scaling of the energy spectra of turbulent channels
,” J. Fluid Mech.
500
, 135
–144
(2004
).4.
T.
Tsukahara
, H.
Kawamura
, and K.
Shingai
, “DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region
,” J. Turbul.
7
, 1
–16
(2006
).5.
S.
Hoyas
and J.
Jimenez
, “Reynolds number effects on the Reynolds-stress budgets in turbulent channels
,” Phys. Fluids
20
, 101511
(2008
).6.
B. F.
Farrell
and P. J.
Ioannou
, “Generalized stability. Part I: Autonomous operators
,” J. Atmos. Sci.
53
, 2025
–2040
(1996
).7.
B. F.
Farrell
and P. J.
Ioannou
, “Generalized stability. Part II: Non-autonomous operators
,” J. Atmos. Sci.
53
, 2041
–2053
(1996
).8.
B. F.
Farrell
, “Optimal excitation of perturbations in viscous shear flow
,” Phys. Fluids
31
, 2093
–2102
(1988
).9.
K. M.
Butler
and B. F.
Farrell
, “Three-dimensional optimal perturbations in viscous shear flows
,” Phys. Fluids
4
, 1637
–1650
(1992
).10.
L. H.
Gustavsson
, “Energy growth of three-dimensional disturbances in plane Poiseuille flow
,” J. Fluid Mech.
224
, 241
–260
(1991
).11.
L. N.
Trefethen
, A. E.
Trefethen
, S. C.
Reddy
, and T. A.
Driscoll
, “Hydrodynamic stability without eigenvalues
,” Science
261
, 578
–584
(1993
).12.
S. C.
Reddy
and D. S.
Henningson
, “Energy growth in viscous shear flows
,” J. Fluid Mech.
252
, 209
–238
(1993
).13.
B. F.
Farrell
and P. J.
Ioannou
, “Stochastic forcing of the linearized Navier-Stokes equations
,” Phys. Fluids A
5
, 2600
–2609
(1993
).14.
B.
Bamieh
and M.
Dahleh
, “Energy amplification in channel flows with stochastic excitation
,” Phys. Fluids
13
, 3258
–3269
(2001
).15.
M. R.
Jovanović
and B.
Bamieh
, “Componentwise energy amplification in channel flows
,” J. Fluid Mech.
534
, 145
–183
(2005
).16.
D. S.
Henningson
, “Comment on “Transition in shear flows. Nonlinear normality versus non-normal linearity” [Phys. Fluids 7, 3060 (1995)]
,” Phys. Fluids
8
, 2257
–2258
(1996
).17.
D. S.
Henningson
and S. C.
Reddy
, “On the role of linear mechanisms in transition to turbulence
,” Phys. Fluids
6
, 1396
–1398
(1994
).18.
K. M.
Butler
and B. F.
Farrell
, “Optimal perturbations and streak spacing in turbulent shear flow
,” Phys. Fluids A
5
, 774
–777
(1993
).19.
J.
Kim
and J.
Lim
, “A linear process in wall bounded turbulent shear flows
,” Phys. Fluids
12
, 1885
–1888
(2000
).20.
F.
Waleffe
, J.
Kim
, and J. M.
Hamilton
, “On the origin of streaks in turbulent shear flows
,” in Turbulent Shear Flows
, edited by F.
Durst
, R.
Friedrich
, B.
Schmidt
, B.
Launder
, F.
Schumann
, and J.
Whitelaw
(Springer-Verlag
, Munich, Germany
, 1993
), Vol. 8
, pp. 37
–49
.21.
M. R.
Jovanović
and B.
Bamieh
, “Modeling flow statistics using the linearized Navier-Stokes equations
,” in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL
(IEEE
, New York, NY
, 2001
), Vol. 5
, pp. 4944
–4949
.22.
B. F.
Farrell
and P. J.
Ioannou
, “Perturbation structure and spectra in turbulent channel flow
,” Theor. Comput. Fluid Dyn.
11
, 237
–250
(1998
).23.
J. C.
del Álamo
and J.
Jiménez
, “Linear energy amplification in turbulent channels
,” J. Fluid Mech.
559
, 205
–213
(2006
).24.
Y.
Hwang
and C.
Cossu
, “Amplification of coherent structures in the turbulent Couette flow: an input-output analysis at low Reynolds number
,” J. Fluid Mech.
643
, 333
–348
(2010
).25.
C.
Cossu
, G.
Pujals
, and S.
Depardon
, “Optimal transient growth and very large scale structures in turbulent boundary layers
,” J. Fluid Mech.
619
, 79
–94
(2009
).26.
R.
Moarref
and M. R.
Jovanović
, “Model-based design of transverse wall oscillations for turbulent drag reduction
,” J. Fluid Mech.
707
, 205
–240
(2012
).27.
L.
Sirovich
, “Turbulence and the dynamics of coherent structures. I - Coherent structures. II - Symmetries and transformations. III - Dynamics and scaling
,” Q. Appl. Math.
45
, 561
–571
(1987
).28.
T. R.
Smith
, J.
Moehlis
, and P.
Holmes
, “Low-dimensional modelling of turbulence using the proper orthogonal decomposition: A tutorial
,” Nonlinear Dyn.
41
, 275
–307
(2005
).29.
J.
Jiménez
, G.
Kawahara
, M. P.
Simens
, M.
Nagata
, and M.
Shiba
, “Characterization of near-wall turbulence in terms of equilibrium and bursting solutions
,” Phys. Fluids
17
, 015105
(2005
).30.
G.
Kawahara
, M.
Uhlmann
, and L.
Van Veen
, “The significance of simple invariant solutions in turbulent flows
,” Annu. Rev. Fluid Mech.
44
, 203
–225
(2012
).31.
M.
Nagata
, “Three-dimensional traveling-wave solutions in plane Couette flow
,” J. Fluid Mech.
217
, 519
–527
(1990
).32.
J. F.
Gibson
, J.
Halcrow
, and P.
Cvitanović
, “Equilibrium and travelling-wave solutions of plane Couette flow
,” J. Fluid Mech.
638
, 243
–266
(2009
).33.
W. C.
Reynolds
and S. C.
Kassinos
, “One-point modelling of rapidly deformed homogeneous turbulence
,” Proc. R. Soc. London, Ser. A
451
, 87
–104
(1995
).34.
K. M.
Bobba
, B.
Bamieh
, and J. C.
Doyle
, “Highly optimized transitions to turbulence
,” in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV
(IEEE
, New York, NY
, 2002
), pp. 4559
–4562
.35.
D. F.
Gayme
, B. J.
McKeon
, A.
Papachristodoulou
, B.
Bamieh
, and J. C.
Doyle
, “A streamwise constant model of turbulence in plane Couette flow
,” J. Fluid Mech.
665
, 99
–119
(2010
).36.
K. J.
Kim
and R. J.
Adrian
, “Very large scale motion in the outer layer
,” Phys. Fluids
11
, 417
–422
(1999
).37.
M.
Guala
, S. E.
Hommema
, and R. J.
Adrian
, “Large-scale and very-large-scale motions in turbulent pipe flow
,” J. Fluid Mech.
554
, 521
–542
(2006
).38.
N.
Hutchins
and I.
Marusic
, “Evidence of very long meandering features in the logarithmic region of turbulent boundary layers
,” J. Fluid Mech.
579
, 1
–28
(2007
).39.
B. J.
McKeon
and A. S.
Sharma
, “A critical-layer framework for turbulent pipe flow
,” J. Fluid Mech.
658
, 336
–382
(2010
).40.
D. F.
Gayme
, B. J.
McKeon
, B.
Bamieh
, A.
Papachristodoulou
, and J. C.
Doyle
, “Amplification and nonlinear mechanisms in plane Couette flow
,” Phys. Fluids
23
, 065108
(2011
).41.
K. M.
Bobba
, “Robust flow stability: Theory, computations and experiments in near wall turbulence
,” Ph.D. thesis (California Institute of Technology
, Pasadena, CA, USA, 2004
).42.
A proof of this fact and the explicit construction of a Lyapunov function based on private communications with A. Papachristodoulou and B. Bamieh is provided in Ref. 69.
43.
B. F.
Farrell
and P. J.
Ioannou
, “Structural stability of turbulent jets
,” J. Atmos. Sci.
60
, 2101
–2118
(2003
).44.
T.
DelSole
and B. F.
Farrell
, “The quasi-linear equilibration of a thermally maintained stochastically excited jet in a quasigeostrophic model
,” J. Atmos. Sci.
53
, 1781
–1797
(1996
).45.
T.
DelSole
, “Stochastic models of quasigeostrophic turbulence
,” Surv. Geophys.
25
, 107
–194
(2004
).46.
J. B.
Marston
, E.
Conover
, and T.
Schneider
, “Statistics of an unstable barotropic jet from a cumulant expansion
,” J. Atmos. Sci.
65
, 1955
–1966
(2008
).47.
S. M.
Tobias
, K.
Dagon
, and J. B.
Marston
, “Astrophysical fluid dynamics via direct numerical simulation
,” Astrophys. J.
727
, 127
(2011
).48.
K.
Srinivasan
and W. R.
Young
, “Zonostrophic instability
,” J. Atmos. Sci.
69
, 1633
–1656
(2012
).49.
B. F.
Farrell
, D. F.
Gayme
, P.
Ioannou
, B.
Lieu
, and M. R.
Jovanović
, “Dynamics of the roll and streak structure in transition and turbulence
,” in Proceedings of the Center for Turbulence Research Summer Program
(Center for Turbulence Research
, Stanford, CA
, 2012
), pp. 34
–54
.50.
S. J.
Kline
, W. C.
Reynolds
, F. A.
Schraub
, and P. W.
Runstadler
, “The structure of turbulent boundary layers
,” J. Fluid Mech.
30
, 741
–773
(1967
).51.
M. T.
Landahl
, “A note on an algebraic instability of inviscid parallel shear flows
,” J. Fluid Mech.
98
, 243
(1980
).52.
J. D.
Swearingen
and R. F.
Blackwelder
, “The growth and breakdown of streamwise vortices in the presence of a wall
,” J. Fluid Mech.
182
, 255
–290
(1987
).53.
H. P.
Bakewell
Jr. and J. L.
Lumley
, “Viscous sublayer and adjacent wall region in turbulent pipe flow
,” Phys. Fluids
10
, 1880
–1889
(1967
).54.
F.
Waleffe
, “Hydrodynamic stability and turbulence: Beyond transients to a self-sustaining process
,” Stud. Appl. Math.
95
, 319
–343
(1995
).55.
J. M.
Hamilton
, J.
Kim
, and F.
Waleffe
, “Regeneration mechanisms of near-wall turbulence structures
,” J. Fluid Mech.
287
, 317
–348
(1995
).56.
F.
Waleffe
, “On a self-sustaining process in shear flows
,” Phys. Fluids
9
, 883
–900
(1997
).57.
P.
Hall
and S.
Sherwin
, “Streamwise vortices in shear flows: Harbingers of transition and the skeleton of coherent structures
,” J. Fluid Mech.
661
, 178
–205
(2010
).58.
W.
Schoppa
and F.
Hussain
, “Coherent structure generation in near-wall turbulence
,” J. Fluid Mech.
453
, 57
–108
(2002
).59.
B. F.
Farrell
and P. J.
Ioannou
, “Optimal excitation of three dimensional perturbations in viscous constant shear flow
,” Phys. Fluids
5
, 1390
–1400
(1993
).60.
B. F.
Farrell
and P. J.
Ioannou
, “Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow
,” J. Fluid Mech.
708
, 149
–196
(2012
).61.
J.
Jiménez
, “Near-wall turbulence
,” Phys. Fluids
25
, 101302
(2013
).62.
J. F.
Gibson
, “Channelflow: A spectral Navier-Stokes simulator in C++
,” Technical Report (University of New Hampshire
, 2014
), see Channelflow.org.63.
J. F.
Gibson
, J.
Halcrow
, and P.
Cvitanović
, “Visualizing the geometry of state space in plane Couette flow
,” J. Fluid Mech.
611
, 107
–130
(2008
); e-print arXiv:0705.3957.64.
R.
Peyret
, Spectral Methods for Incompressible Flows
(Springer-Verlag
, 2002
).65.
C.
Canuto
, M.
Hussaini
, A.
Quateroni
, and T.
Zhang
, Spectral Methods in Fluid Dynamics
(Springer-Verlag
, 1988
).66.
T. A.
Zang
and M. Y.
Hussaini
, “Numerical experiments on subcritical transition mechanism
,” AIAA Paper 85-0296, 1985
.67.
N. C.
Constantinou
, A.
Loranzo-Durán
, M.-A.
Nikolaidis
, B. F.
Farrell
, P. J.
Ioannou
, and J.
Jiménez
, “Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS
,” J. Phys.: Conf. Ser.
506
, 012004
(2014
).68.
U.
Frisch
, Turbulence: The Legacy of A. N. Kolmogorov
(Cambridge University Press
, 1995
).69.
D. F.
Gayme
, “A robust control approach to understanding nonlinear mechanisms in shear flow turbulence
,” Ph.D. thesis (Caltech
, Pasadena, CA, USA, 2010
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.
