Plane Couette flow at the laminar–turbulent transition Paul Manneville Ecole Polytechnique, Palaiseau, 91128, France
[email protected] transition to turbulence ??? ! closed vs. open flows closed flows (e.g. convection) ! confinement effects ! confined vs. extended ! temporal vs. spatio-temporal chaos ! pretty well understood open flows less well understood (even apparently simplest case of parallel flows) 1
• linear stage ! standard stability analysis
! inflectional vs. non-inflectional base profiles !U"
y
U"
2# #(X)
boundary layer
X
Re = ∆U !/ν = (∆U/!) × (!2/ν) = τv /τa " inflectional ! linear instability (inertial), e.g. wake globally super-critical transition to turbulence at low Re " non-inflectional ! no instability at low Re, e.g. boundary layer possible viscous instability at large Re = ReTS ! conditional stability is generic (nonlin. instab. at Re < ReTS) 2
• specific role of advection∗ physical consequence ! interaction mean flow/fluctuation #U(y,z)
U(y)
y z
x
! induction of streaks by streamwise vortices ! lift-up ! universal perturbation amplification mechanism ! transient energy growth even is stable flows (lim. t → ∞) ∗ see,
e.g. P. Schmid, D.S. Henningson, Stability and Transition in Shear Flows (Springer, 2001)
3
• direct transition (by-pass) to turbulence induced by transient perturbation growth in a laminar linearly stable flow ! nucleation of turbulent spots
boundary layer
plane Poiseuille
plane Couette
similar for Poiseuille pipe flow (Ppf)
4
• phenomenology of plane Couette flow (pCf) – no linear instability mode – no overall advection experiments: Saclay group (1992–2002) result: – Re < Reg & 325 ! global stability of base flow ! systematic return to laminar flow when t → ∞ – Re > Reg ! regime at t → ∞ may be turbulent ! nonlinear instability against (localized) finite amplitude perturbation more precisely ! bifurcation diagram Couette flow
transient spots
200
sustained spots/stripes
300
! 280
Rg ! 325
featureless turbulence
400
! 415
500
R 5
vicinity of Reg ??? ! experiments (mostly S.Bottin’s PhD, 1998):∗ (1) nucleation of spots (“S”) ! amplitude of initial perturbation seems to diverge A ∼ 1/(Re − Reg)γ as Re → Reg+ (γ ???) and tend to zero as A ∼ Re−α for Re ) Reg (α ???) 0.6
0.6
0.6
Ru < R < Rg
R < Ru Q
R > Rg Q
Q
0.4
0.4
Ft
0.4
Ft
0.2
Ft
0.2
S1
0.2
S1 0.0
S 0
100
200
t (s)
300
400
0.0
S2 0
100
200
t (s)
300
400
0.0
S2 0
100
200
t (s)
300
(2) lifetime of turbulent state prepared at Re ) Reg quenched ! transients (“Q”) at Re < Reg lifetimes τ ! distribution N (τ * > τ ) ∼ exp(−τ /+τ ,) ! +τ , increases rapidly as Re → Reg− ∗ S.
Bottin, F. Daviaud, P. Manneville, O. Dauchot, Europhys. Lett. 43 (1998) 171–176.
6
400
400 0.05
1192 1220 1248 1260 1272 1284 1292
!1
0.04
!1
300
(s)
log N($’>$)
0
200
0.03 0.02 0.01 0.00 290
300
0
500
1000
0 295
305
$ (s)
early suggestion∗
320
330
R
100
!2
310
315
325
R
+τ , ∼ 1/(Reg − Re)β ,
β∼1
questioned by Hof. et al.† who propose +τ , ∼ exp(bRe) based on (i) analogy with results for Ppf and (ii) re-analysis of data ∗ S.
Bottin & H. Chat´ e, Eur. Phys. J. B 6 (1998) 143–155.
† B.
Hof, J. Westerweel, T.M. Schneider, B. Eckhardt, Nature 443 (2006) 59–62. 7
why ??? Hof et al. ! Ppf transients ≡ chaotic transients associated with homoclinic tangle ! low dim. dynamical systems viewpoint resting on existence of non-trivial unstable periodic orbits (UPOs) " such solutions exist in Ppf : Faisst & Eckhardt; Kerswell et al.∗
as well as in pCf : Nagata, Clever & Busse† ∗ for
a review, see: R.R. Kerswell, Nonlinearity 18 (2005) R17–R44
† M.
Nagata, J. Fluid Mech. 217 (1990) 519–527; R.M. Clever, F.H. Busse, J. Fluid Mech. 344 (1997) 137–153. 8
analogous situation for pCf
lift-up
f k y o ea ilit str tab + ins flow an
not a surprise ! mechanism ? cf. “regeneration” cycle: lift-up + instability propagation ! by-product of instability
me
streaks
streamwise vortices
streamwise modulation
nonlinear feed-back
fugitively observed in experiments see: Hof et al., Science 305 (2004) 1594–1598. 9
• homoclinic tangle ???
unstable periodic orbit with stable and unstable manifolds 1 transverse intersection ⇒ uncountable infinity of intersections (Poincar´ e)
S
S S
H0
U
F
F
H2 H1
U
U U
U
chaotic repellor (invariant set of homoclinic points) ! chaotic transients around it ! exponential distribution of transient lifetimes ! variation of decrement with control parameter ??? 10
• Ppf ! case not completely settled∗ exponential transient length distribution with decrement / 0 – either as (Reg − Re) for Re → Reg− (∼ critical behavior) – or as exp(−bRe) as Re 0 possible origin of discrepancies: – role of experimental conditions (∆P/∆x =cst. or cst. flux) – finite time/size effects ! is the analysis in terms of low dim. dynam. syst. relevant? temporal chaos OK if system is 0D but Ppf is quasi-1D • pCf ! beyond phenomenology ??? modeling in a deliberately spatiotemporal perspective to accounting for quasi-2D feature ! personal work in coll. with M. Lagha (PhD thesis, 2006) ∗ Peixinho
& Mullin, Phys. Rev. 96 (2006) 094501; Willis & Kerswell, Phys. Rev. Lett. 98 (2007) 014501; Hof et al., Nature 443 (2006) 59–62. 11
• modeling ! low dimensional ⇒ freeze all the space dependence ! ODEs governing a small set of amplitudes, cf. Lorenz model similar spirit for open flows ! Waleffe models ! well adapted only to confined systems (or systems with periodic b.c. at “short” distances) ! freeze cross-stream dependence, let in-plane dependence free ! partial differential equations, cf. Swift–Hohenberg model ! adapted to extended systems use Galerkin method to obtain model (2.5D) from primitive (3D) equations previous work ! stress-free b.c. at the plates ! interesting but unrealistic ! realistic no-slip b.c. explicit expression ! last slides (if someone is interested) 12
• a priori relevant general features – non-normal linear terms including lift-up mechanism – linear viscous damping – nonlinear advection terms preserving perturbation kinetic energy – linearly stable base flow for all Re • a posteriori relevant features (from numerical simulations) – extensivity of homogeneous turbulent state – sub-critical “laminar ↔ turbulent” transition (Reg ???) – transient states with exponentially decaying lifetime distribution – turbulent spots resemble what is experimentally observed ! present results relevant to the “critical/exponential” controversy ! define dimensionless system’s size ! aspect ratio Γx = Lx/d, Γz = Lz /d, d ≡ gap, Γ = Γx × Γz , here numerical experiments (periodic b.c.) – at moderate aspect-ratio Γ = 16 × 16 (D = 32 × 32 × 2) – at large aspect ratio Γ = 128 × 64 (D = 256 × 128 × 2)
compare to laboratory experiments ! typically 190 × 35 and to observed internal scale: coherent streak segments ∼ 6 × 3 13
" sub-criticality (Γ = 16 × 16, adiabatic decrease of R) 0.08
mean perturbation energy
0.07
transient 0.06 0.05 0.04
sustained
0.03 0.02 0.01 0 155
160
165
170
175
180
185
190
195
200
205
R
transition transient → “sustained” at Re & 175 & Reg
Remodel ∼ 0.5Relab. ! viscous dissipation and energy transfer to g g cross-stream small scales underestimated (truncation) but qualitative spatio-temporal features are preserved study first the decay transition turbulent → laminar
14
" transients (Γ = 16 × 16)
Q-type experiments: state prepared at Rei = 200 Re decreased to Ref < Reg 2 Rei cumulated histogram (ln scale)
0.0
!0.5
R=174.5 !1.0
!1.5
172
174
!2.0
170
173.5
!2.5
171 !3.0 0
1
173 172.5 2
3
transient length
4
5 4
x 10
! variation of slopes with Re ??? 15
!4
7
!7
x 10
6
slope (log scale)
slope (lin scale)
!8 5
4
3
2
!9
!10
1
0 170
171
172
173
R
174
175
!11 170
171
172
173
174
175
R
– exponential decrease of slopes, hence +τ , ∼ exp(bRe)) – off-aligned points at Re = 174 and 174.5 suggest cross-over to critical behavior very close to Re = 175 statistical improvement beyond reach of numerical means used for that experiment ! explanation ??? visualizations for Γ = 16 × 16 do not discriminate temporal from spatio-temporal behavior 16
temporal is likely in view of size of streak segments compared to Γ ! consider a larger domain ! Γ = 128 × 64 (8 × 4 times larger) result: turbulent state can be maintained over large time periods well below R = Rg = 175 ! expensive to study numerically ! limited number of trials ! no direct statistics (experimentalists do a better job, but with other limitations) •
video of quench at Rf = 167
! nucleation of laminar domains that expand ! late stage is a retraction of the turbulent domain ! suggests that, for Γ = 16 × 16, last stage is also a retraction ! turn the question to “when does the transient begins ?” ! Pomeau’s idea of nucleation expected from the connection between a globally sub-critical bifurcation and a first-order thermodynamic phase transition∗ ∗ in:
Berg´ e, Pomeau, Vidal, L’Espace Chaotique (Hermann, 1998) Chapter IV. 17
test the nucleation idea ?
!
return to Γ = 16 × 16 R=175 sustained (?) turbulence
R=200
12
10 9
10
histrogram (log scale)
histogram (log scale)
8 7 6 5 4 3 2
8
6
4
2
1 0 0.05
0.055
0.06
0.065
0.07
0.075
E +E 0
1
0.08
0.085
0.09
0.095
0 0.02
0.03
0.04
0.05
E0 + E1
0.06
0.07
0.08
– Re = 200 ! Gaussian histogram = incoherent superposition chaotic mixture of laminar and chaotic small structures – Re = 175 & Reg ! max shifts / ; exponential tail at low energy coherent large deviation ! germ that grows if large enough ! irreversible decay to laminar stage when Et < Elim 18
evidence for Elim & 0.025 (ln Elim & −3.7) at moderate Γ mean transient length (log)
!2
turbulent stage !3
log(E t )
initial retractation !4
!5
!6
R=170 %=16x16
!7 0
2000
final viscous decay
4000
6000
8000
10000
12000
8.0
transients R=170, % =16x16 final viscous relaxation
system may return to turbulent state
7.5
7.0
6.5
6.0 0
0.01
0.02
0.03
0.04
energy cut!off
t
back to the wide system ! long time series R=170 0.050
0.045
0.040
4
4.5
5
5.5
t
6
6.5 4
x 10
19
• video 1 ! R = 170, full resolution 59000 < t < 67000 shows existence of large laminar domains that last very long ! wait to see the system decay ??? ! compare to small system (Γ = 16 × 16) • video 2 ! R = 170, “low” resolution 47000 < t < 67000 obtained by binning original large domain into squares 8 × 8 further grouped to give larger rectangular or square sub-domains, i.e. 16 × 16 to be used for comparison with Γ = 16 × 16 system yields individual time series analogous to those of smaller systems ! construct histograms
20
9
9
32 longest transients % = 16x16
8
8 7
log(N)
log(N)
7
6
6
5
5
4
4
3
[16x16]
0
0.01
0.02
0.03
Et
0.04
0.05
0.06
0.07
3 0
0.01
0.02
0.03
Et
0.04
0.05
0.06
0.07
! the 16 × 16 system “dies” at the end of a transient (Et < Elim ) while a given [16 × 16] sub-domain of the 128 × 64 system that become laminar can “resuscitate” by contamination from turbulent neighbors ! first guess : convert frequency of laminar domains of given size in the 128 × 64 system into transient length distribution for the 16 × 16 system (! may need correction due to subtle size effects) 21
size effects ??? ! long-range processes linked to pressure (present in the model 3= more simplified models not directly derived from NS equations, e.g. CMLs) best seen when studying hysteresis at the transition up to now turbulent → laminar transition via nucleation and development of laminar patches now laminar → turbulent ! starting point ? a) localized spot ! two parameters: extension and intensity ! systematic study left for future work b) more or less homogeneous low amplitude “noise” ! to be presented (briefly)
22
relevant “noise” obtained by “attenuating” a turbulent solution
0.06
0.05
R=175
Et
0.04
0.03
R=174 0.02
0.01 0
1000
2000
3000
4000
t
refine to determine “edge of turbulence” ! with this i.c. 174.925 < R < 174.95 change i.c.? 23
change energy contents of i.c. at given R or change R at given i.c. “noisy i.c.” ! rough bifurcation diagram but “basin boundary” depends on i.c. amplitude and homogeneity 0.06
very low energy noisy i.c. ??? turbulent state
0.05
Et
0.04 0.03 0.02
"basin boundary"
0.01
laminar state 0
170
172
174
R
176
178
180
! different transition due to different early stage: initial smoothing ! leaves few germs ! germs grow if R large ! next form transverse bands ! final turbulent invasion stage if R significantly above 200
24
! study laminar–turbulent coexistence and fronts produce a banded i.c. and change R !6
6
x 10
initial front speed
4 2 0 !2 !4 !6 !8
175
180
185
190
195
200
205
R
25
evidence of non-local effects ! speed depends on turbulent fraction 0.08
R=200 0.07
E
t
0.06 0.05 0.04 0.03 fit: Et=3.49 10!14 t3 ! 6.51 10!10 t2 + 5.24 10!6 t + 3.23 10!2
0.02 0
0.5
1
t
1.5
2 4
x 10
interpretation is delicate : periodic b.c. influence band orientation but role of instantaneous turbulent/laminar global pattern is obvious both for onset and decay of turbulence 26
conclusion • context: sub-critical transition to turbulence ! more specifically Ppf & pCf ! similar (same ?) problem • origin of difficulties: nature of the non-trivial solution competing with the base state • answers ? ! dynamical systems and chaos stems from temporal analysis valid for confined system ! classical theory of chaotic transients existence of unstable periodic solutions + tangle these solutions exist (calculated/observed) but is this enough ? – pipe Poiseuille flow ! quasi 1D – plane Couette flow ! quasi-2D ! Pomeau (1986, 1998) ! nucleation problem in connection with first-order (thermodynamic) phase transitions 27
• modeling approach ! dimensional reduction in physical space ! different from standard dynamical-system viewpoint low-order truncation of a Galerkin projection of NS equations – negative feature : energy transfer through cross-stream (small) scales underestimated ! lowered transitional range – positive aspect ! correctly extract energy from base flow through interplay of streamwise vortices and streaks (large in-plane structures) ! qualitatively reproduces hydrodynamical features (e.g. non-local pressure effects) and transition properties • even in absence of firm conclusions, most interesting results : " better appreciation of drawbacks and virtues of dynamical system approach and phase transition viewpoint : ! reinterpretation of transient length distribution " glimpse on origin of complications : size effects and role of topology of laminar/turbulent domains " suggests to look at Ppf along same lines (quasi-1D 3= 0D)
28
• two levels of open questions and perspectives " immediate, concrete, hydrodynamical consequences for other globally sub-critical flows experiencing wild transition to turbulence via streaks, streamwise vortices, spots. . . and for transition control " abstract and general: role of noise and statistics ! nature of the turbulent attractor and thermodynamic approach to far-from-equilibrium systems theory in continuous media Acknowledgments M.Lagha (co-worker), C.Cossu, J.-M.Chomaz, P.Huerre (LadHyX), B.Eckhardt, J.Schumacher (Marburg), S.Bottin, O.Dauchot, F.Daviaud, A.Prigent (Saclay),
L.Tuckerman,
D.Barkley
(ESPCI); IDRIS (Orsay) projects #61462, 72138 29
• the model
! base flow u = ub(y) = y polynomial expansion of perturbations (here lowest order trunc.) !
*
u ,w
*
"
= {U0(x, z, t), W0(x, z, t)} B(1 − y 2) + {U1, W1} Cy(1 − y 2)
v * = V1(x, z, t)A(1 − y 2)2
1
1
1
y0
y0
y0
!1 0
U0 and W0
1
!1 !1
U1 and W1
1
!1 0
V1
1
anticipated to be good enough since – perturbations known to occupy the full gap for Re ∼ Reg – no-slip functions dissipate more than stress-free basis functions – Galerkin expansion possible (but tedious) at higher orders 30
• continuity equation ∂x u* + ∂y v * + ∂z w * = 0 by projection ! " even part (streaks ! {U0(z)}) ∂ x U0 + ∂ z W 0 = 0 " odd part (streamwise vortices ! {V1(z), W1(z)}) √ ∂xU1 − βV1 + ∂z W1 = 0 β = 3 ≈ 1.73
31
• linear momentum
du x 2 * ∂tv* + v* · ∇v* = −∇p*−ub∂xv*−v * dy bˆ + ν ∇ v
" in-plane, even part (streamwise only, spanwise similar) ∂tU0 + NU0 = −∂xP0 − a1∂xU1 − a2V1 + Re−1 (∂xx + ∂zz − γ0) U0
NU0 = α1(U0∂xU0 + W0∂z U0) + α2(U1∂xU1 + W1∂z U1) + α3V1U1 " in-plane, odd part (streamwise only, spanwise similar) ∂tU1 + NU1 = − ∂xP1 − a1∂xU0 + Re−1(∂xx + ∂zz − γ1||)U1
NU1 = α2(U0∂xU1 + U1∂xU0 + W0∂z U1 + W1∂z U0) − α4V1U0
" wall-normal
∂tV1 + NV1 = −βP1 + Re−1(∂xx + ∂zz − γ1⊥ )V1 NV1 = α5(U0∂xV1 + W0∂z V1)
all coefficients combinations of integrals in the form #1 n $m % k & (−1)k 2 m Jn,m = 0 y (1 − y ) dy = k=0 m 2k+n+1
32
