Two possible initial paths of transition to turbulence in simple wall-bounded shear flows are examined by looking at the development in space of infinitesimal disturbances. The first is the—by-now-classical—transient growth scenario which may have an important role in the bypass transition of flows for which traditional eigenmode analysis predicts asymptotic stability. It is studied by means of a simplified parabolic model justified by the underlying physics of the problem; results for optimal disturbances and maximum transient growth are found in excellent agreement with computations based on the full Orr–Sommerfeld/Squire equations. The second path starts with the exponential amplification, in nominally subcritical conditions, of modal disturbances superposed to base flows mildly distorted compared to their idealized counterparts. Such mean flow distortions might arise from the presence of unwanted external forcing related, for example, to the experimental environment. A technique is described that is capable of providing the worst case distortion of fixed norm for any ideal base flow, i.e., that base flow modification capable of maximizing the amplification rate of a given instability mode. Both initial paths considered here provide feasible initial conditions for the transition process, and it is likely that in most practical situations algebraic and exponential growth mechanisms are concurrently at play in destabilizing plane shear flows.

1.
G. I.
Taylor
, “
Some recent developments in the study of turbulence
,” Proceedings of the Fifth International Congress of Applied Mechanics, Cambridge, MA, 12–16 September
1938
, pp.
294
310
(
1939
).
2.
P. S.
Klebanoff
,
K. D.
Tidstrom
, and
L. M.
Sargent
, “
The three-dimensional nature of boundary layer instability
,”
J. Fluid Mech.
12
,
1
(
1962
).
3.
P.
Luchini
, “
Reynolds-number-independent instability of the boundary layer over a flat surface: Optimal perturbations
,”
J. Fluid Mech.
404
,
289
(
2000
).
4.
M.
Matsubara
and
P. H.
Alfredsson
, “
Disturbance growth in boundary layers subjected to free-stream turbulence
,”
J. Fluid Mech.
430
,
149
(
2001
).
5.
B. F.
Farrell
, “
Optimal excitation of perturbations in viscous shear flow
,”
Phys. Fluids
31
,
2093
(
1988
).
6.
K. M.
Butler
and
B. F.
Farrell
, “
Three-dimensional optimal perturbations in viscous shear flow
,”
Phys. Fluids A
4
,
1637
(
1992
).
7.
S. C.
Reddy
and
D. S.
Henningson
, “
Energy growth in viscous channel flow
,”
J. Fluid Mech.
252
,
209
(
1993
).
8.
P. J.
Schmid
and
D. S.
Henningson
, “
Optimal energy density growth in Hagen–Poiseuille flow
,”
J. Fluid Mech.
277
,
197
(
1994
).
9.
P.
Corbett
and
A.
Bottaro
, “
Optimal perturbations for boundary layers subject to streamwise pressure gradient
,”
Phys. Fluids
12
,
120
(
2000
).
10.
P.
Corbett
and
A.
Bottaro
, “
Optimal linear growth in swept boundary layers
,”
J. Fluid Mech.
435
,
1
(
2001
).
11.
E.
Reshotko
and
A.
Tumin
, “
Spatial theory of optimal disturbances in circular pipe flow
,”
Phys. Fluids
13
,
991
(
2001
).
12.
A.
Tumin
and
E.
Reshotko
, “
Spatial theory of optimal disturbances in boundary layer
,”
Phys. Fluids
13
,
2097
(
2001
).
13.
P.
Luchini
and
A.
Bottaro
, “
Görtler vortices: A backward-in-time approach to the receptivity problem
,”
J. Fluid Mech.
363
,
1
(
1998
).
14.
P.
Luchini
, “
Reynolds-number-independent instability of the boundary layer over a flat surface
,”
J. Fluid Mech.
327
,
101
(
1996
).
15.
P.
Andersson
,
M.
Berggren
, and
D. S.
Henningson
, “
Optimal disturbances and bypass transition in boundary layer
,”
Phys. Fluids
11
,
134
(
1999
).
16.
S.
Zuccher
, “
Receptivity and control of flow instabilities in a boundary layer
,” Ph.D. thesis, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Italy, December
2001
.
17.
M. T.
Landahl
, “
A note on an algebraic instability of inviscid parallel shear flows
,”
J. Fluid Mech.
98
,
243
(
1983
).
18.
P.
Schmid
and
H. S.
Henningson
, Stability and Transition in Shear Flows (Springer-Verlag, New York,
2001
).
19.
O.
Dauchot
and
F.
Daviaud
, “
Streamwise vortices in plane Couette flow
,”
Phys. Fluids
7
,
901
(
1995
).
20.
D.
Barkley
and
L. S.
Tuckerman
, “
Stability analysis of perturbed plane Couette flow
,”
Phys. Fluids
11
,
1187
(
1999
).
21.
J.
Lerner
and
E.
Knobloch
, “
The long-wave instability of a defect in a uniform parallel shear
,”
J. Fluid Mech.
189
,
117
(
1988
).
22.
B.
Dubrulle
and
J.-P.
Zahn
, “
Nonlinear instability of viscous plane Couette flow. 1. Analytical approach to a necessary condition
,”
J. Fluid Mech.
231
,
561
(
1991
).
23.
A.
Bottaro
,
P.
Corbett
, and
P.
Luchini
, “
The effect of base flow variation on flow stability
,”
J. Fluid Mech.
476
,
293
(
2003
).
24.
M. I.
Gavarini
, “
Initial stage of transition and optimal control of streaks in Hagen–Poiseuille flow
,” Ph.D. thesis, Department of Mechanical Engineering and Marine Technology, Delft University of Technology, Delft, The Netherlands, September
2004
.
25.
M. I.
Gavarini
,
A.
Bottaro
, and
F. T. M.
Nieuwstadt
, “
Eigenvalues sensitivity to base flow variations in Hagen–Poiseuille flow
,” in Proceedings of ASME FEDSM ’02, Montreal, Quebec, Canada, 14–18 July
2002
, Paper FEDSM2002-31050.
26.
S. J.
Chapman
, “
Subcritical transition in channel flows
,”
J. Fluid Mech.
451
,
35
(
2002
).
27.
A.
Tumin
, “
Receptivity of pipe Poiseuille flow
,”
J. Fluid Mech.
315
,
119
(
1996
).
28.
P. A.
Libby
and
H.
Fox
, “
Some perturbation solutions in laminar boundary-layer theory
,”
J. Fluid Mech.
17
,
433
(
1964
).
29.
W. M. F.
Orr
, “
The stability or instability of the steady motions of a perfect liquid and of a viscous liquid
,”
Proc. R. Ir. Acad., Sect. A
27
,
9
(
1907
).
30.
J. A. C.
Weideman
and
S. C.
Reddy
, AMatLabDifferentiation Matrix Suite, volume 26. ACM Trans. Math. Soft., 2000. Available at http://dip.sun.ac.za/∼weideman/research/differ.html.
31.
P. J.
Schmid
,
A.
Lundbladh
, and
D. S.
Henningson
, “
Spatial evolution of disturbances in plane Poiseuille flow
,” in Transition Turbulence and Combustion, edited by
M. Y.
Hussaini
,
T. B.
Gatski
, and
T. L.
Jackson
(Kluwer Academic, Dordrecht,
1994
).
32.
B. F.
Farrell
and
P.
Ioannou
, “
Stochastic forcing of the linearized Navier–Stokes equations
,”
Phys. Fluids A
5
,
2600
(
1993
).
33.
B.
Bamieh
and
P.
Dahle
, “
Energy amplification in channel flows with stochastic excitation
,”
Phys. Fluids
13
,
3258
(
2001
).
34.
L. N.
Trefethen
,
A. E.
Trefethen
,
S. C.
Reddy
, and
T. A
Driscoll
, “
Hydrodynamic stability without eigenvalues
,”
Science
261
,
578
(
1993
).
35.
O.
Dauchot
and
F.
Daviaud
, “
Finite amplitude perturbation and spots growth mechanism in plane Couette flow
,”
Phys. Fluids
7
,
335
(
1995
).
36.
G. J.
Balas
,
J. C.
Doyle
,
K.
Glover
,
A.
Packard
, and
R.
Smith
, μ-Analysis and Synthesis Toolbox. User’s Guide, Version 3, The MathWorks,
2001
.
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