The potential for transient growth in compressible boundary layers is studied. Transient amplification is mathematically associated with a non‐orthogonal eigenvector basis, and can amplify disturbances although the spectrum of the linearized evolution operator is entirely confined to the stable half‐plane. Compressible boundary layer flow shows a large amount of transient growth over a wide range of parameter values. The disturbance size is here measured by a positive definite energy like quantity that has been derived such that pressure‐related transfer terms in its evolution equation mutually cancel. The maximum of the transient growth is found for structures which are independent of the streamwise direction and is found to scale with R2. This suggests that the transient growth originates from the same lift‐up mechanism found to give large growth in incompressible shear flows. The maximum growth is also found to increase with Mach number. In compressible flow, disturbances that experience optimal transient growth can be excited naturally by a non‐linear interaction of oblique unstable first mode waves. Thus, a triggering of transient growth may account for the difference in timescales between the fast oblique breakdown process and traditional secondary instability.

1.
L. Lees and C. C. Lin, “Investigation of the stability of the laminar boundary layer in a compressible fluid,” NACA Tech. Note No. 1115, 1946.
2.
L. M.
Mack
, “
Stability of the laminar compressible boundary layer according to a direct numerical solution
,” in
AGARDograph
97
, Part I,
329
(
1965
).
3.
L. M.
Mack
, “
Computation of the stability of the laminar compressible boundary layer
,”
Methods Comput. Phys.
4
,
247
(
1965
).
4.
L. M.
Mack
, “
Linear stability theory and the problem of supersonic boundary layer transition
,”
AIAA J.
13
,
278
(
1975
).
5.
L. M. Mack, “Boundary layer stability theory,”AGARD Conference Proceedings No. 224, pp. 1–1–1–22, NATO, Paris, 1984.
6.
D.
Arnal
, “
Stability and transition of two-dimensional laminar boundary layers in compressible flow over an adiabatic wall
,”
Rech. Aerosp.
1988–4
,
15
(
1988
).
7.
M. R. Malik, “Prediction and control of transition in hypersonic boundary layers,” AIAA Paper No. 87–1414, 1987.
8.
M. R.
Malik
, “
Prediction and control of transition in supersonic and hypersonic boundary layers
,”
AIAA J.
27
,
1487
(
1989
).
9.
Z. H. Zurigat, A. H. Nayfeh, and J. A. Masad, “Effect of pressure gradient on the stability of compressible boundary layers,” AIAA Paper No. 90–1451, 1990.
10.
P.
Balakumar
and
M. R.
Malik
, “
Waves produced from a harmonic source in a supersonic boundary layer flow
,”
J. Fluid Mech.
242
,
323
(
1992
).
11.
L.
Boberg
and
U.
Brosa
, “
Onset of turbulence in a pipe
,”
Z. Naturforschung
43a
,
697
(
1988
).
12.
K. M.
Butler
and
B. F.
Farrell
, “
Three-dimensional optimal perturbations in viscous shear flow
,”
Phys. Fluids A
4
,
1637
(
1992
).
13.
L. H.
Gustavsson
, “
Energy growth of three-dimensional disturbances in plane Poiseuille flow
,”
J. Fluid Mech.
224
,
241
(
1991
).
14.
S. C.
Reddy
and
D. S.
Henningson
, “
Energy growth in viscous channel flows
,”
J. Fluid Mech.
252
,
209
(
1993
).
15.
P. J.
Schmid
and
D. S.
Henningson
, “
Optimal energy density growth in Hagen-Poiseuille flow
,”
J. Fluid Mech.
277
,
197
(
1994
).
16.
L. N.
Trefethen
,
A. E.
Trefethen
,
S. C.
Reddy
, and
T. A.
Driscoll
, “
Hydrodynamic stability without eigenvalues
,”
Science
261
,
578
(
1993
).
17.
M. R.
Malik
, “
Numerical methods for hypersonic boundary layer stability
,”
J. Comput. Phys.
86
,
376
(
1990
).
18.
L. S.
Hultgren
and
L. H.
Gustavsson
, “
Algebraic growth of disturbances in a laminar boundary layer
,”
Phys. Fluids
24
,
1000
(
1981
).
19.
P. J.
Schmid
and
D. S.
Henningson
, “
A new mechanism for rapid transition involving a pair of oblique waves
,”
Phys. Fluids A
4
,
1986
(
1992
).
20.
D. S.
Henningson
,
A.
Lundbladh
, and
A. V.
Johansson
, “
A mechanism for bypass transition from localized disturbances in wall bounded shear flows
,”
J. Fluid Mech.
250
,
169
(
1993
).
21.
G.
Kreiss
,
A.
Lundbladh
, and
D. S.
Henningson
, “
Bounds for threshold amplitudes in subcritical shear flows
,”
J. Fluid Mech.
270
,
175
(
1994
).
22.
S.
Berlin
,
A.
Lundbladh
, and
D. S.
Henningson
, “
Spatial simulations of oblique transition
,”
Phys. Fluids
6
,
1949
(
1994
).
23.
H. Fasel, A. Thumm, and H. Bestek, “Direct numerical simulation of transition in supersonic boundary layers: Oblique breakdown,” Transitional and Turbulent Compressible Flows, edited by L. D. Kral and T. A. Zang, ASME FED-Vol. 151, pp. 77–92.
24.
R. J.
Gathmann
,
M.
Si-Ameur
, and
F.
Mathey
, “
Numerical simulations of three-dimensional natural transition in the compressible confined shear layer
,”
Phys. Fluids A
5
,
2946
(
1993
).
25.
C.-L.
Chang
and
M. R.
Malik
, “
Oblique-mode breakdown and secondary instability in supersonic boundary layers
,”
J. Fluid Mech.
273
,
323
(
1994
).
26.
N. D.
Sandham
,
N. A.
Adams
, and
L.
Kleiser
, “
Direct simulation of breakdown to turbulence following oblique instability waves in a supersonic boundary layer
,”
Appl. Sci. Res.
54
,
223
(
1995
).
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