Wave interaction theory can be used as a tool to understand and predict instability in a variety of homogeneous and stratified shear flows. It is, however, most often limited to piecewise-linear profiles of the shear layer background velocity, in which stable vorticity wave modes can be easily identified and their interaction quantified. This approach to understanding shear flow instability is extended herein to smooth shear layer profiles. We describe a method, by which the stable vorticity wave modes can be identified, and show that their interaction results in an excellent description of the stability properties of the smooth shear layer, thus demonstrating the presence of the wave interaction mechanism in smooth shear flows.
REFERENCES
1.
P.
Baines
and H.
Mitsudera
, “On the mechanism of shear flow instabilities
,” J. Fluid Mech.
276
, 327
–342
(1994
).2.
J. R.
Carpenter
, E. W.
Tedford
, E.
Heifetz
, and G. A.
Lawrence
, “Instability of stratified shear flow: Review of a physical mechanism based on interacting waves
,” Appl. Mech. Rev.
64
, 060801
(2013
).3.
A.
Guha
and G. A.
Lawrence
, “A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities
,” J. Fluid Mech.
755
, 336
–364
(2014
).4.
W. D.
Smyth
and J. R.
Carpenter
, Instability in Geophysical Flows
(Cambridge University Press
, 2019
).5.
L.
Redekopp
, “Elements of instability theory for environmental flows
,” in Environmental Stratified Flows
(Kluwer
, Boston
, 2001
).6.
J.
Holmboe
, “On the behavior of symmetric waves in stratified shear layers
,” Geofys. Publ.
24
, 67
–113
(1962
).7.
J.
Carpenter
, N.
Balmforth
, and G.
Lawrence
, “Identifying unstable modes in stratified shear layers
,” Phys. Fluids
22
, 054104
(2010
).8.
K.
Iga
, “Shear instability as a resonance between neutral waves hidden in a shear flow
,” J. Fluid Mech.
715
, 452
–476
(2013
).9.
E.
Heifetz
and J.
Methven
, “Relating optimal growth to counterpropagating Rossby waves in shear instability
,” Phys. Fluids
17
, 064107
(2005
).10.
P. G.
Drazin
and W. H.
Reid
, Hydrodynamic Stability
, 2nd ed. (Cambridge University Press
, 2004
).11.
P.
Schmid
and D.
Henningson
, Stability and Transition in Shear Flows
(Springer Verlag
, 2001
), Vol. 142.12.
P.
Hazel
, “Numerical studies of the stability of inviscid stratified shear flows
,” J. Fluid Mech.
51
, 39
–61
(1972
).13.
P. G.
Drazin
, Introduction to Hydrodynamic Stability
(Cambridge University Press
, 2002
), Vol. 32.14.
P. G.
Baines
, S.
Majumdar
, and H.
Mitsudera
, “The mechanics of the Tollmein-Schlichting wave
,” J. Fluid Mech.
312
, 107
–124
(1996
).15.
J.
Carpenter
, A.
Guha
, and E.
Heifetz
, “A physical interpretation of the wind-wave instability as interacting waves
,” J. Phys. Oceanogr.
47
, 1441
–1455
(2017
).16.
R.
Cairns
, “The role of negative energy waves in some instabilities of parallel flows
,” J. Fluid Mech.
92
, 1
(1979
).17.
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