Motivated by the need for effective vortex control, the character of absolute and convective instabilities (AI/CI) of incompressible and high-Mach number slender vortices with axial-velocity deficit is studied. Attention is focused on the disturbance modes which lead to the maximum absolute growth rate, and their dependence on flow conditions such as axial-flow profile, Reynolds number, and Mach number. A significant difference between the AI/CI and temporal-instability characters of the vortices occurs as the axial velocity deficit reduces. These theoretical results are applied to the flow region where vortex breakdown happens. It is found that the breakdown region is absolutely unstable, where waves are dominated by the spiral disturbance with lowest azimuthal wave number, in reasonable agreement with measurement.
REFERENCES
1.
R. L. Ash and M. R. Khorrami,“Vortex stability,” in Fluid Vortices, edited by S. I. Green (Kluwer, Dorcrecht, 1995), p. 315.
2.
P.
Huerre
and P. A.
Monkewitz
, “Local and global instabilities in spatially developing flows
,” Annu. Rev. Fluid Mech.
22
, 473
(1990
).3.
P. Huerre and M. Rossi, “Hydrodynamic instabilities in open flow,” in Hydrodynamics and Nonlinear Instabilities, edited by C. Godreche and P. Manneville (Cambridge University Press, Cambridge, 1998).
4.
C.
Olendraru
, A.
Sellier
, M.
Rossi
, and P.
Huerre
, “Absolute/convective instability of the Batchelor vortex
,” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron.
323
, 153
(1996
).5.
C.
Olendraru
, A.
Sellier
, M.
Rossi
, and P.
Huerre
, “Inviscid instability of the Batchelor vortex: Absolute-convective transition and spatial branches
,” Phys. Fluids
11
, 1805
(1999
).6.
I.
Delbende
, J. M.
Chomaz
, and P.
Huerre
, “Absolute/convective instabilities in the Batchelor vortex: A numerical study of the linear impulse response
,” J. Fluid Mech.
355
, 229
(1998
).7.
T.
Loiseleux
, J. M.
Chomaz
, and P.
Huerre
, “The effect of swirl on jet and wake: Linear instability of Rankine vortex with axial flow
,” Phys. Fluids
10
, 1120
(1998
).8.
M.-Y. Sun, “Absolute and convective instability of swirling flows,” M.S. thesis, University of Science and Technology of China, Hefei, China, 1995.
9.
J.-Z.
Wu
, X.-Y.
Lu
, A. G.
Denny
, M.
Fan
, and J.-M.
Wu
, “Post-stall flow control on an airfoil by local unsteady forcing
,” J. Fluid Mech.
371
, 21
(1998
).10.
A. Seifert and L. G. Pack, “Oscillatory excitation of unsteady compressible flows over airfoils at flight Reynolds numbers,” AIAA Pap. 99-0925 (1999).
11.
C.-M.
Ho
and P.
Huerre
, “Perturbed free shear layers
,” Annu. Rev. Fluid Mech.
16
, 365
(1984
).12.
M. F.
Yao
, L. B.
Jiang
, J. Z.
Wu
, H. Y.
Ma
, J. Y.
Pan
, and H. J.
Cai
, “Delaying vortex breakdown by waves
,” AIAA Pap.
89
, 1000
(1989
).13.
P. A.
Monkewitz
and K. D.
Sohn
, “Absolute instability in hot jet and their control
,” AIAA Pap.
86
, 1882
(1986
).14.
X. Ming and C. H. Dai, “A new phenomenon of acoustic streaming,” in Proceedings of the International Conference on Fluid Dynamic Measurement and Its Applications, edited by X. Shen and X. Sun, Beijing, China, 25–28 October 1989 (unpublished).
15.
A.
Lifshitz
and D. D.
Holm
, “Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity
,” Phys. Lett. A
157
, 481
(1991
).16.
F.-L.
Zhu
, X.-Y.
Yin
, and J.-Z.
Wu
, “Short-wave instability of strained swirling vortex
,” AIAA Pap.
99
, 0139
(1999
).17.
M. R.
Khorrami
, “On the viscous modes of instability of a trailing line vortex
,” J. Fluid Mech.
225
, 197
(1991
)18.
J. A. K.
Scott
and P. W.
Duck
, “The stability of a trailing-line vortex in compressible flow
,” J. Fluid Mech.
269
, 323
(1994
).19.
X.-Y. Yin, D.-J. Sun, and J.-Z. Wu, “The instability of compressible Burgers and Sullivan vortices” (unpublished).
20.
M. R.
Khorrami
, M. R.
Malik
, and R. L.
Ash
, “Application of spectral collocation techniques to the stability of swirling flows
,” J. Comput. Phys.
81
, 206
(1989
).21.
M. R.
Khorrami
, “Stability of a compressible axisymmetric swirling jet
,” AIAA J.
33
, 650
(1995
).22.
E. W.
Mayer
and K. G.
Powell
, “Viscous and inviscid instabilities of a trailing vortex
,” J. Fluid Mech.
245
, 91
(1992
).23.
R. J.
Deissler
, “The convective nature of instability in plane Poiseulle flow
,” Phys. Fluids
30
, 2303
(1987
).24.
D. O.
Staley
and R. L.
Gall
, “Hydrodynamic instability of small eddies in a tornado vortex
,” J. Atmos. Sci.
41
, 422
(1984
).25.
S.
Leibovich
and K.
Stewartson
, “A sufficient condition for the instability of columnar vortices
,” J. Fluid Mech.
126
, 335
(1983
).26.
K. A.
Emanuel
, “A note on the instability of columnar vortices
,” J. Fluid Mech.
145
, 235
(1984
).27.
S.
Ragab
and M.
Sreedhar
, “Numerical simulation of vortices with axial velocity deficits
,” Phys. Fluids
7
, 549
(1995
).28.
M. R.
Khorrami
, “Behavior of asymmetric unstable modes f a trailing line vortex near the upper neutral curve
,” Phys. Fluids A
4
, 1310
(1992
).29.
M. R. Khorrami, “A study of the temporal stability of multiple cell vortices,” NASA-CR 4261 (1989).
30.
A. K.
Garg
and S.
Leibovich
, “Spectral characteristics of vortex breakdown flow fields
,” Phys. Fluids
22
, 2053
(1979
).31.
J. M.
Delery
, “Aspects of vortex breakdown
,” Prog. Aerosp. Sci.
30
, 1
(1994
).32.
S.
Wang
and Z.
Rusak
, “On the stability of an axisymmetry rotating flow in a pipe
,” Phys. Fluids
8
, 1007
(1996a
).33.
S.
Wang
and Z.
Rusak
, “The stability of noncolumnar swirling flows
,” Phys. Fluids
8
, 1017
(1996b
).34.
J. C.
Tromp
and P. S.
Beran
, “The role of nonunique axisymmetric solutions in 3-D vortex breakdown
,” Phys. Fluids
9
, 992
(1997
).35.
S.
Wang
and Z.
Rusak
, “The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown
,” J. Fluid Mech.
340
, 177
(1997
).36.
S.
Leibovich
, “Vortex stability and breakdown: Survey and extension
,” AIAA J.
22
, 1192
(1984
).37.
C. Y.
Tsai
and S. E.
Widnall
, “Examination of group velocity criterion for breakdown of vortex flow in a divergent duct
,” Phys. Fluids
23
, 864
(1980
).38.
A. J. Bilanin, Ph.D. thesis, MIT, 1973.
39.
W.
Koch
, “Local instability characteristics and frequency determination of self-excited wake flows
,” J. Sound Vib.
99
, 53
(1985
).40.
T. Sarpkaya (private communication).
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.
