Global instabilities in spatially developing flows: Non-normality and nonlinearity

Annual Review of Fluid Mechanics

Volumn 37, Issue , 2005, Pages 357-392

  Chomaz, Jean Marc ^a  

^a ECOLE POLYTECHNIQUE   (France)

Author keywords

Absolute convective instabilites; Fronts; Global modes; Passive and active control; Transient growth

Indexed keywords

BOUNDARY LAYERS; BUBBLES (IN FLUIDS); EIGENVALUES AND EIGENFUNCTIONS; JETS; NONLINEAR SYSTEMS; OPEN CHANNEL FLOW; WAKES;

GLOBAL MODE THEORY; INSTABILITY WAVES; NONLINEAR INSTABILITY; NONLINEARITIES;

FLUID MECHANICS;

FLUID FLOW; FLUID MECHANICS; INSTABILITY;

FLUID FLOW; FLUID MECHANICS; INSTABILITY;

EID: 14544278839     PISSN: 00664189     EISSN: None     Source Type: Book Series    
DOI: 10.1146/annurev.fluid.37.061903.175810     Document Type: Review

References (127)

1. Ahlers G, Cannel DS. 1983. Vortex-front propagation in rotating Couette-Taylor flow. Phys. Rev. Lett. 50:1583-86
2. Aiken CM, Moore AM, Middleton JH. 2003. Non-normal perturbation growth in idealised island and headland wakes. Dyn. Atmos. Oceans 37:171-95
3. Barkley D, Henderson RD. 1996. Floquet stability analysis of the periodic wake of a circular cylinder. J. Fluid Mech. 322:215-41
4. Bender CM, Orszag SA. 1978. Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill
5. Bers A. 1975. Linear waves and instabilities. In Physique des Plasmas ed. C De Witt, J Peyraud, pp. 117-215. New York: Gordon & Breach
6. Brancher P, Chomaz JM. 1997. Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78:658-61
7. Brandt L, Cossu C, Chomaz JM, Huerre P, Henningson DS. 2003. On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485:221-42
8. Brevdo L. 1991. 3-Dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows. Z. Angew. Math. Phys. 48:290-309
9. Brevdo L, Bridges T. 1996. Absolute and convective instabilities of spatially periodic flows. Philos. Trans. R. Soc. London Ser A 354:1027-64
10. Broze G, Hussain F. 1994. Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors. J. Fluid Mech. 263:931-32
11. Büchel PB, Lücke M, Roth D, Schmitz R. 1996. Pattern selection in the absolutely unstable régime as a nonlinear eigenvalue problem: Taylor vortices in axial flow. Phys. Rev. E 53:4764-77
12. Buell JC, Hueffe P. 1988. Inflow/outflow boundary conditions and global dynamics of spatial mixing layers. In Studying Turbul. Using Numerical Simulation Databases-II. NASA Ames/Stanford Cent. Turbul. Res. Rep. CTR-S88:19-27
13. Butler KM, Farrell BE 1992. 3-dimensional optimal perturbations in viscous shear-flow. Phys. Fluids A 4:1637-50
14. Chomaz JM. 1992. Absolute and convective instabilities in nonlinear systems. Phys. Rev. Lett. 69:1931-34
15. Chomaz JM. 2003. Fully nonlinear dynamiques of parallel wakes. J. Fluid Mech. 495:57-75
16. Chomaz JM. 2004. Transition to turbulence in open flows: what linear and fully nonlinear local and global theories tell us. Eur J. Mech. B. 23:385-99
17. Chomaz JM, Couairon A. 1999. Against the wind. Phys. Fluids 11:2977-83
18. Chomaz JM, Couairon A. 2000. Propagating pattern selection and causality reconsidered. Phys. Rev. Lett. 84:1910-13
19. Chomaz JM, Couairon A, Julien S. 1999. Absolute and convective nature of the Eckhaus and zigzag instability with throughflow. Phys. Fluids 11:3369-73
20. Chomaz JM, Huerre P, Redekopp LG. 1988. Bifurcations to local and Global modes in spatially-developing flows. Phys. Rev. Lett. 60:25-28
21. Chomaz JM, Huerre P, Redekopp LG. 1990. Effect of non-linearity and forcing on Global modes. In New Trends in Non-linear Dynamical and Pattern-Forming Phenomena, ed. P Coullet, P Huerre, pp. 259-74. NATO ASISer. B. New York: Plenum
22. Chomaz JM, Huerre P, Redekopp LG. 1991. A frequency selection criterion in spatially-developing flows. Stud. Appl. Math. 84:119-44
23. Cohen-Tannoudji C, Diu B, Laloe F. 1997. Quantum Mechanics. New York: Wiley
24. Cossu C, Chomaz JM. 1997. Global measures of local convective instability. Phys. Rev. Lett. 78:4387-90
25. Cossu C, Chomaz JM, Henningson D. 2001. 2D nonlinear front propagation in the Blasius boundary layer. Bull. Am. Phys. Soc. 46:29
26. Couairon A, Chomaz JM. 1996. Global instabilities in fully nonlinear systems. Phys. Rev. Lett. 77:4015-18
27. Couairon A, Chomaz JM. 1997a. Absolute and convective instabilities front velocities and Global modes in nonlinear systems. Physica D 108:236-76
28. Couairon A, Chomaz JM. 1997b. Pattern selection in the presence of a cross flow. Phys. Rev. Lett. 79:2666-69
29. Couairon A, Chomaz JM. 1999a. Fully nonlinear Global modes in slowly varying flows. Phys. Fluids 11:3688-703
30. Couairon A, Chomaz JM. 1999b. Primary and secondary nonlinear global instability. Physica D 132:428-56
31. Couairon A, Chomaz JM. 2001. Pushed Global modes in weakly inhomogeneous subcritical flows. Physica D 158:129-50
32. Dee G. 1985. Dynamical properties of propagating front solutions of the amplitude equations. Physica D 15:295-304
33. Dee G, Langer J. 1983. Propagating pattern selection. Phys. Rev. Lett. 50:383-86
34. Delbende I, Chomaz JM. 1998. Nonlinear convective/absolute instabilities in parallel two-dimensional wakes. Phys. Fluids 10:2724-36
35. Delbende I, Chomaz JM, Huerre P. 1998. Absolute and convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355:229-54
36. Drazin PG, Reid WH. 1981. Hydrodynamic Stability. Cambridge, UK: Cambridge Univ. Press
37. Edwards WS, Tuckerman LS, Friesner RA, Sorensen DC. 1998. Krylov methods for the incompressible Navier-Stokes equations. J. Comp. Phys. 110:82-102
38. Fauve S. 1998. Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities, ed. C Goldrèche, P Manneville, pp. 387-489. Cambridge, UK: Cambridge Univ. Press See Goldrèche & Manneville 1998, pp. 387-489
39. Fineberg J, Steinberg V. 1987. Vortex-front propagation in Rayleigh-Bénard convection. Phys. Rev. Lett. 58:1332-35
40. Giannetti F, Luchini P. 2003. Receptivity of the circular cylinder's first instability. Proc. 5th Eur. Fluid Mech. Conf., Toulouse
41. Goldrèche C, Manneville P, eds. 1998. Hydrodynamics and Nonlinear Instabilities. Cambridge, UK: Cambridge Univ. Press
42. Gondret P, Ern P, Meignin L, Rabaud M. 1999. Experimental evidence of a nonlinear transition from convective to absolute instability. Phys. Rev. Lett. 82:11442-45
43. Goujon-Durand S, Jenffer P, Wesfreid JE. 1994. Downstream evolution of the Bénard-von Kármán instability. Phys. Rev. E 50:308-13
44. Guckenheimer J, Holmes P. 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York: Springer-Verlag.
45. Hammond D, Redekopp L. 1997. Global dynamics of symmetric and asymmetric wakes. J. Fluid Mech. 331:231-60
46. Harris D, Bassom AP, Soward AM. 2000. An inhomogeneous Landau equation with application to spherical Couette flow in the narrow gap limit. Physica D 137:260-76
47. Hill DC. 1992. A theoretical approach for analyzing the restabilization of wakes. AIAA Pap. 92-0067
48. Ho CM, Huerre P. 1984. Perturbed free shear layers. Annu. Rev. Fluid Mech. 16:365-424
49. Huerre P. 1987. On the absolute/convective nature of primary and secondary instabilities. In Propagation in Systems Far from Equilibrium, ed. JE Wesfreid, HR Brand, P Manneville, G Albinet, N Boccara, pp. 340-33. Berlin: Springer-Verlag
50. Huerre P. 2000. Open shear flow instabilities. In Perspectives in Fluid Dynamics, ed. GK Batchelor, HK Moffatt, MG Worster, pp. 159-229. Cambridge, UK: Cambridge Univ. Press
51. Huerre P, Monkewitz PA. 1985. Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159:151-68
52. Huerre P, Monkewitz PA. 1990. Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22:473-537
53. Huerre P, Rossi M. 1998. Hydrodynamic instabilities in open flows. In Hydrodynamics and nonlinear instabilities, Goldrèche C, Manneville P, ed., pp. 81-294. Cambridge Univ. Press See Goldrèche & Manneville 1998, pp. 81-294
54. Hunt RE, Crighton DG. 1991. Instability of flows in spatially developing media. Proc. R. Soc. London Ser A 435:109-28
55. Jackson CP. 1987. A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182:23-45
56. Kaiktsis L, Karniadakis GE, Orszag SA. 1996. Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. J. Fluid Mech. 321:157-87
57. Koch W. 1985. Local instability characteristics and frequency determination of self-excited wake flows. J. Sound Vib. 99:53-83
58. Koch W. 2002. On the spatio-temporal stability of primary and secondary crossflow vortices in a three-dimensional boundary layer. J. Fluid Mech. 456:85-111
59. Kolmogorov AN, Petrovsky IG, Piskunov NS. 1937. Investigation of a diffusion equation connected to the growth of materials and application to a problem in biology. Bull. Univ. Moscow Ser Int. Sect. A 1: 1-25
60. Kyle DM, Sreenivasan KR. 1993. The instability and breakdown of a round variabledensity jet. J. Fluid Mech. 249:619-64
61. Lauga E, Bewley TR. 2002. The decay of stabilizability with Reynolds number in a linear model of spatially developing flows. Proc. R. Soc. London Ser. A 459:2077-95
62. Lauga E, Bewley TR. 2004. Performance of a linear robust control strategy on a nonlinear model of spatially developing flow. J. Fluid Mech. 512:343-74
63. Le Dizès S, Huerre P, Chomaz JM. 1993. Nonlinear stability analysis of slowly-varying medias: limitations of the weakly nonlinear approach. In Proc. IUTAM Symp. Bluff-Body Wakes, Dyn. Instab., pp. 147-52. Berlin: Springer-Verlag
64. Le Gal P, Croquette V. 2000. Visualization of the space-time impulse response of the subcritical wake of a cylinder Phys. Rev. E 62:4424-26
65. Leu TS, Ho CM. 2000. Control of global instability in a non-parallel near wake. J. Fluid Mech. 404:345-78
66. Lingwood RJ. 1995. Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299:17-33
67. Lingwood RJ. 1996. An experimental study of absolute instability of the rotating-disk boundary-layer fow. J. Fluid Mech. 314:373-405
68. Lingwood RJ. 1997a. Absolute instability of the Ekman layer and related rotating flows. J. Fluid Mech. 331:405-28
69. Lingwood RJ. 1997b. On the impulse response for swept boundary-layer flows. J. Fluid Mech. 344:317-34
70. Mamun CK, Tuckerman LS. 1995. Asymmetry and hopf-bifurcation in spherical Couetteflow. Phys. Fluids 7:80-91
71. Marquillie M, Ehrenstein U. 2003. On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490:169-88
72. Mathis C, Provansal M, Boyer L. 1984. Benard-Von kármán instability-an experimental-study near the threshold. J. Phys. Lett. 45:483-91
73. Maxworthy T. 1999. The flickering candle: transition to a global oscillation in a thermal plume. J. Fluid Mech. 390:297-323
74. Meunier M, Proctor M, Sokoloff D, Soward A, Tobias S. 1997. Asymptotic properties of a nonlinear α-dynamo wave: period, amplitude and latitude dependence. Geophys. Astrophys. Fluid Dyn. 86:249-85
75. Monkewitz P, Huerre P, Chomaz JM. 1993. Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251:1-20
76. Monkewitz PA. 1988. The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers. Phys. Fluids 31:999-1006
77. Monkewitz PA, Bechert DW, Barsikow B, Lehmann B. 1990. Self-excited oscillations and mixing in a heated round jet. J. Fluid Mech. 213:611-39
78. Moore AM, Perez CL, Zavala-Garay J. 2002. A non-normal view of the wind-driven ocean circulation J. Phys. Oceanogr. 32:2681-2705
79. Müller HW, Lücke M, Kamps M. 1989. Convective patterns in horizontal flow. Europhys. Lett. 10:451-54
80. Müller HW, Lücke M, Kamps M. 1992. Transversal convection patterns in horizontal shear flow. Phys. Rev. A 45:3714-26
81. Narayanan S, Hussain F. 1996. Measurements of spatiotemporal dynamics in a forced plane mixing layer. J. Fluid Mech. 320:71-115
82. Nayfeh AH. 1973. Perturbation Methods. New York: Wiley
83. Noack BR, Afanasiev K, Morzynski M, Tadmor G, Thiele F. 2003. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497:335-63
84. Noack BR, Eckelmann H. 1994. A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270:297-330
85. Oertel H. 1990. Wakes behind blunt bodies. Annu. Rev. Fluid Mech. 22:539-64
86. Pier B. 2002. On the frequency selection finite-amplitude vortex shedding in the cylinder wake. J Fluid Mech. 458:407-17
87. Pier B. 2003. Finite-amplitude crossflow vortices secondary instability and transition in the rotating disk boundary layer. J. Fluid Mech. 487:315-43
88. Pier B, Huerre P. 2001. Nonlinear self-sustained structures and fronts in spatially developing wake fows. J. Fluid Mech. 435:145-74
89. Pier B, Huerre P, Chomaz JM. 2001. Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148:49-96
90. Pier B, Huerre P, Chomaz JM, Couairon A. 1998. Steep nonlinear Global modes in spatially developi ng media. Phys. Fluids 10:2433-35
91. Pierrehumbert RT. 1984. Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci. 41:2141-62
92. Pierrehumbert RT, Widnall SE. 1982. The two-and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114:59-82
93. Potylitsin PG, Peltier WR. 1999. Three-dimensional destabilization of Stuart vortices: the influence of rotation and elliptique. J. Fluid Mech. 387:205-26
94. Powell JA, Newell AC, Jones CKRT. 1991. Competition between generic and non-generic fronts in envelope equations. Phys. Rev. A 44:3636-52
95. Provansal M, Mathis C, Boyer L. 1987. Bénard-Von kármán instability-transient and forced régimes. J. Fluid Mech. 182:1-22
96. Reddy SC, Trefethen LN. 1994. Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math. 54:1634-49
97. Roussopoulos K, Monkewitz PA. 1996. Nonlinear modeling of vortex shedding control in cylinder wakes. Physica D 97:264-73
98. Ruith M, Chen P, Meiburg E, Maxworthy T. 2003a. Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486:331-78
99. Ruith M, Meiburg E, Huerre P, Chomaz JM, Gallaire F. 2003b. The role of hydrodynamic instability on the mode and frequency selection of vortex breakdown. Bull. Am. Phys. Soc. 48:126
100. Schmid PJ, Henningson DS. 2001. Stability and Transition in Shear Flows. New York: Springer-Verlag
101. Schmid PJ, Henningson DS. 2002. On the stability of a falling liquid curtain. J. Fluid Mech. 463:163-71
102. Soward A. 2001. Thin aspect ratio w-dynamos in galactic discs and stellar shells. In Advances in Nonlinear Dynamos, ed. M. Nunez, A Ferriz-Mas. The Fluid Mechanics of Astrophysics and Geophysics. London/New York: Taylor & Francis
103. Soward AM, Jones CA. 1983. The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Math. 36:19-42
104. Sreenivasan KR, Raghu S, Kyle D. 1989. Absolute instability in variable density round jets. Exp. Fluids 7:309-17
105. Strykowski PJ, Krothapalli A, Jendoubi S. 1996. The effect of counterflow on the development of compressible shear layers. J. Fluid Mech. 308:63-96
106. Strykowski PJ, Niccum D. 1991. The stability of countercurrent mixing layers in circular jets. J. Fluid Mech. 227:309-43
107. Strykowski PJ, Sreenivasan KR. 1990. On the formation and suppression of vortex 'shedding' at low Reynolds numbers. J. Fluid Mech. 218:71-107
108. Stuart JT. 1967. On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29:417-40
109. Sturrock PA. 1958. Kinematics of growing waves. Phys. Rev. 112:1488-503
110. Theofilis V. 2003. Advances in global linear instability analysis of non-parallel and three-dimensional flows. Prog. Aerosp. Sci. 39:249-315
111. Tobias SM, Proctor MIZE, Knobloch E. 1997. The role of absolute instability in the solar dynamo. Astron. Astrophys. 318:L55-58
112. Tobias SM, Proctor MIZE, Knobloch E. 1998. Convective and absolute instabilities of fluid flows in finite geometry. Physica D 113:4-72
113. Trefethen LN. 1997. Pseudospectra of linear operators. Siam Rev. 39:383-406
114. Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA. 1993. Hydrodynamic stability without eigenvalues. Science 261:578-84
115. Triantafyllou G, Karniadakis G. 1990. Computational reducibility of unsteady viscous flows. Phys. Fluids A 2:653-56
116. Tuckerman LS, Bertagnolio F, Daube O, Le Qur P, Barkley D. 2000. Stokes preconditioning for the inverse Amoldi method. In Continuation Methods for Fluid Dynamics Notes on Numerical Fluid Mechanics, ed. D Henry, A Bergeon, 74:241-55. Vieweg
117. van Saarloos W. 1987. Dynamical velocity selection: marginal stability. Phys. Rev. Lett. 58:2571-74
118. van Saarloos W. 2003. Front propagation into unstable states. Phys. Rep. 386:29-222
119. van Saarloos W, Hohenberg P. 1992. Fronts pulses sources and sinks in generalized complex Ginzburg-Landau equations. Physica D 56:303-67
120. Villermaux E, Hopfinger EJ. 1994. Periodically arranged co-flowing jets. J. Fluid Mech. 263:63-92
121. Woodley B, Peake N. 1997. Global linear stability analysis of thin aerofoil wakes. J. Fluid Mech. 339:239-60
122. Worledge D, Knobloch E, Tobias SM, Proctor MIZE. 1997. Dynamo waves in semi-infinite and finite domains. Proc. R. Soc. London Ser. A 453:119-43
123. Yakubenko PA. 1997. Global capillary instability of an inclined jet. J. Fluid Mech. 346:181-200
124. Yu M-H, Monkewitz PA. 1993. Oscillations in the near field of a heated two-dimensional jet. J. Fluid Mech. 255:323-47
125. Zebib A. 1987. Stability of viscous-flow past a circular-cylinder. J. Eng. Math. 21:155-65
126. Zielinska BJA, Goujon-Durant S, Dusek J, Wesfreid JE. 1997. Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79:3893-96
127. Zielinska BJA, Wesfreid JE. 1995. On the spatial structure of Global modes in wake flow. Phys. Fluids 7:1418-24