×

zbMATH — the first resource for mathematics

A generalized theory of viscous and inviscid flutter. (English) Zbl 1195.76450
Summary: We present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of \(1:1\) resonance. We show this by treating four extreme cases corresponding to viscous and inviscid flows in confined and unconfined flows. To see the common mechanism clearly, we consider the limit when the frequencies of the first few elastic modes are closely clustered and small relative to the convective fluid time scale. This separation of time scales slaves the hydrodynamic force to the instantaneous elastic displacement and allows us to calculate explicitly the dependence of the critical flow speed for flutter on the various problem parameters. We show that the origin of the instability lies in the coincidence of the real frequencies of the first two modes at a critical flow speed beyond which the frequencies become complex, thus making the system unstable to oscillations. This critical flow speed depends on the difference between the frequencies of the first few modes and the nature of the hydrodynamic coupling between them. Our generalized framework applies to a range of elastohydrodynamic systems and further extends the Benjamin-Landahl classification of fluid-elastic instabilities.

MSC:
76Z10 Biopropulsion in water and in air
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] PHYS FLUIDS 20 pp 104106– (2008)
[2] PNAS 102 (6) pp 1829– (2005)
[3] J FLUID MECHANICS 527 pp 353– (2005)
[4] J FLUID MECHANICS 9 pp 513– (1960)
[5] Bertram, Journal of biomechanics 15 (1) pp 39– (1982)
[6] American Journal of Physiology - Gastrointestinal and Liver Physiology 59 pp 118– (1991)
[7] J FLUID STRUCT 16 pp 989– (2002)
[8] J GEOPHYS RES 90 pp 1881– (1985)
[9] J COLLOID INT SCI 278 pp 234– (2004)
[10] Journal of the Acoustical Society of America 93 pp 2172– (1993)
[11] ANNU REV FLU MECH 36 pp 121– (2004)
[12] J FLUIDS STRUCT 12 pp 131– (1998)
[13] J FLUIDS STRUCT 15 pp 1061– (2001)
[14] J FLUID MECHANICS 481 pp 235– (2003)
[15] J GEOPHYS RES 99 pp 11859– (1994)
[16] BULL IMPERIAL MIL MED ACAD 11 pp 356– (1905)
[17] Kumaran, Physical Review Letters 84 (15) pp 3310– (2000)
[18] J FLUID MECHANICS 13 pp 607– (1962)
[19] J WIND ENG 95 pp 1259– (2007)
[20] J FLUID MECHANICS 363 pp 253– (1998)
[21] J WIND ENG IND AERODYN 91 pp 1393– (2003)
[22] Muller, Science 310 (5752) pp 1299– (2005)
[23] GEOLOGICAL SOCIETY OF LONDON SPECIAL PUBLICATIONS 307 pp 45– (2007)
[24] Tarnopolsky, Journal of the Acoustical Society of America 108 (1) pp 400– (2000)
[25] Titze, Journal of the Acoustical Society of America 83 (4) pp 1536– (1988)
[26] J FLUID STRUCT 16 pp 543– (2002)
[27] J FLUID STRUCT 16 pp 529– (2002)
[28] Zhang, Nature; Physical Science (London) 408 (6814) pp 835– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.