Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T22:03:34.063Z Has data issue: false hasContentIssue false

Stability of slender inverted flags and rods in uniform steady flow

Published online by Cambridge University Press:  21 November 2016

John E. Sader*
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
Cecilia Huertas-Cerdeira
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Morteza Gharib
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
*
Email address for correspondence: jsader@unimelb.edu.au

Abstract

Cantilevered elastic sheets and rods immersed in a steady uniform flow are known to undergo instabilities that give rise to complex dynamics, including limit cycle behaviour and chaotic motion. Recent work has examined their stability in an inverted configuration where the flow impinges on the free end of the cantilever with its clamped edge downstream: this is commonly referred to as an ‘inverted flag’. Theory has thus far accurately captured the stability of wide inverted flags only, i.e. where the dimension of the clamped edge exceeds the cantilever length; the latter is aligned in the flow direction. Here, we theoretically examine the stability of slender inverted flags and rods under steady uniform flow. In contrast to wide inverted flags, we show that slender inverted flags are never globally unstable. Instead, they exhibit bifurcation from a state that is globally stable to multiple equilibria of varying stability, as flow speed increases. This theory is compared with new and existing measurements on slender inverted flags and rods, where excellent agreement is observed. The findings of this study have significant implications to investigations of biological phenomena such as the motion of leaves and hairs, which can naturally exhibit a slender geometry with an inverted configuration.

Type
Papers
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J. D. 1991 Fundamentals of Aerodynamics. McGraw-Hill.Google Scholar
Batchelor, G. K. 1974 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Blackburn, H. & Henderson, R. 1996 Lock-in behavior in simulated vortex-induced vibration. Exp. Therm. Fluid Sci. 12, 184189.CrossRefGoogle Scholar
Bollay, W. 1939 A non-linear wing theory and its application to rectangular wings of small aspect ratio. Z. Angew. Math. Mech. 19 (1), 2135.CrossRefGoogle Scholar
Drela, M. 2014 Flight Vehicle Aerodynamics. MIT.Google Scholar
Eloy, C., Lagrange, R., Souilliez, C. & Schouveiler, L. 2008 Aeroelastic instability of cantilevered flexible plates in uniform flow. J. Fluid Mech. 611, 97106.CrossRefGoogle Scholar
Gabbai, R. D. & Benaroya, H. 2005 An overview of modeling and experiments of vortex-induced vibration of circular cylinders. J. Sound Vib. 282, 575616.CrossRefGoogle Scholar
Gilmanov, A., Le, T. B. & Sotiropoulos, F. 2015 A numerical approach for simulating fluid structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains. J. Comput. Phys. 300, 814843.CrossRefGoogle Scholar
Goldstein, S. 1965 Modern Developments in Fluid Dynamics. Dover.Google Scholar
Gurugubelli, P. S. & Jaiman, R. K. 2015 Self-induced flapping dynamics of a flexible inverted foil in a uniform flow. J. Fluid Mech. 781, 657694.CrossRefGoogle Scholar
Jones, J. T. 1990 Wing Theory. Princeton University Press.CrossRefGoogle Scholar
Kim, D., Cossé, J., Huertas Cerdeira, C. & Gharib, M. 2013 Flapping dynamics of an inverted flag. J. Fluid Mech. 736, R1.CrossRefGoogle Scholar
Kornecki, A., Dowell, E. H. & O’Brien, J. 1976 On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J. Sound Vib. 47 (2), 163178.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1970 Theory of Elasticity. Pergamon.Google Scholar
Luhar, M. & Nepf, H. M. 2011 Flow-induced reconfiguration of buoyant and flexible aquatic vegetation. Limnol. Oceanogr. 56, 20032017.CrossRefGoogle Scholar
Paidoussis, M. P., Price, S. J. & De Langre, E. 2010 Fluid-Structure Interactions: Cross-Flow Instabilities. Cambridge University Press.CrossRefGoogle Scholar
Rinaldi, S. & Paidoussis, M. P. 2012 Theory and experiments on the dynamics of a free–clamped cylinder in confined axial air-flow. J. Fluids Struct. 28, 167179.CrossRefGoogle Scholar
Ryu, J., Park, S. G., Kim, B. & Sung, H. J. 2015 Flapping dynamics of an inverted flag in a uniform flow. J. Fluids Struct. 57, 159169.CrossRefGoogle Scholar
Sader, J. E., Cossé, J., Kim, D., Fan, B. & Gharib, M. 2016 Large-amplitude flapping of an inverted-flag in a uniform steady flow – a vortex-induced vibration. J. Fluid Mech. 793, 524555.CrossRefGoogle Scholar
Schmitz, F. W. 1941 Aerodynamics of the Model Airplane. Part 1. Airfoil Measurements. Redstone Scientific Information Center.Google Scholar
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449465.CrossRefGoogle Scholar
Tadrist, L., Saudreau, M. & De Langre, E. 2014 Wind and gravity mechanical effects on leaf inclination angles. J. Theor. Biol. 341, 916.CrossRefGoogle ScholarPubMed
Tang, C., Liu, N.-S. & Lu, X.-Y. 2015 Dynamics of an inverted flexible plate in a uniform flow. Phys. Fluids 27, 073601.CrossRefGoogle Scholar
Taylor, G. I. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214 (1117), 158183.Google Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Rep. 496, 414433.Google Scholar
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413455.CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408, 835839.CrossRefGoogle Scholar

Sader et al. supplementary material

Intermittent dynamics of a slender inverted flag at a wind speed just above bifurcation (κ' = 9.2). Data from figure 9(a) taken from this move.

Download Sader et al. supplementary material(Video)
Video 1.7 MB

Sader et al. supplementary material

Intermittent dynamics of a slender inverted flag at a wind speed well above bifurcation (κ' = 14.3). Data from figure 9(b) taken from this move.

Download Sader et al. supplementary material(Video)
Video 976.1 KB