Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T18:19:18.119Z Has data issue: false hasContentIssue false

The Lighthill–Weis-Fogh clap–fling–sweep mechanism revisited

Published online by Cambridge University Press:  26 April 2011

D. KOLOMENSKIY*
Affiliation:
M2P2–CNRS, Universités d'Aix-Marseille, 38 rue Joliot-Curie, 13451 Marseille CEDEX 20, France
H. K. MOFFATT
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
M. FARGE
Affiliation:
LMD–IPSL–CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris CEDEX 5, France
K. SCHNEIDER
Affiliation:
M2P2–CNRS, Universités d'Aix-Marseille, 38 rue Joliot-Curie, 13451 Marseille CEDEX 20, France CMI, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille CEDEX 13, France
*
Email address for correspondence: [email protected]

Abstract

The Lighthill–Weis-Fogh ‘clap–fling–sweep’ mechanism for lift generation in insect flight is re-examined. The novelty of this mechanism lies in the change of topology (the ‘break’) that occurs at a critical instant tc when two wings separate at their ‘hinge’ point as ‘fling’ gives way to ‘sweep’, and the appearance of equal and opposite circulations around the wings at this critical instant. Our primary aim is to elucidate the behaviour near the hinge point as time t passes through tc. First, Lighthill's inviscid potential flow theory is reconsidered. It is argued that provided the linear and angular accelerations of the wings are continuous, the velocity field varies continuously through the break, although the pressure field jumps instantaneously at t = tc. Then, effects of viscosity are considered. Near the hinge, the local Reynolds number is very small and local similarity solutions imply a logarithmic (integrable) singularity of the pressure jump across the hinge just before separation, in contrast to the ‘negligible pressure jump’ of inviscid theory invoked by Lighthill. We also present numerical simulations of the flow using a volume penalization technique to represent the motion of the wings. For Reynolds number equal to unity (based on wing chord), the results are in good agreement with the analytical solution. At a realistic Reynolds number of about 20, the flow near the hinge is influenced by leading-edge vortices, but local effects still persist. The lift coefficient is found to be much greater than that in the corresponding inviscid flow.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Akhiezer, N. I. 1990 Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, vol. 79. American Mathematical Society (Elementy teorii ellipticheskih funkcij, Nauka, 1970).CrossRefGoogle Scholar
Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalisation method to take into account obstacles in viscous flows. Numer. Math. 81, 497520.CrossRefGoogle Scholar
Barenblatt, G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Chapman, R. F. 1998 The Insects: Structure and Function. Cambridge University Press.CrossRefGoogle Scholar
Cooter, R. J. & Baker, P. S. 1977 Weis-Fogh clap and fling mechanism in Locusta. Nature 269, 5354.CrossRefGoogle Scholar
Crowdy, D. G. 2007 Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Camb. Phil. Soc. 142, 319339.CrossRefGoogle Scholar
Crowdy, D. 2009 The spreading phase in Lighthill's model of the Weis-Fogh lift mechanism. J. Fluid Mech. 641, 195204.CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2007 Computing the Schottky–Klein prime function on the Schottky double of planar domains. Comput. Meth. Funct. Theor. 7 (1), 293308.CrossRefGoogle Scholar
Crowdy, D. G., Surana, A. & Yick, K.-Y. 2007 The irrotational motion generated by two planar stirrers in inviscid fluid. Phys. Fluids 19, 018103.CrossRefGoogle Scholar
Edwards, R. H. & Cheng, H. K. 1982 The separation vortex in the Weis-Fogh circulation-generation mechanism. J. Fluid Mech. 120, 463473.CrossRefGoogle Scholar
Ellington, C. P. 1984 The aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305, 4178.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series, and Products. Academic.Google Scholar
Haussling, H. J. 1979 Boundary-fitted coordinates for accurate numerical solution of multibody flow problems. J. Comput. Phys. 30, 107124.CrossRefGoogle Scholar
Kolomenskiy, D., Moffatt, H. K., Farge, M. & Schneider, K. 2010 Vorticity generation during the clap–fling–sweep of some hovering insects. Theor. Comput. Fluid Dyn. 24 (1–4), 209215.CrossRefGoogle Scholar
Kolomenskiy, D. & Schneider, K. 2009 A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228, 56875709.CrossRefGoogle Scholar
Lehmann, F.-O., Sane, S. P. & Dickinson, M. 2005 The aerodynamic effects of wing–wing interaction in flapping insect wings. J. Exp. Biol. 208, 30753092.CrossRefGoogle ScholarPubMed
Lighthill, M. J. 1973 On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60 (1), 117.CrossRefGoogle Scholar
Maxworthy, T. 1979 Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight. Part 1. Dynamics of the ‘fling’. J. Fluid Mech. 93 (1), 4763.CrossRefGoogle Scholar
Maxworthy, T. 2007 The formation and maintenance of a leading-edge vortex during the forward motion of animal wing. J. Fluid Mech. 587, 471475.CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23, 127153.CrossRefGoogle Scholar
Miller, L. A. & Peskin, C. S. 2005 A computational fluid dynamics of ‘clap and fling’ in the smallest insects. J. Exp. Biol. 208, 195212.CrossRefGoogle ScholarPubMed
Miller, L. A. & Peskin, C. S. 2009 Flexible clap and fling in tiny insect flight. J. Exp. Biol. 212 (19), 30763090.CrossRefGoogle ScholarPubMed
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18 (1), 118.CrossRefGoogle Scholar
Moffatt, H. K. & Duffy, B. R. 1980 Local similarity solutions and their limitations. J. Fluid Mech. 96 (2), 299313.CrossRefGoogle Scholar
Schneider, K. & Farge, M. 2005 Numerical simulation of the transient flow behaviour in tube bundles using a volume penalization method. J. Fluids Struct. 20 (4), 555566.CrossRefGoogle Scholar
Sedov, L. I. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience.CrossRefGoogle Scholar
Spedding, G. R. & Maxworthy, T. 1986 The generation of circulation and lift in a rigid two-dimensional fling. J. Fluid Mech. 165, 247272.CrossRefGoogle Scholar
Sun, M. & Yu, X. 2003 Flows around two airfoils performing fling and subsequent translation and translation and subsequent clap. Acta Mech. Sinica 19, 103117.Google Scholar
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5, 1735.CrossRefGoogle Scholar
Weis-Fogh, T. 1973 Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59, 169230.CrossRefGoogle Scholar
Weis-Fogh, T. 1975 Flapping flight and power in birds and insects, conventional and novel mechanisms. In Swimming and Flying in Nature (ed. Wu, T. Y., Brokaw, C. J. & Brennen, C.), vol. 2, pp. 729762. Plenum.CrossRefGoogle Scholar
Wu, J. C. & Hu-Chen, H. 1984 Unsteady aerodynamics of articulate lifting bodies. AIAA Paper 2184.CrossRefGoogle Scholar

Kolomenskiy et al, supplementary material

Numerical simulation of the flow during fling and during sweep at Reynolds number Re=1. The wings break apart at time t=1.0805. Vorticity, stream function and pressure fields are shown in sequence. Above the hinge, the flow during the whole process is as expected from the local analysis. At t=0.82, when the angle of incidence equals 45°, the vorticity and the pressure just above the hinge point change sign, hence the pressure becomes positive. Below the hinge, the local solution is only valid at t>0.92, when the angle of incidence is larger than 51.3°. However, the pressure below the hinge point changes sign from positive to negative already at t=0.3. When the wings sweep apart at t>1.0805, the dominant contribution to the flow near the trailing edges is due to the outward translation velocity of the wings. The streamlines form closed loops below the trailing edges, but these are not detached eddies, because the vorticity still reaches its extreme values on the wing surfaces. The pressure becomes negative above as well as below the trailing edges.

Download Kolomenskiy et al, supplementary material(Video)
Video 6.6 MB

Kolomenskiy et al, supplementary material

Numerical simulation of the flow during fling and during sweep at Reynolds number Re=1. The wings break apart at time t=1.0805. Vorticity, stream function and pressure fields are shown in sequence. Above the hinge, the flow during the whole process is as expected from the local analysis. At t=0.82, when the angle of incidence equals 45°, the vorticity and the pressure just above the hinge point change sign, hence the pressure becomes positive. Below the hinge, the local solution is only valid at t>0.92, when the angle of incidence is larger than 51.3°. However, the pressure below the hinge point changes sign from positive to negative already at t=0.3. When the wings sweep apart at t>1.0805, the dominant contribution to the flow near the trailing edges is due to the outward translation velocity of the wings. The streamlines form closed loops below the trailing edges, but these are not detached eddies, because the vorticity still reaches its extreme values on the wing surfaces. The pressure becomes negative above as well as below the trailing edges.

Download Kolomenskiy et al, supplementary material(Video)
Video 3.8 MB

Kolomenskiy et al. supplementary material

Flow field during fling and during sweep at Reynolds number Re=20. Vortices are swept from the leading edges, and they influence the flow above the hinge point. However, local effects dominate in a small neighbourhood of the trailing edges

Download Kolomenskiy et al. supplementary material(Video)
Video 6.5 MB

Kolomenskiy et al. supplementary material

Flow field during fling and during sweep at Reynolds number Re=20. Vortices are swept from the leading edges, and they influence the flow above the hinge point. However, local effects dominate in a small neighbourhood of the trailing edges

Download Kolomenskiy et al. supplementary material(Video)
Video 3.7 MB