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Fluid dynamics of flapping wings associated with change of domain topology

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Topological Dynamics in the Physical and Biological Sciences

We re-examine the clap-fling-sweep mechanism employed by some insects to increase lift. As argued by Lighthill (J Fluid Mech 60(1):1-17, 1973), this mechanism can create a circulatory motion even in a totally inviscid fluid, due to a topological change of the solid boundary that represents the wings surfaces. During the stroke, the wings first clap together behind the insect’s back, then open in a fling motion around the `hinge’ formed by the two trailing edges, and finally separate at the hinge and sweep apart.

In a two-dimensional approximation, we use two different conformal mappings in simply and doubly connected domains, respectively, to calculate the complex potential at all stages of the process. The results indicate that circulation (equal in magnitude and opposite round the two wings) can be generated in an inviscid fluid, and that this circulation appears when a solid body immersed in the fluid breaks into two pieces (when fling gives way to sweep). Bound vortex sheets produced during fling are still carried by the just-separated wings. This is accompanied by a continuous time evolution of the velocity everywhere in the fluid, although the pressure field jumps instantaneously at the moment of wing separation.

In a viscous fluid, the flow during the break is essentially different because, locally, the Reynolds number is very low near the hinge point. We describe it by local similarity solutions to the Stokes equation (J Fluid Mech 676:572-606, 2011).

Three-dimensional effects are present in the flow. We study them by performing numerical simulations of the Navier-Stokes equations using a Fourier spectral method with volume penalization. The flow before the break is found to be in a good agreement with the two-dimensional approximation. After the wings move farther than one chord length apart, the three-dimensional nature of the flow becomes essential (J Fluids Struct 27(5-6):784-791, 2011).

This talk is part of the Isaac Newton Institute Seminar Series series.

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