Recent numerical studies indicate that the vortex-boundary interaction is a fundamental mechanism to realize an efficient flight of insects[5] and a slow falling of plant seeds[4]. In slightly viscous fluids, vortex structures are generated due to the formation of boundary layers. They begin interacting with the boundaries and create additional forces acting on the boundaries. Thus theoretical study of such vortex-boundary interaction is a mathematical challenge.
　　　Let us consider the incompressible and inviscid fluid flow in two-dimensional multiply connected domains, which is regarded as a simple mathematical model to represent biofluids such as insect-flights, fish swimming and falling leaves. We further assume that the vorticity is concentrated in a discrete point, which is called a point vortex. The effect of viscosity is neglected in the point-vortex model. On the other hand, however, since the circulation, which is the strength of the point vortex, is a conserved quantity along the path of a fluid particle according to Kelvin’s circulation theorem, we have only to investigate the evolution of point vortices that exist at the initial moment, which makes the theoretical study of vortex-boundary interactions much easier.
　　　In the present talk, I will provide a mathematical formulation to describe the interactions between boundaries and point vortices in multiply connected domains. We first consider a canonical multiply connected domain, called circular domains, inside the unit circle containing circular boundaries. The equation of motion for point vortices in the circular domains has been provided by Sakajo[6] based on an explicit analytic representation of the Green function due to Crowdy and Marshall[3]. For a given multiply connected domain, one can derive the equation for point vortices from the canonical equation by constructing a conformal mapping from the target domain to a canonical circular domain. Since the motion of point vortices is not conformally invariant, we need an explicit representation of the conformal mapping. It is usually difficult to express the conformal mapping analytically, so we make use of a numerical conformal mapping technique based on the particle charge simulation methods for the Laplace equation[1, 2].
　　　Moreover, we show an application of the present mathematical formulation to finding stationary configurations of point vortices behind two parallel plates in the presence of a uniform flow, which would be a mathematical model for a wind power generator with vertical blades. We also give another application of the numerical conformal mapping technique to construct the complex potentials in multiply connected channels[7].

[1] K. Amano, A charge simulation method or the numerical conformal mapping of interior, exterior and doubly-connected domains, J. Comp. Appl. Math. 53 (1994), 353-370.
[2] K. Amano, A charge simulation method for numerical conformal mapping onto circular and radial slit domains, SIAM J. Sci. Comput. 19 (4) (1998), 1169-1187.
[3] D. Crowdy and J. Marshall, Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains, Proc. Roy. Soc. A 461 (2005), pp.2477-2501.
[4] D. Lentink, W. B. Dickson, J. L. van Leeuwen and M. H. Dickinson, Leading-edge vortices elevate lift of autorotating plant seeds, Science 324 (2009), pp.1438-1440.
[5] M. Iima and T. Yanagita, Is a two-dimensional butterfly able to fly by symmetric flapping?, J. Phys. Soc. Japan 70 (2001), pp.5-8.
[6] T. Sakajo, Equation of motion for point vortices in multiply connected circular domains, Proc. Roy. Soc. A 465 (2009), pp. 2589-2611.
[7] T. Sakajo and Y. Amaya, Numerical construction of potential flows in multiply connected channel domains, Computational Methods and Function Theory 11(2) (2011), pp. 415-438.

When fluid flows reach the fully-developed turbulent regime, one observes that the dissipation rate becomes independent of the fluid viscosity. We conjecture that, when the Reynolds number Re tends to infinite, viscous dissipation becomes negligible and turbulent dissipation, triggered by the nonlinear flow dynamics, takes over. To study this we consider the generic case of a jet hitting a wall. We perform direct numerical simulations of the two-dimensional Navier-Stokes equations in the vanishing viscosity limit, using volume penalization method to take into account the wall. We show that the energy dissipation first set up within a very thin vorticity sheet and then detachs from the wall and rolls up into a spiral where dissipation is maximal. We thus propose a new explanation of the d’Alembert’s paradox, stated in 1752, which is based on turbulent dissipation rather than on viscous dissipation. Our observations are compatible with Kato’s theorem, published in 1984, which proved that for dissipation to occur, anywhere in the flow and at any time, at least some dissipation had to occur in the vanishing viscosity limit within a very thin boundary layer whose thickness is proportional to Re^{-1}.

This work is done in collaboration with Romain Nguyen van yen (Freie Universitat Berlin, Germany) and Kai Schneider (Universite d'Aix-Marseille, France). It was published in Physical Review Letters, 106, 184502, 6 May 2011.