Simulation of Butterfy Flapping with the Method of Dipole Domains G.Y. Dynnikova, S.V. Guvernyuk and D.A. Syrovatskiy Simulation of butterfly flapping with the method of dipole domains Galina Ya. Dynnikova*, Sergey V. Guvernyuk*, Dmitry A. Syrovatskiy * * Lomonosov Moscow State University Institue of Mechanics, Michurinsky pr. 1, 119192, Moscow, Russia e-mail: dyn@imec.msu.ru, guv@imec.msu.ru, talismanium@gmail.com ABSTRACT A numerical mesh-free method of dipole domains [1,2] is used for simulation of a butterfly flapping model. This method is based on the representation of a vortex field by the set of dipole particles. The vector function D describes density of dipole moments in accordanse with Navier-Stokes or Euler equations [3]. The butterfly model consists of two flat plates with a common edge performing harmonic oscillations in two planes. New mechanism of the thrust performing is proposed. Key words: Mesh-free numerical method, three dimensional flow, dipole particles, butterfly model, mechanism of the thrust performance. 1. INTRODUCTION The mesh-free particles-based methods are effective for modeling the flows with intensively changing boundaries. In the grid based methods two strategies are usually utilized: morphing grids and overset grids. The first approach is not applicable when the computational domain variation is large. The overset grid technology suffers from the low accuracy of computations which is caused by non conservative character of the interpolation between grids. Mesh-free methods could be a good alternative to grid based techniques for such problems. Among the mesh-free methods, the vortex methods have an advantage in modeling incompressible flows in an unbounded space, since the region with essentially non-zero vorticity has a small volume. In addition, the boundary conditions at infinity are automatically provided. Simulation of 3D vortex flow in 3-D space has the problem of the representation of three- dimensional vortex field by discrete elements. This field must be divergence-free as a curl of velocity field. But when the discrete vortex particles are used, this property can be destroyed. The velocity field which the vortex particle induces in accordance with Biot-Savart formula has non-zero vorticity in the whole space but not only in the localization of the particle. If the set of the vortex particles does not form a divergence-free vector field then the rotor of the induced velocity field does not coincide with this vector field. This leads to errors in the calculation if special measures are not taken. Therefore hybrid methods are often applied with combination of the Eulerian and Lagrangian approaches [4]. After the particles have been moved, their intensities are recalculated at Euler mesh for recovering the solenoidality at each step. This procedure enforces to build grids, and can increase the numerical viscosity. In this work the dipole particles are used for simulating of the 3-D vortex field. This representation provides a solenoidality of the vortex field. The fully lagrangian method of Dipole Domains (DD) is developed in [1]. Dipole distributions are widely used in hydrodynamics to calculate the potential flows (double-layer potential). The idea to construct a numerical method based on the dipole particles was suggested by Yanenko, Veretentsev and Grigoriev [5]. However, numerical implementation hasn’t been performed. Chefranov [6] used the point dipoles to model the vorticity in an ideal fluid for analyzing the mechanisms of turbulence and turbulent viscosity. It has been shown that interaction of the point dipoles in an ideal fluid can lead to explosive growth of localized vorticity. The vortex dipoles were applied in papers [7-9] for the simulation of the inviscid vortex flow and analyzing of the turbulence. In the method of Dipole Domains the smooth dipole particles are used. Viscous interaction of the particles can be taken into account. 168