Abstract:
We present a Lagrangian particle/panel method for incompressible fluid flow. The key variable is the material flow map which is represented by particles and panels [1]. The particle velocity is given by a panel discretization of the Biot-Savart integral. Adaptive refinement, kernel smoothing, and a treecode algorithm are used to enhance accuracy, stability, and efficiency [2]. This type of Lagrangian method avoids explicit discretization of the convective derivative, but remeshing is necessary when the flow map becomes too distorted. We present results on chaotic dynamics in vortex sheet flow in 2D [3], vortex rings in 3D, and the barotropic vorticity equations on a rotating sphere. Possible extension to the shallow water equations will also be discussed.
[1] H. Feng, L. Kaganovskiy, R. Krasny (2009) Azimuthal instability of a vortex ring computed by a vortex sheet panel method, Fluid Dynamics Research 41, 051405.
[2] K. Lindsay, R. Krasny (2001) A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, Journal of Computational Physics 172, 879-907.
[3] R. Krasny, M. Nitsche (2002) The onset of chaos in vortex sheet flow, Journal of Fluid Mechanics 454, 47-69. |