Point Vortices in a Periodic Box Makoto Umeki
نویسنده
چکیده
A statistical approach to a problem of assemblies of point vortices (PVs) goes back to Onsager (1949). A state of negative temperature is considered to be related to clustering of vortices rotating in the same direction and the inverse energy cascade predicted in the two-dimensional Navier-Stokes (2D NS) turbulence. In many numerical simulations, PVs are bounded in a circular wall, since a velocity field due to a PV can be computed by including a single mirror image. Although the axisymmetry with respect to the origin is conserved, the spatial homogeneity is not guaranteed in such a circular system. There has been a numerical difficulty in a simulation of vortices in a box that there emerges an infinite sequence of mirror images. Stremler and Aref (1999) applied a passive particle method and showed the complicated motions of three PVs in a periodic parallelogram. Our objective is to study such turbulent motions of many PVs in a periodic box using Weierstrass elliptic functions and Mathematica. Let us start by representing the 2D NS equation in terms of a complex position z = x+ iy, velocity q = u− iv, pressure p and the kinematic viscosity ν as qt + qqz̄ + q̄qz = −2pz + 4νqzz̄. (1)
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تاریخ انتشار 2006