Dynamics of vortex surfaces in three dimensions: theory and simulations

Abstract The dynamics of vortex surfaces in an ideal fluid is considered. The Hamiltonian and the action are constructed and topological conservation laws are discussed. The axially symmetric case is reduced to an effective 2d problem and studied numerically. There is qualitative correspondence with the results of Moore and Krasny for the purely 2d problem. The general case is approximated by means of a triangulated surface and a corresponding computer model is constructed, taking into account the topological conservation laws. The axially symmetric motion of the triangulated surface agrees with the 2d model, but there are some angular instabilities, which may lead to new vortex structures. The large-scale asymmetric 3d simulations with fairly developed instabilities are reported. The results agree with the general scenario of hierarchy of vortex structures.

[1]  Gregory R. Baker,et al.  The “cloud in cell” technique applied to the roll up of vortex sheets , 1979 .

[2]  Eckart Meiburg,et al.  Experimental and numerical investigation of the three-dimensional transition in plane wakes , 1988, Journal of Fluid Mechanics.

[3]  Morteza Gharib,et al.  Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers , 1987, Journal of Fluid Mechanics.

[4]  E. Siggia,et al.  SCALIAG AND STRUCTURES IN FULLY TURBUI‐ENT FLOWS , 1980 .

[5]  Alexandre J. Chorin,et al.  Discretization of a vortex sheet, with an example of roll-up☆ , 1973 .

[6]  M. Gharib,et al.  Three-dimensional structure of ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers , 1988 .

[7]  J. Higdon,et al.  Nonlinear Kelvin–Helmholtz instability of a finite vortex layer , 1985, Journal of Fluid Mechanics.

[8]  Robert Krasny,et al.  Computation of vortex sheet roll-up in the Trefftz plane , 1987, Journal of Fluid Mechanics.

[9]  E. Novikov.  Hamiltonian description of axisymmetric vortex flows and the system of vortex rings , 1985 .

[10]  Eric D. Siggia,et al.  Collapse and amplification of a vortex filament , 1985 .

[11]  Alain Pumir,et al.  Vortex dynamics and the existence of solutions to the Navier–Stokes equations , 1987 .

[12]  D. I. Meiron,et al.  Difficulties with three‐dimensional weak solutions for inviscid incompressible flow , 1986 .

[13]  H. Kaden.  Aufwicklung einer unstabilen Unstetigkeitsfläche , 1931 .

[14]  D. W. Moore.  The stability of an evolving two-dimensional vortex sheet , 1976 .

[15]  Steven A. Orszag,et al.  Boundary integral methods for axisymmetric and three-dimensional Rayleigh-Taylor instability problems , 1984 .

[16]  E. Siggia,et al.  Point‐vortex simulation of the inverse energy cascade in two‐dimensional turbulence , 1981 .

[17]  Eckart Meiburg,et al.  Three-dimensional shear layers via vortex dynamics , 1988, Journal of Fluid Mechanics.

[18]  Steven A. Orszag,et al.  Generalized vortex methods for free-surface flow problems , 1982, Journal of Fluid Mechanics.

[19]  Pumir,et al.  Incipient singularities in the Navier-Stokes equations. , 1985, Physical review letters.