Two-dimensional nonlinear dynamics of four driven vortices

The interaction of four alternately driven counterrotating vortices in a two-dimensional box, with inpenetrable free-slip boundary conditions in the x direction and periodic boundary conditions in the y direction, has been studied numerically. For viscosity above a critical value the nonlinear state consists of four alternately counterrotating vortices. For a lower value of the viscosity the system evolves to a nonlinear steady state consisting of four vortices and shear flow generated by the ‘‘peeling instability’’ [Drake et al., Phys. Fluids B 4, 447 (1992)]. For a still lower viscosity the steady-state nonlinear state undergoes a Hopf bifurcation. The periodic state is caused by a secondary instability associated with vortex pairing. However, the vorticity of the shear flow, though periodic, has a definite sign. With a further decrease in the viscosity, a global bifurcation gives rise to a periodic state during which the vorticity of the shear flow changes sign. At even lower viscosity, there is a transition to a steady state, involving dominantly shear flow and a two-vortex state. Finally, this state undergoes a bifurcation to a temporally chaotic state, with the further decrease of viscosity. The results are compared to some recent experiments in fluids with driven vortices [P. Tabeling et al., J. Fluid Mech. 215, 511 (1990)].