Scaling of Circulation in Buoyancy Generated Vortices

The temporal evolution of the fluid circulation generated by a buoyancy force when two-dimensional (2D) arrays of 2D thermals are released into a quiescent incompressible fluid is studied through the results of numerous lattice Boltzmann simulations. It is observed that the circulation magnitude grows to a maximum value in a finite time. When both the maximum circulation and the time at which it occurs are non-dimensionalised by appropriately defined characteristic scales, it is shown that two simple Prandtl number (Pr) dependent scaling relations can be devised that fit these data very well over nine decades of Pr spanning the viscous and diffusive regimes and six decades of Rayleigh number (Ra) in the low Ra regime. Also, obtained analytically is the exact result that circulation magnitude continues to grow in time for a single buoyant vortex ring in an infinite unbounded fluid.