Dipole formation in the transient planar wall jet

An initially quiescent quarter-plane of fluid is set into motion by the action of a wall jet, i.e., a jet parallel and adjacent to a wall. The circumstances for which the jet separates from the wall and forms a dipole at the upstream head of the jet are studied numerically. The streamfunction- vorticity formulation is used to track the time evolution of vorticity over a range of Reynolds numbers, 50≤Re≤1000, based on jet velocity and width. By implementing both a no-slip and a slip boundary condition and comparing the results for the time evolution of vorticity, it is found that the boundary condition has a negligible effect on dipole development. By contrast, the relative magnitude of positive ω₊ and negative ω_- vorticity present in the inlet jet flow controls whether dipole formation occurs. In particular, dipole formation requires a sufficiently large magnitude of negative vorticity ω_-/ω ₊≳0.65. The results refine a conjecture by Yushina ("Evolution of the near-wall jet," in General Circulations of the Oceans, Technical Report No. WHOI of the 1989 Summer Study Program in Geophysical Fluid Dynamics, Woods Hole, Oceanographic Institute, Woods Hole, MA, 1989, pp. 470-512) that negative vorticity is necessary for dipole creation. The results further indicate that the criterion for the existence of a moving separation point concurrently serves as a criterion for dipole formation. This suggests that, in some situations in an unsteady boundary layer, dipole generation will be associated with unsteady separation.