Two-dimensional flows with zero net momentum : evolution of vortex quadrupoles and oscillating-grid turbulence

The planar flow arising in an initially quiescent viscous fluid under the action of a localized dipolar-type forcing has been studied analytically and experimentally. The force dipole, with non-dimensional forcing amplitude Re, brings net zero momentum into the fluid and gives rise to the formation of a quadrupolar vortex: a system of two dipolar vortices moving apart. Experimentally, the action of a force dipole was modelled by a vertical cylinder oscillating horizontally in the shallow upper layer of a two-layer fluid. Two cases were studied: single quadrupoles and an array of quadrupoles. It is found that single quadrupoles develop in a self-similar manner: the length L and the translation velocity U of the quadrupolar vortex change with time as L [similar] t1/2 and U [similar] t-1/2. These quantities are characterized by non-dimensional functions a(Re) and ß(Re), respectively, which have been determined theoretically for small Re-values and experimentally for Re-values in the range 160–2200. To produce an array of quadrupoles an array of oscillating vertical rods was used. Two stages in the flow evolution were studied experimentally: the initial stage, when the interactions between the quadrupoles are weak, and the intermediate stage when the interactions play an essential role and the flow is (two-dimensionally) turbulent. It is found that at both stages the width H of the region with intense vortical motions increases with time as H [similar] t1/2. A theoretical explanation of the experimental results is given.