Spontaneous angular momentum generation of two-dimensional fluid flow in an elliptic geometry

Spontaneous spin-up, i.e., the significant increase of the total angular momentum of a flow that initially has no net angular momentum, is very characteristic for decaying two-dimensional turbulence in square domains bounded by rigid no-slip walls. In contrast, spontaneous spin-up is virtually absent for such flows in a circular domain with a no-slip boundary. In order to acquire an understanding of this strikingly different behavior observed on the square and the circle, we consider a set of elliptic geometries with a gradual increase of the eccentricity. It is shown that a variation of the eccentricity can be used as a control parameter to tune the relative contribution of the pressure and viscous stresses in the angular momentum balance. Direct numerical simulations demonstrate that the magnitude of the torque can be related to the relative contribution of the pressure. As a consequence, the number of spin-up events in an ensemble of slightly different initial conditions depends strongly on the eccentricity. For small eccentricities, strong and rapid spin-up events are observed occasionally, whereas the majority of the runs do not show significant spin-up. Small differences in the initial condition can result in a completely different evolution of the flow and an appearance of the end state of the decay process. For sufficiently large eccentricities, all the runs in the ensemble demonstrate strong and rapid spin-up, which is consistent with the flow development on the square. It is verified that the number of spin-up events for a given eccentricity does not depend on the Reynolds number of the flow. This observation is consistent with the conjecture that it is the pressure on the domain boundaries that drives the spin-up processes.