A line vortex in a two-fluid system

Instability of a dense seepage layer on a sloping boundary

“…This is, perhaps, not surprising since for θ > 0 and positive relative density difference S 1 between the two fluids, the bottom fluid 1 will always be in motion relative to upper fluid 2, so that there is effectively a K-H-type flow present; the interface is known to be unstable for any relative motion between the two fluids there; see Drazin & Reid (2004, p. 18). A similar instability was shown to occur also in cylindrical geometry between rotating inviscid fluids by Forbes & Cosgrove (2014). The dimensionless initial density profile in (4.12).…”

Interfacial behaviour in two-fluid Taylor–Couette flow

“…According to linearised theory, any small-amplitude disturbance to the interface will grow exponentially with time if the two fluids move with different speeds. Forbes and Cosgrove (20) recently showed that a similar situation exists in cylindrical geometry, where two adjacent fluids move under the effects of a line vortex. An interface disturbance that grows exponentially, however, eventually violates the linearisation assumption that disturbances are small, so that non-linear effects ultimately become important.…”

“…The interface is plotted in Fig. 2a and its structure is reminiscent of cylindrical Kelvin-Helmholtz instabilities computed by Forbes and Cosgrove (20). Solutions are shown at the two times ωt = 2 and ωt = 4 and it is observed that as time progresses so the interface becomes increasingly distorted.…”

The Formation of Large-Amplitude Fingers in Atmospheric Vortices

“…Since the two fluids either side of the interface are moving with different mean angular speeds, a kind of rotational Kelvin-Helmholtz flow instability exists, and Caflisch et al [2] found over-hanging plumes could develop along the jet. More recently, Forbes and Cosgrove [10] considered planar flow in which a line vortex is present up the z-axis of a cartesian coordinate system, but an initially cylindrical interface centred on the z-axis separated two fluids of possibly differing densities and angular speeds either side. They carried out a linearized inviscid analysis and demonstrated that Kelvin-Helmholtz type instabilities could occur in this cylindrical geometry, in a very similar manner to the classical planar situation discussed by Chandrasekhar [3, page 485].…”

“…Their large-amplitude inviscid results were also subject to the formation of a curvature singularity at the interface within finite time, similar to the result of Moore [16] for planar flow. When viscous effects are included in the model, Forbes and Cosgrove [10] could obtain large-amplitude Kelvin-Helmholtz type fingers and billows arranged around the originally circular interface; however, strong mixing occurred, and as time progressed, some of the finger structures evidently even detached from the inner vortex.…”