For a planar vortex sheet separating equal-density streams of velocities and , a mode of nonzero wavenumber has complex growth exponentOne sign has positive growth rate, so every nonzero wavenumber is unstable in the inviscid zero-thickness model.
Two deep layers of incompressible flow with upper mass density , lower mass density , and tangential velocities have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For the squared departure of the phase velocity from the density-weighted mean velocity is . A negative value gives Kelvin-Helmholtz instability despite stable density stratification.
A top-hat planar jet has one constant streamwise velocity in a layer and another constant velocity outside it. In the inviscid model, the two layer boundaries are parallel vortex sheets.
In a varicose mode, the upper and lower jet interfaces move in opposite directions, so the jet thickness varies while its centreline remains fixed.
In a sinuous mode, the upper and lower jet interfaces move in the same direction, so the jet centreline bends while its thickness is unchanged to first order.
For a jet of velocity occupying in stationary fluid of the same density, disturbances proportional to satisfyfor a varicose mode of a planar jet, andfor a sinuous mode of a planar jet. Both have unstable growth rate
For a top-hat planar jet in stationary fluid of the same mass density, odd and even velocity potentials have the same temporal growth rate. Put . Their dispersion relations use and , but . Their real phase velocities are respectively and . The common growth rate is strictly below for finite , approaching the isolated vortex sheet value when the sheets decouple.
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Kelvin–Helmholtz instability (KHI) is a fluid dynamic phenomenon that occurs when there is a velocity shear in a continuous fluid medium. It arises when two fluid layers with different velocities interact, leading to the formation of waves and vortices at the interface between the layers. This instability can result in the mixing of the fluids and is characterized by the growth of perturbations at the interface.