A vortex sheet is a surface across which tangential velocity is discontinuous while normal velocity remains continuous.
A cylindrical vortex sheet is a vortex sheet supported on a cylindrical surface. Azimuthal Fourier disturbances have harmonic velocity potentials proportional to inside and outside.
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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A vortex sheet is a mathematical concept used primarily in fluid dynamics and aerodynamics to describe the behavior of vortices in a flow field. It represents a discontinuity in the velocity field, particularly for inviscid (non-viscous) flows, where there is a sudden change in the velocity across a thin layer. ### Key Characteristics of a Vortex Sheet: 1. **Definition**: A vortex sheet consists of an infinite number of closely spaced vortices aligned along a line or surface.