Vortex sheet

[1] While the tangential components of the flow velocity are discontinuous across the vortex sheet, the normal component of the flow velocity is continuous.

The discontinuity in the tangential velocity means the flow has infinite vorticity on a vortex sheet.

At high Reynolds numbers, vortex sheets tend to be unstable.

The formulation of the vortex sheet equation of motion is given in terms of a complex coordinate

denote the strength of the sheet, that is, the jump in the tangential discontinuity.

as the integrated sheet strength or circulation between a point with arc length

As a consequence of Kelvin's circulation theorem, in the absence of external forces on the sheet, the circulation between any two material points in the sheet remains conserved, so

The equation of motion of the sheet can be rewritten in terms of

It describes the evolution of the vortex sheet given initial conditions.

Greater details on vortex sheets can be found in the textbook by Saffman (1977).

Once a vortex sheet, it will diffuse due to viscous action.

direction, given by A flat vortex sheet with periodic boundaries in the streamwise direction can be used to model a temporal free shear layer at high Reynolds number.

Let us assume that the interval between the periodic boundaries is of length

Then the equation of motion of the vortex sheet reduces to

The initial condition for a flat vortex sheet with constant strength is

The flat vortex sheet is an equilibrium solution.

However, it is unstable to infinitesimal periodic disturbances of the form

Linear theory shows that the Fourier coefficient

That is, higher the wavenumber of a Fourier mode, the faster it grows.

However, a linear theory cannot be extended much beyond the initial state.

If nonlinear interactions are taken into account, asymptotic analysis suggests that for large

The vortex sheet solution is expected to lose analyticity at the critical time.

See Moore (1979), and Meiron, Baker and Orszag (1983).

The vortex sheet solution as given by the Birkoff-Rott equation cannot go beyond the critical time.

The spontaneous loss of analyticity in a vortex sheet is a consequence of mathematical modeling since a real fluid with viscosity, however small, will never develop singularity.

Viscosity acts a smoothing or regularization parameter in a real fluid.

Using a point vortex approximation and delta-regularization Krasny (1986) obtained a smooth roll-up of a vortex sheet into a double branched spiral.

Since point vortices are inherently chaotic, a Fourier filter is necessary to control the growth of round-off errors.

Continuous approximation of a vortex sheet by vortex panels with arc wise diffusion of circulation density also shows that the sheet rolls-up into a double branched spiral.

In many engineering and physical applications the growth of a temporal free shear layer is of interest.