Vortex Sheets

A vortex sheet is defined as a planar array of parallel vortex filaments. Consider a uniform vortex sheet, lying in the $x$ -$z$ plane, in which the vortex filaments run parallel to the $x$ -axis. (See Figure 9.7.) The vorticity within the sheet can be written

$$\mbox{\boldmath$\omega$} = {\mit\Omega}_x\,\delta(y)\,{\bf e}_x,$$ (9.54)

where $\delta(y)$ is a Dirac delta function. Here, $\Omega$ $= {\mit\Omega}_x\,{\bf e}_x$ is the sheet's vortex intensity per unit length. Let $v_z(y=0_+)$ and $v_z(y=0_-)$ be the $z$ -component of the fluid velocity immediately above and below the sheet, respectively. Consider a small rectangular loop in the $y$ -$z$ plane that straddles the sheet, as shown in the figure. Integration of $\omega$ $=\nabla\times {\bf v}$ around the loop (making use of the curl theorem) yields

$${\mit\Delta} v_z \equiv v_z(y=0_+)-v_z(y=0_-)={\mit\Omega}_x.$$ (9.55)

In other words, a vortex sheet induces a discontinuity in the tangential flow across the sheet. The previous expression can easily be generalized to give

$$\mbox{\boldmath$\Omega$} = {\bf n} \times{\mit \Delta} {\bf v},$$ (9.56)

where $\Omega$ is the sheet's vortex intensity per unit length, ${\bf n}$ is a unit vector normal to the sheet, and ${\mit\Delta}{\bf v}$ is the jump in tangential velocity across the sheet (traveling in the direction of ${\bf n}$ ). Furthermore, it is reasonable to assume that the previous relation holds locally for non-planar and non-uniform vortex sheets.