next up previous
Next: Induced Flow Up: Incompressible Aerodynamics Previous: Zhukovskii's Hypothesis


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

$\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle = {\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

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

$\displaystyle \mbox{\boldmath$\Omega$}$$\displaystyle = {\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.


next up previous
Next: Induced Flow Up: Incompressible Aerodynamics Previous: Zhukovskii's Hypothesis
Richard Fitzpatrick 2016-01-22