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Vortex Lines, Vortex Tubes, and Vortex Filaments

The curl of the velocity field of a fluid, which is generally termed vorticity, is usually represented by the symbol $ \omega$ , so that

$\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle = \nabla\times {\bf v}.$ (4.71)

A vortex line is a line whose tangent is everywhere parallel to the local vorticity vector. The vortex lines drawn through each point of a closed curve constitute the surface of a vortex tube. Finally, a vortex filament is a vortex tube whose cross-section is of infinitesimal dimensions.

Figure 4.9: A vortex filament.
\epsfysize =2.5in

Consider a section $ AB$ of a vortex filament. The filament is bounded by the curved surface that forms the filament wall, as well as two plane surfaces, whose vector areas are $ {\bf S}_1$ and $ {\bf S}_2$ (say), which form the ends of the section at points $ A$ and $ B$ , respectively. (See Figure 4.9.) Let the plane surfaces have outward pointing normals that are parallel (or anti-parallel) to the vorticity vectors, $ \omega$ $ _1$ and $ \omega$ $ _2$ , at points $ A$ and $ B$ , respectively. The divergence theorem (see Section A.20), applied to the section, yields

$\displaystyle \oint$   $\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \cdot d{\bf S} = \int \nabla\cdot$$\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \,dV,$ (4.72)

where $ d{\bf S}$ is an outward directed surface element, and $ d V$ a volume element. However,

$\displaystyle \nabla\cdot$$\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle =\nabla\cdot\nabla\times {\bf v} \equiv 0$ (4.73)

[see Equation (A.173)], implying that

$\displaystyle \oint$   $\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \cdot d{\bf S} =0.$ (4.74)

Now, $ \omega$ $ \cdot d{\bf S}=0$ on the curved surface of the filament, because $ \omega$ is, by definition, tangential to this surface. Thus, the only contributions to the surface integral come from the plane areas $ {\bf S}_1$ and $ {\bf S}_2$ . It follows that

$\displaystyle \oint$   $\displaystyle \mbox{\boldmath$\omega$}$$\displaystyle \cdot d{\bf S} = S_2\,\omega_2-S_1\,\omega_1 = 0.$ (4.75)

This result is essentially an equation of continuity for vortex filaments. It implies that the product of the magnitude of the vorticity and the cross-sectional area, which is termed the vortex intensity, is constant along the filament. It follows that a vortex filament cannot terminate in the interior of the fluid. For, if it did, the cross-sectional area, $ S$ , would have to vanish, and, therefore, the vorticity, $ \omega$ , would have to become infinite. Thus, a vortex filament must either form a closed vortex ring, or must terminate at the fluid boundary.

Because a vortex tube can be regarded as a bundle of vortex filaments whose net intensity is the sum of the intensities of the constituent filaments, we conclude that the intensity of a vortex tube remains constant along the tube.

next up previous
Next: Circulation and Vorticity Up: Incompressible Inviscid Flow Previous: Flow over a Broad-Crested
Richard Fitzpatrick 2016-03-31