Line (point) vortex

This flow is that associated with a straight line vortex. A line vortex can best be described as a string of rotating particles. A chain of fluid particles are spinning on their common axis and carrying around with them a swirl of fluid particles which flow around in circles. A cross-section of such a string of particles and its associated flow shows a spinning point outside of which is streamline flow in concentric circles (Fig. 3.7).

Vortices are common in nature, the difference between a real vortex as opposed to a theoretical line (potential) vortex is that the former has a core of fluid which is rotating as a solid, although the associated swirl outside is similar to the flow outside the point vortex. The streamlines associated with a line vortex are circular and therefore the particle velocity at any point must be tangential only.

Cross-section showing a few of the associated streamlines

Consider a vortex located at the origin of a polar system of coordinates. But the flow is irrotational, so the vorticity everywhere is zero. Recalling that the streamlines are concentric circles, centred on the origin, so that q$ = 0, it therefore follows from Eqn (2.79), that

£ = + ^ = o, i. e. — (r?,) = 0

r dr rdr

So d(rq,)/dr = 0 and integration gives

rq, = C

where C is a constant. Now, recall Eqn (2.83) which is one of the two equivalent definitions of circulation, namely

-f

In the present example, q-T= q, and ds = rdd, so

Г = 2 Trrqi = 2irC.

Thus С = Г/(2тг) and

dy> Г dr 27гг

and

2ттг

Integrating along the most convenient boundary from radius r0 to P(r, ff) which in this case is any radial line (Fig. 3.8):

Г г

= — -—dr (го = radius of streamline,^ = 0)

Jro 2lTr

Circulation is a measure of how fast the flow circulates the origin. (It is introduced and defined in Section 2.7.7.) Here the circulation is denoted by Г and, by convention, is positive when anti-clockwise.

Since the flow due to a line vortex gives streamlines that are concentric circles, the equipotentials, shown to be always normal to the streamlines, must be radial lines emanating from the vortex, and since

qD = 0, ф is a function of 6, and

Id ф Г r &6 2mr

Therefore

Аф = ^-<1в

Y Inr

and on integrating

Г

ф = — 6 + constant

2-к

By defining Ф — 0 when 6 = 0:

Ф = ±е (ЗЛІ)

Compare this with the stream function for a source, i. e.

Ф = ^ (Eqn(3.5))

Also compare the stream function for a vortex with the function for a source. Then consider two orthogonal sets of curves: one set is the set of radial lines emanating from a point and the other set is the set of circles centred on the same point. Then, if the point represents a source, the radial lines are the streamlines and the circles are the equipotentials. But if the point is regarded as representing a vortex, the roles of the two sets of curves are interchanged. This is an example of a general rule: consider the streamlines and equipotentials of a two-dimensional, continuous, irrotational flow. Then the streamlines and equipotentials correspond respectively to the equi­potentials and streamlines of another flow, also two-dimensional, continuous and irrotational.

Since, for one of these flows, the streamlines and equipotentials are orthogonal, and since its equipotentials are the streamlines of the other flow, it follows that the streamlines of one flow are orthogonal to the streamlines of the other flow. The same is therefore true of the velocity vectors at any (and every) point in the two flows. If this principle is applied to the source-sink pair of Section 3.3.6, the result is the flow due to a pair of parallel line vortices of opposite senses. For such a vortex pair, therefore the streamlines are the circles sketched in Fig. 3.17, while the equipotentials are the circles sketched in Fig. 3.16.