Vortex

We now consider flows which go in a circumferential direction with no radial flow. These are Vortex flows as shown in Fig. 4.20.

Figure 4.20: A Vortex Flow

The velocity potential and stream function are given by,

$$\phi = K \theta$$

$$\psi = -K \ln r$$ (4.76)

The velocity components are given by

$$v_\theta = {1 \over r}{\partial \phi \over {\partial \theta}} = {K \over r}$$

$$v_r = 0$$ (4.77)

It is seen that $v_\theta$ is infinite at the origin and decreases as r increases and becomes zero as r approaches infinity.

A question arises now as to whether we are contradicting ourselves? How is it that a vortex flow is irrotational? We should note that the term "Irrotational" refers to the behaviour of a fluid element and not to the path taken by it. At an elemental level the flow is still irrotational. Such a vortex is called a Free Vortex. A good and familiar example is that of a bath tub vortex. Contrary to this we have a Forced Vortex which behaves like a solid body. These have their velocity given by $v_\theta = K r$, with a zero velocity at the origin. The velocity increases as one moves away from the origin. A water filled tank is a good example.

Subsections

• Circulation around a Vortex

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney