Next: Circulation and point vortices Up: Vorticity Previous: Evolution of vorticity

# Vorticity in 2D and the Streamfunction

Let

 (6.9)

The constraint can be satisfied by using a vector potential for (c.f., .

 (6.10)

For two dimensional flow, need only have a non-zero -component, e.g., .

 (6.11)

Note that . Thus and has the components

 (6.12)

[check that is incompressible: ]

The scalar is called streamfunction since a contour const. is aligned with the velocity - i.e. it is a streamline. See Figure (6.1).

In 2D, .

Thus , and

 (6.13)

given , we can solve (6.11) for , then use (6.9) to find .

Consequently, equation (6.8) reduces, in 2D to

 (6.14)

in a given fluid element remains constant, since the convective derivative is zero, as illustrated in Figure (6.2).

 (6.15)

Example 6.1 Rankine Vortex.

Use coordinates, rather than : , still. The geometry of the Rankine Vortex is illustrated in Figure (6.3) and is defined to be

since is axisymmetric we look for solutions of the form

 (6.16)

( const)

Dividing by , followed by integration we find

since must be regular at

Now find using (6.16):

(This is similar to solid body'' rotation with angular velocity .)

(6.13)

(6.16) .

is continuous at , i.e.,

.

Hence the full solution for is and (see Figure (6.4))

 (6.17)

Next: Circulation and point vortices Up: Vorticity Previous: Evolution of vorticity
Andrew Wright
2002-09-16