Question

6. Consider a potential vortex of strength m, located at a distance h above a stationary wall. Determine the complex potential function and derive the velocity and pressure fields. Is the vortex stationary?

Answer 1

Let's assume the origin at center of the wall and the vortex at .

Let the velocity and pressure far from the vortex is and .

The potential of uniform flow will be

Since, the rotation of the vortex is not given, so we will account for both rotation -

for counterclockwise rotation of flow,the potential function is

for clockwise rotation, the potential function will be-

where, and in cartesian coordinate is given as -

Thus the complex potential function for the vortex will be -

The x component of the velocity field is given as -

The y component of the velocity will be zero as no flow throgh the wall can occur.

The pressure field across the wall can be calculated using Bernoulli's equation -

The difference in pressure results a net force along y direction to the vortex, which will keep oscillating the vortex, depending on its distance from wall.