Velocity-Potential Integrals

2.2.1 3D potentials

Velocity-Potential Integrals Подпись: (2.43) (2.44) (2.45) (2.46)

The velocity fields of the

Velocity-Potential Integrals

Velocity-Potential Integrals

One complication here is that the arctan( ) polar angle can contain some arbitrary multiple of 2n. This requires introduction of a branch cut extending from the vortex point out to infinity in some direction, as shown in Figure 2.8. The angle jumps by 2n and the potential jumps by Г across the branch cut.

Подпись: 1 2vr Velocity-Potential Integrals Подпись: (2.56)

The branch cut also appears for a doublet sheet, which has the following potential in 2D.

Подпись: Figure 2.9: Potential of a 2D vortex of strength Г on the left (same as in Figure 2.8), and of a constant-strength 2D doublet sheet on the right which is equivalent to two equal and opposite vortices ±Г. For the doublet sheet, the branch cut is restricted to the sheet itself.

However, the branch cut now is only on the doublet sheet itself. It does not need to extend to infinity like with a vortex, unless the doublet sheet itself extends to infinity. Figure 2.9 compares the branch cuts of a point vortex and a doublet sheet, the latter being equivalent to two point vortices of opposite sign.