2D point vortex

Another particular solution: $\phi = a\theta + b$ (verify $\nabla ^2\phi = 0$ except at $r = 0$)

Potential function for a point vortex of circulation $\Gamma$ at $r = 0$:

$$\phi = \frac{\Gamma }{2\pi }\theta \notag$$

Stream function:

$$\psi = -\frac{\Gamma}{2\pi}\ln r \notag$$

1.

Question: Evaluate the radial velocity $V_{r}$, the tangential velocity $V_{\theta}$ and the vorticity $\omega_{z}$.

(a)

Hint: velocity vector in polar coordinates (use the keyword utility: coordinate system - velocity vector).

(b)

Hint: write the vorticity $\omega_{z} = \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$ in polar coordinates.

2.

Question: Evaluate the circulation $\Gamma$ along an arbitrary closed contour containing the 2D vortex.

(a)

Hint: deform the closed contour to a circle containing the 2D vortex at the origin.

(b)

Hint: evaluate the velocity tangent to the circle.

(c)

Hint: integrate the tangent velocity along the circle.

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