# How to describe the profile of a free vortex?

I was reading about free vortexes and came across this equation that describes the profile of a free vortex $$z_{r}= z_{\infty} + \dfrac{k^2}{2gr^2}$$

how do you get this relationship? I can't get the expression.

I would be very grateful if you recommend bibliography for this relationship and the free vortices.

Source: equation

You haven't written the equation from the source you give correctly, the $$z_\infty$$ and $$z_r$$ should swap.

Consider a free vortex flow, where fluid is draining through a hole in the base of an infinite reservoir. We will follow a streamline from a point far away from the hole, to a closer point.

By a continuity argument, the velocity at a distance $$r$$ from the hole is

$$v=\frac{k}{r},$$

where $$k$$ is a constant. This means that very far away from the hole we can say that the velocity is effectively equal to 0. Using Bernoulli along our streamline from a point at infinity to a closer point, we have

$$\rho g z_\infty=\rho g z_r + \frac{\rho v^2}{2},$$

where $$z_\infty$$ is the height of the free surface very far from the hole and $$z_r$$ is the height at a closer point. Substituting for $$v$$ using our first equation and rearranging, we find that

$$z_\infty = z_r + \frac{k^2}{2gr^2}.$$

• Thank you very much! Commented Mar 17, 2021 at 21:29