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