The Problem

Suppose I have an open window with an area $A$, and a uniform wind source (a fan) pointing towards the window, positioned at a distance $s$ away from the window.

At the exit of the fan, let the velocity of the air be, uniformly, equal to $u$. How can one calculate the average velocity $v$ across the window? If not calculate, how can one even model this system?

What I've found so far:

Bernoulli's principle tells us that in an area of high velocity in a fluid, there is also an area of proportionally lower pressure. This means that air from the surroundings is drawn in, and contributes some velocity to the air that makes it through the window. This is called entrainment.

What I'm missing:

Everywhere I've looked online, there is only a written explanation of entrainment like the one you see here. Nowhere can I actually find an equation or even a proportionality between the distance of the fan from the opening and the contribution of entrained air to the final velocity. Is there some equation that models entrainment? How can I even get started at modelling this system mathematically rather than intuitively?

I would really, really, really appreciate it if anyone could help me out with this.