I'm not very good with fluid physics, and need some help. Imagine the following setup with water contained in-front of a wall with an opening on the bottom:
How do I calculate the water flow $Q$?. I have made some re-search and found I need to (partially) calculate the pressure across the opening (orifice). But I don't know the pressure on the back side of the orifice. Can this be solved in any way?
Note: I'm not saying "please give me the solution, I'm lazy". I want to figure it out myself. But since, in this case, I only found formulas involving calculating pressure drop, I canno't use them to solve the problem. Therefore I'm turning my face to you, to see if there's another way to solve this problem.
Update: The "tank" holding the water is actually a big lake, and the opening is how much the water gate have opened. I need to very precisely calculate how much water flows through the opening.
Best Answer
First assume that $h$ doesn't change very much because you have a large body of water (we can relax this condition later). Let's also assume that the hole is small compared to the depth ($d \ll h$) - we'll relax this too. For this case, the answer is straightforward, you'd use Bernoulli's equations and simply set the static pressure ($\rho g h$) equal to the dynamic pressure ($\frac{1}{2}\rho v^2$). Then you'd pull out $v$ and multiply it by the area $A$ of the hole to get $Q$, since $Q$ is the volumetric flow rate.
Now, let's relax the condition that $d \ll h$. Since the pressure at the hole varies with depth, the velocity will vary too. You can treat this like a calculus problem where you calculate the incremental change in velocity as a function of height. To calculate $Q$, you'd need to integrate $w \int v(x) \,\mathrm{d}x$ for $x = 0$ to $x = d$. Note $w$ would be the width of your hole into the page (assuming a square hole).
Once you obtain the expression above ($Q$ as a function of $h$), you could then relax the condition that $h$ be constant by noting that $h$ will depend on the volumetric flow rate and the geometry of the lake. Once you have $Q(h)$ from the previous step you can use that to calculate $h(t)$ and back-substitute that into your equation from the previous step.