How to find the downstream flow depth when the discharge is not given (2 unknowns)?

Question

Example problem:

case 1 is simple as the discharge is given and all I have to do is use some formulas. But case number 2 is different as discharge is not given. How do I go about to solve this?

So the general approach is to use the bernoulli equation and the continuity equation for open channel flow. Here we have 2 unknowns, the discharge $Q$ and $d_2 = d_c$, the critical depth, which is assumed to occur on the crest of the weir. $v_2 = v_c$ for critical flow. I need to solve for $d_2$ in terms of the unknown $Q$. How do I do this with iteration and not solve with a cubic equation?

Answer 1

Scale it from a similar scenario such as case 1.

Case1 area = 0.25m*W
Case2 area = 0.9m*W

Case1 pressure = 0.25m*Blagh
Case2 pressure = 0.9m*Blagh

So flow increases by factor of (0.9/.25)^2 = 12.96. (one factor from decreased impedance by way of larger area, one from increased pressure)

Volume discharge = 12.96*3.195m3/s = 41.407 m3/s

If you need iteration on some other flow model beyond that, start with that as the initial guess and see if it converges. Such a model should overconstrain which should let you solve for discharge in terms of discharge which in turn leaves you with trying to guess where discharge = discharge (guessing in between your input discharge and your derived discharge should reduce the difference if model would converge).