Example problem: enter image description here

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? enter image description here

enter image description here

  • $\begingroup$ I don't even understand how to find the downstream flow depth without further info. $\endgroup$
    – mart
    Commented Aug 8, 2022 at 13:31
  • $\begingroup$ @mart I am not sure, but thats all that is required to solve for $d_2$, I believe I need to iterate (trial and error) $\endgroup$
    – CountDOOKU
    Commented Aug 13, 2022 at 15:38
  • $\begingroup$ Try assuming a flow velocity profile (velocity as a function of height- [email protected], [email protected]). Can even assume same shape as the 1.45m one and scale $\endgroup$
    – Abel
    Commented Aug 14, 2022 at 2:20

1 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).


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