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I want to define the flow through a top hinge flap gate on an inclined wall. The water can flow out of the bottom of the gate as well as the side and, the top.

I want to have an expression of the flow such that it is dependent on the angle of the hinge. However, I haven't been able to derive that myself. Thus I hope someone here can help me out.

I made an illustration of the gate: enter image description here

Both upstream and down stream is open to atmospheric pressure and the flow towards the gate can be neglected (only gravity is causing the flow).

So far I've tried to use the orifice equation for the top and bottom as:

$Q_{bottom} = Cd_{bottom} \cdot A_{bottom} \cdot \sqrt{2\cdot g \cdot (h-h_1)}$

$Q_{top} = Cd_{top} \cdot A_{top} \cdot \sqrt{2\cdot g\cdot (h_2)}$

Where the areas are $A_{bottom} = L1 \cdot W$ and $A_{top} = L2 \cdot W$ (W is the width of the gate)

My problem is I don't think this way is the correct one. My thought was if I could define a control volume along the entire flap or something like that. Does anyone have an idea?

Nil
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    This problem will not have any closed form solution and what you did makes sense, although for Area I would use a line perpendicular to the flap, not a horizontal line. FWIW, you actually have 3 obstructions it flows through, Q through the main opening and then Qbottom and Qtop around the flap, with Q = Qbottom + Qtop. It depends how detailed you want the model, since the Cd terms are all just plugs to make the equation fit empirical data anyway. You could actually just use a single Q and A and always find a Cd that works given measured data. – Poisson Aerohead Dec 12 '22 at 05:27
  • If I should use a single area I won't get the effect of varying angle. or should I take that empirically? – Nil Dec 12 '22 at 06:13
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    Right, if you want to try to find Cd's that are independent of the angle, you need to vary the area and so do what you are doing. That is a complicated flow. Q flows through the main area, leading to a pressure drop. The Q divides into Qbottom and Qtop based on the two areas and the difference with atmospheric pressure. The pressure is actually variable between the top and bottom of the flap. For the lower one, the relevant height to subtract from h is not h1, but the level height downstream of the flap. That is the real pressure differential driving the flow. – Poisson Aerohead Dec 12 '22 at 06:20
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    So seems a common question: https://engineering.stackexchange.com/q/53480/10902 – Solar Mike Dec 12 '22 at 07:11
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    I have a suspicion that alpha is a relatively trivial detail, and you'll get much the same answer if you took alpha as 90 and just had two orifices size(theta, h) spaced apart vertically. – Greg Locock Dec 12 '22 at 23:53

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