At what flow rate will water be pushed through a hole in a dam?

[rcoopster]

If I have a 1 ft tapered round hole through a 5 ft thick dam wall at 34 ft deep as shown in the attached diagram, how fast or at what volumetric flow rate will the pressure at that level push the water through the hole?

I know that the answer will depend upon the viscosity of the water and the pressure generated by the 34 ft of water, but I can't seem to find a formula that calculates what I feel to be a reasonable answer.

Thanks,

Rcoopster

 

[Simon Bridge]

You'd normally be expected to do this by conservation of energy.

If the hole is of sufficient length to be considered a short pipe instead then you'd also need the formulas for viscous flow and the difference in pressure each side.

These are pretty standard so this description should point you in the right direction.

 

[haruspex]

The energy equation will give a reasonable estimate of the velocity of the emerging jet (Torricelli's Theorem: v² = 2gh). But it doesn't tell you the volumetric flow. This is because the cross-sectional area of the jet (at its narrowest) is typically less than the area of the hole. For a circular hole in a thin wall the ratio is about 0.62. For a thicker wall it can drop to 0.5. You can increase the flow rate for a given external hole size by attaching a suitably shaped mouthpiece to the inside of the tank. This helps funnel the fluid more efficiently into the hole.
(I always thought the ideal funnel shape was known as a Borda mouthpiece, but looking it up now I see that this is a general term for a tube attached to the inside of the hole, and is usually considered to be cylindrical. E.g. http://mysite.du.edu/~jcalvert/tech/fluids/bernoul.htm. I haven't found anything on the net re the ideal shape.)