Some months ago, I posted an engineering challenge: figure out the water rate over an non-standard V-weir dam during a high flow period (a storm) based on measurements made on the weir during a low flow period. My solution follows. Weir dams of this sort are erected mostly to prevent flooding. They provide secondary benefits, though by improving the water and providing a way to measure the flow.
A series of compound, rectangular weir dams in Maine.
The problem was stated as follows: You’ve got a classic V weir on a dam, but it is not a knife-edge weir, nor is it rectangular or compound as in the picture at right. Instead it is nearly 90°, not very tall, and both the dam and weir have rounded leads. Because the weir is of non-standard shape, thick and rounded, you can not use the flow equation found in standard tables or on the internet. Instead, you decide to use a bucket and stopwatch to determine the flow during a relatively dry period. You measure 0.8 gal/sec when the water height is 3″ in the weir. During the rain-storm some days later, you measure that there are 12″ of water in the weir. Give a good estimate of the storm-flow based on the information you have.
A V-notch weir, side view and end-on.
I also gave a hint, that the flow in a V weir is self-similar. That is, though you may not know what the pattern will be, you can expect it will be stretched the same for all heights.
The upshot of this hint is that, there is one, fairly constant flow coefficient, you just have to find it and the power dependence. First, notice that area of flow will increase with height squared. Now notice that the velocity will increase with the square root of hight, H because of an energy balance. Potential energy per unit volume is mgH, and kinetic energy per unit volume is 1/2 mv2 where m is the mass per unit volume and g is the gravitational constant. Flow in the weir is achieved by converting potential height energy into kinetic, velocity energy. From the power dependence, you can expect that the average v will be proportional to √H at all values of H.
Combining the two effects together, you can expect a power dependence of 2.5 (square root is a power of 0.5). Putting this another way, the storm height in the weir is four times the dry height, so the area of flow is 16 times what it was when you measured with the bucket. Also, since the average height is four times greater than before, you can expect that the average velocity will be twice what it was. Thus, we estimate that there was 32 times the flow during the storm than there was during the dry period; 32 x 0.8 = 25.6 gallons/sec., or 92,000 gal/hr, or 3.28 cfs.
The general equation I derive for flow over this, V-shaped weir is
Flow (gallons/sec) = Cv gal/hr x(feet)5/2.
where Cv = 3.28 cfs. This result is not much different to a standard one in the tables — that for knife-edge, 90° weirs with large shoulders on either side and at least twice the weir height below the weir (P, in the diagram above). For this knife-edge weir, the Bureau of Reclamation Manual suggests Cv = 2.49 and a power value of 2.48. It is unlikely that you ever get this sort of knife-edge weir in a practical application. Be sure to measure Cv at low flow for any weir you build or find.
Robert Buxbaum, vote for me for water commissioner. Here are some thoughts on other problems with our drains.