If dam breaches really looked like the flume model that generated the data for the 6% fit, how would the 6% fit look applied to a laboratory reservoir model?
Consider a reservoir 44 feet by 44 feet with an initial depth of water behind the dam of 2 feet. The 44 feet behind the dam is measured from the middle depth (1 ft) of the upstream slope. This way, the upstream slope can be varied to any value of H:V and the initial volume of water behind the dam will be the same, because the cross-sectional the triangles of water above and below the 1 ft depth on the upstream slope add and subtract equally.
This perspective view of the model for the 1:1 upstream slope also shows the 1:1 breach side side slopes, and a headcut 3.014 ft wide by 1.781 ft deep. The crest of the headcut and the reservoir floor are at the same level. The grid squares are 1 ft by 1 ft in the horizontal. This image comes from a graphics program called Blender (www.blender.org), which has a fluid simulation capability. In the Blender model, the reservoir has sides and a dropoff into a fluid sink at the downstream end of the headcut. This will differ from the flume model a bit because of the velocity of approach due to the 10 ft width of the flume. But because no velocities were measured in the flume experiment, this factor is not determined here.
The calculations are done in sheets D & E in the Lotus 97 1-2-3 spreadsheet, fe2208b.123, for a time, t, of 0 to 120 seconds. At t = 0, the water is held at the upstream slope above the breach, and then let go, as if the breach appeared instantaneously in the dam. That way the different volumes of the 1:1 and 6:1 upstream slope breaches have no effect on the initial volume of water behind the dam. The dam breach, since it and the 6% fit are based only upon a plywood flume model, does not change during the flow as the reservoir empties. Flows (Q, cubic feet per second, ft^3/s) are calculated by the 6% fit, and simple first-order difference equations update the reservoir volume, V (ft^3) and upstream head, he (ft). The reservoir volume, V, begins at 3872 cubic feet, and is updated by a time step (delta-t) of 0.04 seconds, using:
The volume of the reservoir is a function of side length, 44 ft, head, he, and upstream slope, mu (6 for 6:1, and 1 for 1:1), as:
which allows the updating differential, delta-he.
These plots show the upstream heads (ft) and breach outflows (cubic feet per second), and the ratio of the breach flow for the 6:1 upstream slope to the flow for the 1:1 upstream slope. Obviously there are some big differences due to upstream slope. These are investigated further here in a graphics and fluid simulation model called Blender, based upon the Lattice Boltzman method for caculating flow.
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