Water surface profile calculations
The calculation of water surface profiles and associated hydraulic parameters is a common task of hydraulic engineers. In natural, gradually varied channels, velocity and depth change from cross section to cross section. However, the energy and mass are conserved. The energy and continuity equations can be used to step from a water surface elevation at one cross section to a water surface at another cross section that is a given distance upstream (subcritical) or downstream (supercritical). Programs, such as HEC–RAS, use the one dimensional energy equation, with energy losses due to friction evaluated with Manning’s equation, to compute water surface profiles. Equation 9 becomes:
This one dimensional energy equation can be restated as:
The water surface profile determination is accomplished with an iterative computational procedure called the standard step method. This is graphically illustrated in figure 17.
The energy loss includes friction losses (usually evaluated with Manning’s equation) and losses associated with changes in cross-sectional areas and velocities. This is represented in equation 56:
Friction loss is evaluated as the product of the friction slope and the discharge weighted reach length. This is shown in equation 57:
Example problem: Backwater from a log drop
Problem: Determine the maximum crest level of a log weir set all the way across the channel that would cause no backwater, and the crest level required to cause 1 foot of backwater just upstream of the weir (fig. 18). Assume a discharge of 491.5 cubic feet per second, depth of 4 feet, and uniform flow conditions without the weir.
Solution: To create no backwater, the log weir would have to pass the same discharge at the same water surface. The evaluation should be between the log crest (section 2) and a point (section 1) not very far upstream (fig. 19).
This can be evaluated using the energy approach with Bernoulli’s equation. An assumption can be made that there is very little friction loss between the two points. The difference in the channel bottom elevation is also negligible over this short distance. So,
where:
hL = head loss (assumed negligible) becomes:
where:
D = height of the log weir
If the flow is high enough, the log weir will be drowned out by the normal depth tail water and would not cause backwater. At lower discharges, the flow over the log will pass through critical depth (as shown in fig. 19).
At critical depth, the velocity head is equal to half the hydraulic depth:
Substituting back into Bernoulli’s equation, since V2 is Vcr,
To determine whether the log causes backwater, compare the y1 calculated to the flow depth without the log (4 ft). That is:
where:
V1 = velocity upstream of the weir
Using the critical depth formula (where d is hydraulic depth and T is top width) along with the continuity equation, Q = VA, the following can be derived:
To find the maximum log crest before backwater is created, a log crest must be chosen and checked with a trial and error approach. For this example, suppose D = 1. Choose a depth and calculate the flow area by the dimensions of the cross section. Then compare with the Acr. When the two flow areas are the same, this is the critical depth for that Q (given as 491.5 ft3/s).
However, the velocity head must still be calculated to assure that there is no backwater. Note that the velocity head is negligible as long as the velocity is not too large. For example, a velocity of 5 feet per second results in a velocity head of 0.39 feet.
If this velocity head term is neglected, then given ycr = 1.9, T = 41.4, A = 67.8, the above formula solves as:
Since this solution is less than the clear channel depth of 4, it may be possible to raise the weir.