88 ANALYSIS OF FLOW PROBLEMS 



profiles (Z) and (m). Let us consider profiles (/) and (w) first. An 

 approximate value of the discharge can be computed on the basis of 

 assumptions that there is no loss of energy up to the crest, and that 

 flow over the crest is at the critical depth. This approximate value of 

 the discharge may be used to compute the value of the normal depth on 

 the downstream slope, to check whether the slope is actually critical or 

 steep, as assumed. (Until the discharge is known, it may not be cer- 

 tain whether the slope is mild or steep.) Having tentatively deter- 

 mined that the flow is as shown in (/) or (m), and not (k), the A 2 curve 

 should be computed upstream to the source of supply at the upper end 

 of the adverse slope. The pool level thus computed, which should 

 include allowance for the velocity head at the entrance to the adverse 

 slope, will be above the known pool elevation. If the difference is 

 appreciable, a lower value should be assumed for the critical depth at 

 the downstream end of the adverse slope, and the pool elevation recom- 

 puted. The computed pool elevations will lie on each side of the 

 given pool elevation, if the second critical depth was chosen carefully, 

 so that the discharge corresponding to the given pool elevation may be 

 determined by interpolation. 



To determine the discharge when the profile is as shown in Fig. 

 801 {k), first find the value of the discharge corresponding to the normal 

 depth for which the total head at the crest is equal to the height of the 

 pool level above the crest, and also the value corresponding to another 

 slightly smaller normal depth. Then proceed as before, figuring back 

 to the pool level and interpolating to find the correct discharge. Other 

 methods of computation may be used for these cases, but the procedure 

 outlined here is as simple as any, and has definite advantages when the 

 discharge must be known for a range of pool elevations. 



Figure 801 shows all of the possibilities for steady flow over a break 

 in grade in an infinitely long channel of constant cross section. No 

 changes of grade to or from the horizontal are included, for steady flow 

 would be impossible if the flow came from, or discharged into, an infi- 

 nitely long channel with horizontal bottom. 



The degree of certainty with which the profile to be expected under any 

 given condition can be predicted, depends upon two factors: (1) accu- 

 racy of the evaluation of the friction losses, and (2) nearness of the 

 depth of flow to the critical depth. Where the depth of flow is well 

 above or well below the critical depth, the effect of inaccuracy in the 

 evaluation of the friction losses is not great, but where the depth of 

 flow is near the critical a small uncertainty in the friction loss corre- 

 sponds to a large uncertainty in the depth of flow. For example, a slope 

 estimated to be a steep slope, with depth of flow slightly less than the 



