86 ANALYSIS OF FLOW PROBLEMS 



Figure 801 (/) shows the change from critical slope to steep slope. 

 An S2 curve forms over the steep slope, starting from the critical depth 

 over the brink. Here again the vertical components of velocity 

 become of importance; the sharp corner called for by the backwater 

 curve theory would be rounded, the effect extending a short distance 

 up and down stream. 



Figure 801 (g) shows a change of grade from steep to mild. Here dif- 

 ferent profiles may form depending upon the relative steepness of the 

 two grades. If the normal depth on the mild slope is comparatively 

 small, the swiftly flowing stream on the steep slope will continue flow- 

 ing at uniform depth right up to the change of grade, where an M3 

 curve begins. This curve continues on down the mild slope, increasing 

 in depth and decreasing in velocity until a hydraulic jump forms, after 

 which flow continues at the normal depth on the mild slope. The 

 location of the jump can be found by the method explained in Chapter 

 HI. As the mild slope becomes flatter, the normal depth on it increases, 

 and the jump moves upstream. When the normal depth becomes com- 

 paratively large, the jump forms upstream from the change of grade, 

 and is followed by an Si curve joining the downstream normal depth 

 over the change of grade. 



Figure 801 ih) shows a change of grade from steep to critical. Flow 

 continues down the steep slope at its normal depth to the change of 

 grade, where a C3 curve forms, extending horizontally across to its inter- 

 section with the normal depth on the critical slope. This is another 

 transition case, the existence of which depends upon a delicate balance 

 between the roughness and slope of the downstream portion of the 

 channel. 



Figures 801 (i) and (j) show the two possibilities when there is a break 

 in grade with steep slopes on each side. The flow is normal down to 

 the change of grade, after which an S3 curve or an SZ curve forms, 

 depending upon whether the downstream grade is flatter or steeper than 

 the upstream grade. 



Figures 801 ik), (/), and (w) show changes from adverse slope to mild, 

 critical, and steep slopes, respectively. An A2 curve forms over the 

 adverse slope in each of them. This curve joins the normal depth line 

 for the mild slope, but reaches the critical depth at the crest for the 

 critical slope and the steep slope. An S2 curve forms on the steep 

 slope, while above the critical slope the flow is at the normal depth 

 from very near the crest. The sharp intersection of the normal depth 

 line and the A2 curve would of course be rounded. 



A point of particular interest about the profiles shown in Figs. 

 801 ()fe), (/), and (m), is that the discharge is not fixed by upstream 



