Locate control points and sketch profiles

The figure is not available here, so the method is given first and then worked for the three arrangements that these figures normally show: (a) a steep reach joining a mild reach, (b) a mild reach with a sluice gate, (c) a mild reach ending in a free overfall.

Method

1. Draw the NDL and CDL on every reach: mild $y_n > y_c$, steep $y_n < y_c$, horizontal $y_n = \infty$, adverse $y_n$ does not exist.
2. Control points (where depth is known):
   • critical depth at a break from mild to steep, and at a free overfall (taken as $y_c$ slightly upstream of the brink);
   • the depth set by a sluice gate (vena contracta downstream, backwater depth upstream);
   • the depth at a reservoir or pool.
3. Subcritical flow is controlled from downstream; supercritical flow from upstream. Build the profile from each control point, using the family M1, M2, M3, S1, S2, S3, H2, H3.
4. Where a supercritical profile must join a subcritical one, put a hydraulic jump, with the supercritical depth and subcritical depth conjugate.

(a) Steep reach then mild reach

• Control point: uniform supercritical flow upstream at $y_n$ (steep); uniform subcritical flow far downstream at $y_n$ (mild).
• Profile: uniform flow on the steep reach, an M3 curve rising on the mild reach, then a hydraulic jump to the mild normal depth.

 STEEP                MILD
 ---- NDL  yn(steep) .  M3   J  ______ yn(mild)
 ____________________/'''' |-'
                       (jump)

(b) Mild reach with a sluice gate

• Control points: gate opening gives the depth $a C_c$ downstream (supercritical) and a deeper pool upstream; uniform subcritical flow far downstream.
• Upstream of the gate: M1 backwater curve (depth above $y_n$) rising towards the gate.
• Downstream of the gate: M3 curve from the vena contracta, ending with a hydraulic jump to the NDL.

        M1                  gate
 NDL ......___---'''''|G|          J  ___ NDL
                       |_|__ M3 __/|--'

(c) Mild reach ending at a free overfall

• Control point: depth $\approx y_c$ at the brink.
• Profile: an M2 drawdown curve from the NDL (asymptotic far upstream) falling to $y_c$ at the end. There is no jump.

 NDL ......___
              ''--.__  M2
 CDL - - - - - - - -  '--.   (free overfall)

Every profile is tangent to the NDL where it approaches uniform flow, meets the CDL steeply (except at a jump or overfall), and a jump is shown wherever supercritical flow meets subcritical flow of greater depth.