Surface profiles on mild slope

Mild slope

A channel slope is mild when the normal depth is greater than the critical depth, $y_n > y_c$ (equivalently $S_0 < S_c$). Uniform flow on it is subcritical.

Analysis

Use the dynamic equation in the wide-channel form

$$\frac{dy}{dx} = S_0\,\frac{1 - (y_n/y)^{10/3}}{1 - (y_c/y)^3}$$

with $y_n > y_c$. The NDL and CDL divide the flow into three zones.

Zone 1: $y > y_n > y_c$ (M1 curve)

Numerator $> 0$, denominator $> 0$, so $dy/dx > 0$ (depth increases downstream).

• Upstream end: $y \to y_n$, so $dy/dx \to 0$. The curve is asymptotic to the NDL.
• Downstream end: $y \to \infty$, so $dy/dx \to S_0$. The water surface becomes horizontal, as in a reservoir.
• Backwater curve, caused by dams or weirs.

Zone 2: $y_n > y > y_c$ (M2 curve)

Numerator $< 0$, denominator $> 0$, so $dy/dx < 0$ (depth falls downstream).

• Upstream end: $y \to y_n$, $dy/dx \to 0$: asymptotic to the NDL.
• Downstream end: $y \to y_c$, denominator $\to 0$, so $dy/dx \to -\infty$: the curve meets the CDL at a steep angle.
• Drawdown curve, as at a free overfall or a break to a steeper slope.

Zone 3: $y < y_c < y_n$ (M3 curve)

Numerator $< 0$, denominator $< 0$, so $dy/dx > 0$ (depth rises downstream).

• Upstream end: starts from a control at small depth (e.g. below a gate).
• Downstream end: $y \to y_c$, $dy/dx \to \infty$: the curve meets the CDL steeply, and in practice a hydraulic jump forms before it.