Surface profiles on steep slope

Steep slope

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

Analysis

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

Zone 1: $y > y_c > y_n$ (S1 curve)

Numerator $> 0$, denominator $> 0$, so $dy/dx > 0$.

• Upstream end: $y \to y_c$, $dy/dx \to \infty$. The curve meets the CDL at a steep angle, i.e. it begins at a hydraulic jump.
• Downstream end: $y \to \infty$, $dy/dx \to S_0$; the surface becomes horizontal.
• It is a backwater curve behind a dam, weir or sluice gate on a steep slope.

Zone 2: $y_c > y > y_n$ (S2 curve)

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

• Upstream end: $y \to y_c$, $dy/dx \to -\infty$: steep departure from the CDL (e.g. from a break from mild to steep slope).
• Downstream end: $y \to y_n$, $dy/dx \to 0$: asymptotic to the NDL.

Zone 3: $y < y_n < y_c$ (S3 curve)

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

• Upstream end: starts at a control at small depth (below a sluice gate).
• Downstream end: $y \to y_n$, $dy/dx \to 0$: asymptotic to the NDL.