Subcritical Flow Over a Bump: 22-Agric-A4 Fluid Flow, December 2013, Q2

Question 1b: Subcritical Flow Over a Bump (choose 1a or 1b — equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A wide brick channel carries uniform (normal) flow at a very mild longitudinal slope $S_0 = 0.02^{\circ}$ (an angle, not a percent grade — see the callout below), Manning roughness $n=0.015$. A smooth bump of height $\Delta z = 0.10\ \text{m}$ sits on the bed; the minimum water depth over its crest is $y_2 = 0.50\ \text{m}$, and the source states the resulting surface profile is a slight depression.

Quantity	Value
Manning's $n$ (brickwork)	0.015
Bed slope, $S_0$	$0.02^{\circ}$ ($\tan S_0 = 3.491\times10^{-4}$)
Bump height, $\Delta z$	0.10 m
Minimum depth over bump, $y_2$	0.50 m

Check: the source prints the slope with a degree symbol, "$0.02^{\circ}$" — taken literally (a very flat channel, $\tan(0.02^\circ)=3.49\times10^{-4}$) rather than as a 2% grade. Only this reading gives subcritical upstream flow ($Fr_1<1$), consistent with the problem's own statement that the bump causes a depression in the water surface (a 2% grade would make the approach flow supercritical, and a positive bump in supercritical flow raises the surface instead).

Find. The velocity over the bump crest, $V_2$, and the flow rate per unit width, $q$.

Figure 1b — Subcritical approach flow ($y_1$, normal depth) accelerates over a positive bump; the free surface dips even though the bed rises, the classic (non-intuitive) open-channel Venturi effect.

Approach. The upstream depth $y_1$ is unknown, but two independent relations pin it down together with the discharge $q$: Manning's equation ties $q$ to the normal depth $y_1$ at the given slope, and the frictionless energy equation across the short bump ties $y_1$ to the known crest depth $y_2$. Solve the two simultaneously (iteratively), keeping the subcritical root.

Manning's equation for the approach flow (wide channel, $R_h\approx y_1$). $$V_1 = \frac{1}{n}y_1^{2/3}S_0^{1/2} \;\Rightarrow\; q = V_1y_1 = \frac{S_0^{1/2}}{n}\,y_1^{5/3} = \frac{(3.491\times10^{-4})^{1/2}}{0.015}\,y_1^{5/3} = 1.246\,y_1^{5/3}.$$
Frictionless energy equation across the bump. Taking the upstream bed as datum ($z_1=0$, $z_2=\Delta z$), $$y_1+\frac{V_1^2}{2g} = \Delta z + y_2 + \frac{V_2^2}{2g}, \qquad V_2 = \frac{q}{y_2}.$$
Solve simultaneously (iterating on $y_1$). Substituting $q=1.246\,y_1^{5/3}$ into the energy equation and searching for the subcritical root ($Fr_1<1$) between $y_1=0.6$–$0.65$ m gives $$\boxed{y_1 \approx 0.623\ \text{m}}, \qquad q = 1.246(0.623)^{5/3} \approx \boxed{q \approx 0.567\ \text{m}^2/\text{s}}.$$ Check: $V_1=q/y_1=0.909\ \text{m/s}$, $Fr_1=V_1/\sqrt{gy_1}=0.37<1$ (subcritical, consistent with a surface depression), and the crest Froude number $Fr_2 = V_2/\sqrt{gy_2}$ stays below 1 as well (crest depth $y_2=0.50\ \text{m}$ exceeds the local critical depth $y_c\approx0.32\ \text{m}$), so the bump does not choke the flow.
Velocity over the bump. $$V_2 = \frac{q}{y_2} = \frac{0.567}{0.50} = \boxed{V_2 \approx 1.13\ \text{m/s}}.$$

Quantity	Result
Normal (upstream) depth, $y_1$	0.623 m
(a) Velocity over the bump, $V_2$	1.13 m/s
(b) Flow rate per unit width, $q$	0.567 m²/s

Concept & Theory

Topic. Gradually varied open-channel flow over a bed obstruction, combined with normal (Manning) flow to fix the approach conditions.

Key relations. Manning's equation $V=\tfrac1n R_h^{2/3}S_0^{1/2}$; specific-energy (Bernoulli) balance across a short, frictionless bump $y_1+V_1^2/2g=\Delta z+y_2+V_2^2/2g$; critical depth $y_c=(q^2/g)^{1/3}$.

Why this works. A positive bump under subcritical flow is the open-channel analogue of a Venturi throat: as the bed rises, the flow must speed up to keep specific energy available for the higher bed elevation, and for $Fr<1$ the specific-energy curve says a smaller depth carries the same (slightly reduced local) energy at higher velocity, so the free surface drops. The two governing equations (Manning far upstream, energy across the bump) are needed together because neither alone fixes both the discharge and the upstream depth.

Common pitfall. Assuming $R_h\approx y_1$ is the only place the "wide channel" idealization enters — it also means the same per-unit-width discharge $q$ applies on both sides of the bump, so continuity is $q=V_1y_1=V_2y_2$, not $Q=$ const with an implied channel width.

Source. White, Fluid Mechanics, Ch. 10 (Open-Channel Flow) — Frictionless Flow over a Bump; Uniform Flow, the Manning Equation.

Question 2 of 6: Subcritical Flow Over a Bump

Question 1b: Subcritical Flow Over a Bump (choose 1a or 1b — equal value)