16-Civ-A5 Hydraulic Engineering · December 2014

Question 5 of 6: River blockage — unsteady hydraulics

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Exams, December 2014 — 98-Civ-A5 Hydraulic Engineering, 3 hours, closed book (one aid sheet). Six questions of equal value (20 marks each); candidates complete any five. All six are solved here as a study resource. ‘cms’ = m³/s; water ρ = 1000 kg/m³, ν = 1.31×10⁻⁶ m²/s; local losses and velocity head neglected unless stated.

Reference texts. Mays, Water Resources Engineering, 3rd ed. (pipe systems, valves, parallel/loop networks); Chow, Open-Channel Hydraulics (Manning uniform flow, critical depth, specific energy); Crowe, Elger & Roberson, Engineering Fluid Mechanics (pipe force balance, St-Venant equations). Exam-supplied relations: Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with $S = h_f/L$ (SI), Manning $Q = \tfrac{1}{n}A\,R^{2/3}\,S^{1/2}$, Darcy–Weisbach, and $\text{TDH} = H_s + H_f$. In inverted form the friction loss carried by a known discharge is $h_f = L\left(\dfrac{Q}{0.278\,C\,D^{2.63}}\right)^{1/0.54}$.

Question 5: River blockage — unsteady hydraulics (20 marks)

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 river in steady flow at some approach depth and velocity is instantaneously and completely dammed by a landslide. This is a rapid, transient, one-dimensional open-channel problem.

Find. (a) the upstream and downstream hydraulic response framed by continuity, momentum, and energy; (b) the St-Venant (dynamic wave) equations with each term explained.

(a) Conditions immediately after blockage. The instant the channel is sealed, the river faces a moving boundary that stops the flow, and two opposite waves are launched.

Upstream of the blockage — a positive surge (bore). Incoming water arrives with nowhere to go and must decelerate to rest against the dam. By continuity, the volume that keeps arriving has to be stored, so the water surface rises and a positive surge (a moving hydraulic bore) propagates upstream, its front advancing against the approach flow. By momentum, stopping the moving column requires a force: the blockage supplies a large reaction, and across the abrupt surge front there is a momentum jump exactly analogous to a hydraulic jump translated upstream — the specific-force balance sets the height and celerity of the bore. By energy, the flow’s kinetic energy is converted to potential energy (rising stage) with substantial dissipation in the turbulent surge front; the energy grade line drops across the bore, so this is a rapidly-varied, non-conservative transition. The result upstream is a rapidly rising, ponding pool with near-zero velocity behind the advancing bore.

Downstream of the blockage — a negative surge (drawdown). With inflow cut to zero, the water already downstream keeps moving away under its own momentum and gravity while nothing replaces it. By continuity, the depletion of storage lowers the surface, sending a negative surge (a drawdown or depression wave) travelling downstream; depth and discharge fall, and in an extreme case the bed just below the slide is exposed. By momentum, the downstream column decelerates gradually as gravity and boundary friction act without the upstream push; unlike the sharp upstream bore, a negative wave spreads and flattens (it cannot steepen into a shock). By energy, the drawdown is comparatively gentle and closer to gradually-varied, so energy losses are dominated by ordinary boundary friction rather than a concentrated jump. The result downstream is a receding, thinning flow — a “starvation” wave.

(b) St-Venant equations. The unsteady, non-uniform, one-dimensional flow is governed by the continuity and momentum (dynamic-wave) equations. In terms of flow area $A$, discharge $Q$, depth $y$ and mean velocity $V$:

Continuity: $$\dfrac{\partial A}{\partial t} + \dfrac{\partial Q}{\partial x} = 0.$$

Momentum (dynamic wave): $$\dfrac{\partial V}{\partial t} + V\dfrac{\partial V}{\partial x} + g\dfrac{\partial y}{\partial x} = g\left(S_0 - S_f\right).$$

Term by term:

$\dfrac{\partial A}{\partial t}$ — rate of change of storage (how fast the cross-sectional area, hence depth, changes with time); the term that captures the rising pool upstream and the falling stage downstream.
$\dfrac{\partial Q}{\partial x}$ — spatial change of discharge along the channel; nonzero because flow is non-uniform. Together with the storage term it enforces mass conservation.
$\dfrac{\partial V}{\partial t}$ — local acceleration: the velocity change with time at a fixed section; large during the sudden transient.
$V\dfrac{\partial V}{\partial x}$ — convective acceleration: velocity change from moving to a section of different area/depth; the non-uniform (advective) inertia term.
$g\dfrac{\partial y}{\partial x}$ — pressure gradient: the net hydrostatic force from the sloping water surface, which drives the surge fronts.
$gS_0$ — gravity (bed slope) component, the streamwise pull of gravity along the channel bottom.
$gS_f$ — friction slope: boundary-resistance losses (e.g. Manning $S_f = n^2V^2/R^{4/3}$), always opposing motion.

Retaining every term gives the full dynamic wave, appropriate here because the abrupt blockage makes the local and convective accelerations and the pressure-gradient term all important; dropping them yields the diffusion-wave or kinematic-wave simplifications, which cannot represent the sharp bore.

Topic. Unsteady open-channel flow; positive and negative surges; St-Venant equations.

Key relations. Continuity $\partial A/\partial t + \partial Q/\partial x = 0$; dynamic-wave momentum $\partial V/\partial t + V\partial V/\partial x + g\,\partial y/\partial x = g(S_0 - S_f)$; surge celerity $c \approx \sqrt{gy}$ relative to the water.

Why this works. A complete blockage is a moving-boundary condition that mass conservation must accommodate on both sides: storage grows upstream (a shock-like positive surge that can steepen) and depletes downstream (a spreading negative surge). The St-Venant equations are simply depth-averaged mass and momentum conservation, so each transient feature maps directly onto a term — the storage term to the changing pool, the acceleration and pressure-gradient terms to the surge dynamics.

Common pitfall. Treating the situation as steady (it is inherently transient), or claiming a hydraulic jump “forms” at a fixed point — here the analogous feature is a translating bore. Another is asserting the downstream wave can steepen into a bore; negative surges flatten.

Source. Chow, Open-Channel Hydraulics (unsteady flow, surges); Crowe, Elger & Roberson, Engineering Fluid Mechanics (control-volume continuity/momentum).

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