16-Civ-A5 Hydraulic Engineering · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2016 — 98-Civ-A5 Hydraulic Engineering. Closed book, 3 hours, one aid sheet permitted. Six questions of equal value (20 marks each); candidates answer any five. All six are solved here as a study resource. Take water density ρ = 1000 kg/m³, kinematic viscosity ν = 1.31 × 10−6 m²/s; local losses and velocity head are negligible unless stated.
Reference texts. Mays, Water Resources Engineering (Wiley) — pipe/pump systems & distribution networks; Chow, Open-Channel Hydraulics (McGraw-Hill) — normal/critical depth, specific energy, unsteady flow; Transportation Association of Canada, Geometric Design Guide for Canadian Roads — gutter/roadway drainage. Permitted equation set (from the exam cover): Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with $S = h_f/L$; Manning $Q = \tfrac{1}{n}A R^{2/3} S^{1/2}$; total dynamic head $\text{TDH} = H_s + H_f$.
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 / observed. After the gate slams shut, the upstream profiles at $t=0,\ \Delta t,\ 2\Delta t$ show a steep-fronted wave (a positive surge, or moving bore) that climbs and propagates upstream — against the flow direction — and steepens with time.
The correct choice is the dynamic wave model (the full Saint-Venant equations). The reasoning rests on which physical terms the two models keep, and on what the observed profiles demand of those terms.
The governing momentum equation. One-dimensional unsteady open-channel flow is described by continuity together with the Saint-Venant momentum equation, $$\underbrace{\frac{1}{g}\frac{\partial V}{\partial t}}_{\text{local (inertial) accel.}} \;+\; \underbrace{\frac{V}{g}\frac{\partial V}{\partial x}}_{\text{convective accel.}} \;+\; \underbrace{\frac{\partial y}{\partial x}}_{\text{pressure gradient}} \;+\; \big(S_f - S_o\big) \;=\; 0 .$$ The kinematic wave model discards the first three terms and keeps only $S_f = S_o$ — friction exactly balances gravity, so the flow is treated as locally uniform and the wave is simply advected downstream at the kinematic celerity. The dynamic wave model retains every term, including the inertial (local + convective acceleration) and pressure-gradient contributions.
Why the observed flow is dynamic, not kinematic. Immediately after closure the discharge at the gate drops to zero while water keeps arriving from upstream. Mass piles up and the depth rises abruptly, forming a steep wave front. Several features of that front rule out the kinematic model:
Unsteady and non-uniform. The profile changes with time at every section ($\partial V/\partial t \neq 0$, $\partial y/\partial t \neq 0$), so the flow is strongly unsteady; and the depth varies sharply along the channel at the front ($\partial y/\partial x$ large), so it is non-uniform. The kinematic assumption $S_f=S_o$ presumes a nearly uniform, gradually varied flow — precisely what a sudden bore is not.
Inertia and pressure gradient dominate. The rapid deceleration of the water column makes the local acceleration term $\tfrac1g\partial V/\partial t$ first-order, and the steep depth rise makes the pressure-gradient term $\partial y/\partial x$ first-order. A model that deletes both cannot represent the surge at all. The bore is essentially a moving hydraulic jump whose position and height are governed by a momentum balance across the front — a purely dynamic phenomenon.
Upstream propagation. This is the decisive observation. A kinematic wave can travel in one direction only — downstream, with the flow — because it carries no pressure information and has a single characteristic. The disturbance here moves upstream, which requires the two characteristics of the full dynamic model travelling at $V \pm c$ with wave celerity $c=\sqrt{gy}$; since the surge advances against the current, $c \gt V$ and the backward characteristic carries the front upstream. Only the dynamic wave admits this behaviour.
Momentum, not just continuity. The steepening of the front reflects the nonlinearity of the momentum equation (faster-moving, deeper water overtaking the shallower water ahead), which forms and sharpens a shock. Water is treated as incompressible, so the “compressibility” that steepens acoustic shocks is absent; here it is the free-surface (gravity-wave) analogue — the celerity increases with depth, so the higher back of the wave overtakes its toe and a bore forms. Capturing that needs the inertial and pressure terms the kinematic model throws away.
Conclusion. Because the post-closure flow is unsteady, non-uniform, inertia- and pressure-dominated, and propagates a steepening surge upstream against the flow, the dynamic (Saint-Venant) wave model is required. The kinematic wave model — valid for slow, gradually varied flood routing on steep channels where $S_f\approx S_o$ — would miss the surge entirely, since it can neither propagate a disturbance upstream nor represent the abrupt pressure gradient of the bore.
