16-Civ-A5 Hydraulic Engineering · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Worked solutions. Closed-book format, 3 hours; six questions, candidates complete any five — all six are solved here as a study resource. All questions are of equal value (20 marks).
Reference texts. Mays, Water Resources Engineering, 2nd ed. (pipe systems, equivalent pipes, valves, network analysis, rigid water-column / mass-oscillation model); Chow, Open-Channel Hydraulics (Manning uniform flow, critical depth, specific energy); Crowe, Elger & Roberson, Engineering Fluid Mechanics (unsteady flow, St. Venant equations). Exam-supplied relations: Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with $S = h_f/L$ (SI units), Manning $Q=\tfrac{1}{n}A R^{2/3}S^{1/2}$, and Total Dynamic Head $\mathrm{TDH}=H_S+H_f$.
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.
The one-dimensional St. Venant momentum equation for open-channel flow states, per unit weight,
$$\underbrace{\frac{1}{g}\frac{\partial V}{\partial t}}_{\text{local inertia}}+\underbrace{\frac{V}{g}\frac{\partial V}{\partial x}}_{\text{convective inertia}}+\underbrace{\frac{\partial y}{\partial x}}_{\text{pressure}}=\underbrace{S_0-S_f}_{\text{gravity}-\text{friction}}.$$
The kinematic wave model retains only the right-hand side, $S_0=S_f$: it assumes the water surface stays parallel to the bed, the flow is essentially uniform at every instant, and inertia and the pressure-gradient (backwater) term are negligible. It can translate a flood wave downstream but it cannot steepen a front, cannot propagate a disturbance upstream, and cannot represent a discontinuity, because it carries no momentum or pressure physics — only a single-valued stage–discharge relation with celerity $c=dQ/dA$.
The observed profiles tell a different story. Slamming the gate shut brings the discharge at the gate abruptly to zero; the water piles up and a steep-fronted positive surge (a moving hydraulic bore) travels upstream, against the flow, as seen in the successive Figure 5 profiles. This front has a very large $\partial y/\partial x$ (a near-discontinuity in depth), a strong local acceleration $\partial V/\partial t$ as the flow is arrested, and it advances at the surge celerity $c=\sqrt{gy}\pm V$ set by momentum conservation across the front. The flow is emphatically unsteady and non-uniform, and it is inertia- and pressure-dominated — precisely the terms the kinematic model discards. (Water is treated as incompressible, so the surge is a gravity/inertia wave, not an acoustic/compressibility effect.)
Therefore the dynamic wave model — the full St. Venant equations, retaining both inertia terms and the pressure term — is the appropriate description. Only it can reproduce the upstream-propagating bore, its celerity, and the abrupt depth rise. The kinematic simplification would be valid only for the opposite situation: a slow, gradually varied flood wave on a steep channel where the surface stays nearly parallel to the bed and inertia is genuinely negligible. Here, the sharp surge produced by sudden gate closure violates every one of those assumptions, so the dynamic wave model must be used.
