16-Civ-A5 Hydraulic Engineering · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — 98-Civ-A5 Hydraulic Engineering, December 2015. Full 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); where a question has parts, each part is of equal value.
Reference texts. Mays, Water Resources Engineering, 2nd ed. (pipe systems, equivalent pipes, valves, quasi-steady network simulation); Chow, Open-Channel Hydraulics (Manning uniform flow, gutter/triangular sections); Crowe, Elger & Roberson, Engineering Fluid Mechanics (unsteady open-channel flow, St. Venant equations, kinematic vs. dynamic waves). Exam-supplied relations: Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with $S = h_f/L$ (SI units), Manning $Q=\tfrac{1}{n}A R^{2/3}S^{1/2}$, valve law $Q=\tau E_s\sqrt{H_{u/s}-H_{d/s}}$, and Total Dynamic Head $\mathrm{TDH}=H_S+H_f$. Unless stated, local losses and velocity head are neglected, $\rho=1000\ \text{kg/m}^3$, $\nu=1.31\times10^{-6}\ \text{m}^2/\text{s}$.
Convention. Inverting Hazen–Williams gives a single-pipe resistance $h_f = k\,Q^{1.852}$ with $k=L\big/(0.278\,C\,D^{2.63})^{1.852}$ (SI, $Q$ in m³/s). Identical pipes in parallel between two nodes share the flow equally; pipes in series add their losses at a common flow. This collapses these "N identical pipe" networks by equivalent-pipe reduction, so no Hardy–Cross iteration is needed.
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 observed profiles decide the question directly. Immediately after the gate shuts, water can no longer leave, so it piles up at the gate; a distinct step in the water surface forms and travels upstream, against the direction of flow, growing steeper at each successive time. A disturbance that moves against the current and sharpens into a near-vertical front is a positive surge (a moving hydraulic bore), and only the dynamic wave model — the full St. Venant equations — can represent it.
The Saint-Venant momentum equation, written as a slope balance, is $$\underbrace{\frac{1}{g}\frac{\partial V}{\partial t}}_{\text{local inertia}}+\underbrace{\frac{V}{g}\frac{\partial V}{\partial x}}_{\text{convective inertia}}+\underbrace{\frac{\partial y}{\partial x}}_{\text{pressure}}+\underbrace{(S_f-S_0)}_{\text{friction }-\text{ gravity}}=0 .$$ The kinematic wave model keeps only the last group, i.e. it assumes $S_f\approx S_0$ (friction exactly balances the bed slope) and discards the local acceleration, convective acceleration and the water-surface (pressure) gradient. That reduces the dynamics to a single continuity relation carrying the flow monotonically downstream at the kinematic celerity $c=dQ/dA$; a kinematic wave cannot propagate a disturbance upstream and cannot steepen a front — exactly the two features that Figure 5 shows.
Physically, the sudden closure creates a strong unsteady ($\partial V/\partial t\neq0$), non-uniform ($\partial y/\partial x\neq0$) flow in which inertia and the pressure-gradient (backwater) term dominate. The rapid deceleration at the gate is a large local-inertia effect; the rising water surface is a large pressure term; and momentum conservation across the moving front is what fixes the surge height and celerity $c=\sqrt{g\,\bar y}$ (relative to the water). Retaining these inertial and pressure terms is precisely what distinguishes the dynamic wave from the kinematic and diffusion approximations. Water is treated as incompressible throughout — the surge is a free-surface gravity wave, not an acoustic/compressibility wave, so it is the open-channel momentum balance, not compressibility, that carries the disturbance.
In short: because the surge moves upstream and steepens, friction and gravity are not in balance and the acceleration and pressure terms cannot be dropped. The dynamic (full St. Venant) wave model is required; the kinematic wave model is appropriate only for gradual, downstream-moving flood waves on steep channels where $S_f\approx S_0$, which is not the situation here.
