16-Civ-A5 Hydraulic Engineering · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper: National Exams — 16-Civ-A5 Hydraulic Engineering — December 2019 · 3 hours · closed book (one aid sheet) · six questions, complete any five (all six solved here) · each question 20 marks, equal-value parts.
Reference texts: Chin, Water-Resources Engineering (Pearson); Chow, Open-Channel Hydraulics (McGraw-Hill); Chaudhry, Open-Channel Flow (Springer); Wylie & Streeter, Fluid Transients in Systems (Prentice-Hall); Munson, Young & Okiishi, Fundamentals of Fluid Mechanics (Wiley); Roberson, Cassidy & Chaudhry, Hydraulic Engineering.
Governing relations supplied on the exam cover sheet. Hazen–Williams $Q = 0.278\,C\,D^{2.63}\,S^{0.54}$ with slope $S=\Delta h/L$ (SI, $Q$ in m³/s, $D$ in m); Manning $Q=\tfrac{1}{n}A\,R^{2/3}\,S^{1/2}$; Darcy–Weisbach $\Delta h = 0.0826\,\tfrac{fL}{D^5}Q^2$. Unless stated, local losses and velocity head are neglected, diameters are nominal, and water has $\rho=1000\ \text{kg/m}^3$, $\nu=1.31\times10^{-6}\ \text{m}^2/\text{s}$.
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.
Answer. The one-dimensional St-Venant equations are the depth-averaged statements of mass and momentum conservation for unsteady, non-uniform open-channel flow. For a prismatic channel of cross-sectional area $A$, discharge $Q$, top width $T$ and depth $y$, they are the continuity equation
$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=0,$$and the momentum equation
$$\frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g\,(S_0-S_f),$$where $V=Q/A$, $S_0$ is the bed slope and $S_f$ the friction slope (e.g. from Manning). The four terms of the momentum equation are, in order, the local acceleration ($\partial V/\partial t$), the convective acceleration ($V\,\partial V/\partial x$), the pressure/depth-gradient force ($g\,\partial y/\partial x$) and the net of gravity and friction ($g(S_0-S_f)$). Retaining all of them is the full dynamic wave; dropping the two acceleration terms gives the diffusion wave, and keeping only $S_0=S_f$ gives the kinematic wave.
Continuity. When the slide dams the river, the discharge downstream of the blockage is driven abruptly to zero at the plug while inflow continues from upstream. The imbalance $\partial Q/\partial x\neq 0$ forces $\partial A/\partial t\gt0$ just upstream: water piles up and the depth rises rapidly behind the dam. Downstream of the plug the reverse holds — flow drains away with no replacement, so $\partial A/\partial t\lt0$ and the depth falls.
Momentum. The sudden stoppage creates a steep water-surface gradient at the plug, so the pressure term $g\,\partial y/\partial x$ and the local acceleration $\partial V/\partial t$ dominate; bed slope and friction ($g(S_0-S_f)$) are secondary over the first instants. Upstream this launches a positive surge (a moving bore / hydraulic-jump front) that propagates upstream against the flow at celerity $c\approx\sqrt{gA/T}$ relative to the water, raising the depth as it passes. Downstream a negative wave (drawdown) propagates away, lowering the depth. Because these fronts are steep and inertia-driven, only the full dynamic St-Venant system (not the kinematic simplification, which cannot propagate a disturbance upstream) can represent them.
Energy. Across the abrupt positive surge, mechanical energy is not conserved — like a hydraulic jump, the bore dissipates energy through turbulence, so specific energy drops across the front even though mass and momentum are conserved. This is exactly why the surge is analysed with the momentum equation rather than the Bernoulli/energy equation: energy loss is finite and unknown a priori, whereas the momentum flux across the front is balanced. Away from the surge, on the gently varying reaches upstream and downstream, energy grade-line concepts again apply and the flow relaxes toward a new gradually-varied profile (a rising backwater pool upstream, a receding limb downstream) as the terms re-balance and friction reasserts itself.
Immediately after the slide, then, the river is governed by the full dynamic-wave St-Venant equations: continuity converts the flow stoppage into a rising pool upstream and a falling stage downstream; momentum sends a positive surge upstream and a drawdown downstream; and energy is dissipated across the surge front while being conserved along the smoother reaches. As time proceeds and gradients ease, the acceleration terms decay and the solution tends to a quasi-steady backwater behind the (now ponding) blockage.
