16-Civ-A5 Hydraulic Engineering · May 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper: National Exams — 16-Civ-A5 Hydraulic Engineering — May 2018 · 3 hours · closed book (one aid sheet) · six questions, complete any five (all 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); 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 $\Delta h = 0.0826\,\tfrac{fL}{D^5}Q^2$; total dynamic head $TDH=H_s+H_f$. 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 question text repeats “$t=\Delta t$”; Figure 5 labels the third profile $t=2\Delta t$, which is the reading used here.) The observed profiles show water piling up and steepening at the downstream (gate) end and the disturbance travelling upstream with time — the classic signature of a positive surge (a moving hydraulic bore) generated by sudden gate closure. This behaviour can only be captured by the full dynamic wave model (the complete Saint-Venant equations); the kinematic wave model is inadequate here.
The one-dimensional Saint-Venant momentum equation can be written, in slope form, as
$$S_f = S_0 - \underbrace{\frac{\partial y}{\partial x}}_{\text{pressure}} - \underbrace{\frac{V}{g}\frac{\partial V}{\partial x}}_{\text{convective inertia}} - \underbrace{\frac{1}{g}\frac{\partial V}{\partial t}}_{\text{local inertia}} .$$The kinematic wave model keeps only $S_f = S_0$: it assumes friction balances gravity at every instant, i.e. locally uniform, steady flow, and discards the pressure-gradient and both inertia terms. A kinematic wave can only translate downstream at the kinematic celerity and it cannot steepen into a front, cannot propagate a disturbance upstream, and cannot represent backwater from a downstream control. None of these are compatible with Figure 5.
After a sudden closure the flow is strongly unsteady ($\partial V/\partial t$ large — the local-inertia term is dominant) and markedly non-uniform ($\partial y/\partial x$ and $\partial V/\partial x$ large as the surge face steepens). The rapid deceleration converts momentum/kinetic energy into a rise in stage, so inertia and the pressure-gradient (water-surface slope) terms are first-order and must be retained; only the dynamic wave keeps them. The surge propagates upstream because its absolute celerity, $c = V \pm \sqrt{gA/T}$, is directed against the flow when the gravity-wave speed exceeds the (now small) flow velocity — i.e. the flow is subcritical (Froude number $\lt 1$), which is precisely the regime in which downstream disturbances travel upstream. Water is treated as effectively incompressible, so the "wave" is a free-surface gravity wave (a change in depth/momentum), not an acoustic/compressibility wave; the storage is accommodated by the rising surface rather than by fluid compression.
In summary: the steepening, upstream-moving front demands the terms the kinematic model throws away. The dynamic wave model — full continuity plus the momentum equation with its local-inertia, convective-inertia and pressure-gradient terms — is the appropriate description of the transient following the sudden sluice-gate closure.
