16-Civ-A5 Hydraulic Engineering · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts (subject):
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. A rockslide instantaneously and completely dams a flowing river.
Find. (a) a physical description of the flow just upstream and downstream of the dam in terms of continuity, momentum and energy; (b) the St-Venant (dynamic-wave) equations with each term explained.
a) Conditions immediately after the blockage. The event is inherently unsteady and rapidly varied, so the two sides behave very differently.
Just upstream of the dam. The approaching flow is abruptly stopped at the barrier. By continuity, the incoming discharge can no longer pass, so water is stored: the depth rises sharply at the face and a steep-fronted positive surge (a moving hydraulic bore) propagates upstream against the current, its celerity set by the abrupt depth increase. By momentum, the barrier must supply the force that destroys the incoming momentum flux; the large imbalance between the raised hydrostatic pressure behind the front and the lower pressure ahead of it drives the bore upstream, exactly as in a moving hydraulic jump. By energy, the kinetic energy of the approaching stream is converted into potential energy (rising pool level) together with a substantial turbulent-head loss concentrated in the surge front. The pool deepens until an equilibrium (or overtopping) level is reached.
Just downstream of the dam. The supply of water is cut off. By continuity, the reach continues to drain downstream but receives no replacement, so the depth falls and a negative surge (drawdown wave) travels downstream. By momentum, the driving pressure gradient weakens as the depth drops and gravity/friction decelerate the residual flow; by energy, the reduced depth and velocity lower both the potential and kinetic energy of the flow. The bed may become locally exposed as the wave passes and the discharge decays toward zero.
b) The St-Venant equations. Unsteady, gradually-varied open-channel flow is governed by the continuity and dynamic-momentum equations:
$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=0,$$ $$\frac{1}{g}\frac{\partial V}{\partial t}+\frac{V}{g}\frac{\partial V}{\partial x}+\frac{\partial y}{\partial x}=S_0-S_f.$$Term by term, in the momentum equation: $\tfrac{1}{g}\,\partial V/\partial t$ is the local (temporal) acceleration — the flow speeding up or slowing down in time, which dominates in a sudden surge; $\tfrac{V}{g}\,\partial V/\partial x$ is the convective acceleration — velocity change along the channel; $\partial y/\partial x$ is the pressure (depth-gradient) term — the water-surface slope; $S_0$ is the bed slope (gravity forcing); and $S_f$ is the friction slope (boundary resistance, e.g. from Manning). In the continuity equation, $\partial A/\partial t$ is the rate of change of cross-sectional storage and $\partial Q/\partial x$ the streamwise change in discharge. Because a dam-break/blockage generates a sharp-fronted surge, the local- and convective-acceleration and pressure terms are all significant — the full dynamic wave is required; the simplified kinematic ($S_0 = S_f$) or diffusion approximations cannot reproduce a surge that steepens and propagates upstream.
