Question 2 of 6: St. Venant equations after a sudden channel blockage

Question 2: St. Venant equations after a sudden channel blockage (20 marks)

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 river carrying an initially near-uniform flow is suddenly and completely blocked by a landslide dam. The event is unsteady (time-varying) and non-uniform (depth and velocity vary along the reach), so the flow can only be described by the full one-dimensional shallow-water (St. Venant) equations rather than by any steady backwater equation.

Find. The two St. Venant equations, a term-by-term physical reading of the momentum equation, and a discussion tying the two equations to the conservation of mass, momentum, and energy in the moments after the blockage.

The St. Venant equations. For one-dimensional flow in a prismatic channel with cross-sectional area $A$, discharge $Q$, mean velocity $v=Q/A$, depth $y$, bed slope $S_0$ and friction slope $S_f$, the conservation forms are

Reading the momentum equation term by term as the blockage takes effect:

• $\dfrac{\partial v}{\partial t}$ — local acceleration: the velocity at a fixed point changes with time. At the dam it plunges from the river velocity to zero within seconds, so this term is large and dominant right after the slide.
• $v\dfrac{\partial v}{\partial x}$ — convective acceleration: velocity changes with position as fluid decelerates approaching the wall of impounded water; it steepens the front of the wave.
• $g\dfrac{\partial y}{\partial x}$ — pressure (depth-gradient) force: the rising water behind the block creates a strong adverse depth gradient that pushes back on the incoming flow and drives the surge upstream.
• $gS_0$ — gravity / bed-slope force: the downslope pull of gravity, essentially unchanged by the slide.
• $gS_f$ — friction (resistance) slope: boundary shear, usually written $S_f=n^2v^2/R^{4/3}$ (Manning); it dissipates energy and damps the transient over time.

Elaboration — continuity, momentum, energy. Continuity is the first equation: because the total river inflow cannot suddenly vanish while the outflow at the dam is forced to zero, $\partial A/\partial t$ must be strongly positive just upstream of the block — water accumulates and the depth rises. Integrated across the moving wave front, continuity gives the surge (bore) celerity relative to the still water, $c=\sqrt{g\,A/T}\approx\sqrt{gy}$ for a wide channel, so the disturbance propagates upstream at a finite speed rather than instantaneously. Momentum is the second equation: the imbalance between the pressure force of the impounded water and the incoming flow’s momentum flux is what decelerates the river and forms a hydraulic bore; across the bore, momentum (specific force $M=Q^2/(gA)+\bar{h}A$) is conserved even though the surface is discontinuous. Energy is not conserved across the bore — a moving hydraulic jump dissipates mechanical energy through turbulence, so specific energy decreases through the front; this is precisely why the momentum principle, not the energy (Bernoulli) principle, must be used to relate the depths and velocities on either side. Away from the bore, in the gradually varied reaches, energy methods remain valid and the friction slope $S_f$ accounts for the steady dissipation that eventually brings the impounded pool to rest. The dynamic-wave (full St. Venant) form is required here because the local- and convective-acceleration terms and the depth-gradient term are all first-order — the kinematic ($S_0=S_f$) and diffusion-wave simplifications drop exactly the terms that let a front steepen and travel upstream, so they cannot represent a sudden blockage.