A river has a mean velocity of 5 m/s and an average depth of 5 m. A student classifies it as supercritical because 5 m/s is 'fast flow.' Evaluate this classification.
ACorrect — 5 m/s exceeds the threshold speed for supercritical flow in most natural channels
BIncorrect — the Froude number Fr = V/√(gD) = 5/√(9.81 × 5) ≈ 0.71 < 1, so the flow is subcritical despite the high velocity
CIncorrect — velocity alone never determines flow regime; you need to know the channel slope
DCorrect — depth only matters for laminar flow; at turbulent velocities, Fr is not the relevant criterion
Fr = 5 / √(9.81 × 5) = 5 / √49.05 = 5 / 7.0 ≈ 0.71 < 1 → subcritical. The denominator √(gD) is the speed of small surface gravity waves in water of depth D. At depth 5 m, gravity waves travel at ~7 m/s, faster than the flow velocity of 5 m/s, so disturbances can still propagate upstream — subcritical. A flow at 5 m/s in a 0.1 m deep channel would be supercritical (Fr = 5/√0.98 ≈ 5.1). The critical insight is that both velocity and depth determine the Froude number; high velocity in deep water can still be subcritical.
Question 2 Multiple Choice
A sluice gate releases water at velocity 8 m/s and depth 0.5 m into a downstream stilling basin (Fr = 8/√(9.81 × 0.5) ≈ 3.6). An engineer wants to design a hydraulic jump to dissipate this energy. Which statement correctly describes the required conditions?
AThe jump will form spontaneously as long as the downstream channel is deeper, transitioning from subcritical to supercritical flow
BThe supercritical flow must be forced to transition to subcritical flow; the jump proceeds from supercritical to subcritical, never the reverse
CThe engineer can design the jump to run in either direction by adjusting the downstream depth
DA hydraulic jump requires Fr > 5; at Fr = 3.6, the flow will simply decelerate gradually
A hydraulic jump is an irreversible transition from supercritical (Fr > 1) to subcritical (Fr < 1) flow — always in this direction, never the reverse. The thermodynamic reason is entropy: a hydraulic jump dissipates energy (head loss is positive), making it physically possible. A transition from subcritical to supercritical would require energy addition, not dissipation. The engineer designs the stilling basin to provide a tailwater (downstream) depth equal to the conjugate depth, which forces the transition at the desired location. The incoming Fr ≈ 3.6 indicates a 'strong' jump that will dissipate roughly 30–40% of the kinetic energy.
Question 3 True / False
In a subcritical river reach, constructing a dam downstream will raise the water surface upstream for a considerable distance (backwater effect), but this effect would not occur if the river were supercritical.
TTrue
FFalse
Answer: True
The Froude number determines whether disturbances can propagate upstream. In subcritical flow (Fr < 1), surface gravity waves travel faster than the flow velocity, so information about the dam (a downstream boundary condition) propagates upstream as a backwater curve. In supercritical flow (Fr > 1), the flow outruns all disturbances — no wave can travel faster than the current against the flow. A dam placed downstream of a supercritical reach would cause a hydraulic jump immediately at the dam face, but the surface profile upstream of the jump would be unaffected by the dam's presence.
Question 4 True / False
Manning's equation Q = (1/n)·A·R_h^(2/3)·S^(1/2) is dimensionally consistent and can be applied with any unit system by simply substituting the appropriate values.
TTrue
FFalse
Answer: False
Manning's equation is empirical and dimensionally inconsistent — Manning's n carries implicit dimensions to make the equation balance. In SI units, n has effective units of s/m^(1/3); in English units (feet, seconds), n has different effective dimensions. The equation was fitted to data in specific unit systems and produces correct results only when the unit system matches the convention used. Mixing SI and English units, or using the SI form of the equation with English measurements without correction, gives errors of roughly 50%. The Darcy-Weisbach equation for pipe flow is dimensionally consistent and unit-agnostic; Manning's equation is not.
Question 5 Short Answer
Explain the analogy between the Froude number in open channel flow and the Mach number in compressible gas dynamics — what physical quantities are being compared in each case, and why does the analogy hold?
Think about your answer, then reveal below.
Model answer: Both the Froude number and the Mach number compare a flow velocity to the speed of small disturbance propagation in that medium. The Mach number Ma = V/c compares flow velocity to the speed of sound (pressure waves) in a compressible gas. The Froude number Fr = V/√(gD) compares flow velocity to the speed of surface gravity waves in a channel of depth D. In both cases, when the flow velocity exceeds the wave speed (Ma > 1 or Fr > 1), disturbances generated at a point cannot propagate against the flow — the flow outruns its own signals. This produces identical physical phenomena: shock waves in gas dynamics correspond to hydraulic jumps in open-channel flow, both being abrupt transitions from supersonic/supercritical to subsonic/subcritical conditions with irreversible energy dissipation. The analogy holds because both cases involve a critical wave speed that divides disturbance-transmitting regimes from disturbance-blocking regimes.
The analogy is not merely pedagogical — it reflects a deep mathematical similarity. The governing equations for shallow water flow and one-dimensional compressible gas flow have the same mathematical form, with the ratio of specific heats in the gas equations corresponding to a factor of 2 in the shallow water equations. This means solutions, stability criteria, and wave phenomena in one domain translate directly to the other. Engineers and physicists studying one domain routinely draw on intuition from the other.