Critical depth, specific energy and hydraulic jumps on the PE Civil WRE exam

Manning's equation tells you how deep water runs in a long, uniform channel. Specific energy explains what it does when something changes: a gate, a step in the bed, a narrowing, a drop. On the Water Resources & Environmental exam these show up as critical depth, Froude number, alternate depths and hydraulic jumps. The Handbook covers them under Total Head and Specific Energy, Normal and Critical Flow, the momentum-depth relationship, and Rapidly Varied Flow and Hydraulic Jump.

What you actually need to own

• Specific energy: E = y + v²/2g = y + Q²/(2gA²), measured from the channel bottom. For a given flow, plotting E against y gives the specific-energy curve. Its minimum is critical depth.
• Critical depth, rectangular channel: work in unit discharge q = Q/b, so y_c = (q²/g)^(1/3). At critical, the velocity head is y_c/2 and E_min = 1.5·y_c.
• Critical depth, any other shape: solve Q²/g = A³/T at critical, where T is top width. For a trapezoid or circle that means trial and error, or the Handbook's approximate critical-depth expressions for those shapes, within their stated ranges.
• Froude number: Fr = v / √(g·D), with hydraulic depth D = A/T. Fr < 1 is subcritical, Fr > 1 is supercritical, Fr = 1 is critical. In a rectangle D equals y. In other shapes it doesn't.
• Alternate depths: a subcritical and a supercritical depth with the same specific energy. Given one, find the other by solving the energy equation for the root on the other limb.
• Transitions: carry E across a rise or contraction and see what's left. A rise lowers the water surface in subcritical flow and raises it in supercritical flow. If the rise asks for more than E − E_min, the section chokes and the upstream depth has to change.
• Hydraulic jumps (rectangular, horizontal): sequent depths share the same momentum, not the same energy. y₂/y₁ = ½·(√(1 + 8·Fr₁²) − 1). Energy loss is ΔE = (y₂ − y₁)³ / (4·y₁·y₂).
• Jump location: compare the tailwater to the depth required as the sequent depth. If the tailwater is too low, the jump moves downstream. If it's too high, the jump is pushed upstream and can be drowned. This is the logic behind stilling basins at drops and culvert outlets.

Where people lose points

• Using Q instead of q in the rectangular critical-depth formula. Divide by width first.
• Using hydraulic radius, or adding a 2g, in the Froude number. The denominator is √(g·A/T).
• Picking the wrong alternate depth. Both roots satisfy the equation. The upstream regime tells you which limb you're on, and in a smooth transition the flow stays on that limb.
• Reading jump height as energy loss. y₂ − y₁ is the height of the jump. The loss is the cubic expression above.
• Treating a jump as energy-conserving. Across the jump, momentum is conserved and energy is lost. Use the energy equation on either side of it, or across it only with the jump loss as a head-loss term.
• Applying the rectangular jump relation to a non-rectangular section without saying so. Other shapes need the momentum function M = Q²/(gA) + A·ȳ.

How to study it

Sketch the specific-energy curve before every problem and mark where the flow sits. Most regime-reversal errors disappear once you can see which limb you're on. Drill critical depth in a rectangle and a trapezoid, then alternate depths under a sluice gate, then a jump below a spillway or a culvert outlet. If you get stuck, CastorPrep's tutor can walk through the problem you're working on.

Part of the Hydraulics–Open Channel area of the PE Civil WRE exam. → Start practicing free — the whole practice bank, free with an account.