Bernoulli, Pipe Friction & Open-Channel Flow
Energy heads, Darcy-Weisbach vs Hazen-Williams, minor losses, specific energy and the Froude-number story of open channels.
Key formulas
hf = f · (L/D) · (V²/2g)
Darcy-Weisbach friction loss: The workhorse of pipe-network design; iterate f with the Moody chart when velocity is unknown.
yc = (q²/g)^(1/3), Ec_min = (3/2)·yc
Critical depth (rectangular channel): Sequent depths of a hydraulic jump satisfy the momentum function; upstream supercritical depth plus jump relation gives the sequent depth and energy dissipated.
Cheat sheet
- ▸Bernoulli (ideal, along a streamline): p/γ + v²/2g + z = constant; real flows add head loss hf to the downstream side.
- ▸Darcy-Weisbach: hf = f·(L/D)·(V²/2g) — most general pipe-loss formula; f depends on Reynolds number and relative roughness (Moody chart).
- ▸Minor losses: entrance K≈0.5, sudden expansion (V1−V2)²/2g, sudden contraction K≈0.5 on the downstream velocity, bends/valves K from tables. Long pipelines ignore them; short pump suction lines never do.
- ▸Laminar vs turbulent: Re < 2000 laminar (f = 64/Re), Re > 4000 turbulent; most water-supply flows are turbulent and in the transition zone.
- ▸Specific energy E = y + V²/2g: critical flow at E_min with yc = (q²/g)^(1/3); Froude number Fr = V/√(gy) — Fr<1 subcritical ( tranquil), Fr>1 supercritical (rapid).
- ▸Hydraulic jump: supercritical → subcritical transition with energy dissipation; forms downstream of spillways and sluice gates to protect the channel bed.
- ▸Manning's equation: V = (1/n)·R^(2/3)·S^(1/2) — the standard open-channel velocity formula; n is surface roughness (concrete ≈ 0.013, earthen ≈ 0.025).
Subcritical vs supercritical flow
| Aspect | Subcritical (Fr < 1) | Supercritical (Fr > 1) |
|---|---|---|
| Depth vs critical | Deeper than yc | Shallower than yc |
| Gravity vs inertia | Gravity dominates | Inertia dominates |
| Control from | Downstream (backwater curves) | Upstream (drawdown/rapidly varied) |
| Disturbances | Propagate upstream | Swept downstream |
| Typical location | Canals, natural rivers | Spillway chutes, steep channels |
Top 4 campus interview questions
MCQ mastery quiz
1. The three heads in Bernoulli's equation are…
2. In Darcy-Weisbach, halving the pipe diameter (same velocity) multiplies hf per metre by…
3. Flow is critical when the Froude number equals…
4. A hydraulic jump converts…
5. Manning's n for a smooth concrete-lined channel is roughly…