Printable cheat sheet

Open Channel Flow Cheat Sheet — FE/PE Civil Exam (Printable)

Q: What is the difference between normal depth and critical depth?

Normal depth is the uniform-flow depth you get from Manning's equation — it depends on roughness n and slope S as well as discharge. Critical depth is the depth where the Froude number equals 1 — it depends only on discharge and channel geometry. They coincide only at the critical slope; on a mild slope normal depth is deeper than critical, on a steep slope it is shallower.

Q: When is R = D/4 valid for a circular pipe?

Only for a full circular pipe or one flowing exactly half full — those are the two depths where the geometry collapses to R = D/4. At any other partly-full depth, recompute area and wetted perimeter from the central angle θ = 2·arccos(1 − 2y/D) before finding R = A/P.

Q: Do I use 1.486 or 1.0 in Manning's equation?

1.486 with US customary units (feet, ft/s), 1.0 with SI units (metres, m/s). Manning's equation is empirical, so the constant carries the unit conversion — it is not optional. A velocity that looks ten times too small usually means the 1.486 was dropped.

Q: How do I tell subcritical from supercritical flow?

Compute the Froude number Fr = V/√(g·Dh), where hydraulic depth Dh = A/T uses the top width T, not the wetted perimeter. Fr 1 is supercritical (shallow, fast), Fr = 1 is critical. For a rectangular channel Dh equals the flow depth y.

Every open-channel formula the FE Civil and PE Civil exams test, on one scannable page — Manning's equation in both unit systems, channel geometry, Froude number, critical depth, specific energy, and best hydraulic sections — with the exam trap noted on each line and four worked problems below.

Last reviewed: 2026-10-09. All worked examples were independently solved and cross-checked; the calculator pages reproduce each answer.

Printing this sheet: press Ctrl+P (or ⌘P on a Mac) — navigation, the quiz and the call-to-action boxes are hidden automatically, so you get a clean reference printout.

1 · Manning's equation — uniform flow

V = (k/n) · R^2/3 · S^1/2 Q = A · V

V	mean velocity (ft/s with k = 1.486, m/s with k = 1.0)
Q	discharge = velocity × area
k	1.486 for US customary units, 1.0 for SI — the most-tested constant on the page
n	Manning's roughness (concrete ≈ 0.013, clean earth ≈ 0.022, natural channel ≈ 0.03–0.05)
R = A/P	hydraulic radius = area / wetted perimeter (the free surface is not wetted)
S	bed slope for uniform flow, dimensionless (ft/ft or m/m) — not percent unless converted

Manning's describes uniform flow: constant depth, with the energy grade line, water surface and bed all parallel. It does not apply to pressurised pipes, hydraulic jumps, weirs, or backwater curves. The equation is empirical, which is exactly why the constant changes with the unit system.

2 · Channel geometry — area and wetted perimeter

Rectangular: A = b·y, P = b + 2y

Trapezoidal: A = (b + z·y)·y, P = b + 2y√(1+z²)

Circular, full or exactly half-full: R = D/4 (no other depth)

Circular, partly full: θ = 2·arccos(1 − 2y/D), A = (D²/8)(θ − sinθ), P = D·θ/2

b, y	bottom width, flow depth
z	side slope as horizontal:vertical (z = 2 means 2H:1V)
θ	central angle of the wetted arc, in radians
Do NOT use R = D/4 for a partly full pipe — recompute A and P for the actual depth.

3 · Froude number — which regime

Fr = V / √(g·D_h) with D_h = A/T

Fr < 1	subcritical — deep, slow; disturbances travel upstream
Fr = 1	critical — the minimum-energy state for the discharge
Fr > 1	supercritical — shallow, fast; disturbances swept downstream
D_h = A/T	hydraulic depth uses the top width T, not the wetted perimeter — for a rectangle, D_h = y

The exam's favourite Froude trap is computing D_h with P instead of T. A second one: asking for the regime of flow on a "mild" vs "steep" slope — mild slope gives subcritical normal depth, steep slope gives supercritical.

4 · Critical depth and specific energy

Rectangular channels: y_c = (q²/g)^1/3 and E_min = 3y_c/2

Specific energy: E = y + V²/2g

q = Q/b	discharge per unit width — using total Q in the y_c formula is a classic error
y_c	critical depth — depends only on discharge and geometry, not on n or S
E_min	minimum specific energy, occurring exactly at critical depth
Alternate depths	for any E > E_min there are two depths with the same E — one subcritical, one supercritical

5 · Best hydraulic sections — maximum conveyance

Trapezoidal: z = 1/√3 ≈ 0.577, R = y/2 Rectangular: b = 2y, R = y/2 Triangular: z = 1

Idea	the "best" section minimises wetted perimeter for a given area — less friction, more flow
Trapezoidal optimum	side slopes of 1/√3 form half a regular hexagon; at the optimum the hydraulic radius is y/2
Rectangular optimum	width exactly twice the depth (also R = y/2) — a square-ish section is never optimal
The exam use	usually a one-step identification ("which section carries the most flow for this area?"), not a derivation

Worked examples — the four the exam keeps repeating

Worked example Normal depth by trial (US units)

Given: a rectangular channel, bottom width b = 8 ft, must carry Q = 120 cfs, Manning's n = 0.013, bed slope S = 0.002. Find the normal depth y_n.

Solution: y appears in both A and R, so iterate Manning's equation Q = (1.486/n)·A·R^2/3·S^1/2:

y = 2.00 ft: A = 16.0 ft², P = 12.00 ft, R = 1.333 ft → Q = 99.1 cfs (too low)
y = 2.20 ft: A = 17.6 ft², P = 12.40 ft, R = 1.419 ft → Q = 113.6 cfs (too low)
y = 2.30 ft: A = 18.4 ft², P = 12.60 ft, R = 1.460 ft → Q = 121.1 cfs (a touch high)
Narrowing between 2.20 and 2.30: y ≈ 2.29 ft gives Q = 120.0 cfs.

Answer: y_n ≈ 2.29 ft. Sanity check: V = 120/18.3 = 6.56 ft/s, Fr = 6.56/√(32.2×2.29) = 0.77 < 1 — subcritical, as expected on a mild 0.002 slope.

Verify it yourself on our Manning's equation calculator.

Worked example Froude number and flow regime

Given: a rectangular channel 10 ft wide carries 240 cfs at a depth of 3.0 ft. Classify the flow.

Solution:

Area: A = 10 × 3.0 = 30.0 ft²; velocity: V = Q/A = 240/30.0 = 8.0 ft/s
Hydraulic depth (rectangle): D_h = y = 3.0 ft
Fr = V/√(g·D_h) = 8.0/√(32.2 × 3.0) = 8.0/9.83 = 0.81

Answer: Fr = 0.81 < 1 — subcritical flow. The trap answer is 1.23 (the reciprocal): always divide velocity by the wave speed, not the reverse.

Worked example Critical depth and minimum specific energy

Given: a wide rectangular channel carries q = 8 ft²/s per unit width. Find the critical depth and the minimum specific energy.

Solution:

y_c = (q²/g)^1/3 = (64/32.2)^1/3 = (1.9876)^1/3 = 1.26 ft
E_min = 3y_c/2 = 1.5 × 1.26 = 1.89 ft

Answer: y_c ≈ 1.26 ft, E_min ≈ 1.89 ft. At any other depth the same 8 ft²/s needs more than 1.89 ft of specific energy — the excess splits into a subcritical and a supercritical alternate depth.

Worked example Partly-full pipe — the R = D/4 trap

Given: a 24-inch (D = 2.0 ft) concrete pipe, n = 0.013, slope S = 0.005, flowing at 40% of its diameter (y/D = 0.40). Find Q.

Solution:

Central angle: θ = 2·arccos(1 − 2×0.40) = 2·arccos(0.20) = 2.7389 rad
Area: A = (D²/8)(θ − sinθ) = 0.5 × (2.7389 − 0.3919) = 1.1735 ft²
Wetted perimeter: P = D·θ/2 = 2.7389 ft; R = A/P = 0.4285 ft — not D/4 = 0.50 ft
Velocity: V = (1.486/0.013) × (0.4285)^2/3 × √0.005 = 114.31 × 0.5683 × 0.07071 = 4.594 ft/s
Discharge: Q = 1.1735 × 4.594 = 5.39 cfs

Answer: Q ≈ 5.39 cfs. Using R = D/4 = 0.50 ft (valid only at full or half-full) gives 5.97 cfs — about 11% too high. For reference, the full-pipe capacity is 16.0 cfs, so the pipe carries 0.337 of full flow at 40% depth.

Common exam traps

The 1.486. US units get 1.486, SI gets 1.0. A velocity that looks an order of magnitude off usually means the constant was dropped or swapped.
R = D/4 has exactly two valid depths. Full pipe or exactly half-full. Any other partly-full depth needs the θ computation first.
The free surface is not wetted. A rectangular channel's wetted perimeter is b + 2y — counting the water surface as a fourth side inflates P and shrinks R.
D_h uses top width T, not wetted perimeter P. Fr = V/√(g·A/T). Swapping in P is the standard Froude-number distractor.
q vs Q. Critical depth uses discharge per unit width q = Q/b. Feeding total Q into (q²/g)^1/3 is always wrong for a wide channel.
Normal depth ≠ critical depth. They meet only at the critical slope. Mild slope: normal depth is deeper than critical (subcritical). Steep slope: shallower (supercritical).
Slope units. Manning's wants S dimensionless. A "2% grade" is S = 0.02, not 2.

Frequently asked questions

What is the difference between normal depth and critical depth?

Normal depth is the uniform-flow depth you get from Manning's equation — it depends on roughness n and slope S as well as discharge. Critical depth is the depth where the Froude number equals 1 — it depends only on discharge and channel geometry. They coincide only at the critical slope; on a mild slope normal depth is deeper than critical, on a steep slope it is shallower.

When is R = D/4 valid for a circular pipe?

Only for a full circular pipe or one flowing exactly half full — those are the two depths where the geometry collapses to R = D/4. At any other partly-full depth, recompute area and wetted perimeter from the central angle θ = 2·arccos(1 − 2y/D) before finding R = A/P.

Do I use 1.486 or 1.0 in Manning's equation?

1.486 with US customary units (feet, ft/s), 1.0 with SI units (metres, m/s). Manning's equation is empirical, so the constant carries the unit conversion — it is not optional. A velocity that looks ten times too small usually means the 1.486 was dropped.

How do I tell subcritical from supercritical flow?

Compute the Froude number Fr = V/√(g·D_h), where hydraulic depth D_h = A/T uses the top width T, not the wetted perimeter. Fr < 1 is subcritical (deep, slow), Fr > 1 is supercritical (shallow, fast), Fr = 1 is critical. For a rectangular channel D_h equals the flow depth y.

What is specific energy, and why does it have a minimum?

Specific energy E = y + V²/2g is the energy per unit weight measured from the channel bed. It has a minimum at critical depth: at very small depths the velocity head term blows up, and at large depths the depth term dominates. For any E above the minimum there are two alternate depths — one subcritical, one supercritical.

Is a partly-full storm sewer open-channel flow?

Yes. Any conduit with a free surface is open-channel flow, even inside a pipe — use Manning's equation with the filled portion's area and wetted perimeter. Manning's does not apply to a full, pressurised pipe.

Manning (uniform flow)	V = (k/n)·R^2/3·S^1/2, Q = A·V — k = 1.486 US / 1.0 SI
Rectangular	A = b·y, P = b + 2y (free surface not wetted)
Trapezoidal	A = (b + z·y)·y, P = b + 2y√(1+z²), z = H:V side slope
Circular	Full or half-full: R = D/4. Other depths: θ = 2·arccos(1 − 2y/D), then A and P from θ.
Froude number	Fr = V/√(g·D_h), D_h = A/T — < 1 subcritical, = 1 critical, > 1 supercritical
Critical depth (rectangle)	y_c = (q²/g)^1/3, q = Q/b — and E_min = 3y_c/2
Specific energy	E = y + V²/2g — minimum at critical depth; two alternate depths for larger E
Best hydraulic section	Trapezoidal: z = 1/√3 ≈ 0.577, R = y/2 · Rectangular: b = 2y · Triangular: z = 1

Open Channel Flow Cheat Sheet — FE/PE Civil Exam (Printable)

1 · Manning's equation — uniform flow

2 · Channel geometry — area and wetted perimeter

3 · Froude number — which regime

4 · Critical depth and specific energy

5 · Best hydraulic sections — maximum conveyance

Worked examples — the four the exam keeps repeating

Worked example Normal depth by trial (US units)

Worked example Froude number and flow regime

Worked example Critical depth and minimum specific energy

Worked example Partly-full pipe — the R = D/4 trap

Test the sheet

1. A rectangular channel 6 ft wide carries 90 cfs at a depth of 2.5 ft. What is the Froude number and the flow regime?

2. A rectangular channel carries q = 4 ft²/s per unit width. The critical depth is closest to:

3. A 36-inch concrete pipe flows exactly half full. A student computes discharge with R = D/4. What is wrong with this?

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