Open-Channel Flow & Manning Equation

Open-Channel Flow & Manning Equation

Quick Answer: Open-channel uniform flow → Manning: $V = \frac{1.49}{n} R_h^{2/3} S^{1/2}$ (US). Compute A, P, R_h, then Q = VA. Check Froude number if the stem mentions critical depth or hydraulic jump. Open-channel flow appears in storm sewers, culverts, irrigation ditches, and natural streams. The NCEES Handbook lists Manning's equation in environmental and civil sections — locate it quickly.

Manning Equation US customary (open channel): $V = \frac{1.49}{n} R_h^{2/3} S^{1/2}$ SI form uses K = 1.0 instead of 1.49 — identify system before calculating. | Symbol | Meaning | |--------|---------| | V | Average velocity (ft/s or m/s) | | n | Manning roughness (empirical) | | R_h | Hydraulic radius A/P (ft or m) | | S | Energy slope (ft/ft) ≈ bed slope for uniform flow | | Q | Discharge = V × A |

Roughness Coefficient n | Channel | Typical n | |---------|-----------| | Concrete finished | 0.012–0.015 | | Earth, straight | 0.022–0.030 | | Natural stream | 0.03–0.05 | | Weedy | 0.06+ | Higher n → lower velocity for same geometry. Exam provides n or asks you to pick from a table.

Trapezoidal Channel Geometry Bottom width b, side slope zH:1V, depth y: [ A = y(b + zy) ] $P = b + 2y\sqrt{1+z^2}$ $R_h = \frac{A}{P}$ Worked example: b = 8 ft, z = 2, y = 4 ft, n = 0.025, S = 0.001. A = 4(8 + 8) = 64 ft²; P = 8 + 2×4×√5 = 8 + 17.9 = 25.9 ft; R_h = 2.47 ft. $V = \frac{1.49}{0.025} (2.47)^{2/3} (0.001)^{1/2} = 59.6 \times 1.76 \times 0.0316 \approx 3.3 \text{ ft/s}$ Q = 3.3 × 64 ≈ 211 cfs.

Solving for Normal Depth When Q, b, z, n, S are known, y is unknown — trial-and-error, bisection, or calculator solver. Pick the physically realistic positive root below bank height. Exam strategy: Substitute answer choices for y and check which Q matches.

Circular Conduit Partially Full Ratio y/D from geometry tables (Handbook) gives A and R_h. Storm sewers often flow part full — do not assume full pipe unless stated.

Froude Number and Flow Regime $Fr = \frac{V}{\sqrt{g D_h}}$ D_h = hydraulic depth A/T (T = top width). | Regime | Fr | Behavior | |--------|-----|----------| | Subcritical | < 1 | Deep, slow; disturbances travel upstream | | Critical | = 1 | Minimum specific energy for given Q | | Supercritical | > 1 | Shallow, fast; hydraulic jump possible | Hydraulic jump dissipates energy when supercritical flow meets subcritical — raised floor or block in stilling basin.

Specific Energy and Critical Depth (Conceptual) For rectangular channel, critical depth: $y_c = \left(\frac{q^2}{g}\right)^{1/3}$ q = Q/b unit discharge. Used for spillway and channel transition problems.

Non-Uniform Flow Notes Gradually varied flow (backwater curves) when depth changes along channel — FE usually sticks to uniform Manning unless profile is described qualitatively.

Practical Applications - Size drainage channel for design storm Q from rational/SCS. - Check freeboard — design depth below top of bank. - Evaluate scour at culvert outlets (velocity-linked).

Common Errors - Using pipe diameter as R_h without geometry for partial flow. - Wrong exponent: 2/3 on R_h, 1/2 on S. - Mixing SI and US Manning constant. > Exam trap: Hazen-Williams is for pressurized pipe flow, not open channels — use Manning when the water has a free surface. Manning problems are high-frequency FE calculations. Drill trapezoid geometry plus one partial-full pipe example before exam day.

Compound Channels and Overbank Flow Floodplains carry flow above bankfull in compound sections — wetted area and perimeter change nonlinearly with stage. Handbook may provide composite methods; exam usually uses simple trapezoid or rectangle.

Culvert vs. Open Channel Culverts may flow part full (open channel) or full (pressurized). Inlet control vs. outlet control governs capacity — distinguish from simple Manning reach problems.

Energy and Momentum (Brief) Specific energy diagram shows alternate depths for same discharge. Hydraulic jump dissipates energy when supercritical flow transitions to subcritical — stilling basins below dam spillways.

Calculator Tips Store R_h^(2/3) and S^(1/2) separately when comparing multiple depths in multiple-choice Manning problems — substitute four depth options quickly. Sanity check: Doubling channel depth more than doubles Q because both A and R_h increase. Sanity check: Doubling channel depth more than doubles Q because both A and R_h increase. On the FE Environmental exam, confirm units before substituting into handbook equations; distractors often differ only by conversion factors. Practice locating handbook sections by keyword during timed drills so lookup takes under one minute.

Rectangular Channel — Depth Trial Q = 120 cfs, b = 12 ft, n = 0.020, S = 0.002. Try y = 3 ft: A = 36, P = 18, R_h = 2.0, V = (1.49/0.02)(2)^(2/3)(0.002)^0.5 = 74.5 × 1.59 × 0.0447 ≈ 5.3 cfs/ft² × 36 ft² → Q ≈ 191 cfs (too high). Try y = 2 ft: A = 24, R_h = 1.33, V ≈ 3.6, Q ≈ 86 cfs (too low). Interpolate → y ≈ 2.4 ft.

Partial-Full Pipe Quick Check 18-in pipe flowing 0.5 full — area fraction ≈ 0.39 of full (Handbook table). If full Q would be 8 cfs, part-full Q ≈ 8 × 0.39 × (velocity ratio) — velocity ratio also from table; exam often gives composite chart.

Chezy vs. Manning $V = C\sqrt{R_h S}$ with $C = \frac{R_h^{1/6}}{n}$ in US units linking to Manning. Same physics — do not mix Chezy C from one table with Manning n from another without conversion.

Channel Sizing for Freeboard Design depth 3.2 ft with 1 ft freeboard → top of bank at 4.2 ft. Check shear stress on liners qualitatively:

Study Structure

• Skim the Quick Answer, then the table above
• Close with one practice quiz item before moving on