How to compute open channel discharge: fundamental hydraulic parameter for designing agricultural canals, sewers, rivers, drainage ditches, road runoff, irrigation systems. Robert Manning's empirical formula (Ireland 1889): V = (1/n) * R^(2/3) * S^(1/2) and Q = V * A, where V is mean velocity (m/s), n is Manning's roughness coefficient (glass 0.010; plastic 0.011; concrete 0.013; earth 0.025; dense vegetation 0.045), R = A/P hydraulic radius (m, wetted area / wetted perimeter), S = So bed slope (m/m). Essential tool for hydraulic engineers, civil engineers, agronomists, sizing irrigation works (USDA NRCS, USGS), urban sewer networks (NYC DEP, Chicago MWRD, LA DPW), river barrages USACE, road drainage AASHTO, FEMA floodplain. Froude number Fr = V/sqrt(g*Dh) classifies regime: subcritical Fr1 (fast, hydraulic jumps possible).

Manning and Strickler open channel formulas

Manning equation 1889: V = (1/n) * R^(2/3) * S^(1/2). Robert Manning Irish port engineer, SI metric formulation. Strickler 1923 uses K=1/n: V = K * R^(2/3) * S^(1/2). Q = V * A volumetric discharge (m^3/s, cfs USA). Example concrete rectangular b=3 m, y=1.5 m, n=0.013, So=0.001: A=4.5 m^2, P=6 m, R=0.75. V=2.0 m/s, Q=9.0 m^3/s = 318 cfs.

Standard geometric sections: rectangular A=b*y, P=b+2y. Trapezoidal A=(b+zy)y, P=b+2y*sqrt(1+z^2), z side slope. Triangular A=zy^2, P=2y*sqrt(1+z^2). Circular (sewer): theta=2*arccos(1-y/r), A=(theta-sin theta)*r^2/2, P=theta*r. Most efficient section maximum discharge: rectangular y=b/2 (semicircle), trapezoidal z=1/sqrt(3)=0.577 (60° slope).

Froude number: Fr = V / sqrt(g * Dh), Dh=A/T hydraulic depth (T top water surface width). Fr1 supercritical (fast, torrential, upstream control). Hydraulic jump when super->sub transition: dissipates energy, controlled stilling basin downstream of dam spillways.

Typical Manning n coefficients: smooth glass/plastic 0.009-0.011; smooth concrete 0.011-0.013; rough concrete 0.015-0.017; smooth earth 0.018-0.025; earth with short vegetation 0.025-0.030; rocky channel 0.030-0.045; river dense vegetation 0.040-0.070; flood-prone with obstructions 0.050-0.150. Reference Chow 1959 "Open Channel Hydraulics" McGraw-Hill, USDA NRCS National Engineering Handbook.

Velocity limits: minimum to prevent sedimentation (V_min=0.3-0.6 m/s clear water, 0.9 m/s loaded water). Maximum to prevent erosion: earth 1.5 m/s, concrete 5-6 m/s, rock 10-15 m/s. ASCE Manual 37 USA standard, AASHTO Highway Drainage Guidelines USA.

History and context

Open channel hydraulics has ancient origins: Roman aqueducts Aqua Appia 312 BC, Aqua Virgo 19 BC (today Trevi Rome), Aqua Augusta Naples (95 km longest Roman), Chinese Han dynasty canals 600 BC. Modern mathematical study: Antoine Chezy (France 1770) formula V=C*sqrt(R*S) empirical coefficient C, classical foundation. Robert Manning (Dublin 1816-1897) Irish port engineer, published 1889 "On the flow of water in open channels and pipes" Transactions Institution Civil Engineers Ireland Volume 20, metric formulation n. Albert Strickler (Switzerland 1887-1963) ETH Zurich 1923 proposed K=1/n European notation. Hsing Chow (China-USA 1919-1981) "Open Channel Hydraulics" 1959 McGraw-Hill modern standard text. USA: USACE U.S. Army Corps of Engineers Hydraulic Engineering Center HEC-RAS software industry standard. USGS U.S. Geological Survey stream gaging. USDA NRCS Natural Resources Conservation Service agricultural drainage. Famous structures: Hoover Dam Nevada-Arizona 1936, Grand Coulee Dam Washington 1942, Glen Canyon Arizona 1966, Three Gorges China 2008. USA: Bechtel Engineering San Francisco, MWH Global (Stantec), AECOM. Universities: MIT Boston, Stanford CA, UT Austin, Caltech, Colorado State Fort Collins hydrology research, USC Los Angeles. Disasters: Johnstown Flood Pennsylvania 1889 (2209 deaths South Fork dam), Saint Francis Dam California 1928 (431 deaths), Buffalo Creek WV 1972 (125 deaths). FEMA Federal Emergency Management 1979 floodplain regulation.

Manning n coefficient and velocity limits table

• Smooth glass, PVC, plastic: n=0.009-0.011 (laboratory flumes)
• Steel pipes (sewer pipe): n=0.011-0.013
• Smooth concrete (sewer canal): n=0.013
• Rough concrete (road canal): n=0.015-0.017
• Smooth earth (rural drainage): n=0.018-0.025
• Earth with short vegetation: n=0.025-0.030
• Rocky channel (natural bed): n=0.030-0.045
• River with dense vegetation: n=0.040-0.070
• Flood-prone with tree obstructions: n=0.070-0.150
• Minimum velocity (no sedimentation): 0.3-0.9 m/s
• Maximum earth velocity (erosion): 1.5 m/s
• Maximum concrete velocity: 5-6 m/s
• Maximum rock velocity: 10-15 m/s
• Subcritical Froude: Fr < 0.95 (calm fluvial)
• Critical: 0.95 < Fr < 1.05 (unstable transition)
• Supercritical: Fr > 1.05 (rapid torrential)

Worked numerical examples

Example 1 - American concrete sewer main NYC DEP. Section b=3 m, y=1.5 m. n=0.013 (smooth concrete). So=0.001 m/m. A=3*1.5=4.5 m^2. P=3+2*1.5=6 m. R=0.75 m. V=(1/0.013)*0.75^(2/3)*sqrt(0.001)=2.01 m/s = 6.6 ft/s. Q=9.04 m^3/s = 319 cfs = 9040 L/s. Fr=0.524 subcritical calm. Adequate for NYC DEP main service 100,000 residents.

Example 2 - American agricultural canal trapezoidal Iowa. b=5 m, y=2 m, z=2 (1:2 side slope earth stability). n=0.025 (earth with short vegetation). So=0.0005 m/m. A=18 m^2. P=13.94 m. R=1.29 m. V=1.06 m/s = 3.5 ft/s. Q=19.1 m^3/s = 675 cfs. Fr=0.239 subcritical. Iowa State USDA NRCS irrigation system corn soybean 100,000 acres.

Example 3 - American highway drainage triangular ditch Texas DOT. y=0.4 m, z=1.5 (1:1.5 safety AASHTO). n=0.030 (low vegetation ditch). So=0.005 m/m (5‰ highway). A=0.24 m^2. P=1.44 m. R=0.167 m. V=0.71 m/s. Q=0.17 m^3/s = 6 cfs. Fr=0.506 subcritical. Texas Department of Transportation Highway Drainage Manual TxDOT.

Example 4 - American urban sewer Chicago MWRD circular. D=0.6 m (24"), y=0.4 m (67% fill). n=0.013 concrete. So=0.002 m/m. theta=4.012 rad. A=0.215 m^2. P=1.204 m. R=0.179 m. V=1.09 m/s = 3.6 ft/s. Q=0.234 m^3/s = 8.3 cfs = 234 L/s. Capacity standard Chicago Metropolitan Water Reclamation District residential sector 5000 residents.

Example 5 - American natural river Mississippi River. b=200 m main channel (low water), y=10 m flood, z=3 (broad floodplain). n=0.040 (river dense vegetation banks). So=0.0001 m/m (very gentle). A=(200+30)*10=2300 m^2. P=200+63.25=263.25 m. R=8.74 m. V=1.11 m/s = 3.6 ft/s. Q=2553 m^3/s = 90,200 cfs (low end Mississippi). Fr=0.083 subcritical fluvial. USGS Vicksburg gauge, USACE Mississippi River Commission stage forecasting.

Limits and considerations

1. Manning empirical formula valid steady uniform turbulent flow: temporal variations (floods, drawdown) require non-steady Saint-Venant 1D.
2. n coefficient not constant: varies with depth (different submerged vegetation), season (leaves), discharge (regime).
3. Composite sections (channel+floodplain): separate calculation per parts, different n per segment.
4. Plan curves: increase effective n +10-20%, energy losses turbulence.
5. Constrictions, expansions, drops: require water surface profile analysis (M1, M2, S1, S2 standard step method).
6. Sediment transport: solid discharge cuts different bedforms (dunes, antidunes), n changes.
7. Subcritical vs supercritical: different dynamics, hydraulic control, hydraulic jumps dissipate energy.
8. Cavitation: high-velocity channels (>10 m/s) erode concrete, damage gates.
9. Air entrainment: high discharges over weirs trap air, foam reduces water density.
10. Does not consider sedimentation: design requires V_min>V_sed (Camp 1946 USA municipal engineering).

Common errors

1. Confusing Manning n with Chezy C coefficient (n=R^(1/6)/C different relationship).
2. Neglecting n update for seasonal vegetation or flow debris.
3. Calculating R=A/P neglecting submerged vegetated banks wetting.
4. Using Manning for laminar regime (Re<2000): valid only turbulent.
5. Confusing So bed slope with Sf energy slope (Se>So if accelerating).
6. Neglecting regime: subcritical vs supercritical changes dramatically.
7. Composite sections channel+floodplain with single n: 30-50% errors.
8. Excessive slope: beyond rock 1:5 (20%) Manning fails, tumbling regime.
9. Sizing only Q max without considering monthly V_min sedimentation.
10. Neglecting meander effects, obstacles (rocks, fallen trees) in n calculation.

Numbered applications

1. Urban sewers: NYC DEP, LA DPW, Chicago MWRD, Houston PWD sizing pipes 8"-60".
2. Agricultural irrigation: USDA NRCS, USGS, California Central Valley Project, Idaho potato.
3. Highway drainage: TxDOT, Caltrans, AASHTO Highway Drainage Manual, MoT cunette.
4. Natural rivers: USGS gaging Mississippi, Colorado, Hudson, USACE Galveston.
5. Hydroelectric power: TVA Tennessee Valley Authority, BPA Bonneville Power Administration.
6. Dam spillways: USACE, Bureau of Reclamation Hoover Glen Canyon, FERC inspection.
7. Water supply aqueducts: NYC DEP Catskill Croton, LA Aqueduct Owens Valley, Boston MWRA.
8. Stream restoration: USACE, NRCS National Engineering Handbook USA stream design.
9. Flood control: FEMA floodplain regulation, USACE levee certification, ARS rebuilds.
10. Educational: civil engineering MIT, Stanford, UT Austin, Caltech, Colorado State.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary hydraulic sizing. For professional design consult Chow "Open Channel Hydraulics" 1959 McGraw-Hill, Henderson "Open Channel Flow" 1966, ASCE Manual 37 USA, USACE HEC-RAS software, Manning RW 1889 ICE Ireland, AASHTO Highway Drainage Manual USA, USDA NRCS National Engineering Handbook, FHWA HEC-22 Drainage Design. USACE Hydrologic Engineering Center Davis CA. USGS Water Science.

Updated July 2026