Open channel flow

In short

Open channel flow is the flow of liquid in a conduit with a free surface at atmospheric pressure - canals, rivers, storm drains, part-full sewers and flumes. It differs fundamentally from pipe flow because the depth itself is an unknown that adjusts with discharge, slope and boundary resistance, so gravity rather than pressure difference drives the motion. It is a high-weightage section in GATE Civil (Fluid Mechanics and Hydraulics), ESE, state PSC/AE/JE papers and university exams, and is daily working knowledge for irrigation and canal design (CWC, state irrigation departments, WAPCOS), urban drainage (AMRUT storm-water projects), hydropower waterways and consultancy design offices (L&T, AFCONS, Tata Consulting Engineers).

Explained

Classification and basic geometry

• Open channel vs pipe flow: free surface present; piezometric line coincides with water surface; hydraulic grade line = water surface; flow caused by bed slope component of gravity.
• Channel types: natural (rivers, streams - irregular, non-prismatic) and artificial (canals, flumes, culverts, part-full sewers). Prismatic = constant cross-section and bed slope along length; non-prismatic otherwise.
• Rigid boundary channels (lined canal, sewer) keep shape; mobile boundary (alluvial river, unlined canal) adjust by scour and deposition - handled by Kennedy/Lacey theory in the canal-design topic.
• Geometric elements: depth of flow y, area A, wetted perimeter P, hydraulic radius R = A/P, top width T, hydraulic depth D = A/T, section factor Z = A√D.
• Flow classification by time: steady (depth constant in time) / unsteady. By space: uniform (depth and velocity same at every section) / non-uniform, the latter split into gradually varied flow (GVF) and rapidly varied flow (RVF).
• By Reynolds number Re = ρVR/μ: laminar Re < 500, transitional 500-2000, turbulent Re > 2000 (based on R). Practically all field flows are turbulent.
• By Froude number Fr = V/√(gD): subcritical (tranquil) Fr < 1, critical Fr = 1, supercritical (rapid, shooting) Fr > 1. Fr is the ratio of flow velocity to the speed of a small gravity (surface) wave, so in subcritical flow a disturbance travels upstream and the control lies downstream; in supercritical flow it cannot, and the control lies upstream.
• Velocity distribution: maximum velocity occurs 0.05-0.25 y below the surface (air drag and secondary currents); mean velocity ≈ velocity at 0.6 y from the surface, or ½(v₀.₂ + v₀.₈) - the basis of current-meter gauging.
• Pressure distribution is hydrostatic only if streamlines are straight and nearly parallel; on steep slopes p = γy cos²θ, and curvilinear flow (over a spillway crest, in a jump) departs strongly from hydrostatic.

Uniform flow and resistance equations

• Uniform flow exists when gravity component driving the flow exactly balances boundary shear: bed slope S₀ = water-surface slope = energy slope Sf. The depth is then the normal depth y₀.
• Chezy (1769): V = C√(RS). C is dimensional (m^½/s), typically 30-80. Obtained from Bazin, Kutter or Ganguillet-Kutter formulae.
• Manning (1889): V = (1/n) R^2/3 S^1/2 - the standard formula in Indian practice. n is Manning's roughness coefficient (s/m^1/3). Relation: C = R^1/6/n.
• Typical n values: smooth cement plaster 0.011-0.013; concrete lined 0.013-0.017; brick 0.015; earthen canal clean 0.022-0.025; weedy earthen 0.030-0.040; natural stream with boulders 0.040-0.060; corrugated metal 0.024. IS 7112 and CWC canal manuals tabulate design values.
• Conveyance K = (1/n)AR^2/3 so Q = K√S₀; useful in compound sections where K of each sub-area is added.
• Darcy-Weisbach form: Sf = fV²/(8gR) using 4R as equivalent diameter; links to Moody-type analysis and shows Manning's n is valid only in the fully rough turbulent regime.
• Compound/natural channel: divide into main channel and berms/flood plains, compute Q of each with its own n and R, add. Composite roughness by Horton-Einstein equation when roughness varies around one perimeter.
• Normal depth computation is implicit: solve AR^2/3 = nQ/√S₀ by trial, or use non-dimensional curves of Q n/(S₀^½ B^8/3) vs y/B.
• Second (steeper) root: for closed conduits running part full (circular sewers), Q is maximum at y/D ≈ 0.938 and V maximum at y/D ≈ 0.81, not at full depth - a classic exam/interview point.

Most economical (efficient) section

• For a given A, n and S₀, Q is maximum when P is minimum (maximum R); for a given Q, this needs least area and least lining cost.
• Rectangle: b = 2y, i.e. y = b/2; R = y/2.
• Trapezoid: sloping side length = half the top width; R = y/2; the best of all trapezoids is the half-hexagon (side slope 60° to horizontal, m = 1/√3).
• Triangle: vertex angle 90° (side slopes 1:1); R = y/(2√2).
• Circular section: maximum discharge at y = 0.938 D, maximum velocity at y = 0.81 D.
• Semi-circle is the absolute most efficient shape of all, but hard to construct; hence trapezoids with practical side slopes (1.5H:1V in ordinary soil, 0.25:1 in rock, 2:1 in sandy soil) are used.
• Limitation: "most economical" ignores earthwork balance, freeboard, permissible velocity (non-scouring, non-silting), land cost and construction practicality, so field sections rarely match it exactly.

Specific energy, critical flow and channel transitions

• Specific energy E = y + V²/2g = y + Q²/(2gA²) - energy per unit weight measured from the channel bed.
• The E-y diagram at constant Q is a curve with two limbs and a minimum: for every E > Emin there are two alternate depths (one subcritical, one supercritical) carrying the same Q. At Emin the flow is critical.
• Condition for critical flow: Q²T/(gA³) = 1, i.e. Fr = 1. For a rectangular channel: yc = (q²/g)^1/3 where q = Q/B, and Emin = 1.5 yc, Vc = √(g yc).
• Specific force (momentum function) M = A ȳ + Q²/(gA): two depths with the same M are conjugate (sequent) depths - these govern the hydraulic jump, not alternate depths.
• Channel transitions - upward step: as a hump is raised in subcritical flow, depth over the hump falls; at the critical hump height ΔZc = E₁ − Emin the flow turns critical over the crest; a higher hump chokes the flow and raises the upstream level (afflux).
• Width reduction (contraction): for subcritical flow the depth falls through the contraction; the limiting width Bc is that at which critical flow occurs; narrower than this causes choking. Basis of the venturi flume and standing-wave flume.
• Channel slope classification by bed slope: mild (y₀ > yc), critical (y₀ = yc), steep (y₀ < yc), horizontal (S₀ = 0) and adverse (S₀ < 0). Note that "steep" is hydraulic, not geometric - it depends on Q, n and shape too.
• Controls: a sluice gate, weir, free overfall or a change of slope fixes a depth-discharge relation and is where GVF computation starts. At a free overfall, critical depth occurs about 3-4 yc upstream of the brink.

Gradually varied flow (GVF)

• Assumptions: hydrostatic pressure (nearly parallel streamlines), Manning/Chezy valid for the friction slope at a section, channel prismatic, α ≈ 1, slope small.
• Dynamic equation: dy/dx = (S₀ − Sf)/(1 − Fr²). Sign of numerator compares actual depth with normal depth, sign of denominator compares it with critical depth.
• Profile naming: the zone above both y₀ and yc is 1, between them 2, below both 3; combined with the slope letter gives M1, M2, M3, S1, S2, S3, C1, C3, H2, H3, A2, A3 (12 practical profiles in all; C2 would be uniform critical flow).
• Common profiles: M1 - backwater curve upstream of a weir or dam on a mild slope, asymptotic to horizontal far upstream; M2 - drawdown to a free overfall or sudden steepening; M3 - downstream of a sluice gate on a mild slope, ending in a hydraulic jump; S1 - behind a dam on a steep slope; S2 - drawdown after a mild-to-steep break; S3 - below a gate on a steep slope.
• Asymptotic behaviour: the profile approaches y₀ asymptotically, meets yc at a steep (theoretically 90°) angle - the GVF equation fails there because streamline curvature invalidates the assumptions - and approaches the bed vertically.
• Computation methods: (a) direct step method - choose Δy, compute Δx (for prismatic channels); (b) standard step method - choose Δx, solve for y by trial using the energy equation (used for natural rivers, HEC-RAS); (c) graphical integration; (d) Bresse's and Bakhmeteff's varied-flow functions.
• Applications: backwater at bridge piers and culverts, afflux and land submergence upstream of a barrage, length of canal affected by a cross regulator, delivery of a canal fall.

Rapidly varied flow - the hydraulic jump and measuring structures

• A hydraulic jump is an abrupt rise of depth when supercritical flow is forced to become subcritical; energy is dissipated in the roller by intense turbulence. Energy equation cannot be used across it (unknown losses) - the momentum equation is used.
• Sequent depth ratio (rectangular, horizontal bed): y₂/y₁ = ½(√(1 + 8Fr₁²) − 1).
• Jump height = y₂ − y₁; energy loss EL = (y₂ − y₁)³/(4y₁y₂); efficiency E₂/E₁; length Lj ≈ 5-7 y₂ for Fr₁ > 4.5.
• Classification by Fr₁: 1-1.7 undular; 1.7-2.5 weak; 2.5-4.5 oscillating (damaging waves - avoided in stilling basin design); 4.5-9.0 steady jump (best for energy dissipators, 45-70% loss); > 9.0 strong/choppy (up to 85% loss but rough water surface).
• Uses: energy dissipation below spillways, sluices and canal falls (IS 4997 covers hydraulic design of stilling basins); mixing of chemicals; aeration; raising downstream level for irrigation offtake; and as a flow-measuring device.
• Surges and waves: a moving hydraulic jump is a surge (positive/negative, e.g. gate operation in a canal or power channel); celerity of a small gravity wave c = √(gy), so the absolute wave speed is V ± c. Unsteady flow generally is governed by the Saint-Venant equations (continuity + momentum), used in flood routing and dam-break analysis.
• Flow measurement: broad-crested weir Q = 1.705 Cd b H^3/2 (free flow, critical depth on crest); sharp-crested rectangular weir Q = (2/3)Cd √(2g) L H^3/2; triangular (V-notch) Q = (8/15) Cd √(2g) tan(θ/2) H^5/2; venturi flume and standing-wave flume (free flow when a jump forms downstream); Parshall flume; sluice-gate (free and submerged) equations. Field discharge in canals is commonly measured by current meter (area-velocity) and by standing-wave flumes as per IS 9108 / IS 6063.

Key points

• Open channel flow - flow with a free surface at atmospheric pressure, driven by gravity; depth is an unknown that adjusts itself.
• Hydraulic radius R = A/P - the single geometric parameter governing resistance; replaces diameter of pipe flow (D ≈ 4R).
• Normal depth y₀ - the uniform-flow depth where bed slope exactly balances friction slope; depends on Q, n, S₀ and shape.
• Manning's formula V = (1/n)R^2/3S^1/2 - the Indian design standard; n is roughness, independent of depth but valid only in rough turbulent flow.
• Chezy's C = R^1/6/n - dimensional and depth-dependent, hence less convenient than n.
• Froude number Fr = V/√(gD) - decides subcritical (control downstream) or supercritical (control upstream) flow.
• Critical depth yc - the depth at which specific energy is minimum for a given Q (or Q is maximum for a given E); Fr = 1.
• Rectangular channel - yc = (q²/g)^1/3, Emin = 1.5yc, Vc = √(g yc).
• Alternate depths - two depths with the same specific energy and discharge; conjugate/sequent depths - two depths with the same specific force (jump).
• Most economical section - minimum wetted perimeter; rectangle b = 2y, best trapezoid is a half-hexagon, both with R = y/2.
• Circular conduit part full - Qmax at y = 0.938D and Vmax at y = 0.81D; important for sewer design.
• GVF equation dy/dx = (S₀ − Sf)/(1 − Fr²) - gives 12 profiles over M, S, C, H and A slopes.
• M1 backwater curve - rise upstream of a weir/barrage; its length decides land submergence and afflux.
• Hydraulic jump - supercritical to subcritical transition with large energy loss; solved by momentum, never energy.
• Sequent depth ratio y₂/y₁ = ½(√(1 + 8Fr₁²) − 1); EL = (y₂ − y₁)³/(4y₁y₂).
• Choking - raising a hump or narrowing a channel beyond the critical limit forces critical flow and raises the upstream water level.
• Wave celerity c = √(gy) - the physical meaning of Fr = 1 and the basis of surge analysis.
• Permissible velocity - canals are designed non-silting and non-scouring (roughly 0.6-2.0 m/s in unlined earth canals, up to 3-6 m/s in concrete-lined ones, as per IS 7112/IS 10430 and departmental manuals).

Formulas and examples

Example 1 - Uniform flow in a trapezoidal lined canal. A trapezoidal canal has bed width b = 6 m, side slopes 1.5H:1V, bed slope 1 in 2000, n = 0.016, depth of flow y = 2 m. Find Q and check the flow regime.

• A = (6 + 1.5 × 2) × 2 = (6 + 3) × 2 = 18 m²
• P = 6 + 2 × 2 × √(1 + 1.5²) = 6 + 4 × 1.8028 = 13.21 m
• R = 18/13.21 = 1.3626 m; R^2/3 = 1.3626^0.6667 ≈ 1.2290
• √S = √(1/2000) = √0.0005 = 0.02236
• V = (1/0.016) × 1.2290 × 0.02236 = 62.5 × 0.027481 = 1.718 m/s
• Q = 18 × 1.718 = 30.9 m³/s
• T = 6 + 2 × 1.5 × 2 = 12 m → D = 18/12 = 1.5 m; Fr = 1.718/√(9.81 × 1.5) = 1.718/3.836 = 0.448 → subcritical. Velocity 1.72 m/s is within the non-scouring limit for a concrete-lined canal.

Example 2 - Critical depth and specific energy. A rectangular channel 5 m wide carries 20 m³/s at a depth of 2.0 m.

• q = 20/5 = 4 m³/s per m; yc = (q²/g)^1/3 = (16/9.81)^1/3 = (1.6309)^1/3 = 1.177 m
• Emin = 1.5 × 1.177 = 1.766 m
• At y = 2.0 m: V = 20/(5 × 2) = 2.0 m/s; E = 2.0 + 2²/(2 × 9.81) = 2.0 + 0.2039 = 2.204 m; Fr = 2/√(9.81 × 2) = 2/4.429 = 0.452 → subcritical
• Alternate depth y′ solves y′ + 0.8155/y′² = 2.204 (since q²/2g = 16/19.62 = 0.8155). Try y′ = 0.70: 0.70 + 1.664 = 2.364 (high); y′ = 0.75: 0.75 + 1.450 = 2.200 ≈ 2.204 → alternate depth ≈ 0.749 m (supercritical).
• Hump check: maximum hump without choking ΔZc = E − Emin = 2.204 − 1.766 = 0.438 m. A hump of 0.30 m gives E₂ = 1.904 m over the crest; solving y + 0.8155/y² = 1.904 subcritically gives y ≈ 1.575 m, i.e. the surface drops by about 0.13 m over the hump (bed rises 0.30, depth falls 0.43).

Example 3 - Hydraulic jump below a sluice gate. Water issues under a gate into a 4 m wide rectangular channel at y₁ = 0.4 m with Q = 16 m³/s.

• V₁ = 16/(4 × 0.4) = 10 m/s; Fr₁ = 10/√(9.81 × 0.4) = 10/1.981 = 5.048 → steady jump
• y₂/y₁ = ½(√(1 + 8 × 5.048²) − 1) = ½(√(1 + 203.9) − 1) = ½(14.314 − 1) = 6.657 → y₂ = 6.657 × 0.4 = 2.663 m
• EL = (2.663 − 0.4)³/(4 × 0.4 × 2.663) = (2.263)³/4.261 = 11.587/4.261 = 2.719 m
• E₁ = 0.4 + 10²/19.62 = 0.4 + 5.097 = 5.497 m; E₂ = 5.497 − 2.719 = 2.778 m → efficiency = 2.778/5.497 = 50.5%, so about 49.5% of the energy is destroyed.
• Power dissipated = ρgQEL = 1000 × 9.81 × 16 × 2.719 ≈ 426.7 kW; stilling basin length ≈ 6 × 2.663 ≈ 16 m.

Example 4 - Most economical rectangular section. Design the most economical rectangular concrete channel (n = 0.015) to carry 10 m³/s on a slope of 1 in 1000.

• For the best rectangle b = 2y, so A = 2y², P = 4y, R = y/2.
• Q = (1/n) A R^2/3 √S → 10 = (1/0.015) × 2y² × (y/2)^2/3 × √0.001
• √0.001 = 0.031623; (1/2)^2/3 = 0.6300 → 10 = 66.667 × 2 × 0.6300 × 0.031623 × y^8/3 = 2.6560 y^8/3
• y^8/3 = 3.7651 → y = 3.7651^3/8 = 1.644 m → b = 3.29 m
• Check: A = 2 × 1.644² = 5.41 m², V = 10/5.41 = 1.85 m/s (safe for concrete lining). Add freeboard (about 0.5 m for this size as per IS 10430) → total depth ≈ 2.14 m.

Example 5 - Direct step (one step of a backwater curve). A rectangular channel 6 m wide, n = 0.018, S₀ = 0.0004, carries Q = 12 m³/s. A weir raises the depth to 2.5 m. Find the distance to the section where depth is 2.4 m.

• Section 1 (y = 2.5): A = 15 m², P = 6 + 5 = 11 m, R = 1.3636; V = 0.8 m/s; E₁ = 2.5 + 0.8²/19.62 = 2.5326 m; Sf1 = (nV/R^2/3)² = (0.018 × 0.8/1.2297)² = (0.011710)² = 1.3713 × 10⁻⁴
• Section 2 (y = 2.4): A = 14.4 m², P = 10.8 m, R = 1.3333; V = 0.8333 m/s; E₂ = 2.4 + 0.8333²/19.62 = 2.4354 m; Sf2 = (0.018 × 0.8333/1.2114)² = (0.012383)² = 1.5334 × 10⁻⁴
• S̄f = 1.452 × 10⁻⁴; Δx = (E₂ − E₁)/(S₀ − S̄f) = (2.4354 − 2.5326)/(0.0004 − 0.0001452) = (−0.0972)/(0.0002548) = −381.6 m
• The negative sign means section 2 lies 382 m upstream of the weir - an M1 backwater curve (normal depth here is about 1.70 m, so y > y₀ > yc confirms zone 1 on a mild slope).

Where it is used

• Irrigation and canal design: main canals, branch canals and distributaries designed with Manning's equation, permissible velocities and freeboard (IS 7112 for canal cross-sections, IS 10430 for lined canal design, IS 4745 for canal linings); CWC, NWDA, state irrigation departments, WAPCOS, river-linking projects.
• Urban drainage and sewerage: storm-water drains and part-full sewers under CPHEEO manuals - self-cleansing velocity (about 0.6-0.8 m/s) and the y/D = 0.8 design rule; AMRUT and Smart City drainage projects.
• Hydraulic structures: energy dissipators and stilling basins below spillways, canal falls, head regulators and cross regulators (IS 4997, IS 4410 terminology); afflux and waterway design for road and rail bridges (IRC SP:13, IRC 5).
• Hydropower and water conveyance: power channels, head race and tail race channels, forebays, surge and rejection-surge analysis in NHPC/THDC/SJVN projects.
• River engineering and flood studies: 1-D models (HEC-RAS, MIKE 11) applying the standard step and Saint-Venant equations for flood levels, embankment heights and CWC flood forecasting.
• Job roles: hydraulic design engineer, irrigation AE/JE, water resources modeller, drainage/WASH consultant, and lab/field roles doing current-meter gauging and flume calibration.

Interview questions

• Q: How does open channel flow differ from pipe flow? A: It has a free surface at atmospheric pressure and is driven by gravity, with the water surface acting as the hydraulic grade line; the depth is an additional unknown that varies with discharge, slope and roughness.
• Q: What is normal depth and when does it occur? A: The depth of uniform flow, where the gravity component along the bed exactly balances boundary resistance, so bed slope, water-surface slope and energy slope are equal; it needs a long prismatic channel of constant slope and roughness.
• Q: What does the Froude number physically mean? A: It is the ratio of flow velocity to the celerity √(gD) of a small surface wave; if Fr < 1 disturbances can travel upstream, so control is downstream, and if Fr > 1 they cannot, so control is upstream.
• Q: Why is the momentum equation used for a hydraulic jump instead of the energy equation? A: The energy loss in the turbulent roller is large and unknown, but the external forces (hydrostatic pressure on the two faces) are known, so momentum gives a usable relation between the depths.
• Q: Difference between alternate and sequent depths? A: Alternate depths have the same specific energy and discharge (no loss, as across a smooth transition); sequent or conjugate depths have the same specific force and occur across a jump, where energy is lost.
• Q: What happens if a hump in a subcritical channel is made higher than the critical height? A: The flow chokes - critical depth is held on the crest and the upstream water level rises (afflux) to supply the extra specific energy needed.
• Q: Which is the most efficient channel section, and what is used in practice? A: The semi-circle has the least wetted perimeter for a given area; practically a trapezoid close to a half-hexagon, or a rectangle with b = 2y, is used because of construction and stability constraints.
• Q: At what depth does a circular sewer carry its maximum discharge? A: At about 0.938 of the diameter; maximum velocity occurs at about 0.81D, which is why sewers are designed to run part full rather than full.
• Q: What is a backwater curve and why does it matter in a barrage project? A: It is the M1 rise of the water surface upstream of an obstruction; its length and afflux decide land submergence, embankment heights and land-acquisition cost.
• Q: Why does a GVF profile approach critical depth at a steep angle? A: Because 1 − Fr² in the denominator tends to zero, dy/dx tends to infinity; physically the assumption of nearly parallel streamlines breaks down and the flow becomes rapidly varied.
• Q: How would you measure discharge in an existing unlined field canal? A: By the area-velocity method with a current meter (mean velocity from the 0.6-depth or the 0.2/0.8-depth average), or by installing a standing-wave or Parshall flume that gives discharge from a single head reading.
• Q: A canal must carry a fixed Q; would you increase the slope to reduce the section? A: Only within limits - a steeper slope raises velocity and may exceed the non-scouring limit and waste command-area head; design is a balance between velocity limits, available ground slope and excavation cost.

Common mistakes

• Using the pipe diameter or depth in place of hydraulic radius R = A/P, or forgetting that D ≈ 4R when converting formulae.
• Confusing alternate depths with sequent depths, and applying the energy equation across a hydraulic jump.
• Taking the critical-depth formula yc = (q²/g)^1/3 as universal - it is valid only for rectangular channels; others need Q²T/(gA³) = 1.
• Mixing up hydraulic depth D = A/T with the depth of flow y when computing Fr in trapezoidal or circular sections.
• Assuming a steep bed geometrically means supercritical flow - "mild" and "steep" are defined by comparing y₀ with yc, which depends on Q and n as well.
• Starting a GVF computation from the wrong end: subcritical profiles must be computed upstream from a downstream control, supercritical profiles downstream from an upstream control.
• Unit slips - taking S as "1 in 2000" instead of 0.0005, using cm with m, or forgetting that n has units and must not be scaled like a dimensionless factor.

Related topics

Fluid properties and statics · Irrigation engineering · Dams and spillways · Flood control and river engineering · Sewer network design · Hydrology