Open-Channel Flow

Introduction and Context

Open-channel flow is the movement of liquid with a free surface exposed to the atmosphere. Rivers, canals, drainage channels, spillways, culverts flowing partially full, and irrigation systems are common examples. Unlike pressurized pipe flow, the driving force is strongly associated with gravity and the water-surface slope. The free surface can change in depth and shape, making open-channel analysis both a hydraulic and a geometric problem.

The central engineering tasks are to determine discharge, flow depth, velocity, energy losses, flow regime, and the response of the water surface to changes in channel geometry or control structures. This chapter develops the concepts of uniform flow, specific energy, critical flow, the Froude number, hydraulic jumps, gradually varied flow, resistance laws, and hydraulic controls.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain the distinction between uniform, gradually varied, and rapidly varied open-channel flow.
2. Calculate discharge and mean velocity using continuity and resistance equations.
3. Use specific energy to determine critical depth and alternate depths.
4. Classify flow using the Froude number.
5. Analyze hydraulic jumps and estimate energy dissipation.
6. Explain gradually varied flow profiles and the role of the normal and critical depths.
7. Select appropriate equations and controls for practical channel design.
8. Recognize the limitations of one-dimensional open-channel models.

1. Geometry and Fundamental Variables

For a channel cross-section, the principal geometric quantities are:

For a rectangular channel of width b and depth y,

For a very wide rectangular channel, where b is much greater than y, R≈y and D≈y. These approximations should not be used for narrow channels or deep sections.

The continuity equation for steady flow is

For steady incompressible flow, Q is constant along a reach unless there is lateral inflow, withdrawal, or branching. The mean velocity is an area average; local velocity varies over the cross-section because of boundary friction.

2. Classification of Open-Channel Flow

Flow is steady when conditions at a fixed location do not change with time, and unsteady when depth, velocity, or discharge changes with time. Flow is uniform when depth, area, and mean velocity remain constant along the channel. Uniform flow requires a constant channel geometry, roughness, slope, and discharge over the reach. Flow is non-uniform or varied when depth changes longitudinally.

A useful spatial classification is:

• Gradually varied flow (GVF): depth changes over a relatively long distance, so pressure is approximately hydrostatic and the flow remains close to one-dimensional.
• Rapidly varied flow (RVF): depth changes over a short distance, as at a hydraulic jump or sharp control. Hydrostatic pressure and gradual-flow assumptions may fail.

Flow is also classified by the Froude number:

where g is gravitational acceleration, approximately 9.81 m/s², and D = A/T is hydraulic depth. Thus:

• Fr<1: subcritical or tranquil flow;
• Fr=1: critical flow;
• Fr>1: supercritical or rapid flow.

The Froude number compares inertial effects with gravity effects. It is dimensionless because both V and √(gD) have units of m/s. For a rectangular channel, D = y, so Fr=V/√(gy).

3. Energy Concepts and Specific Energy

Taking the channel bed as the elevation datum, the specific energy is the energy per unit weight measured relative to the channel bottom:

Here E is specific energy in metres of water, y is flow depth, V is mean velocity, g is gravitational acceleration, and α is the kinetic-energy correction coefficient. In many engineering calculations α is taken as 1.0; a measured or computed value may be used when velocity distribution is strongly non-uniform.

For a rectangular channel with unit discharge q = Q/b,

This expression assumes α = 1 and constant discharge per unit width. It shows that, for a given q, specific energy is large when depth is very small because velocity is high, and also large when depth is very large because elevation head is high. The minimum energy condition is critical flow.

The critical-flow condition for a general channel is

Equivalently, Fr=1 when the kinetic-energy coefficient is taken as unity. For a rectangular channel this becomes

where y_c is critical depth. The formula applies to a prismatic rectangular channel with steady discharge and α = 1.

The minimum specific energy in a rectangular channel is

At a given specific energy greater than E_c, two possible depths may exist: a shallow supercritical depth and a deep subcritical depth. These are called alternate depths. At E = E_c, the two depths merge into the critical depth. If E < E_c, the specified discharge cannot pass through the section without a change in upstream conditions or a change in the control geometry.

4. Worked Example: Critical Depth and Flow Regime

A rectangular channel 4 m wide carries 12 m³/s. Determine the critical depth, critical velocity, and flow regime if the actual depth is 0.80 m.

The discharge per unit width is

Critical depth is

At the actual depth,

For a rectangular section, D = y = 0.80 m. Therefore,

The flow is supercritical because Fr > 1. Since the actual depth is less than critical depth, the same conclusion follows from the depth criterion for a rectangular channel.

5. Uniform Flow and Channel Resistance

Uniform flow occurs when the gravitational component of the channel weight is balanced by boundary resistance. The most widely used empirical relation for open channels is Manning’s equation:

or

where n is Manning’s roughness coefficient, R is hydraulic radius, and S₀ is the bed slope, assumed equal to the friction slope for uniform flow. In SI units, V is in m/s, A in m², and R in m; n has the corresponding empirical SI dimensions.

Manning’s equation is appropriate for steady, gradually varied or uniform flow in channels where roughness and hydraulic conditions can be represented by a calibrated coefficient. It is not a universal law: n depends on bed material, vegetation, irregularity, sediment, alignment, stage, and maintenance condition.

The depth corresponding to uniform flow is called the normal depth, y_n. It is found by solving Manning’s equation together with the channel geometry. For a rectangular channel,

This equation is generally solved by trial-and-error, a numerical root finder, or a spreadsheet.

Other resistance formulations include the Darcy–Weisbach equation expressed through a friction slope and the Chezy equation:

where C is the Chezy coefficient and S_f is the friction slope. Manning’s equation is common in civil and water-resources practice, while Darcy–Weisbach is attractive when friction-factor methods and Reynolds-number effects are important.

Example: Discharge in a Trapezoidal Channel

A trapezoidal channel has bottom width 3 m, side slope 2 horizontal to 1 vertical, depth 1.2 m, Manning coefficient n = 0.025, and bed slope S₀ = 0.001. Determine the approximate discharge.

For side slope z = 2,

The wetted perimeter is

Thus,

Manning’s equation gives

Therefore,

The result should be reported as approximate because the roughness coefficient is empirical and the channel is assumed prismatic and uniform.

6. Critical Flow and Hydraulic Controls

A hydraulic control is a section where the discharge–depth relationship is governed by a known energy or momentum condition. Broad-crested weirs, flumes, channel contractions, and changes in bed elevation can produce critical flow. At the control section, the depth is often close to critical depth, enabling discharge measurement or flow regulation.

A raised bed or hump changes the local elevation head. If upstream specific energy is sufficient, the flow accelerates over the hump and may become critical at its crest. If the hump is raised beyond the available specific head, upstream water level rises and the discharge becomes limited. This condition is called choking.

For a rectangular channel with α = 1, the specific-energy relation is especially useful for analyzing contractions and transitions. In real channels, energy losses due to turbulence, separation, and bed friction must be included when the transition is long or abrupt.

7. Hydraulic Jumps

A hydraulic jump is a short, strongly turbulent transition from supercritical to subcritical flow. It commonly occurs downstream of a spillway, sluice gate, or steep channel reach. The jump dissipates mechanical energy and is deliberately used in stilling basins to protect downstream channels.

Because the jump is short, the momentum equation is more useful than the energy equation. For a rectangular horizontal channel, the conjugate or sequent-depth relation is

where y₁ is the supercritical upstream depth, y₂ is the subcritical downstream depth, and Fr₁ = V₁/√(gy₁). The equation assumes a rectangular prismatic channel, hydrostatic pressure distributions at sections sufficiently away from the roller, negligible external forces, and negligible bed-elevation change through the jump.

The energy loss per unit weight is

for a rectangular channel under the same assumptions. This loss is always nonnegative for a true jump.

Jumps are classified approximately by upstream Froude number: undular jumps occur near Fr₁ = 1 to 1.7; weak jumps occur at higher values; oscillating, steady, and strong jumps occur as Fr₁ increases. Exact boundaries vary with geometry and turbulence, so these categories are descriptive rather than universal design limits.

Worked Example: Stilling-Basin Jump

A rectangular channel has an upstream depth y₁ = 0.20 m and discharge per unit width q = 2.0 m²/s. Determine the downstream conjugate depth and energy loss.

The upstream velocity is

The upstream Froude number is

Hence the flow is strongly supercritical. The conjugate depth is

The energy loss is

A stilling basin must provide sufficient tailwater depth and structural protection to contain this turbulent roller and prevent scour.

8. Gradually Varied Flow

In gradually varied flow, the depth changes slowly enough for pressure to remain approximately hydrostatic. The governing equation for a prismatic channel is

where x is measured in the downstream direction, S₀ is bed slope, S_f is the friction slope, and Fr is the local Froude number. The sign convention assumes y is measured vertically upward and x downstream. Different texts may use an equivalent equation with both numerator and denominator signs reversed; the physical interpretation is unchanged if the convention is used consistently.

For a channel described by Manning’s equation,

The equation becomes singular at Fr = 1, reflecting the special nature of critical flow and the difficulty of smoothly passing through critical conditions in the standard GVF model.

Profile Classification

The relative positions of normal depth y_n and critical depth y_c divide the channel into three zones:

• Zone 1: depth above both y_n and y_c.
• Zone 2: depth between y_n and y_c.
• Zone 3: depth below both y_n and y_c.

The bed slope is classified as mild when y_n > y_c, steep when y_n < y_c, critical when y_n = y_c, horizontal when S₀ = 0, and adverse when the bed rises in the downstream direction. Profile labels such as M1, M2, S2, and H2 combine slope type with zone. For example, an M1 profile occurs on a mild slope with depth above normal depth; it commonly develops upstream of a downstream dam or backwater control.

Boundary conditions determine the profile direction. Subcritical flow is controlled from downstream, whereas supercritical flow is controlled from upstream. This principle is essential when deciding where a boundary condition should be imposed in numerical models.

Common solution methods include direct-step methods, standard-step methods, finite-difference schemes, and gradually varied flow computer programs. The direct-step method is convenient for prismatic channels when the water-surface profile is calculated between known depths. The standard-step method is more flexible for irregular sections and backwater computations.

9. Flow-Control Structures and Applications

Open-channel structures alter depth, velocity, energy, or discharge. Important examples include:

• Sluice gates: create supercritical flow beneath the gate and often a hydraulic jump downstream.
• Broad-crested weirs: establish critical flow over a sufficiently long, smooth crest.
• Sharp-crested weirs: measure discharge using a calibrated head–discharge relation, with attention to approach velocity and ventilation.
• Parshall and other flumes: accelerate flow through a shaped throat to provide a measurable control with relatively low head loss.
• Drop structures: dissipate excess elevation head and reduce channel slope.
• Stilling basins: contain hydraulic jumps and limit downstream scour.
• Culverts: may operate as open channels, submerged conduits, or mixed-flow systems depending on inlet and outlet conditions.

Design must consider freeboard, debris, sediment deposition, cavitation, aeration, downstream scour, structural stability, access for maintenance, and safe conditions for operators and the public. A mathematically correct hydraulic calculation is not sufficient if the structure is vulnerable to blockage or erosion.

10. Illustrative Case: Rehabilitation of an Irrigation Canal

Background

An illustrative irrigation canal conveys water from a regulating reservoir to agricultural outlets. The original earthen reach has vegetation, local sediment deposits, and an undersized downstream transition. Farmers report insufficient delivery during high-demand periods and erosion below a control gate.

Problem

The design discharge is 7 m³/s. Field measurements indicate a variable roughness equivalent to approximately n = 0.030 in the maintained reach and n = 0.045 in vegetated sections. The bed slope is 0.0008. The proposed solution must carry the design flow, avoid excessive upstream flooding, and control downstream erosion.

Analysis

Engineers first survey the cross-section, bed levels, roughness zones, and tailwater rating. Manning calculations estimate normal depth for each reach. A backwater model then determines the water-surface profile upstream of the gate. The gate produces supercritical flow locally, so a hydraulic-jump calculation is made using the measured tailwater depth. Because the calculated conjugate depth is greater than the available tailwater depth, the jump would sweep downstream and cause scour.

Uncertainty is addressed by testing roughness values, sediment deposition scenarios, and several tailwater levels. The model is checked against observed water levels rather than accepted solely because it produces numerically stable results.

Approach and Solution

The selected design includes a graded transition, improved vegetation management, a stilling basin with baffle elements, and a downstream riprap apron. A measuring flume is installed at a location where critical control is reliable and debris access is manageable. The gate is operated with a specified minimum opening to prevent unstable oscillation and excessive local velocities.

Outcome

The redesigned reach conveys the target discharge with adequate freeboard under the design condition. The stilling basin contains the hydraulic jump for the expected tailwater range, while the apron provides additional erosion resistance during unusual events. Periodic surveys are required because sediment and vegetation can change both roughness and geometry.

Learning Points

1. Normal-depth calculations alone do not predict a backwater profile.
2. Gate discharge and downstream erosion must be analyzed together.
3. Tailwater is a critical boundary condition for hydraulic-jump location.
4. Roughness is a field parameter, not merely a handbook constant.
5. Monitoring and maintenance are part of hydraulic design.

11. Limitations, Safety, and Contemporary Practice

One-dimensional equations use cross-sectional averages and cannot fully represent secondary currents, three-dimensional separation, strong curvature, air entrainment, or rapidly changing turbulence. Empirical coefficients may vary with stage and sediment. Hydraulic models can also be sensitive to inaccurate topographic data and boundary conditions.

Physical models, acoustic Doppler measurements, remote sensing, and two- or three-dimensional computational fluid dynamics can supplement one-dimensional analysis. Artificial intelligence may assist with roughness calibration, flood forecasting, anomaly detection, or automated control, but it does not remove the need for physically based equations, field validation, uncertainty analysis, and professional accountability. Training data may be biased toward ordinary conditions and may not represent rare floods or debris-blockage events.

Ethically and legally, designers must account for public safety, downstream consequences, environmental flows, fish passage, water rights, and floodplain regulation. Designs should follow applicable national standards, agency guidance, and permit requirements. Emergency spillways, warning systems, inspection schedules, and failure-mode planning are essential where uncontrolled releases could threaten life or property.

Best practice is to:

• define the design discharge and return-period event clearly;
• survey geometry and levels accurately;
• calibrate roughness using field observations;
• distinguish steady from unsteady conditions;
• check both energy and momentum where appropriate;
• test sensitivity to tailwater, blockage, sediment, and roughness;
• provide freeboard and erosion protection;
• validate important predictions through measurements or physical modeling;
• document assumptions, units, equations, and operating limits.

Chapter Summary

Open-channel flow is governed by gravity, geometry, resistance, and the free-surface boundary. Continuity relates discharge, area, and mean velocity. Manning’s equation estimates resistance-controlled uniform flow and allows normal depth to be found. Specific energy explains alternate depths and identifies critical flow, where the Froude number equals one. Subcritical flow is tranquil and generally controlled downstream; supercritical flow is rapid and generally controlled upstream.

Hydraulic jumps are rapidly varied transitions that conserve momentum approximately while dissipating energy. Their sequent-depth and energy-loss equations are particularly useful in stilling-basin design. Gradually varied flow is described by the water-surface differential equation involving bed slope, friction slope, and Froude number. Normal depth, critical depth, and channel slope define profile zones and determine the response to downstream controls.

Practical design requires more than selecting a formula. Roughness, sediment, vegetation, tailwater, debris, erosion, safety, environmental effects, and model uncertainty must be considered. Flow-control structures should be designed, operated, inspected, and maintained as integrated hydraulic systems.

Key Terms

Specific energy: Energy per unit weight measured relative to the channel bed.

Critical depth: Depth at which specific energy is minimum for a specified discharge.

Normal depth: Depth associated with uniform flow for a given channel, slope, roughness, and discharge.

Froude number: Dimensionless ratio comparing inertial and gravity effects.

Subcritical flow: Flow with Fr < 1, generally deeper and controlled downstream.

Supercritical flow: Flow with Fr > 1, generally shallower and controlled upstream.

Hydraulic radius: Flow area divided by wetted perimeter.

Hydraulic depth: Flow area divided by top width.

Uniform flow: Flow whose depth and velocity do not change along the channel.

Gradually varied flow: Flow in which depth changes slowly over distance.

Rapidly varied flow: Flow with a large depth change over a short distance.

Hydraulic jump: Turbulent transition from supercritical to subcritical flow.

Conjugate depths: Upstream and downstream depths related by momentum across a hydraulic jump.

Manning coefficient: Empirical coefficient representing channel resistance.

Friction slope: Energy gradient associated with boundary resistance.

Hydraulic control: Section that governs the discharge–depth relationship.

Alternate depths: Two depths having the same specific energy and discharge.

Freeboard: Vertical distance between the design water surface and channel top.

Stilling basin: Structure designed to contain and dissipate hydraulic-jump energy.

Backwater curve: Water-surface profile produced by a downstream control or obstruction.

Review Questions

Short Answer Questions

1. Define hydraulic radius and hydraulic depth, and distinguish between them.
2. What physical effects are compared by the Froude number?
3. State the assumptions behind Manning’s equation for uniform flow.
4. Why is momentum rather than energy commonly used to analyze a hydraulic jump?
5. Distinguish normal depth from critical depth.

Long Answer Questions

1. Explain specific energy in a rectangular channel and derive the rectangular-channel critical-depth relation.
2. Describe the classification of open-channel flow by steadiness, uniformity, and Froude number.
3. Explain the gradually varied flow equation and the significance of the normal-depth and critical-depth zones.
4. Discuss the design purpose, operation, and limitations of hydraulic jumps and stilling basins.
5. Compare Manning, Chezy, and Darcy–Weisbach approaches to estimating channel resistance.

Analytical / Application Questions

1. A 5 m wide rectangular channel carries 15 m³/s at a depth of 0.75 m. Calculate velocity, Froude number, and critical depth, and classify the flow.
2. A hydraulic jump occurs in a rectangular channel with upstream depth 0.30 m and discharge per unit width 3.0 m²/s. Determine the upstream Froude number, conjugate depth, and energy loss per unit weight.
3. An irrigation canal has a downstream gate and shows rising upstream water levels after vegetation growth. Explain how increased roughness changes normal depth, friction slope, and the likely backwater profile. State what field measurements should be collected before redesign.

Multiple Choice Questions with Answers

Question 1

For a rectangular channel, the critical-depth condition is represented by which relation?

A. Fr = 0

B. Fr = 1

C. Fr > 2 only

D. S₀ = 0

Correct Answer: B. Fr = 1

Question 2

A channel has discharge 8 m³/s, area 4 m², and hydraulic depth 1 m. What is its Froude number? Use g = 9.81 m/s².

A. 0.40

B. 0.81

C. 1.28

D. 2.55

Correct Answer: C. 1.28

Question 3

Which depth is associated with uniform flow for a specified channel, slope, roughness, and discharge?

A. Critical depth

B. conjugate depth

C. Normal depth

D. Alternate depth

Correct Answer: C. Normal depth

Question 4

Why is the momentum equation preferred for an ordinary hydraulic jump?

A. The jump is long and frictionless

B. The jump is short and dissipates substantial mechanical energy

C. The flow is always uniform through the jump

D. Specific energy remains constant through the jump

Correct Answer: B. The jump is short and dissipates substantial mechanical energy

Question 5

For subcritical flow, a boundary condition normally influences the flow primarily from the:

A. Upstream direction

B. Downstream direction

C. Channel bed only

D. Atmosphere only

Correct Answer: B. Downstream direction

Question 6

In Manning’s equation, increasing the roughness coefficient n while holding other variables constant generally:

A. Increases velocity

B. Decreases velocity

C. Makes the Froude number exactly one

D. Eliminates the friction slope

Correct Answer: B. Decreases velocity

Question 7

A flow with Froude number 0.65 is classified as:

A. Supercritical

B. Critical

C. Subcritical

D. Pressurized

Correct Answer: C. Subcritical

Question 8

What is the principal purpose of a stilling basin?

A. To increase supercritical velocity indefinitely

B. To contain a hydraulic jump and dissipate energy

C. To eliminate all sediment transport

D. To create vacuum pressure

Correct Answer: B. To contain a hydraulic jump and dissipate energy

Question 9

For a rectangular hydraulic jump with upstream Froude number greater than one, the downstream conjugate depth is generally:

A. Less than the upstream depth

B. Equal to zero

C. Greater than the upstream depth

D. Always equal to critical depth

Correct Answer: C. Greater than the upstream depth

Question 10

Which quantity is defined as flow area divided by wetted perimeter?

A. Hydraulic depth

B. Top width

C. Hydraulic radius

D. Freeboard

Correct Answer: C. Hydraulic radius