Characteristics of Open-Channel Flow

Free-surface flow at atmospheric pressure, hydraulic radius and depth, the steady/uniform/varied classifications, and the Froude number as the governing group.

Open-Channel FlowMechanical EngineeringFree preview
⏱️ About 14 min

Every river, canal, and drainage ditch you have ever seen shares a single defining feature that sets them apart from the pressurized pipes of the earlier modules: the water moves under a free surface that is open to the air. That surface is everywhere at atmospheric pressure, and because it is free it can deform — it rises, it falls, it can steepen into a standing wave or crash into a churning hydraulic jump. Open-channel flow is the bread and butter of civil and hydraulic engineering (rivers, spillways, sewers running half full), and it is included here as an optional breadth module — a chance to see the same fluid mechanics you already know, now applied to a flow whose shape is set by gravity rather than a pipe wall. The new governing number is the Froude number, and it will reorganize everything.

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The big idea: Open-channel flow is gravity-driven flow of a liquid with a free surface at atmospheric pressure (so the pressure at the surface is zero gauge, and the pressure variation with depth is hydrostatic across any given section). Two geometric quantities replace the pipe diameter. The hydraulic radius R_h = A/P — flow area divided by the wetted perimeter P (the length of solid boundary touching the water) — sets the frictional resistance and drives the Manning equation. The hydraulic depth D = A/T — area divided by the top width T (the free-surface width) — sets the speed of a surface wave and drives the Froude number. For a rectangular channel of width b and depth y, both simplify: R_h is by/(b + 2y) and D = y. Flow is classified in two independent ways: by time (steady vs unsteady) and by space (uniform vs non-uniform, with non-uniform further split into gradually varied and rapidly varied). But the classification that matters most is by the Froude number Fr = V/√(gD): Fr < 1 is subcritical (tranquil, deep, slow — surface waves can travel upstream), Fr = 1 is critical, and Fr > 1 is supercritical (rapid, shallow, fast — waves are swept downstream). Where pipe flow was governed by Reynolds number, open-channel flow is governed by Froude number.
🎯 By the end, you'll be able to
  • Define open-channel flow and state why the free surface is at atmospheric (zero-gauge) pressure
  • Compute the hydraulic radius R_h = A/P and the hydraulic depth D = A/T for a given section
  • Classify open-channel flow as steady/unsteady and as uniform/non-uniform (gradually vs rapidly varied)
  • Compute the Froude number Fr = V/√(gD) and classify the flow as subcritical, critical, or supercritical
  • Explain why surface waves can travel upstream in subcritical flow but not in supercritical flow
📎 Helpful to know first

A free surface changes everything

Through the first nine modules the fluid was almost always confined — a pipe, a duct, a channel fully closed so that pressure could build up above atmospheric. Open-channel flow is different. The liquid (almost always water) flows under a free surface exposed to the atmosphere, so the pressure right at the surface is fixed at atmospheric pressure — which is the zero of gauge pressure. There is no longer an unknown pressure to solve for at the boundary; instead, the surface itself can move, and gravity — not a pressure gradient imposed by a pump — is what drives the flow downhill.

This single change ripples through the analysis. Because the pressure distribution through the depth is hydrostatic (the streamlines are essentially straight and parallel), the pressure force is absorbed into the depth y itself, and the energy and momentum equations are rewritten in terms of depths and velocities rather than pressures. And because the surface is free, the flow can change shape: it can accelerate and thin out (supercritical flow), slow and deepen (subcritical flow), or pass through the abrupt, dissipative transition called a hydraulic jump. None of that is possible inside a rigid pipe.

Open-channel flow is the home territory of civil and hydraulic engineering — river hydraulics, irrigation canals, road drainage, and the stilling basins below dam spillways. For a mechanical engineer it is a breadth topic, which is why this module is labelled optional. But the tools are the same energy and momentum equations you have already met, applied to a new boundary condition.

\[ p\big|_{\text{surface}}=p_{\text{atm}}\;(=0\ \text{gauge})\qquad R_{h}=\frac{A}{P}\qquad D=\frac{A}{T} \]
The free surface sits at atmospheric pressure (zero on the gauge). The two section properties that replace the pipe diameter are the hydraulic radius R_h = A/P (area over wetted perimeter) and the hydraulic depth D = A/T (area over top width). For a rectangular channel of width b and depth y: R_h = by/(b+2y) and D = y.
A trapezoidal open-channel cross-section. Water fills the channel below a horizontal free surface labelled at atmospheric pressure. The flow area A is shaded. The wetted perimeter P is the total length of solid boundary in contact with the water, shown as a thick accent line along the bed and both side slopes. The top width T spans the free surface, the depth y is shown as a vertical double arrow, and the bed width b is along the bottom. free surface — p = p_atm (gauge 0) flow area A bed width b wetted perimeter P top width T depth y flow into page
An open-channel cross-section. The flow area A is the water-filled region; the wetted perimeter P is the length of solid boundary touching the water (bed plus the wetted parts of the side slopes — note the surface itself is NOT part of P); the top width T is the free-surface width; and y is the depth. R_h = A/P governs friction; D = A/T governs the Froude number.

Classifying open-channel flow

Open-channel flow is described by two independent sets of adjectives, one about time and one about space. By time: steady flow has a depth and velocity at each point that do not change with time (the usual textbook assumption), while unsteady flow does (a flood wave arriving, a tide reversing). By space: uniform flow has a constant depth and velocity along the channel — the water surface is parallel to the bed — while non-uniform (or varied) flow has a depth that changes along the channel.

Varied flow comes in two flavours with very different physics. In gradually varied flow (GVF) the depth changes slowly over a long distance — the backwater behind a dam, the drawdown toward a free overfall. The streamlines stay nearly straight and parallel, the pressure stays hydrostatic, and the analysis is a gentle differential balance (the subject of an advanced course). In rapidly varied flow (RVF) the depth changes abruptly over a short distance — a hydraulic jump, flow under a sluice gate, over a weir. Streamlines curve sharply, and the transitions are analyzed with the energy or momentum equation applied across the short region. This module's worked tools — specific energy and the hydraulic jump — are the rapidly-varied cases; the Manning equation is the uniform-flow case.

✨ Hydraulic radius vs hydraulic depth — do not mix them up

Both have area in the numerator, and beginners often treat them as the same thing. They are not. The hydraulic radius Rh = A/P belongs to friction: it appears in the Manning equation and measures how much wetted boundary is dragging on the flow relative to how much water there is. The hydraulic depth D = A/T belongs to waves and the Froude number: it is the average depth seen by a surface disturbance, and the speed of a small gravity wave is c = √(gD). For a wide rectangular channel (width much greater than depth) both happen to be approximately equal to the depth y, which is the source of the confusion — but in a trapezoidal or irregular section they differ, and using the wrong one gives the wrong friction and the wrong Froude number.

The Froude number governs open-channel flow

In pipe flow the Reynolds number told you whether the flow was laminar or turbulent — a question about viscosity versus inertia inside a closed conduit. In open-channel flow the analogous question is about gravity versus inertia, and the answer is the Froude number:

Fr = V / √(gD),

where D is the hydraulic depth and √(gD) is the speed of a small surface (gravity) wave. The Froude number is therefore the flow speed measured in surface-wave speeds, and that is exactly why it controls the flow regime. (You met this group in the Module 7 lesson on dimensionless numbers; here it does real work.)

  • Subcritical (Fr < 1): the flow is slower than a surface wave. A disturbance — a thrown stone, a downstream obstruction — sends waves that travel upstream and are felt throughout the reach. The flow is governed by downstream conditions. This is tranquil, relatively deep, relatively slow flow — a placid river.
  • Critical (Fr = 1): the flow speed exactly equals the wave speed. This is a singular, sensitive state (Lesson 2) at which the specific energy is a minimum for the given discharge.
  • Supercritical (Fr > 1): the flow outruns its own surface waves. Disturbances cannot travel upstream; the flow is governed entirely by upstream conditions. This is rapid, shallow, fast flow — water shooting out from under a sluice gate or down a steep spillway.

Whether disturbances can propagate upstream is not a detail — it decides which end of the channel controls the flow, and it is why a supercritical flow cannot 'know' about an obstruction downstream until it slams into it and forms a hydraulic jump (Lesson 4).

\[ Fr=\frac{V}{\sqrt{gD}}\qquad c_{\text{wave}}=\sqrt{gD}\qquad\begin{cases}Fr<1:&\text{subcritical — waves travel upstream}\\Fr=1:&\text{critical}\\Fr>1:&\text{supercritical — waves swept downstream}\end{cases} \]
The Froude number is the flow speed in units of the surface-wave speed c = √(gD). Below 1 (subcritical) waves propagate upstream and downstream conditions control the flow; above 1 (supercritical) waves are swept downstream and only upstream conditions matter. The Reynolds number, by contrast, governs the laminar/turbulent split and is usually large in open channels.
📝 Worked example: Water flows at Q = 10 m³/s in a rectangular channel of width b = 3 m at a depth y = 1.5 m. Compute the hydraulic radius R_h and the Froude number Fr, and classify the flow.
  1. Flow area: A = by = (3)(1.5) = 4.5 m². Wetted perimeter: P = b + 2y = 3 + 2(1.5) = 6 m.
  2. Hydraulic radius: R_h = A/P = 4.5/6 = 0.75 m.
  3. Velocity: V = Q/A = 10/4.5 = 2.22 m/s.
  4. For a rectangular channel D = y = 1.5 m, so Fr = V/√(gD) = 2.22/√(9.81×1.5) = 2.22/3.84 = 0.58.
  5. Fr = 0.58 < 1, so the flow is SUBCRITICAL (tranquil) — downstream conditions control it.
✓ R_h = 0.75 m; Fr ≈ 0.58 (subcritical).
✏️ Practice: A rectangular channel has bed width b = 6 m and carries water at depth y = 1.5 m. Compute the hydraulic radius R_h = A/P. Give your answer in metres.
m
Solution
  1. A = by = (6)(1.5) = 9.0 m².
  2. P = b + 2y = 6 + 2(1.5) = 9.0 m.
  3. R_h = A/P = 9.0/9.0 = 1.0 m.
✏️ Practice: Water flows at Q = 8 m³/s in a rectangular channel of width b = 4 m at depth y = 1.2 m. Compute the Froude number Fr = V/√(gy) and classify the flow. Give Fr to two decimal places.
Solution
  1. A = by = (4)(1.2) = 4.8 m²; V = Q/A = 8/4.8 = 1.67 m/s.
  2. Fr = V/√(gy) = 1.67/√(9.81×1.2) = 1.67/3.43 = 0.486 ≈ 0.49.
  3. Fr < 1, so the flow is subcritical.

Check your understanding

1. In open-channel flow the pressure at the free surface is:
The free surface is open to the atmosphere, so the pressure there is atmospheric — the zero of gauge pressure. With the streamlines nearly straight and parallel, the pressure then varies hydrostatically with depth below the surface.
2. The hydraulic radius R_h of a channel section is defined as:
R_h = A/P, the flow area divided by the wetted perimeter (the length of solid boundary in contact with the water). It governs frictional resistance and appears in the Manning equation. A/T is the hydraulic depth D, which governs the Froude number instead.
3. A rectangular channel flow has Fr = 2.3. This flow is:
Fr > 1 is supercritical: the flow is faster than a surface wave, so disturbances are swept downstream and the flow cannot respond to downstream conditions. Subcritical is Fr < 1; critical is Fr = 1. (Laminar/turbulent is a Reynolds-number question, not a Froude-number one.)
🔎 Related lessons
Specific Energy & Critical Depth · soon Uniform Flow — the Manning Equation · soon Common Dimensionless Groups (the Froude number) · soon
✅ Key takeaways
  • Open-channel flow is gravity-driven flow beneath a free surface at atmospheric (zero-gauge) pressure; the pressure varies hydrostatically with depth.
  • Section properties: hydraulic radius R_h = A/P (drives friction/Manning) and hydraulic depth D = A/T (drives the Froude number). For a rectangle, D = y.
  • Classify by time (steady/unsteady) and space (uniform/varied; varied splits into gradually and rapidly varied).
  • The Froude number Fr = V/√(gD) governs the regime: subcritical (Fr<1, waves go upstream), critical (Fr=1), supercritical (Fr>1, waves swept downstream).
➡️ You can now describe a channel and classify its flow by the Froude number. The next lesson introduces the single most useful diagram in open-channel hydraulics — the specific-energy diagram — which plots depth against energy and makes the subcritical and supercritical regimes visible as two branches of one curve, meeting at the critical depth. It is the key to weirs, sluice gates, and the hydraulic jump to come.
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