Fluid Mechanics · Chapter 13
Fluid Mechanics · Chapter 13

Open-Channel Flow

Open-channel flow has a free surface and is driven by gravity — rivers, canals, spillways, and partly filled pipes. This chapter develops the Froude number that classifies the flow, specific energy and critical depth, the Manning equation for uniform flow, the most efficient channel sections, and the varied-flow phenomena that culminate in the hydraulic jump.

Chapter Objectives

  • Classify open-channel flows by spatial variation, Reynolds regime, and Froude regime.
  • Define a channel control volume, datum, flow direction, and gauge-pressure convention.
  • Apply continuity, energy, momentum, Manning, and gradually varied flow models appropriately.
  • Calculate critical depth, normal depth, and hydraulic-jump conjugate depth.
  • Audit assumptions, units, signs, regime limits, and sensitivity to supplied coefficients.
Section 13–1

Classification of Open-Channel Flows

Open-channel flow is liquid flow with a free surface exposed to the atmosphere — rivers, canals, spillways, and partly filled pipes. Unlike pipe flow, the cross-section and depth are not fixed by walls but adjust freely, and gravity, not a pressure difference, drives the motion.

Flow Classifications

By variation in space, flow is uniform when depth and mean velocity do not change downstream, gradually varied when hydrostatic pressure remains a useful approximation, and rapidly varied near gates, jumps, and spillways. Steady describes variation in time, not distance. Reynolds number assesses viscous regime; Froude number assesses gravity-wave communication.

Physical meaning

The free surface fixes local pressure at atmospheric pressure, so gauge pressure is zero there and depth replaces the pressure-head variable used in a full pipe. A useful control volume spans the wetted section, with \(x\) positive downstream and \(z\) positive upward from a declared datum.

Section 13–2

Froude Number & Wave Speed

The Froude Number Governs the Regime

The Froude number compares section-average speed with shallow gravity-wave speed. For a general section use hydraulic depth \(D_h=A/T\), where \(T\) is top width: \(\mathrm{Fr}=V/\sqrt{gD_h}\). The familiar \(D_h=y\) form is exact only for a rectangular section under the shallow-water model.

Froude Number & Wave Speed
\[ \mathrm{Fr} = \frac{V}{\sqrt{gy}}, \qquad c_0 = \sqrt{gy} \]
Three regimes: \(\mathrm{Fr}<1\) is subcritical (tranquil) flow — disturbances travel both upstream and downstream; \(\mathrm{Fr}=1\) is critical; \(\mathrm{Fr}>1\) is supercritical (rapid) flow — disturbances cannot travel upstream, the exact analogue of supersonic flow.
EXAMPLE 1Wave communication in a test flume
Water at \(20^\circ\text{C}\) flows steadily through a \(1.20\ \text{m}\)-wide rectangular flume at \(Q=0.90\ \text{m}^3/\text{s}\) and depth \(y=0.40\ \text{m}\). Determine wave speed and Froude regime.

Recognition strategy

A free surface, known depth, and requested wave behavior identify a Froude calculation rather than a pipe Reynolds or Mach calculation.

Given information

  • Supplied \(b=1.20\ \text{m}\), \(Q=0.90\ \text{m}^3/\text{s}\), \(y=0.40\ \text{m}\), and \(g=9.81\ \text{m}/\text{s}^2\).
  • Pressure is atmospheric at the surface, hence zero gauge; water properties are referenced to \(20^\circ\text{C}\), although density cancels.

Model and interpretation

Choose the wetted section as the control surface, \(x\) positive downstream, and use steady incompressible one-dimensional shallow-water equations. The relations are approximate because vertical acceleration is neglected.

Step-by-step solution
1. Form the area. The rectangular flow area is \(A=by=(1.20)(0.40)=0.480\ \text{m}^2\), establishing the section used by continuity.
2. Enforce mass conservation. Steady incompressible continuity gives \(V=Q/A=0.90/0.480=1.875\ \text{m}/\text{s}\).
3. Find wave speed. The shallow-wave approximation gives \(c=\sqrt{gy}=\sqrt{(9.81)(0.40)}=1.981\ \text{m}/\text{s}\).
4. Classify. \(\mathrm{Fr}=V/c=1.875/1.981=0.946\); because this is below one, the flow is subcritical but near critical.

Verification

Units: \(Q/A\) produces \(\text{m}/\text{s}\), while \(\sqrt{gy}\) also produces \(\text{m}/\text{s}\), leaving Froude number dimensionless as required.

Signs and direction: Positive \(Q\) and \(V\) point downstream by convention; wave speed is a positive magnitude, and \(\mathrm{Fr}<1\) permits an upstream-traveling disturbance.

Plausibility: Mean speed is only \(5.4\%\) below wave speed, so the near-critical classification is credible; no energy, momentum, Mach, or friction correlation is needed for this local regime check.

Final result: \(V=1.88\ \text{m}/\text{s}\), \(c=1.98\ \text{m}/\text{s}\), and \(\mathrm{Fr}=0.946\), subcritical.

Engineering interpretation

An upstream control can still influence this station, but the small margin from critical means depth uncertainty could change the classification and should be measured carefully.

Section 13–3

Specific Energy

Specific Energy

The mechanical energy of the flow measured relative to the channel bottom — the sum of flow depth and velocity head. For a given flow rate it is a function of depth alone, and it has a minimum.

Specific Energy
\[ E_s = y + \frac{V^2}{2g} \]

Critical Depth

The depth at which specific energy is minimum for a given flow rate; at this depth \(\mathrm{Fr}=1\). For a rectangular channel it depends only on the flow rate per unit width \(q\).

Critical Depth (rectangular channel)
\[ y_c = \left(\frac{q^2}{g}\right)^{1/3} \]

For any specific energy above the minimum there are two possible depths — a deep subcritical and a shallow supercritical — called alternate depths. The flow can carry the same energy and discharge at either.

Physical meaning

At fixed unit discharge, reducing depth accelerates the flow by continuity. Specific energy therefore rises at both very small and very large depths, with a minimum at critical depth. Crossing that minimum smoothly requires a control such as a crest or contraction.

Section 13–4

Continuity & Energy Equations

Conservation Laws for Channels

For a steady incompressible control volume, mass conservation gives \(Q=A_1V_1=A_2V_2\). With gauge pressure zero at both free surfaces, the mechanical-energy balance contains bed elevation, depth, kinetic-energy head, machine head if present, and irreversible head loss. Hydrostatic pressure and nearly uniform section velocity are approximations that must be checked.

Energy Equation (between sections 1 and 2)
\[ z_1 + y_1 + \frac{V_1^2}{2g} = z_2 + y_2 + \frac{V_2^2}{2g} + h_L \]

Problem-solving strategy

  1. Sketch the reach, set \(x\) downstream and \(z\) upward, choose a bed datum, and mark all controls.
  2. Use continuity first, then classify each section with \(\mathrm{Fr}\) before selecting energy or momentum.
  3. Use energy for gradual reaches with a defensible loss model; use momentum across a short hydraulic jump.
  4. For normal depth solve Manning iteratively because area and hydraulic radius both depend on depth.
  5. Check mass conservation, units, energy loss direction, regime, and sensitivity to supplied roughness or discharge coefficients.
Section 13–5

Uniform Flow in Channels & the Manning Equation

Normal Depth

In uniform flow the depth, velocity, and cross-section stay constant along the channel; gravity exactly balances friction. The constant depth is the normal depth \(y_n\), set by the flow rate, slope, roughness, and channel shape.

Manning Equation (SI)
\[ V = \frac{1}{n}R_h^{2/3}S_0^{1/2}, \qquad R_h = \frac{A}{P_w} \]

Here \(n\), with SI units \(\text{s}/\text{m}^{1/3}\), is an empirical roughness supplied from project data, \(S_0\) is dimensionless bed slope, and \(P_w\) excludes the free surface. Manning is an approximate uniform-flow correlation, not a universal friction law.

EXAMPLE 2Iterated normal depth in a lined drain
A rectangular drain \(2.00\ \text{m}\) wide must carry \(3.00\ \text{m}^3/\text{s}\) on slope \(S_0=0.0025\). Given \(n=0.015\ \text{s}/\text{m}^{1/3}\), estimate normal depth and regime.

Recognition strategy

Constant geometry, slope, roughness, and requested normal depth identify an implicit Manning problem followed by a Froude classification.

Given information

  • Supplied \(b=2.00\ \text{m}\), \(Q=3.00\ \text{m}^3/\text{s}\), \(S_0=0.0025\), \(n=0.015\ \text{s}/\text{m}^{1/3}\), and \(g=9.81\ \text{m}/\text{s}^2\).
  • Water is treated incompressible at \(20^\circ\text{C}\); pressure is atmospheric, or zero gauge, along the surface.

Model and interpretation

The reach is a steady, uniform control volume with \(x\) positive downslope. Manning is approximate and its supplied coefficient represents the entire wetted boundary; no fully developed pipe or friction-factor correlation is invoked.

Step-by-step solution
1. Write depth functions. For trial depth \(y\), \(A=2y\), \(P_w=2+2y\), and \(R_h=2y/(2+2y)\).
2. Form the residual. Solve \(f(y)=nQ/[A R_h^{2/3}S_0^{1/2}]-1=0\), retaining the supplied SI Manning basis.
3. Iterate. At \(y=0.76\ \text{m}\), Manning gives \(Q=2.895\ \text{m}^3/\text{s}\); at \(y=0.78\ \text{m}\), it gives \(Q\approx3.00\ \text{m}^3/\text{s}\).
4. Classify the solution. \(V=Q/(by)=3.00/(2.00\times0.78)=1.92\ \text{m}/\text{s}\), so \(\mathrm{Fr}=1.92/\sqrt{9.81(0.78)}=0.695\).

Verification

Units: \(n^{-1}AR_h^{2/3}S_0^{1/2}\) reduces to \(\text{m}^3/\text{s}\), while the calculated Froude number is dimensionless.

Signs and direction: Positive bed slope and discharge are downstream; gravity supplies exactly the modeled friction loss in uniform flow, so depth has no downstream gradient.

Plausibility: Substitution returns the target discharge within rounding, and \(\mathrm{Fr}<1\) indicates a deep tranquil solution; uncertainty in empirical \(n\) should be propagated before design.

Final result: \(y_n=0.78\ \text{m}\), \(V=1.92\ \text{m}/\text{s}\), and \(\mathrm{Fr}=0.695\), based on the approximate Manning model.

Engineering interpretation

The drain is comfortably subcritical, but freeboard, debris, roughness aging, transition losses, and surveyed slope remain project inputs rather than consequences of Manning’s equation.

Section 13–6

Best Hydraulic Cross Sections

Most Efficient Section

For a given area and slope, discharge is maximized when the hydraulic radius is largest — that is, when the wetted perimeter is smallest. The best hydraulic section minimizes excavation and lining cost. Among all shapes the semicircle is optimal; among rectangles, the one with depth equal to half the width; among trapezoids, the half-hexagon.

Hydraulic efficiency alone does not establish a preferred design: excavation, lining, bank stability, sediment transport, maintenance access, and freeboard can dominate the choice.

Section 13–7

Gradually Varied Flow

Surface Profiles

When depth changes slowly along the channel, friction and gravity no longer balance exactly, and the free surface follows a smooth gradually varied flow profile. These profiles are classified by the channel slope (mild, steep, critical) and by whether the depth lies above or below the normal and critical depths — the basis for backwater computations behind dams and weirs.

Gradually varied depth equation
\[ \frac{dy}{dx}=\frac{S_0-S_f}{1-\mathrm{Fr}^2} \]

This differential model assumes steady, one-dimensional flow, hydrostatic pressure, gradual depth change, and a local uniform-flow estimate of \(S_f\). Numerical integration must not step blindly through \(\mathrm{Fr}=1\), where the denominator vanishes.

Section 13–8

Rapidly Varied Flow & the Hydraulic Jump

Hydraulic Jump

When supercritical flow must slow to subcritical, it does so abruptly through a turbulent hydraulic jump — the open-channel analogue of a normal shock. Depth rises suddenly, and a substantial fraction of the mechanical energy is dissipated as turbulence and heat, which is exactly why jumps are engineered into spillway stilling basins.

Depth Ratio across a Jump (rectangular channel)
\[ \frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8\,\mathrm{Fr}_1^2} - 1\right) \]
supercritical (Fr > 1) shallow, fast subcritical (Fr < 1) deep, slow jump (energy dissipated)
Figure 13–1. A hydraulic jump abruptly raises the depth as flow transitions from supercritical to subcritical, dissipating energy — the channel-flow counterpart of a shock wave. (Original schematic.)
EXAMPLE 3Stilling-basin jump estimate
A \(4.00\ \text{m}\)-wide rectangular spillway delivers steady water flow \(Q=12.0\ \text{m}^3/\text{s}\) at upstream jump depth \(y_1=0.50\ \text{m}\). Estimate conjugate depth and energy loss.

Recognition strategy

An abrupt supercritical-to-subcritical transition requires the momentum relation; applying Bernoulli across the dissipative roller would be the wrong method.

Given information

  • Supplied \(b=4.00\ \text{m}\), \(Q=12.0\ \text{m}^3/\text{s}\), \(y_1=0.50\ \text{m}\), \(g=9.81\ \text{m}/\text{s}^2\).
  • Water is incompressible at \(20^\circ\text{C}\); both free surfaces are at atmospheric, zero-gauge pressure.

Model and interpretation

Enclose the short jump in a steady control volume, take \(x\) downstream and bed horizontal, assume hydrostatic end sections, and neglect bed shear over the short length. The conjugate-depth equation is approximate.

Step-by-step solution
1. Audit continuity. \(V_1=Q/(by_1)=12.0/(4.00\times0.50)=6.00\ \text{m}/\text{s}\).
2. Confirm inlet regime. \(\mathrm{Fr}_1=6.00/\sqrt{(9.81)(0.50)}=2.71>1\), so a hydraulic jump is admissible.
3. Apply momentum. The rectangular conjugate ratio is \(y_2/y_1=[\sqrt{1+8\mathrm{Fr}_1^2}-1]/2=3.36\), giving \(y_2=1.68\ \text{m}\).
4. Evaluate dissipation. \(h_L=(y_2-y_1)^3/(4y_1y_2)=(1.68-0.50)^3/[4(0.50)(1.68)]=0.489\ \text{m}\).
5. Check downstream state. \(V_2=12.0/[4.00(1.68)]=1.79\ \text{m}/\text{s}\) and \(\mathrm{Fr}_2=1.79/\sqrt{9.81(1.68)}=0.440\).

Verification

Units: The depth ratio and Froude numbers are dimensionless; the cubic-depth loss expression reduces to metres of mechanical-energy head.

Signs and direction: Both velocities are positive downstream, while \(h_L=0.489\ \text{m}>0\) correctly represents irreversible energy removal; no pump or turbine is present.

Plausibility: Continuity gives identical \(Q\) at both sections, momentum changes a supercritical inlet to a subcritical outlet, and specific energy decreases rather than increases.

Final result: \(y_2=1.68\ \text{m}\) and \(h_L=0.489\ \text{m}\), based on the approximate conjugate-depth model.

Engineering interpretation

This first estimate locates the required tailwater level; basin length, fluctuating loads, aeration, scour protection, and uncertainty in approach conditions still require hydraulic design review.

Section 13–9

Flow Control & Measurement

Weirs, Sluice Gates & Flumes

Because depth and discharge are tightly linked in channel flow, a structure that fixes one measures the other. Weirs (overflow obstructions) and sluice gates (underflow openings) relate the upstream head to a well-defined discharge, and the Parshall flume measures flow without ponding sediment.

Rectangular Sharp-Crested Weir (discharge)
\[ \dot{V} = C_d\,\frac{2}{3}\sqrt{2g}\;b\,H^{3/2} \]
Instructor Note

Lean on the gas-dynamics analogy: Froude number is the channel-flow Mach number, critical depth is the throat, and the hydraulic jump is a shock. If you understood Chapter 12, most of this chapter is the same logic with a free surface instead of compressibility. Always classify the regime by \(\mathrm{Fr}\) before reaching for an equation.

Chapter recap

  • Classify with Reynolds and Froude numbers before choosing a balance.
  • Use mass and energy for gradual reaches, momentum for a short jump, and Manning only with supplied roughness.
  • Declare atmospheric gauge pressure, datum, downstream sign, property basis, and approximation limits.

Before moving on

You should be able to compute section geometry, critical or normal depth, identify flow-control direction, solve a conjugate-depth estimate, and verify mass conservation, units, signs, regimes, and energy loss.