13:2 Open Channel Flows - Gradually Varying Flows, Energy, Critical Depth and Froude Number

Overview

Derek Elsworth explains how open-channel flow changes from uniform conditions to gradually varied flow using the energy equation, Manning’s formula, and specific energy per unit width. He connects the specific-energy curve to critical depth and the Froude number, showing why deep, slow subcritical flow can communicate upstream while shallow, fast supercritical flow cannot.

Key takeaways

Manning’s formula estimates uniform open-channel flow using bed slope, hydraulic radius, and roughness; its English-units form uses the factor 1.49.
In a rectangular channel, specific energy per unit weight is E = y + q²/(2gy²), with q as discharge per unit width.
A fixed specific-energy value above the minimum can correspond to both a deep, slow flow and a shallow, fast flow because velocity equals q/y.
Critical flow occurs at Fr = V/√(gy) = 1 and at minimum specific energy; for a rectangular channel, E_min = 3/2 yc.
Subcritical flow has Fr < 1 and allows surface-wave information to travel upstream, whereas supercritical flow has Fr > 1 and prevents upstream communication.
For compound-channel Manning calculations, sum subsection discharges and exclude imaginary internal boundaries from each wetted perimeter.

Chapters

0:00 Uniform, Gradually Varied, and Rapidly Varied Open-Channel Flow

Uniform flow keeps the water depth approximately constant downstream, as in a roadside ditch along College Avenue in State College.
Gradually varied flow changes depth over distance, while a hydraulic jump is a rapidly varied transition from shallow, fast flow to deep, slow flow.
For a constant-width flume, continuity requires velocity multiplied by depth to remain constant across the transition.

5:27 Manning’s Formula and the Role of Channel Roughness

Manning’s formula estimates uniform-flow velocity from bed slope, hydraulic radius, and roughness coefficient; the SI coefficient differs from the English-units value of 1.49.
Hydraulic radius is flow area divided by wetted perimeter, and Manning’s roughness coefficient increases from smooth surfaces to rougher channels.
For a wide rectangular channel, hydraulic radius approaches flow depth as channel width becomes much larger than depth.

8:30 Compound Channels, Wetted Perimeter, and Efficient Sections

A compound channel can be divided into subsections with separate Manning coefficients, and the subsection discharge rates are summed.
When calculating each subsection’s hydraulic radius, count wetted boundaries but exclude imaginary dividing lines between subsections.
Standard efficient flow sections include a rectangle with depth half its width, a semicircle, and a regular hexagonal section.

12:00 Worked Manning Calculation for a Trapezoidal Canal

The example uses a smooth-concrete canal with a 12-foot bottom, 5-foot depth, 40-degree sides, and a bed slope of 1.4 feet per 1,000 feet.
The flow area combines the bottom rectangle with two triangular side areas, while wetted perimeter includes the bottom and both sloping sides.
Using area, wetted perimeter, Manning’s coefficient, and the English-units factor gives discharge in cubic feet per second.

17:29 Why Gradually Varied Flow Needs a New Energy Relation

Unlike uniform flow, gradually varied flow can have different upstream and downstream depths and velocities.
The bed slope, free-surface slope, and energy-grade-line slope need not be equal when depth changes downstream.
The analysis starts with the Bernoulli energy equation, including elevation head, pressure head, velocity head, and head loss.

18:38 Deriving the Gradually Varied Flow Energy Equation

At a free surface, pressure head is represented by water depth, so the pressure terms become y₁ and y₂.
Bed slope is the elevation drop divided by channel length, while friction slope is head loss divided by that length.
The resulting relation compares upstream and downstream specific energy and accounts for the difference between friction slope and bed slope.

27:25 Discharge per Unit Width and Specific Energy

For a rectangular channel, discharge per unit width is q = Q/B = Vy, so mean velocity can be written as V = q/y.
Substituting velocity into the energy equation gives specific energy E = y + q²/(2gy²) for flow depth y.
Continuity keeps q constant between sections of a channel with constant width, even when depth and velocity change.

33:42 Reading the Specific-Energy Curve: Deep and Shallow Flow

For fixed discharge per unit width, plotting E = y + q²/(2gy²) against depth produces a curve with a minimum.
At many energy values above the minimum, the curve permits two positive depths: a deep, slow state and a shallow, fast state.
Because V = q/y, decreasing depth increases velocity, while increasing depth decreases velocity.

42:00 Critical Depth and the Froude Number

The minimum-energy point corresponds to critical flow, where the Froude number Fr = V/√(gy) equals 1.
Depths on the deep branch are subcritical, with Fr < 1; depths on the shallow branch are supercritical, with Fr > 1.
For a rectangular channel, the minimum specific energy is E_min = 3/2 yc, where yc is critical depth.

46:10 Wave Propagation Explains Upstream Communication

Small surface disturbances travel at approximately √(gy), so the Froude number compares flow velocity with wave speed.
In subcritical flow, waves can move upstream against the current; in supercritical flow, the current carries disturbances downstream.
Elsworth compares the Fr = 1 boundary to a Mach cone, where flow speed matches the speed at which information propagates.

50:21 Using Critical Depth and Energy Curves at a Weir

Critical depth can be found by locating minimum energy, where the slope of the energy-depth curve is zero.
For a flat-bottom, negligible-loss example, equal energy and continuity relate the upstream and downstream depths around a weir.
Selecting a discharge per unit width and reading its energy curve identifies the possible depths on either side of the obstruction.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Derek Elsworth.