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

Overview

Derek Elsworth moves from uniform-flow resistance calculations using Manning’s coefficient to gradually varying open-channel flow, deriving how Bernoulli energy, bed elevation, friction loss, and changing depth relate. He uses specific energy and the Froude number to explain critical depth, the deep-slow subcritical and shallow-fast supercritical states, hydraulic jumps, and how gravity waves carry information through a channel.

Key takeaways

For rectangular-channel flow, unit discharge q = Q/B makes average velocity V = q/y, so decreasing depth necessarily increases velocity when discharge and width stay fixed.
Specific energy E = y + q²/(2gy²) has a minimum at critical depth; for a rectangular channel, y_c = (q²/g)^(1/3).
The two positive depths possible at a given discharge and energy correspond to distinct regimes: deep, slow subcritical flow and shallow, fast supercritical flow.
The Froude number compares water velocity with gravity-wave speed: Fr below 1 allows upstream wave communication, while Fr above 1 prevents it.
Manning resistance calculations depend on the wetted perimeter and hydraulic radius; in compound sections, calculate subsection geometry carefully and sum the subsection discharges.
A hydraulic jump converts shallow, fast supercritical flow into deeper, slower subcritical flow, demonstrating how open channels can switch between flow states.

Chapters

0:00 Water Transfers, Drought, and the Engineering Context for Open Channels

Derek Elsworth connects open-channel hydraulics to water supply and disputes over transfers involving California and the Colorado River.
He invokes the Owens Valley water history depicted in Chinatown to illustrate how water infrastructure creates long-running allocation conflicts.
He previews the key distinction from pipe flow: a free-surface channel can carry water deep and slowly or shallowly and quickly.

3:00 Hydraulic Jumps and the Deep-Slow to Shallow-Fast Transition

Uniform flow has the same upstream and downstream depth, while gradually varying flow changes depth along the channel.
Continuity links depth and speed: for a given discharge, a larger cross-section carries slower flow and a smaller one carries faster flow.
A hydraulic jump is a rapid transition from shallow, fast upstream flow to deeper, slower downstream flow.

5:00 Turbulent Open-Channel Resistance and the Manning–Chezy Framework

Elsworth reviews friction losses in pipes, external flows, and open channels, noting that typical open-channel flows are treated as turbulent.
The Chézy and Manning approaches relate average velocity to channel geometry, slope, and resistance; Manning’s n is larger for rough, brush-lined channels than smooth concrete.
For Manning calculations, the discharge is average velocity multiplied by cross-sectional area; the English-unit coefficient is 1.49 versus 1.0 in SI.

7:00 Compound-Channel Subdivision and Efficient Cross-Sections

For a very wide, shallow channel, the hydraulic radius approaches the flow depth because area divided by wetted perimeter approaches depth.
A compound channel can be divided into subsections and the subsection discharges summed; each hydraulic radius uses its own area and boundary perimeter.
The most efficient shapes discussed include a rectangular channel with depth equal to half its width, a right-angle triangular section, and a semicircular section.

11:00 Worked Manning Calculation for a Trapezoidal Concrete Channel

The example uses a trapezoidal channel with a 5-foot flow depth, 12-foot bottom width, and side slopes set at 40 degrees.
The bed gradient is 1.4 feet per 1,000 feet, and smooth concrete is assigned a Manning roughness near 0.012–0.014.
The trapezoid’s area comes from its rectangular base plus the two side triangles; its wetted perimeter includes the bottom and sloping sides.
With area, hydraulic radius, slope, and Manning n specified, the discharge or mean velocity follows by substitution into the Manning relation.

16:30 From Uniform Flow to Gradually Varying Water-Surface Profiles

In uniform flow, bed slope, water-surface slope, and energy-grade-line slope coincide because depth and velocity remain constant.
In gradually varying flow, upstream and downstream depths differ, so water-surface and energy slopes need not be parallel to the bed.
The same energy framework can also help describe rapid transitions, even though a weir or hydraulic jump is not gradually varying.

20:00 Bernoulli Energy Balance with Bed Slope and Friction Loss

Elsworth applies Bernoulli’s equation between upstream and downstream sections, including elevation head, pressure head, velocity head, and head loss.
At the channel bed, pressure head equals the local water depth, reducing the pressure terms to y₁ and y₂.
Bed slope is defined as S₀ = ΔZ/L, while friction slope is S_f = h_L/L; their contributions determine how energy changes along the reach.

27:00 Unit Discharge and the Specific-Energy Equation

For a channel of width B, unit discharge is lowercase q = Q/B; it remains constant downstream when total discharge and width are unchanged.
In a rectangular channel, average velocity is V = q/y, so specific energy is E = y + q²/(2gy²).
Comparing upstream and downstream sections shows how specific energy, bed elevation, and friction loss combine; energy is not generally constant when losses or elevation changes matter.

34:00 Specific-Energy Curve, Critical Depth, and Alternate Flow States

For fixed unit discharge q, plotting E = y + q²/(2gy²) against depth produces a curve with a minimum specific energy.
At that minimum, the flow is critical; for a rectangular channel, critical depth satisfies y_c = (q²/g)^(1/3).
Above the critical depth, the flow is subcritical—deeper and slower; below it, flow is supercritical—shallower and faster.
For energy above the minimum, the curve permits two physically meaningful positive depths, while the third root of the cubic energy equation is negative and impossible for water depth.

43:00 Froude Number, Gravity-Wave Propagation, and Upstream Information

For a wide rectangular channel, the Froude number is Fr = V/√(gy); it compares flow speed with the shallow-water gravity-wave speed.
Fr < 1 identifies subcritical flow, where surface disturbances can travel upstream; Fr = 1 is critical, and Fr > 1 is supercritical, where disturbances cannot propagate upstream against the flow.
Elsworth illustrates the transition with ripples from a pebble: at critical speed, waves cannot move upstream relative to the channel, while faster flow sweeps them downstream.
The same wave-speed comparison is related to ship bow waves and, by analogy, acoustic waves around supersonic aircraft.

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.