HYD-29

Specific energy E = y + v²/2g

E = y + q²/(2 g y²) for a rectangular channel. Minimum at critical depth.

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Open channelSpecific energy

Governing equation

E=y+\dfrac{v^2}{2g}

$y \in \left[0.1, 6\right]$

where

y: Depth (m)
v: Velocity (m/s)
E: Specific energy (m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

E=y+\dfrac{v^2}{2g}

$y \in \left[0.1, 6\right]$

$y = 0.1$ $E = 0.215$

Along the curve

②

Domain of the variable — blue

y_{\min} = 0.1,\quad y_{\max} = 6

$y \in \left[0.1, 6\right]$

$y = 0.1$ $y = 0.1$

Along the curve

③

Response curve — green

E=y+\dfrac{v^2}{2g}

$y \in \left[0.1, 6\right]$

$y = 0.1$ $E = 0.215$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}E}{\mathrm{d}y}

$y \in \left[0.1, 6\right]$

$y = 0.137$ $E^{\prime} = 1$

Along the curve

Lecture brief

Historical brief

Open-channel and pipe flow were written by Chezy, Manning, Darcy and Weisbach in the nineteenth century, then Bakhmeteff and Bélanger on the hydraulic jump. The sheets compute conveyance, head loss and gradually varied profiles. This sheet (HYD-29 — Specific energy E = y + v²/2g) is the form associated with Specific energy. Working symbols: $y$ , $v$ $\rightarrow$ $E$ . Energy relative to the bed. Two depths share one E — the subcritical and supercritical pair.

Purpose

Purpose: compute

E

from

y

v

in Hydraulics via

E=y+\dfrac{v^2}{2g}

E = y + q²/(2 g y²) for a rectangular channel. Minimum at critical depth. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

y = 1.200\,\mathrm{m}

v = 1.500\,\mathrm{m/s}

, the governing relation

E=y+\dfrac{v^2}{2g}

yields

E = 1.315\,\mathrm{m}

. A depth, a velocity head, an E-y curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Depth y1.2 mVelocity v1.5 m/s

Outputs

Specific energy E1.315 m

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Narration of this film

A depth, a velocity head, an E-y curve.

Energy relative to the bed. Two depths share one E — the subcritical and supercritical pair.

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