HYD-26

Chézy uniform flow

V = C √(R S). Ancestor of Manning; C ≈ R^{1/6}/n.

Reading speed

Open channelChézy

Governing equation

V=C\sqrt{RS}

$S \in \left[1\times 10^{-4}, 0.02\right]$

where

C: Chézy C (m^{1/2}/s)
R: Hydraulic radius (m)
S: Slope (—)
V: Velocity (m/s)

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

V=C\sqrt{RS}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $V = 0.447$

Along the curve

②

Domain of the variable — blue

S_{\min} = 1\times 10^{-4},\quad S_{\max} = 0.02

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $S = 1\times 10^{-4}$

Along the curve

③

Response curve — green

V=C\sqrt{RS}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 1\times 10^{-4}$ $V = 0.447$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}V}{\mathrm{d}S}

$S \in \left[1\times 10^{-4}, 0.02\right]$

$S = 2.25\times 10^{-4}$ $V^{\prime} = 1557.3$

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-26 — Chézy uniform flow) is the form associated with Chézy. Working symbols: $C$ , $R$ , $S$ $\rightarrow$ $V$ . Chézy balanced gravity against a quadratic wall shear. Manning's n maps onto C = R^{1/6}/n (SI).

Purpose

Purpose: compute

V

from

C

R

S

in Hydraulics via

V=C\sqrt{RS}

V = C √(R S). Ancestor of Manning; C ≈ R^{1/6}/n. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C = 50.000\,\mathrm{m^1/2/s}

R = 0.800\,\mathrm{m}

S = 0.001\,\mathrm{—}

, the governing relation

V=C\sqrt{RS}

yields

V = 1.414\,\mathrm{m/s}

. A prismatic channel, a Chézy C, a uniform V. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Chézy C C50 m^{1/2}/sHydraulic radius R0.8 mSlope S0.001 —

Outputs

Velocity V1.414 m/s

Reading speed

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Narration of this film

A prismatic channel, a Chézy C, a uniform V.

Chézy balanced gravity against a quadratic wall shear. Manning's n maps onto C = R^{1/6}/n (SI).

Reading speed

Watch on YouTube