HYD-01

Manning discharge

Uniform-flow discharge Q = (1/n) A R^{2/3} S^{1/2} (SI).

Reading speed

Open channelManning 1890ISO 748

Governing equation

Q=\dfrac{1}{n}A R^{2/3} S^{1/2},\quad R=A/P

$y \in \left[0.1, 8\right]$

where

n: Manning n (s/m^{1/3})
b: Width (m)
y: Depth (m)
S: Bed slope (—)
R: Hydraulic radius (m)
Q: Discharge (m³/s)
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

Q=\dfrac{1}{n}A R^{2/3} S^{1/2},\quad R=A/P

$y \in \left[0.1, 8\right]$

$y = 0.1$ $R = 0.179$

Along the curve

②

Domain of the variable — blue

y_{\min} = 0.1,\quad y_{\max} = 8

$y \in \left[0.1, 8\right]$

$y = 0.1$ $y = 0.1$

Along the curve

③

Response curve — green

Q=\dfrac{1}{n}A R^{2/3} S^{1/2},\quad R=A/P

$y \in \left[0.1, 8\right]$

$y = 0.1$ $R = 0.179$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}R}{\mathrm{d}y}

$y \in \left[0.1, 8\right]$

$y = 0.15$ $R^{\prime} = 3.791$

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-01 — Manning discharge) is the form associated with Manning 1890 · ISO 748. Working symbols: $n$ , $b$ , $y$ , $S$ $\rightarrow$ $R$ , $Q$ , $V$ . Manning recast Chézy's C as R^{1/6}/n. The formula is empirical yet the open-channel standard for uniform flow.

Purpose

Purpose: compute

R

Q

V

from

n

b

y

S

in Hydraulics via

Q=\dfrac{1}{n}A R^{2/3} S^{1/2},\quad R=A/P

Uniform-flow discharge Q = (1/n) A R^{2/3} S^{1/2} (SI). Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 0.030\,\mathrm{s/m^1/3}

b = 8.000\,\mathrm{m}

y = 1.500\,\mathrm{m}

S = 0.001\,\mathrm{—}

, the governing relation

Q=\dfrac{1}{n}A R^{2/3} S^{1/2},\quad R=A/P

yields

R = 1.091\,\mathrm{m}

Q = 13.405\,\mathrm{m^{3}/s}

V = 1.117\,\mathrm{m/s}

. Wide rectangle: A = b y, P = b+2y, R = A/P. SI n. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Manning n n0.03 s/m^{1/3}Width b8 mDepth y1.5 mBed slope S0.001 —

Outputs

Hydraulic radius R1.091 m
Discharge Q13.405 m³/s
Velocity V1.117 m/s

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Narration of this film

Wide rectangle: A = b y, P = b+2y, R = A/P. SI n.

Manning recast Chézy's C as R^{1/6}/n. The formula is empirical yet the open-channel standard for uniform flow.

Reading speed

Watch on YouTube