HYD-30

Froude number and critical depth

Fr = v/√(g y), yc = (q²/g)^{1/3} for a rectangle. Critical when Fr = 1.

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Open channelFroude

Governing equation

\mathrm{Fr}=\dfrac{v}{\sqrt{gy}},\quad y_c=(q^2/g)^{1/3}

$v \in \left[0.2, 10\right]$

where

v: Velocity (m/s)
y: Depth (m)
q: Unit discharge (m²/s)
\mathrm{Fr}: Froude (—)
y_c: Critical depth (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

\mathrm{Fr}=\dfrac{v}{\sqrt{gy}},\quad y_c=(q^2/g)^{1/3}

$v \in \left[0.2, 10\right]$

$v = 0.2$ $\mathrm{Fr} = 0.064$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0.2,\quad v_{\max} = 10

$v \in \left[0.2, 10\right]$

$v = 0.2$ $v = 0.2$

Along the curve

③

Response curve — green

\mathrm{Fr}=\dfrac{v}{\sqrt{gy}},\quad y_c=(q^2/g)^{1/3}

$v \in \left[0.2, 10\right]$

$v = 0.2$ $\mathrm{Fr} = 0.064$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mathrm{Fr}}{\mathrm{d}v}

$v \in \left[0.2, 10\right]$

$v = 0.262$ $\mathrm{Fr}^{\prime} = 0.319$

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-30 — Froude number and critical depth) is the form associated with Froude. Working symbols: $v$ , $y$ , $q$ $\rightarrow$ $\mathrm{Fr}$ , $y_c$ . Fr is the ratio of flow speed to shallow-water wave speed. Control sections sit at Fr = 1.

Purpose

Purpose: compute

\mathrm{Fr}

y_c

from

v

y

q

in Hydraulics via

\mathrm{Fr}=\dfrac{v}{\sqrt{gy}},\quad y_c=(q^2/g)^{1/3}

Fr = v/√(g y), yc = (q²/g)^{1/3} for a rectangle. Critical when Fr = 1. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

v = 2.000\,\mathrm{m/s}

y = 1.000\,\mathrm{m}

q = 2.000\,\mathrm{m^{2}/s}

, the governing relation

\mathrm{Fr}=\dfrac{v}{\sqrt{gy}},\quad y_c=(q^2/g)^{1/3}

yields

\mathrm{Fr} = 0.639\,\mathrm{—}

y_c = 0.742\,\mathrm{m}

. A depth, a speed, a Fr tick and a yc. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Velocity v2 m/sDepth y1 mUnit discharge q2 m²/s

Outputs

Froude \mathrm{Fr}0.639 —
Critical depth y_c0.742 m

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Narration of this film

A depth, a speed, a Fr tick and a yc.

Fr is the ratio of flow speed to shallow-water wave speed. Control sections sit at Fr = 1.

Reading speed

Watch on YouTube