HYD-05

Bélanger hydraulic jump

Sequent depth y2/y1 = ½ (−1 + √(1+8 Fr₁²)).

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Rapidly varied flowBélanger 1828USBR

Governing equation

\dfrac{y_2}{y_1}=\dfrac12\left(-1+\sqrt{1+8\mathrm{Fr}_1^2}\right),\ \mathrm{Fr}_1=\dfrac{V_1}{\sqrt{gy_1}}

$V_1 \in \left[0.5, 15\right]$

where

y_1: Upstream depth (m)
V_1: Upstream velocity (m/s)
\mathrm{Fr}_1: Upstream Froude (—)
y_2: Sequent 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

\dfrac{y_2}{y_1}=\dfrac12\left(-1+\sqrt{1+8\mathrm{Fr}_1^2}\right),\ \mathrm{Fr}_1=\dfrac{V_1}{\sqrt{gy_1}}

$V_1 \in \left[0.5, 15\right]$

$V_1 = 0.5$ $\mathrm{Fr}_1 = 0.046$

Along the curve

②

Domain of the variable — blue

V_1_{\min} = 0.5,\quad V_1_{\max} = 15

$V_1 \in \left[0.5, 15\right]$

$V_1 = 0.5$ $V_1 = 0.5$

Along the curve

③

Response curve — green

\dfrac{y_2}{y_1}=\dfrac12\left(-1+\sqrt{1+8\mathrm{Fr}_1^2}\right),\ \mathrm{Fr}_1=\dfrac{V_1}{\sqrt{gy_1}}

$V_1 \in \left[0.5, 15\right]$

$V_1 = 0.5$ $\mathrm{Fr}_1 = 0.046$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\mathrm{Fr}_1}{\mathrm{d}V_1}

$V_1 \in \left[0.5, 15\right]$

$V_1 = 0.591$ $\mathrm{Fr}_1^{\prime} = 0.184$

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-05 — Bélanger hydraulic jump) is the form associated with Bélanger 1828 · USBR. Working symbols: $y_1$ , $V_1$ $\rightarrow$ $\mathrm{Fr}_1$ , $y_2$ . Momentum and continuity across a free jump on a horizontal floor close on Bélanger's sequent-depth relation.

Purpose

Purpose: compute

\mathrm{Fr}_1

y_2

from

y_1

V_1

in Hydraulics via

\dfrac{y_2}{y_1}=\dfrac12\left(-1+\sqrt{1+8\mathrm{Fr}_1^2}\right),\ \mathrm{Fr}_1=\dfrac{V_1}{\sqrt{gy_1}}

Sequent depth y2/y1 = ½ (−1 + √(1+8 Fr₁²)). 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 = 0.400\,\mathrm{m}

V_1 = 6.000\,\mathrm{m/s}

, the governing relation

\dfrac{y_2}{y_1}=\dfrac12\left(-1+\sqrt{1+8\mathrm{Fr}_1^2}\right),\ \mathrm{Fr}_1=\dfrac{V_1}{\sqrt{gy_1}}

yields

\mathrm{Fr}_1 = 3.029\,\mathrm{—}

y_2 = 1.525\,\mathrm{m}

. Rectangular frictionless floor, no slope, aerated jump. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Upstream depth y_10.4 mUpstream velocity V_16 m/s

Outputs

Upstream Froude \mathrm{Fr}_13.029 —
Sequent depth y_21.525 m

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Narration of this film

Rectangular frictionless floor, no slope, aerated jump.

Momentum and continuity across a free jump on a horizontal floor close on Bélanger's sequent-depth relation.

Reading speed

Watch on YouTube