HYD-24

Hydraulic jump sequent depth

y2/y1 = ½ (−1 + √(1+8 Fr1²)). Momentum across a rapid jump.

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Open channelBélanger

Governing equation

\dfrac{y_2}{y_1}=\tfrac12\bigl(-1+\sqrt{1+8\mathrm{Fr}_1^2}\bigr)

$\mathrm{Fr}_1 \in \left[1.1, 9\right]$

where

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

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}=\tfrac12\bigl(-1+\sqrt{1+8\mathrm{Fr}_1^2}\bigr)

$\mathrm{Fr}_1 \in \left[1.1, 9\right]$

$\mathrm{Fr}_1 = 1.1$ $y_2 = 0.454$

Along the curve

②

Domain of the variable — blue

\mathrm{Fr}_1_{\min} = 1.1,\quad \mathrm{Fr}_1_{\max} = 9

$\mathrm{Fr}_1 \in \left[1.1, 9\right]$

$\mathrm{Fr}_1 = 1.1$ $\mathrm{Fr}_1 = 1.1$

Along the curve

③

Response curve — green

\dfrac{y_2}{y_1}=\tfrac12\bigl(-1+\sqrt{1+8\mathrm{Fr}_1^2}\bigr)

$\mathrm{Fr}_1 \in \left[1.1, 9\right]$

$\mathrm{Fr}_1 = 1.1$ $y_2 = 0.454$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}y_2}{\mathrm{d}\mathrm{Fr}_1}

$\mathrm{Fr}_1 \in \left[1.1, 9\right]$

$\mathrm{Fr}_1 = 1.15$ $y_2^{\prime} = 0.541$

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-24 — Hydraulic jump sequent depth) is the form associated with Bélanger. Working symbols: $y_1$ , $\mathrm{Fr}_1$ $\rightarrow$ $y_2$ , $y_2/y_1$ . Bélanger: horizontal hydrostatic force plus momentum flux, friction neglected on a short apron.

Purpose

Purpose: compute

y_2

y_2/y_1

from

y_1

\mathrm{Fr}_1

in Hydraulics via

\dfrac{y_2}{y_1}=\tfrac12\bigl(-1+\sqrt{1+8\mathrm{Fr}_1^2}\bigr)

y2/y1 = ½ (−1 + √(1+8 Fr1²)). Momentum across a rapid jump. 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}

\mathrm{Fr}_1 = 3.500\,\mathrm{—}

, the governing relation

\dfrac{y_2}{y_1}=\tfrac12\bigl(-1+\sqrt{1+8\mathrm{Fr}_1^2}\bigr)

yields

y_2 = 1.790\,\mathrm{m}

y_2/y_1 = 4.47\,\mathrm{—}

. A supercritical jet, a roller, a sequent depth. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Upstream depth y_10.4 mUpstream Froude \mathrm{Fr}_13.5 —

Outputs

Sequent depth y_21.790 m
Ratio y_2/y_14.47 —

Reading speed

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Narration of this film

A supercritical jet, a roller, a sequent depth.

Bélanger: horizontal hydrostatic force plus momentum flux, friction neglected on a short apron.

Reading speed

Watch on YouTube