HYD-25

Gradually varied flow dy/dx

dy/dx = (S0 − Sf) / (1 − Fr²). Backwater when Fr < 1 and Sf ≠ S0.

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Open channelGVFChow

Governing equation

\dfrac{dy}{dx}=\dfrac{S_0-S_f}{1-\mathrm{Fr}^2}

where

S_0: Bed slope (—)
S_f: Friction slope (—)
\mathrm{Fr}: Froude (—)
dy/dx: Surface slope (—)

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-25 — Gradually varied flow dy/dx) is the form associated with GVF · Chow. Working symbols: $S_0$ , $S_f$ , $\mathrm{Fr}$ $\rightarrow$ $dy/dx$ . Differentiate specific energy along a prismatic channel. Sf from Manning or Chezy. Singular at Fr = 1.

Purpose

Purpose: compute

dy/dx

from

S_0

S_f

\mathrm{Fr}

in Hydraulics via

\dfrac{dy}{dx}=\dfrac{S_0-S_f}{1-\mathrm{Fr}^2}

dy/dx = (S0 − Sf) / (1 − Fr²). Backwater when Fr < 1 and Sf ≠ S0. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

S_0 = 0.001\,\mathrm{—}

S_f = 6.000e-4\,\mathrm{—}

\mathrm{Fr} = 0.400\,\mathrm{—}

, the governing relation

\dfrac{dy}{dx}=\dfrac{S_0-S_f}{1-\mathrm{Fr}^2}

yields

dy/dx = 4.762e-4\,\mathrm{—}

. A sloping channel, a water-surface slope, a backwater curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Bed slope S_00.001 —Friction slope S_f0.0006 —Froude \mathrm{Fr}0.4 —

Outputs

Surface slope dy/dx0.000476 —

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Narration of this film

A sloping channel, a water-surface slope, a backwater curve.

Differentiate specific energy along a prismatic channel. Sf from Manning or Chezy. Singular at Fr = 1.

Reading speed

Watch on YouTube