HYD-35

Moody laminar / turbulent f

f = 64/Re if Re < 2300, else Haaland. A two-regime Moody snapshot.

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Pipe flowMoody

Governing equation

f=\begin{cases}64/\mathrm{Re}&\mathrm{Re}<2300\\f_{\mathrm{Haaland}}&\mathrm{Re}\ge2300\end{cases}

$\mathrm{Re} \in \left[200, 1\times 10^{6}\right]$

where

\mathrm{Re}: Reynolds (—)
\varepsilon/D: Relative roughness (—)
f: Friction factor (—)

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

f=\begin{cases}64/\mathrm{Re}&\mathrm{Re}<2300\\f_{\mathrm{Haaland}}&\mathrm{Re}\ge2300\end{cases}

$\mathrm{Re} \in \left[200, 1\times 10^{6}\right]$

$\mathrm{Re} = 200$ $f = 0.32$

Along the curve

②

Domain of the variable — blue

\mathrm{Re}_{\min} = 200,\quad \mathrm{Re}_{\max} = 1\times 10^{6}

$\mathrm{Re} \in \left[200, 1\times 10^{6}\right]$

$\mathrm{Re} = 200$ $\mathrm{Re} = 200$

Along the curve

③

Response curve — green

f=\begin{cases}64/\mathrm{Re}&\mathrm{Re}<2300\\f_{\mathrm{Haaland}}&\mathrm{Re}\ge2300\end{cases}

$\mathrm{Re} \in \left[200, 1\times 10^{6}\right]$

$\mathrm{Re} = 200$ $f = 0.32$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f}{\mathrm{d}\mathrm{Re}}

$\mathrm{Re} \in \left[200, 1\times 10^{6}\right]$

$\mathrm{Re} = 6488.1$ $f^{\prime} = -2.31\times 10^{-5}$

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-35 — Moody laminar / turbulent f) is the form associated with Moody. Working symbols: $\mathrm{Re}$ , $\varepsilon/D$ $\rightarrow$ $f$ . Hagen–Poiseuille gives 64/Re. Above transition the log-law + roughness takes over.

Purpose

Purpose: compute

f

from

\mathrm{Re}

\varepsilon/D

in Hydraulics via

f=\begin{cases}64/\mathrm{Re}&\mathrm{Re}<2300\\f_{\mathrm{Haaland}}&\mathrm{Re}\ge2300\end{cases}

f = 64/Re if Re < 2300, else Haaland. A two-regime Moody snapshot. Use it when a real hydraulics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\mathrm{Re} = 1500.000\,\mathrm{—}

\varepsilon/D = 5.000e-4\,\mathrm{—}

, the governing relation

f=\begin{cases}64/\mathrm{Re}&\mathrm{Re}<2300\\f_{\mathrm{Haaland}}&\mathrm{Re}\ge2300\end{cases}

yields

f = 0.04267\,\mathrm{—}

. A Re slider, a jump from 64/Re onto the turbulent carpet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Reynolds \mathrm{Re}1500 —Relative roughness \varepsilon/D0.0005 —

Outputs

Friction factor f0.04267 —

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Narration of this film

A Re slider, a jump from 64/Re onto the turbulent carpet.

Hagen–Poiseuille gives 64/Re. Above transition the log-law + roughness takes over.

Reading speed

Watch on YouTube