FLD-03

Reynolds number

Re = ρ v D / μ. Laminar below ~2300 in a pipe, turbulent above.

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FluidsReynolds

Governing equation

Re=\dfrac{\rho v D}{\mu}

$v \in \left[0.01, 30\right]$

where

\rho: Density (kg/m³)
v: Speed (m/s)
D: Diameter (m)
\mu: Viscosity (Pa·s)
Re: Reynolds (—)

The figure, part by part

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①

Governing law — red

Re=\dfrac{\rho v D}{\mu}

$v \in \left[0.01, 30\right]$

$v = 0.01$ $Re = 500$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0.01,\quad v_{\max} = 30

$v \in \left[0.01, 30\right]$

$v = 0.01$ $v = 0.01$

Along the curve

③

Response curve — green

Re=\dfrac{\rho v D}{\mu}

$v \in \left[0.01, 30\right]$

$v = 0.01$ $Re = 500$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Re}{\mathrm{d}v}

$v \in \left[0.01, 30\right]$

$v = 0.199$ $Re^{\prime} = 50000$

Along the curve

Lecture brief

Historical brief

Bernoulli, Navier–Stokes, Reynolds (1883) and Stokes drag made continuum flow a dimensionless craft. The sheets compute head, drag, Re and a pedagogical NS snapshot. This sheet (FLD-03 — Reynolds number) is the form associated with Reynolds. Working symbols: $\rho$ , $v$ , $D$ , $\mu$ $\rightarrow$ $Re$ . Inertia over viscous stress. The single most important dimensionless group in fluids.

Purpose

Purpose: compute

Re

from

\rho

v

D

\mu

in Fluid physics via

Re=\dfrac{\rho v D}{\mu}

Re = ρ v D / μ. Laminar below ~2300 in a pipe, turbulent above. Use it when a real fluid physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho = 1000.000\,\mathrm{kg/m^{3}}

v = 1.500\,\mathrm{m/s}

D = 0.050\,\mathrm{m}

\mu = 0.001\,\mathrm{Pa·s}

, the governing relation

Re=\dfrac{\rho v D}{\mu}

yields

Re = 75000.0\,\mathrm{—}

. A pipe, a dye filament that stays or breaks. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density \rho1000 kg/m³Speed v1.5 m/sDiameter D0.05 mViscosity \mu0.001 Pa·s

Outputs

Reynolds Re75000.0 —

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Narration of this film

A pipe, a dye filament that stays or breaks.

Inertia over viscous stress. The single most important dimensionless group in fluids.

Reading speed

Watch on YouTube