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The Reynolds number

Air and fluids · level 8, lesson 1 of 6

A question to start

A bee and an airliner both fly, but the air feels very different to each: to the bee it is thick and sticky, to the airliner thin and slippery. On Mars, even a helicopter rotor feels the air as sticky. How can we put a number on that?

The idea

A moving fluid has two kinds of force inside it. Inertia keeps each bit of fluid moving; viscosity, the fluid's internal friction, drags neighbouring bits together. Their ratio is the Reynolds number:

\[Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu},\]

where \(L\) is a size (a wing's width, a pipe's diameter), \(\mu\) the viscosity in Pa s, and \(\nu = \mu/\rho\) the kinematic viscosity.

  • Small \(Re\): viscosity wins. The flow is smooth and syrupy; drag is relatively large.
  • Large \(Re\): inertia wins. Flow tends to break into eddies; wings work well.
Air at the ground Mars Earth
Density \(\rho\) 0.016 kg/m³ 1.217 kg/m³
Viscosity \(\mu\) \(1.1\times10^{-5}\) Pa s (carbon dioxide, 217 K) about \(1.8\times10^{-5}\) Pa s
Kinematic viscosity \(\nu\) \(6.8\times10^{-4}\) m²/s \(1.5\times10^{-5}\) m²/s

Mars air has about the same viscosity as Earth air, but 75 times less density. So its kinematic viscosity is about 46 times bigger, and every Reynolds number is about 46 times smaller for the same speed and size.

1001,00010,000100,0001 million blade on Marspigeonsame blade on Earth

Worked examples

1. Ingenuity's blade tip. Speed 151 m/s; suppose the blade is 0.1 m wide. Mars: \(Re = 0.016 \times 151 \times 0.1/(1.1\times10^{-5}) = 22{,}000\). Earth: \(1.217 \times 151 \times 0.1/(1.8\times10^{-5}) = 1.0\times10^6\).

2. A pigeon's wing. 15 m/s, 0.1 m wide, on Earth: \(Re = 1.217 \times 15 \times 0.1/(1.8\times10^{-5}) = 100{,}000\). Ingenuity's blades, moving ten times faster, worked at a fifth of a pigeon's Reynolds number.

3. A dust grain 3 µm across falling at 1 cm/s on Mars: \(Re = 0.01 \times 3\times10^{-6}/(6.8\times10^{-4}) \approx 4\times10^{-5}\). Pure syrup.

Try it

  1. What does a small Reynolds number tell you about a flow?
  2. Find \(Re\) for a 2 m wide car at 30 m/s on Earth.
  3. Find \(Re\) for a 0.5 m wide rover mast in a 10 m/s Mars wind.
  4. Why did Ingenuity's blades work at such a low Reynolds number, though they moved so fast?
Answers
  1. Viscosity matters a lot: the flow is smooth and drag is relatively large.
  2. \(1.217 \times 30 \times 2/(1.8\times10^{-5}) \approx 4\times10^6\).
  3. \(0.016 \times 10 \times 0.5/(1.1\times10^{-5}) \approx 7{,}300\).
  4. Mars air's density is so low that its kinematic viscosity is about 46 times Earth air's.

Common mistake

"Thin air has low viscosity." Viscosity hardly depends on density. Mars air is as viscous as Earth air; it is the low density, and so the low inertia, that makes viscosity matter more.

Where it leads