Skip to content

Reynolds Number

Intuition

Every flow is a contest between inertia (the tendency of moving fluid to keep going, amplifying disturbances) and viscosity (internal friction smoothing them out). The Reynolds number is the scoreboard. Small \(Re\): honey-like, orderly, reversible motion. Large \(Re\): eddies, wakes, chaos — turbulence.

Definition

\[\boxed{\,Re = \frac{UL}{\nu} = \frac{\rho U L}{\mu}\,}\]
  • \(U\) — characteristic velocity
  • \(L\) — characteristic length (pipe diameter, chord, body size)
  • \(\nu = \mu/\rho\) — kinematic viscosity

It is the ratio of the convective term to the viscous term in the Navier–Stokes equation:

\[\frac{|(\mathbf{v}\cdot\nabla)\mathbf{v}|}{|\nu\nabla^2\mathbf{v}|} \sim \frac{U^2/L}{\nu U/L^2} = \frac{UL}{\nu}\]

Regimes

\(Re\) Character
\(\ll 1\) creeping (Stokes) flow — viscosity rules; bacteria, microfluidics
\(\sim 1\)–\(10^3\) smooth laminar flow; steady wakes appear
\(\sim 2300\) classic transition threshold in pipes
\(\gg 10^4\) turbulent — inertia rules; aircraft, rivers, atmosphere

Since \(\nu \propto 1/Re\) in nondimensional form, high-\(Re\) flow behaves almost inviscidly — except in boundary layers and in the smallest eddies of the energy cascade, where viscosity always collects its due.

Why it matters

  • Dynamic similarity: matching \(Re\) (and geometry) between model and full scale makes wind-tunnel testing valid — see Buckingham Pi.
  • Regime prediction: whether a design sees laminar or turbulent flow changes drag, heat transfer, and mixing by orders of magnitude.
  • The turbulence switch: \(Re\) is the control parameter in most stability analyses.

Worked example

A car (\(L \approx 4\) m) at 100 km/h in air (\(\nu \approx 1.5\times10^{-5}\ \text{m}^2/\text{s}\)):

\[Re = \frac{27.8 \times 4}{1.5\times10^{-5}} \approx 7\times10^6\]

Deeply turbulent. A paramecium (\(L\sim 10^{-4}\) m, \(U\sim 10^{-3}\) m/s, water \(\nu = 10^{-6}\)): \(Re \sim 0.1\) — it lives in your honey-world, where coasting is impossible and swimming strategies must be non-reciprocal (the "scallop theorem").

Common mistakes

  • Ambiguous \(L\). State your length scale; \(Re\) based on radius vs diameter differs by 2. Conventions matter when quoting thresholds.
  • Treating 2300 as universal. It's the pipe-flow transition under ordinary disturbance levels; carefully quiet experiments stay laminar far higher.
  • Assuming high \(Re\) = inviscid everywhere. Boundary layers, separation and dissipation are viscous phenomena that persist at any \(Re\).

Knowledge graph position

Prerequisites: Viscosity, Dimensional analysis. Leads to: Turbulence, Hydrodynamic stability.

Quiz

Q1 (computational). Water (\(\nu = 10^{-6}\ \text{m}^2/\text{s}\)) flows at 2 m/s through a 5 cm pipe. Laminar or turbulent?

Answer

\(Re = 2\times0.05/10^{-6} = 10^5 \gg 2300\) — turbulent, comfortably.

Q2 (conceptual). Why is it impossible to test a full-scale-Reynolds airliner with a 1:50 model in the same wind tunnel air at the same speed?

Answer

\(Re \propto UL\): shrinking \(L\) by 50 drops \(Re\) by 50. You'd need 50× the speed (compressibility ruins it), a pressurized/cryogenic tunnel (raising \(\rho\), lowering \(\nu\)), or acceptance of scale effects.

Q3 (multiple choice). The Reynolds number compares:

  • (a) pressure to gravity (b) inertia to viscosity (c) velocity to sound speed (d) kinetic to potential energy
Answer

(b). (a) is Froude-like, (c) is the Mach number.