Reynolds Number — External Flow

Flat Plate — Air at 20°C, V = 10 m/s, L = 1 m

Aerofoil — Air at 20°C, V = 30 m/s, chord c = 0.5 m

Cylinder — Water at 20°C, V = 0.5 m/s, D = 50 mm

Formula (same as internal flow, different characteristic length):

Regime thresholds by geometry:

Flat plate / aerofoil — transition position:

The boundary layer is laminar for x < x_cr and turbulent for x > x_cr. Re_cr ≈ 5×10⁵ is the standard engineering value for a smooth surface with low free-stream turbulence. For aerofoils, x_cr is expressed as a percentage of chord c.

Aerofoil — why Re_c matters:

• Re_c < 10⁴ — insects and micro-vehicles; viscosity so dominant that conventional aerofoil sections perform poorly
• 10⁴ – 10⁵ — laminar separation bubble forms on suction surface; thin, highly-cambered sections (e.g. Eppler, Selig) needed
• 10⁵ – 5×10⁵ — glider and UAV range; bubble may burst → sudden stall; turbulators sometimes added to fix transition
• > 5×10⁵ — BL transitions early; classical NACA sections work well; thin-aerofoil theory reliable

The transition position x_tr is estimated using the flat-plate criterion Re_x = 5×10⁵ — a useful approximation for zero-pressure-gradient sections. Actual x_tr shifts forward with increasing angle of attack or adverse pressure gradient.

Cylinder drag crisis:

Around Re ≈ 2–5×10⁵ the laminar boundary layer trips to turbulent before separating, moving the separation point from ~80° to ~140° from the stagnation point. This dramatically narrows the turbulent wake and causes C_D to drop from ~1.2 to ~0.3 — the so-called drag crisis. Golf ball dimples intentionally trigger this transition at lower Re to reduce drag in flight.

Where:

• ρ = fluid density [kg/m³]
• V∞ = free-stream velocity [m/s]
• L = plate length from leading edge [m]
• c = aerofoil chord length [m]
• D = cylinder or sphere outer diameter [m]
• μ = dynamic viscosity [Pa·s]
• ν = kinematic viscosity [m²/s] = μ / ρ
• x_cr / x_tr = transition position [m]