Viscous effects on airfoils: stall and profile drag

Why it matters

Inviscid theory predicts zero drag and lift rising without limit. Real airfoils have profile drag, which sets cruise efficiency, and a maximum lift coefficient, which sets stall speed and landing performance. Both come from the thin viscous boundary layer and its tendency to separate, so understanding it explains airfoil choice, Reynolds-number effects in wind tunnels and the very different ways airfoils stall.

Key ideas

Boundary layer and separation. Viscosity enforces no-slip at the surface; the velocity rises to the outer (potential) value across a thin boundary layer. Where the outer pressure rises downstream (adverse pressure gradient — behind the suction peak on the upper surface), the slow fluid near the wall is decelerated further. If the pressure rise is too strong, the near-wall flow reverses and the boundary layer separates, leaving a thick wake. Separation is favoured by: a strong adverse gradient (high α, thick or highly cambered rear sections), a laminar boundary layer (less near-wall momentum), and low Reynolds number.

Laminar versus turbulent. A laminar boundary layer has low skin friction but separates easily; a turbulent one has 3–5 times higher skin friction but its fuller velocity profile resists separation much better. Transition depends on Re, pressure gradient, surface roughness and free-stream turbulence. A laminar layer that separates may become turbulent in the free shear layer and reattach, forming a laminar separation bubble.

Profile drag of a 2-D section = skin-friction drag (integral of wall shear) + pressure (form) drag (the pressure distribution no longer cancels because the boundary layer and wake displace the flow and prevent full pressure recovery at the rear). At low α a streamlined airfoil's profile drag is mostly skin friction; near stall it is mostly pressure drag. For a finite wing, profile drag adds to induced drag; for a whole aircraft, "parasite drag" adds the fuselage, nacelles, interference and so on.

Estimating profile drag.

• Flat-plate skin friction (both sides) is the baseline; laminar and turbulent correlations are given below.
• Thickness raises local velocities and adds pressure drag; an empirical form factor such as 1 + 2(t/c) + 60(t/c)⁴ multiplies the turbulent flat-plate value (one of several in data books).
• Experimentally, drag is measured by a wake survey: the momentum deficit in the wake equals the drag, c_d = (2/c)∫(u/V∞)(1 − u/V∞) dy (for a survey where static pressure has recovered to p∞).
• Laminar-flow (NACA 6-series) airfoils keep favourable pressure gradients over the front 40–60 % and show a low-drag "bucket" in the polar over a limited c_l range.

Stall. As α rises, the separation point on the upper surface moves forward, the lift curve bends over, and c_l reaches c_l,max at the stalling angle (typically 12–18° for conventional sections at high Re). Three classical types:

• Trailing-edge stall (thick sections, t/c ≳ 12 %): turbulent separation starts at the trailing edge and creeps forward; gentle, rounded lift peak.
• Leading-edge stall (moderate thickness, about 9–12 %): a short laminar bubble near the nose suddenly bursts; abrupt loss of lift.
• Thin-airfoil stall (t/c ≲ 9 %): a long bubble grows steadily from the leading edge; lift slope falls early and c_l,max is low. Past stall, drag rises sharply and the pitching moment usually breaks nose-down.

Reynolds-number effects. Higher Re makes the boundary layer turbulent earlier and more resistant to separation, so c_l,max and the stall angle generally rise and c_d falls. Wind-tunnel models run at lower Re than full scale, which is why trip strips are used and why scale corrections are needed.

Formulas

• Re = ρ∞V∞c/μ — chord Reynolds number; μ in Pa·s.
• C_f = 1.328/√Re — laminar flat plate (Blasius), one side, whole plate.
• C_f = 0.074/Re^0.2 — turbulent flat plate, one side, roughly 5 × 10⁵ < Re < 10⁷.
• c_d ≈ 2C_f·[1 + 2(t/c) + 60(t/c)⁴] — empirical airfoil estimate at low α (form factor from your data book).
• c_d = c_d,friction + c_d,pressure — profile drag.
• c_d = (2/c)∫(u/V∞)(1 − u/V∞) dy — wake survey (momentum deficit).
• D′ = q∞·c·c_d (N/m).

Worked examples

Example 1 (standard). A 12 %-thick airfoil of chord 1.5 m flies at 40 m/s in sea-level air (ρ = 1.225 kg/m³, μ = 1.79 × 10⁻⁵ Pa·s). Estimate c_d from fully laminar and fully turbulent flat-plate friction, then apply the form factor to the turbulent value, and find the drag per unit span.

1. Re = 1.225 × 40 × 1.5/1.79 × 10⁻⁵ = 4.11 × 10⁶.
2. Laminar, both sides: 2 × 1.328/√(4.11 × 10⁶) = 2 × 1.328/2026 = 0.00131.
3. Turbulent, both sides: 2 × 0.074/(4.11 × 10⁶)^0.2 = 0.148/21.02 = 0.00704.
4. Form factor: 1 + 2(0.12) + 60(0.12)⁴ = 1 + 0.24 + 0.0124 = 1.252; c_d ≈ 0.00704 × 1.252 = 0.00882.
5. q∞ = 0.5 × 1.225 × 40² = 980 Pa; D′ = 980 × 1.5 × 0.00882 = 13.0 N/m.

Answer: Re ≈ 4.1 × 10⁶; c_d ≈ 0.0013 (all laminar) to 0.0070 (all turbulent, flat plate); about 0.0088 for the turbulent airfoil, D′ ≈ 13 N/m. A real section with some laminar run lies between these.

Example 2 (GATE level). A wake survey 1 chord behind an airfoil of chord 1 m shows a symmetric, triangular velocity deficit: u/V∞ = 0.8 on the centre line, rising linearly to 1.0 at y = ±0.05 m. Static pressure has recovered to p∞. Find c_d.

1. Let η = |y|/h with h = 0.05 m; u/V∞ = 1 − s with s = 0.2(1 − η).
2. (u/V∞)(1 − u/V∞) = (1 − s)s = s − s².
3. ∫ over the wake = 2h ∫₀¹ (s − s²) dη = 2h[0.2/2 − 0.04/3] = 2 × 0.05 × (0.1 − 0.01333) = 0.008667 m.
4. c_d = (2/c) × 0.008667 = 2 × 0.008667/1 = 0.0173.

Answer: c_d ≈ 0.017. (A relatively high value — this airfoil is at a high α or has a thick wake.)

Common mistakes

• Calling profile drag "parasite drag"; parasite drag is the aircraft-level non-lift-dependent drag.
• Using one-side flat-plate C_f for an airfoil without doubling for the two surfaces.
• Assuming turbulent boundary layers are always worse. They have more friction but delay separation and stall.
• Applying the linear lift curve near and beyond stall.
• Forgetting that wind-tunnel c_l,max at low Re is usually lower than in flight.
• Omitting the factor 2/c in the wake-survey formula, or integrating only one half of the wake.

For GATE AE

Expect: Reynolds number; laminar and turbulent skin-friction coefficients and drag; momentum-deficit (wake) drag for simple profiles; MCQs on separation, adverse pressure gradient, stall types versus thickness, the laminar separation bubble, and Re effects on c_l,max. Learn which statements are true for laminar versus turbulent layers — they appear repeatedly as options.

Quick check

1. Compute Re for c = 2 m, V = 60 m/s, ρ = 1.225 kg/m³, μ = 1.81 × 10⁻⁵ Pa·s.
2. What are the two components of profile drag?
3. Which stall type is typical of thick airfoils?
4. Why does a turbulent boundary layer resist separation better?

Answers: 1. 8.12 × 10⁶. 2. Skin-friction drag and pressure (form) drag. 3. Trailing-edge stall. 4. Its fuller velocity profile carries more momentum close to the wall.