Shock-boundary layer interaction

Why it matters

Inviscid theory treats a shock as a clean jump, but on a real wing, intake ramp or control flap the shock meets a boundary layer. The interaction can thicken or separate the boundary layer, smear the pressure rise, cause buffet, intake unstart, control-surface loss and local heating peaks several times the undisturbed value. Many transonic and supersonic design limits are really shock–boundary-layer interaction (SBLI) limits.

Key ideas

Why a shock is felt upstream. Outside the boundary layer the flow is supersonic and cannot carry signals upstream. But near the wall the boundary-layer flow is subsonic (the sonic line lies inside it). The shock's pressure rise travels upstream through this subsonic layer, so the boundary layer starts to thicken some distance ahead of the inviscid shock position. The thickening turns the outer supersonic flow, producing compression waves that coalesce into a "separation shock" ahead of the main shock. The wall pressure therefore rises over a distance of several boundary-layer thicknesses instead of abruptly.

Basic interactions.

• Normal shock on a transonic wing: the shock foot spreads into a lambda (λ) shock — a forward oblique leg and a rear leg — and if strong enough separates the flow behind it. This causes drag rise, buffet and shock oscillation.
• Incident oblique shock reflecting from a wall (intakes, wind tunnels): the reflected shock forms ahead of the inviscid impingement point; with a strong enough incident shock a separation bubble forms, with an expansion fan over it and reattachment compression behind.
• Compression corner (ramp, flap): separation occurs ahead of the corner when the ramp angle exceeds the incipient-separation angle; the separated region reduces flap effectiveness.

Laminar vs turbulent. A laminar boundary layer has low momentum near the wall and separates under small pressure ratios (roughly 1.1–1.5); its upstream influence extends much farther. A turbulent layer has a fuller velocity profile and withstands a considerably larger rise, of order p₂/p₁ ≈ 2 at M ≈ 2 (increasing with Mach number) before separating. Wind-tunnel models are often tripped to make their boundary layers turbulent. Incipient-separation and plateau-pressure values are empirical — take them from correlations in your data book.

Effect of Reynolds number. Higher Reynolds number gives a thinner boundary layer (relative to body length) and a smaller interaction region; whether the layer is laminar or turbulent matters more than Re itself.

Heating. At reattachment the boundary layer is thin and the flow is compressed, giving local heat-transfer peaks — critical for hypersonic control flaps and intakes.

Unsteadiness. Separated interactions oscillate at low frequency, giving fluctuating loads (buffet, panel fatigue).

Control methods.

• Boundary-layer suction or bleed ahead of or under the shock (standard on supersonic intakes).
• Vortex generators that mix high-momentum fluid towards the wall.
• Shock control bumps that split a normal shock into weaker waves.
• Shaping to keep shocks weak (supercritical airfoils, multi-shock intakes) — the best fix is a weaker shock.

Formulas

Inviscid pressure rise through the interaction (outside the boundary layer) is set by the shock system:

p₂/p₁ = 1 + 2γ/(γ+1)·(M₁²·sin²β − 1) — each oblique shock (β from the θ–β–M relation).

p₃/p₁ = (p₂/p₁)·(p₃/p₂) — incident plus reflected shock at a wall.

Re_x = ρ·V·x/μ — Reynolds number, decides laminar or turbulent state.

a = √(γ·R·T), V = M·a, ρ = p/(R·T) — to build Re from tunnel conditions.

Separation criterion (empirical): separation occurs when the imposed p₂/p₁ exceeds the incipient-separation value for the boundary-layer state, Mach number and Reynolds number (from correlations or data).

Symbols: p pressure (Pa); M Mach number; β shock angle; γ ratio of specific heats; ρ density (kg/m³); V velocity (m/s); x distance from the leading edge (m); μ dynamic viscosity (Pa·s, from a data book or Sutherland's law); R gas constant (J/(kg·K)).

Worked examples

Example 1 (standard). A wind-tunnel model plate 0.3 m long sits in a Mach 2 test section with T = 166.7 K, p = 20 kPa. Given μ = 1.13 × 10⁻⁵ Pa·s at this temperature (data-book value), find the Reynolds number at the trailing edge and say whether the boundary layer is likely to be turbulent there.

1. ρ = p/(RT) = 20 000/(287 × 166.7) = 0.418 kg/m³.
2. a = √(1.4 × 287 × 166.7) = 258.8 m/s; V = 2 × 258.8 = 517.6 m/s.
3. Re = ρVx/μ = 0.418 × 517.6 × 0.3/(1.13 × 10⁻⁵) = 5.7 × 10⁶.

Answer: Re ≈ 5.7 × 10⁶ — high enough that the layer is normally turbulent over most of the plate (tripping makes sure), so it resists separation better than a laminar layer.

Example 2 (GATE level). An oblique shock generated by an 8° wedge in a Mach 2 stream strikes a flat wall and reflects. Find the inviscid pressure rise across the incident and reflected shocks, and compare with a normal shock at M = 2.

1. Incident shock (θ = 8°, M = 2): β = 37.2°, Mₙ₁ = 2 sin 37.2° = 1.209, p₂/p₁ = 1 + 1.1667 × (1.4628 − 1) = 1.540, M₂ = 1.714.
2. Reflected shock must turn the flow back 8° at M₂ = 1.714: β = 44.1°, Mₙ = 1.714 × sin 44.1° = 1.193, p₃/p₂ = 1 + 1.1667 × (1.4225 − 1) = 1.493.
3. Total: p₃/p₁ = 1.540 × 1.493 = 2.30.
4. Normal shock at M = 2: p₂/p₁ = 4.5.

Answer: p₃/p₁ ≈ 2.30 imposed on the wall boundary layer — enough to separate a laminar layer, and close to the incipient-separation range for a turbulent one, so bleed or a turbulent, energised layer would be needed.

Common mistakes

• Treating the shock as felt only at its inviscid position; the interaction starts upstream through the subsonic part of the boundary layer.
• Assuming a laminar layer is more robust because it has lower skin friction — it separates far more easily.
• Ignoring the reflected shock: the wall sees the combined pressure rise.
• Thinking higher Mach number always means separation; it is the pressure ratio relative to the boundary layer's tolerance that matters.
• Quoting incipient-separation pressure ratios as exact; they are empirical.

For GATE AE

Questions here are mainly conceptual: why upstream influence exists, laminar vs turbulent behaviour, lambda shocks on transonic wings, effects (drag, buffet, unstart, heating) and control methods (bleed, vortex generators, bumps). Numerical parts reuse oblique-shock and normal-shock relations for the imposed pressure rise and the Reynolds number to decide the boundary-layer state.

Quick check

1. Through which part of the boundary layer does the shock's pressure rise travel upstream?
2. Which separates more easily under a shock, a laminar or a turbulent boundary layer?
3. What shape does a normal shock foot take on a transonic wing?
4. Name two methods of controlling SBLI.
5. Where is heat transfer highest in a separated interaction?

Answers: 1. the subsonic layer near the wall; 2. laminar; 3. a lambda shock; 4. any two of suction/bleed, vortex generators, shock control bumps, shaping for weaker shocks; 5. near reattachment.