Shock-expansion theory for supersonic airfoils

Shock-expansion theory: panel-by-panel oblique shocks and Prandtl–Meyer fans for flat-plate and diamond airfoils, with lift and wave drag.

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Why it matters

Supersonic wings, fins and intake ramps are made of flat or nearly flat panels with sharp corners. Shock-expansion theory puts together the two exact building blocks you already know — the oblique shock and the Prandtl–Meyer fan — to give the pressure on each panel, and hence lift, wave drag and moment, without any small-angle approximation.

Key ideas

The method. Divide the airfoil surface into straight segments. Starting from the free stream, march along each surface:

  • where the surface turns into the flow (concave corner, or a leading edge inclined to the stream), apply an oblique shock with the turning angle;
  • where it turns away (convex corner), apply a Prandtl–Meyer expansion;
  • on each straight segment the flow is uniform, so the pressure is constant there.

Then integrate the panel pressures to get the resultant force: lift perpendicular and drag parallel to the free stream.

Assumptions. Steady, two-dimensional, inviscid flow of a perfect gas; sharp leading edge with attached shocks (turning below θmax); supersonic flow everywhere on the surface. The method is exact for the flow next to the surface of a flat plate and of a diamond to the first wave interaction; on curved surfaces it neglects the weak reflections of expansion waves from the shock, which is acceptable for slender shapes. Viscous effects (skin friction, shock–boundary-layer interaction, separation at the trailing edge) are not included.

Flat plate at angle of attack α. On the upper surface the flow turns away from itself by α at the leading edge → expansion fan, lower pressure. On the lower surface it turns into itself by α → oblique shock, higher pressure. At the trailing edge, a shock on the upper side and a fan on the lower side bring the two streams to a common direction and pressure (the slipline between them carries the entropy difference). The pressure difference acts normal to the plate, so the resultant has both lift and wave drag, even with zero thickness: cd = cl·tan α.

Diamond (double-wedge) airfoil at zero α. A leading-edge shock on each front face raises pressure; an expansion at mid-chord lowers it on the rear faces. The front faces push backwards and the rear faces "suck" backwards, giving pure wave drag and no lift.

Comparison with linear theory. For thin sections at small angles, the results agree closely with Ackeret theory (cl = 4α/√(M² − 1)); shock-expansion is needed when angles or Mach numbers make the nonlinear terms important, and it gives the asymmetry between compression and expansion that linear theory misses.

Trailing-edge waves do not affect surface pressures in supersonic flow because disturbances cannot travel upstream.

Formulas

Pressure coefficient: Cp = (p − p∞)/q∞, with q∞ = ½ρ∞V∞² = (γ/2)·p∞·M∞².

Compression panel (turn θ into the flow): solve tan θ = 2·cot β·(M²sin²β − 1)/(M²(γ + cos 2β) + 2) for β, then p₂/p₁ = 1 + 2γ/(γ+1)·(M²sin²β − 1) and M₂ = Mₙ₂/sin(β − θ).

Expansion panel (turn θ away): ν(M₂) = ν(M₁) + θ, p₂/p₁ = [(1 + (γ−1)/2·M₁²)/(1 + (γ−1)/2·M₂²)]^(γ/(γ−1)).

Flat plate: cn = Cp,lower − Cp,upper, cl = cn·cos α, cd = cn·sin α.

Diamond of half-angle ε at α = 0, thickness ratio t/c = tan ε: cd = (Cp,front − Cp,rear)·(t/c).

Symbols: p pressure (Pa); p∞, M∞ free-stream values; q∞ dynamic pressure (Pa); θ panel turning angle; β shock angle; ν Prandtl–Meyer function (degrees); α angle of attack; ε wedge half-angle; cl, cd, cn lift, drag, normal-force coefficients per unit span based on chord c.

Worked examples

Example 1 (standard). A flat plate is at α = 5° in air at M∞ = 3. Find cl and cd by shock-expansion theory. γ = 1.4.

  1. Upper surface (expansion by 5°): ν(3) = 49.76°, so ν = 54.76° and M_u = 3.273. p_u/p∞ = (2.8/(1 + 0.2 × 10.713))^3.5 = 0.668.
  2. Lower surface (shock with θ = 5°): weak solution β = 23.13°, Mₙ₁ = 3 sin 23.13° = 1.178, p_l/p∞ = 1 + 1.1667 × (1.3888 − 1) = 1.454.
  3. q∞/p∞ = 0.7 × 9 = 6.3. cn = (1.454 − 0.668)/6.3 = 0.1248.
  4. cl = 0.1248 × cos 5° = 0.1244; cd = 0.1248 × sin 5° = 0.0109.
  5. Ackeret check: cl = 4 × 0.08727/√8 = 0.1234 — within 1%.

Answer: cl ≈ 0.124, cd ≈ 0.011.

Example 2 (GATE level). A symmetric diamond airfoil with half-angle ε = 5° (t/c = 0.0875) is at α = 0 in air at M∞ = 2. Find the wave-drag coefficient.

  1. Front faces: oblique shock with θ = 5° at M = 2 gives β = 34.30°, Mₙ₁ = 2 sin 34.30° = 1.127, p₂/p∞ = 1.315, M₂ = 1.821.
  2. Mid-chord: the surface turns 2ε = 10° away. ν(1.821) = 21.34°, so ν₃ = 31.34° and M₃ = 2.185.
  3. p₃/p₂ = (1 + 0.2 × 1.821²)^3.5/(1 + 0.2 × 2.185²)^3.5 = 5.936/10.441 = 0.5685, so p₃/p∞ = 1.315 × 0.5685 = 0.748.
  4. q∞/p∞ = 0.7 × 4 = 2.8. cd = (1.315 − 0.748)/2.8 × 0.0875 = 0.0177.
  5. Linear theory: cd = 4(t/c)²/√(M² − 1) = 4 × 0.00765/1.732 = 0.0177 — almost identical at this thinness.

Answer: cd ≈ 0.0177.

Common mistakes

  • Applying a shock where the surface turns away (or a fan where it turns into the flow): look at the turning relative to the local flow direction.
  • Using the free-stream Mach number for the second corner instead of the Mach number on the preceding panel.
  • Forgetting that a flat plate at incidence has wave drag; cd = cl·tan α.
  • Using q∞ = ½ρV² with ρ unknown — use q∞ = 0.7·p∞·M∞² for air.
  • Using the strong-shock root or ignoring detachment when θ exceeds θmax.
  • Expecting trailing-edge waves to change the surface pressures.

For GATE AE

Expect: pressure ratio on the upper and lower surfaces of a flat plate at a given α and M; lift and drag coefficients from given panel pressures; diamond-airfoil wave drag; comparison with linear (Ackeret) values; identifying where shocks and fans occur on a given shape. Tables of θ–β–M and ν(M) are typically given or the shock angle is supplied, so practise the bookkeeping: panel by panel, local Mach number first, then pressure.

Quick check

  1. On a flat plate at incidence in supersonic flow, which surface has an expansion fan at the leading edge?
  2. Why does a zero-thickness flat plate have drag in supersonic flow?
  3. For air, express q∞ in terms of p∞ and M∞.
  4. At mid-chord of a diamond airfoil at α = 0, through what angle does the flow turn?
  5. Do trailing-edge shocks affect the surface pressure on the plate?

Answers: 1. the upper (leeward) surface; 2. the pressure difference acts normal to the plate and has a component along the stream (wave drag); 3. q∞ = 0.7·p∞·M∞²; 4. 2ε, an expansion; 5. no — information cannot travel upstream in supersonic flow.

Interview questions

All Compressible Aerodynamics interview questions

Try answering each one aloud before you open it.

  1. 1.What is shock-expansion theory in the context of supersonic airfoils?Concept

    It is a method for finding surface pressures on sharp-edged supersonic airfoils made of straight panels by applying exact oblique-shock relations wherever the surface turns into the flow and exact Prandtl–Meyer expansions wherever it turns away. The pressure is uniform on each panel, so integrating gives lift, wave drag and moment. Unlike Ackeret theory it is not linearised, so it remains accurate at larger angles; it assumes inviscid, 2-D flow with attached shocks and supersonic flow everywhere.

  2. 2.Explain the role of shock waves in shock-expansion theory.Concept

    In shock-expansion theory, shock waves are responsible for sudden changes in flow properties such as pressure, temperature, and density. When supersonic flow encounters a shock wave, it decelerates and compresses, leading to an increase in pressure and temperature. This is a key aspect of the theory as it helps in determining the aerodynamic forces on the airfoil.

  3. 3.How do expansion waves differ from shock waves in supersonic flow?Concept

    Expansion waves, unlike shock waves, cause the flow to accelerate and expand, resulting in a decrease in pressure and temperature. They occur when the flow turns away from itself, such as around a convex corner. In shock-expansion theory, expansion waves are used to model the flow over the rear part of the airfoil, complementing the effects of shock waves.

  4. 4.Why is shock-expansion theory important for designing supersonic airfoils?Application

    Shock-expansion theory is crucial for designing supersonic airfoils because it provides a framework to predict the aerodynamic forces and pressure distribution on the airfoil. This information is essential for optimizing the airfoil shape to achieve desired performance characteristics, such as minimizing drag and maximizing lift, at supersonic speeds.

  5. 5.What happens if the angle of attack of a supersonic airfoil is increased?Application

    The lower-surface leading-edge shock gets stronger and the upper-surface expansion gets larger, so the pressure difference and lift grow — roughly linearly, cl ≈ 4α/√(M² − 1), at small α. Wave drag grows faster, roughly as α², because the force is normal to the plate (cd ≈ cl·α). At larger angles the lower-surface turning can exceed θmax and the shock detaches, and in a real flow the upper-surface shock at the trailing edge can separate the boundary layer, so the inviscid theory stops applying.

  6. 6.How does the Mach number affect the shock-expansion theory analysis?Application

    The Mach number significantly affects the shock-expansion theory analysis as it determines the strength and angle of shock waves and expansion fans. Higher Mach numbers typically result in stronger shock waves and more pronounced changes in flow properties. The theory assumes supersonic conditions, so the Mach number must be greater than 1 for the analysis to be valid.

  7. 7.On a symmetric diamond airfoil at zero angle of attack in supersonic flow, where do the shocks and expansions form, and why is there drag but no lift?Numerical

    Oblique shocks form at the sharp leading edge on both surfaces because the front faces turn the flow into itself, giving pressure above free stream on the front faces. At the shoulder the surface turns away by twice the half-angle, so a Prandtl–Meyer fan lowers the pressure on the rear faces below free stream; shocks at the trailing edge turn the flow back. Upper and lower surfaces are symmetric, so lift is zero, but the high front-face pressure and low rear-face pressure both push backwards, giving wave drag, cd ≈ 4(t/c)²/√(M² − 1) for a thin section.

  8. 8.Air at Mach 3 expands around a 10° convex corner on an airfoil surface. What is the Mach number after the expansion?Numerical

    Use the Prandtl–Meyer function: ν(3) = 49.76° for γ = 1.4. After the turn ν = 49.76° + 10° = 59.76°, and inverting gives M ≈ 3.58. The pressure falls isentropically by p₂/p₁ = [(1 + 0.2 × 9)/(1 + 0.2 × 3.58²)]^3.5 ≈ 0.43.

  9. 9.What assumptions are made in shock-expansion theory?Concept

    The flow is steady, two-dimensional and inviscid, the gas is calorically perfect, and the flow is supersonic everywhere along the surface. The leading edge must be sharp with an attached shock (turning less than θmax), and the surface is approximated by straight panels with uniform flow on each. For curved surfaces the weak reflections of expansion waves from the shock are neglected. Unlike linear theory it does not require thin sections or small angles.

  10. 10.Explain how shock-expansion theory can be used to predict the lift and drag on a supersonic airfoil.Application

    Shock-expansion theory predicts the lift and drag on a supersonic airfoil by analyzing the pressure distribution over the airfoil surface. The theory calculates the changes in pressure due to shock waves and expansion fans, which are then integrated over the airfoil surface to determine the aerodynamic forces. The lift is the component of the aerodynamic force perpendicular to the flow direction, while the drag is the component parallel to the flow direction.

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