Prandtl-Glauert and other compressibility corrections

Why it matters

Most airfoil data are measured or computed at low speed, but airliners cruise at M ≈ 0.8. Compressibility corrections convert low-speed pressure coefficients, lift-curve slopes and moments to high subsonic Mach numbers in one line, and they are the first step in estimating the critical Mach number of a wing.

Key ideas

Origin. For subsonic flow the linearised equation (1 − M∞²)φxx + φyy = 0 can be turned into Laplace's equation by stretching the y-coordinate: ξ = x, η = β·y, with β = √(1 − M∞²). Solving the incompressible problem for the same airfoil and transforming back gives the Prandtl–Glauert rule: at the same airfoil and angle of attack, every pressure coefficient is the incompressible value divided by β.

Consequences.

• Cp = Cp,0/β, cl = cl,0/β, cm = cm,0/β, and the lift-curve slope a = a₀/β. Compressibility increases the magnitude of pressure coefficients and lift (the factor 1/β is greater than 1).
• The location of the aerodynamic centre and of the centre of pressure does not change (both lift and moment scale by the same factor).
• Inviscid subsonic drag remains zero (d'Alembert), as long as no shocks form.
• At M∞ = 0.7, 1/β = 1.40; at M∞ = 0.8, 1/β = 1.67. As M∞ → 1 the rule predicts infinite Cp, a sign it has failed.

Accuracy and range. Because it comes from linear theory, Prandtl–Glauert is good for thin airfoils at small angles up to about M∞ ≈ 0.7, and it underestimates the correction for large suction peaks. Two improved rules are used:

• Kármán–Tsien: derived from the hodograph method with a tangent-gas approximation; it includes the dependence on the local Cp itself and agrees better with experiment up to the critical Mach number.
• Laitone: includes the local Mach number variation and gives an even larger correction than Kármán–Tsien. For a given Cp,0, the corrections satisfy |Prandtl–Glauert| < |Kármán–Tsien| < |Laitone| in magnitude for suction peaks. None of them is valid once shocks form (transonic flow).

Finite wings. For a wing of aspect ratio A, the lift-curve slope uses the Prandtl–Glauert factor on the section slope inside the lifting-line or Helmbold formula; for high-aspect-ratio wings a ≈ a₀/(β + a₀/(πeA)) (approximate) — take the exact form your textbook uses.

Göthert's rule extends the idea to 3-D bodies by stretching all lateral coordinates and comparing with an incompressible body of reduced thickness.

Formulas

β = √(1 − M∞²)

Cp = Cp,0 / √(1 − M∞²) — Prandtl–Glauert.

cl = cl,0 / √(1 − M∞²); a = a₀ / √(1 − M∞²) (per rad or per degree, same as a₀).

Cp = Cp,0 / [ √(1 − M∞²) + (M∞² / (1 + √(1 − M∞²)))·(Cp,0 / 2) ] — Kármán–Tsien.

Cp = Cp,0 / [ √(1 − M∞²) + (M∞²·(1 + (γ−1)/2·M∞²) / (2√(1 − M∞²)))·Cp,0 ] — Laitone.

Cp(M₂) = Cp(M₁)·β₁/β₂ — converting between two subsonic Mach numbers (Prandtl–Glauert).

Symbols: Cp,0, cl,0, a₀ incompressible pressure coefficient, lift coefficient and lift-curve slope; Cp, cl, a values at free-stream Mach number M∞; γ ratio of specific heats. Valid for thin airfoils at small angles, subsonic flow with no shocks (roughly M∞ < 0.7 for Prandtl–Glauert).

Worked examples

Example 1 (standard). At a point on a thin airfoil the low-speed pressure coefficient is Cp,0 = −0.5. Find Cp at M∞ = 0.6 by the three rules. γ = 1.4.

1. β = √(1 − 0.36) = 0.8.
2. Prandtl–Glauert: Cp = −0.5/0.8 = −0.625.
3. Kármán–Tsien: M∞²/(1 + β) = 0.36/1.8 = 0.2; denominator 0.8 + 0.2 × (−0.25) = 0.75; Cp = −0.5/0.75 = −0.667.
4. Laitone: M∞²(1 + 0.2M∞²)/(2β) = 0.36 × 1.072/1.6 = 0.2412; denominator 0.8 + 0.2412 × (−0.5) = 0.6794; Cp = −0.5/0.6794 = −0.736.

Answer: Cp ≈ −0.625 (P–G), −0.667 (K–T), −0.736 (Laitone).

Example 2 (GATE level). A wind-tunnel test at M∞ = 0.5 gives a minimum Cp of −0.60 on a thin airfoil, and a section lift-curve slope of 0.11 per degree at the same Mach number. Estimate the minimum Cp and the lift coefficient at α = 3° for the same (symmetric) airfoil at M∞ = 0.7, using Prandtl–Glauert.

1. β₁ = √(1 − 0.25) = 0.8660, β₂ = √(1 − 0.49) = 0.7141.
2. Cp,min(0.7) = −0.60 × 0.8660/0.7141 = −0.728.
3. a(0.7) = 0.11 × 0.8660/0.7141 = 0.1334 per degree.
4. The section is symmetric, so zero lift is at α = 0 (P–G does not move it): cl = 0.1334 × 3 = 0.400.

Answer: Cp,min ≈ −0.73, cl ≈ 0.40 at α = 3°.

Common mistakes

• Multiplying by β instead of dividing — compressibility increases |Cp| and cl.
• Using Prandtl–Glauert at M∞ = 0.9 or with shocks present.
• Correcting from a non-zero Mach number as if it were incompressible; use β₁/β₂.
• Applying the factor to drag; inviscid subsonic drag stays zero, and wave drag needs shocks.
• Expecting the aerodynamic centre or zero-lift angle to move under the Prandtl–Glauert rule.

For GATE AE

Expect one-step numericals: corrected Cp or cl at a given Mach number, conversion between two Mach numbers, ranking of the three corrections, and statements on validity and what the rule predicts (lift increases, aerodynamic centre unchanged). These results feed directly into critical Mach number calculations, so practise them together.

Quick check

1. What is the Prandtl–Glauert factor 1/β at M∞ = 0.85?
2. Does compressibility increase or decrease subsonic lift-curve slope?
3. Cp,0 = 0.4; find Cp at M∞ = 0.7 by Prandtl–Glauert.
4. Which gives the largest |Cp| for a suction peak: P–G, K–T or Laitone?
5. Does the aerodynamic centre move under the P–G rule?

Answers: 1. 1.898; 2. increase; 3. 0.560; 4. Laitone; 5. no.