Critical Mach number, drag divergence and supercritical airfoils

Why it matters

Transport aircraft cruise just below the Mach number where drag shoots up. Every hundredth of a Mach number of cruise speed is worth fuel and time, so designers fight to raise the critical and drag-divergence Mach numbers — with thin sections, sweep and supercritical airfoils. Estimating the critical Mach number is a classic GATE AE calculation.

Key ideas

Critical Mach number M_cr. As a body speeds up, the flow accelerates over its upper surface, so the local Mach number there exceeds the free-stream value. The critical Mach number is the free-stream Mach number at which the local Mach number first reaches 1 somewhere on the surface (at the point of minimum pressure). Below M_cr the flow is everywhere subsonic; above it a supersonic pocket forms, usually terminated by a shock.

Finding M_cr (two curves).

1. The pressure coefficient at the point where the local Mach number is exactly 1 depends only on M∞ (from isentropic flow): Cp,cr(M∞). It falls in magnitude as M∞ increases (−1.29 at 0.6, −0.78 at 0.7, −0.43 at 0.8).
2. The actual minimum pressure coefficient grows in magnitude with M∞, e.g. by Prandtl–Glauert Cp,min = Cp,min,0/√(1 − M∞²). M_cr is where the two curves cross. A thicker or more highly loaded section has a larger |Cp,min,0|, hence a lower M_cr.

Drag divergence. Just above M_cr the supersonic pocket and its shock are weak and drag barely changes. As M∞ rises further the shock strengthens, causing wave drag and shock-induced boundary-layer separation, and the drag coefficient rises steeply. The drag-divergence Mach number M_dd is defined by convention, commonly where dCd/dM = 0.1 (Douglas) or where Cd has risen by 0.002 (Boeing). Typically M_cr < M_dd < 1; the region between M_cr and M_dd is usable. Beyond M_dd buffet, loss of lift and pitch changes (Mach tuck) can occur.

Raising M_cr and M_dd.

• Thinner sections — smaller suction peaks.
• Sweep — only the velocity component normal to the leading edge, M∞ cos Λ, governs the pressures (simple sweep theory, an upper bound); real wings gain less.
• Area rule — smooth the total cross-sectional area distribution of wing plus fuselage to cut transonic wave drag.
• Supercritical airfoils (Whitcomb) — a large leading-edge radius, a relatively flat upper surface and strong aft camber (a cusp on the lower surface near the trailing edge). The flat top keeps the supersonic pocket at nearly uniform, modest Mach number so that it ends in a weak shock, or none at all. This raises M_dd for a given thickness — or allows a thicker, lighter wing at the same M_dd. The aft loading produces a larger nose-down pitching moment, which the tail must trim.

Empirical estimate. The Korn relation M_dd + cl/10 + t/c = κ (κ ≈ 0.87 for conventional and about 0.95 for supercritical sections) is a useful design estimate; the constant is empirical, so take it from your reference.

Formulas

Cp,cr = (2/(γ·M∞²))·{ [ (2 + (γ−1)·M∞²)/(γ+1) ]^(γ/(γ−1)) − 1 } — Cp where local M = 1.

Cp,min = Cp,min,0 / √(1 − M∞²) — compressible minimum Cp (Prandtl–Glauert; Kármán–Tsien gives a slightly lower M_cr).

M_cr: solve Cp,cr(M∞) = Cp,min(M∞).

M_n = M∞·cos Λ — Mach number normal to a swept leading edge (simple sweep theory).

M_dd + cl/10 + t/c ≈ κ — Korn relation (empirical).

Symbols: Cp,cr critical pressure coefficient; Cp,min,0 incompressible minimum pressure coefficient of the section; M∞ free-stream Mach number; γ ratio of specific heats; Λ sweep angle; cl section lift coefficient; t/c thickness ratio; κ technology factor (dimensionless).

Worked examples

Example 1 (standard). Find the critical pressure coefficient at M∞ = 0.7 for air.

1. (2 + 0.4 × 0.49)/2.4 = 2.196/2.4 = 0.9150.
2. 0.9150^3.5 = 0.7327.
3. Cp,cr = (2/(1.4 × 0.49)) × (0.7327 − 1) = 2.9155 × (−0.2673) = −0.779.

Answer: Cp,cr ≈ −0.78 — if the minimum Cp on the airfoil at M = 0.7 is more negative than this, the flow is already supercritical.

Example 2 (GATE level). A thin airfoil has a low-speed minimum pressure coefficient of −0.43. Estimate its critical Mach number using Prandtl–Glauert.

1. Try M∞ = 0.73: Cp,min = −0.43/√(1 − 0.5329) = −0.43/0.6834 = −0.629; Cp,cr(0.73) = −0.662. Cp,min is less negative — not yet critical.
2. Try M∞ = 0.74: Cp,min = −0.43/0.6726 = −0.639; Cp,cr(0.74) = −0.626. Now Cp,min is more negative — past critical.
3. Interpolating: M_cr ≈ 0.737.

Answer: M_cr ≈ 0.74 (Kármán–Tsien gives about 0.72).

Common mistakes

• Confusing M_cr (sonic flow first appears locally) with M_dd (drag rises sharply) or with the sonic condition M = 1 itself.
• Treating M_cr as a property of the gas only; it depends on the shape and the angle of attack.
• Using Cp,cr with the wrong sign — it is negative (a suction).
• Thinking supercritical airfoils avoid supersonic flow; they allow it but keep the shock weak.
• Applying full cos Λ gains to real swept wings without caveat.

For GATE AE

Expect: Cp,cr at a given Mach number, M_cr from a low-speed Cp,min using Prandtl–Glauert (graphical or trial), the effect of thickness, camber, angle of attack and sweep on M_cr, and conceptual questions on drag divergence, supercritical airfoil features and the area rule. Practise a two-point trial-and-interpolate solution; it is fast and accurate enough.

Quick check

1. What is Cp,cr at M∞ = 0.8 for air?
2. Is M_dd higher or lower than M_cr?
3. Which surface of a supercritical airfoil is relatively flat?
4. Does increasing thickness raise or lower M_cr?
5. A wing has 30° sweep and flies at M∞ = 0.85. What is the normal Mach number by simple sweep theory?

Answers: 1. −0.435; 2. higher; 3. the upper surface; 4. lowers it; 5. 0.85 × cos 30° = 0.736.