Prandtl-Meyer Expansion Fan: How Supersonic Flow Accelerates Around a Corner

Why supersonic flow speeds up around a corner

In subsonic flow, a convex corner would let information travel upstream and the streamlines would smoothly bend with a local pressure adjustment. In supersonic flow (M > 1) disturbances cannot outrun the stream, so a corner communicates itself only through Mach waves — infinitesimally weak pressure lines inclined at the Mach angle µ = asin(1/M) to the local velocity. When the wall turns away from the flow (a convex, or expansive, corner), the flow must expand to fill the widening space.

The key result is counter-intuitive: for M > 1 the area-velocity relation dA/A = (M² − 1)·dV/V flips sign relative to subsonic flow. An expanding stream tube therefore demands higher velocity, lower pressure, lower density, and lower temperature. Each incremental Mach wave turns the flow by a tiny dθ and raises the Mach number by dM; the corner is resolved by a centred fan of these waves — the Prandtl-Meyer expansion fan. Because every wave is infinitesimally weak, the process crosses no shock and generates no entropy: it is isentropic and fully reversible. Run the same air back through a matching compression corner and you recover exactly the upstream state.

The Prandtl-Meyer function ν(M)

Integrating the incremental turning dθ = √(M²−1)·dV/V across the fan yields the Prandtl-Meyer function, the angle through which a flow starting at M = 1 must expand to reach Mach number M:

ν(M) = √((γ+1)/(γ−1)) · atan√((γ−1)(M²−1)/(γ+1)) − atan√(M²−1)

with γ the ratio of specific heats (1.4 for air, ~1.2 for rocket exhaust). ν is measured in degrees or radians and is a monotonically increasing function of M. Its value at M = 1 is exactly 0, and as M → ∞ it asymptotes to a finite maximum:

• ν_max = (π/2)·(√((γ+1)/(γ−1)) − 1) = 130.45° for γ = 1.4.
• This ceiling means a flow can only be turned about 130.45° before, in the limit, reaching infinite Mach number and zero pressure and temperature — a vacuum. Turn beyond that and the flow simply cannot follow the wall; it separates.

The design rule for a finite corner of angle Δθ is simply ν(M₂) = ν(M₁) + Δθ. You look up ν(M₁), add the geometric turn angle, and invert ν to get M₂. Because ν has no closed-form inverse, M₂ is found from tables, a chart, or a two-line Newton iteration.

Sizing an expansion: a Mach 2 corner worked end to end

Take air (γ = 1.4) at M₁ = 2.00, static pressure p₁ = 100 kPa, static temperature T₁ = 250 K, expanding around a Δθ = 10° convex corner. The procedure:

• Step 1 — upstream ν: ν(2.00) = 26.38°.
• Step 2 — add the turn: ν(M₂) = 26.38° + 10° = 36.38°.
• Step 3 — invert ν: M₂ ≈ 2.385.
• Step 4 — apply isentropic relations (stagnation quantities are conserved because Δs = 0). With p₀/p = (1 + 0.2M²)^3.5 and T₀/T = (1 + 0.2M²): p₀/p₁ = 7.824, p₀/p₂ = 14.28, so p₂/p₁ = 7.824/14.28 = 0.548 → p₂ ≈ 54.8 kPa.
• Step 5 — temperature: T₀/T₁ = 1.80, T₀/T₂ = 2.137, so T₂/T₁ = 0.842 → T₂ ≈ 210 K, a 40 K drop.

The fan itself spans from the leading Mach wave at µ₁ = asin(1/2.00) = 30.0° to the trailing wave at µ₂ = asin(1/2.385) = 24.8°, both measured from the local flow direction. The whole expansion is contained in a wedge of open angle roughly (µ₁ − µ₂ + Δθ) ≈ 15°. Density falls with ρ₂/ρ₁ = (p₂/p₁)/(T₂/T₁) = 0.651 by the ideal-gas law.

Method of characteristics: turning theory into nozzle geometry

The single most important engineering use of ν(M) is the method of characteristics (MOC), the workhorse for designing shock-free supersonic nozzle and inlet contours. In a 2-D isentropic supersonic field the flow properties are constant along characteristic lines — Mach waves — and the Riemann invariants are conserved:

• Along a left-running characteristic (C⁻): θ + ν(M) = constant = K⁻.
• Along a right-running characteristic (C⁺): θ − ν(M) = constant = K⁺.

At any grid intersection, θ = ½(K⁻ + K⁺) and ν = ½(K⁻ − K⁺), which instantly gives the local Mach number by inverting ν. A minimum-length supersonic nozzle is built by first over-expanding the flow at the throat through a sharp-cornered expansion section whose total turn equals ½·ν(M_exit), then straightening it in a cancellation section where the wall is shaped so that each reflected characteristic is absorbed rather than reflected — producing a shock-free, uniform, parallel exit flow. This is exactly how a Mach 3 wind-tunnel nozzle or a rocket bell contour is laid out before CFD refinement.

Where expansion fans show up in real hardware

Prandtl-Meyer expansions are everywhere in high-speed flow:

• Supersonic airfoils and diamond/biconvex wings: the flow accelerating over the shoulder of a double-wedge section expands, dropping surface pressure and contributing to lift and wave drag. Linearized supersonic (Ackeret) theory gives surface pressure coefficient C_p = 2θ/√(M²−1); exact answers come from stringing together shocks on windward faces and expansion fans on leeward faces.
• Over-expanded and under-expanded nozzles: a rocket nozzle whose exit pressure exceeds ambient (under-expanded) dumps the pressure difference in a Prandtl-Meyer fan at the lip, producing the barrel-shocks and expanding plume you see on a Falcon 9 upper-stage or a launch at altitude.
• Shock-cell / diamond patterns: in an imperfectly expanded jet, expansion fans reflect off the constant-pressure free boundary as compression waves, coalesce into shocks, reflect again as fans, and repeat — the periodic shock-diamond pattern and its screech tone in afterburner plumes.
• Supersonic inlets and scramjets: corner expansions around cowl lips and isolator geometry are designed with MOC to avoid unstart.
• Wind-tunnel nozzles: every blow-down supersonic tunnel throat-to-test-section contour is an MOC expansion-cancellation design.

Trade-offs, limits, and failure modes

The expansion is beautifully clean, but the physics has hard boundaries:

• The ν_max ceiling: you cannot turn more than ν_max − ν(M₁). Ask a nozzle wall to turn the flow past this and the stream detaches from the surface, leaving a separated region and re-circulation — a real limit on how far an over-expanded plume or a leeward wing surface can follow geometry.
• The isentropic assumption breaks with viscosity: real expansions still form a boundary layer; a strong favourable pressure gradient thins it, but downstream re-compression (shock–boundary-layer interaction) can separate the flow. The inviscid fan is only the core solution.
• Condensation shocks: because expansion drops T sharply, moist or vapour-laden flow can cross saturation and condense, releasing latent heat and forming a condensation shock — the visible white cone over a transonic wing, and a genuine error source in humid wind tunnels.
• Non-centred vs centred fans: a sharp corner gives a centred fan (all waves from one point, infinite local gradient); a gradual convex curve spreads the same total turn over a finite arc, gentler on the boundary layer but geometrically longer.
• Best practice: chain expansions and shocks correctly in sequence — an oblique shock is not the reverse of an expansion fan because the shock adds entropy. Turning a flow out and back by equal angles through a shock then a fan (or vice-versa) does not return the original state; total pressure is permanently lost at the shock.

Expansion fan versus oblique shock for the same 10° flow deflection at Mach 2.0 (air, γ = 1.4)

Property	Expansion fan (convex turn)	Oblique shock (concave turn)
Corner geometry	Flow turns away from itself	Flow turns into itself
Mach number	Increases 2.00 → 2.39	Decreases 2.00 → 1.64
Static pressure ratio p₂/p₁	≈ 0.55 (drop)	≈ 1.71 (rise)
Static temperature	Falls ~40 K	Rises ~43 K
Entropy change Δs	0 (isentropic)	> 0 (loss, non-isentropic)
Wave structure	Continuous fan of Mach waves	Single thin discontinuity