Visualize a supersonic expansion fan around a convex corner with the Prandtl-Meyer function. Adjust the upstream Mach, downstream Mach, specific-heat ratio and inlet pressure to see how the turning angle and the downstream pressure and temperature ratios respond.
Parameters
Upstream Mach M_1
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Downstream Mach M_2
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Specific-heat ratio γ
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Upstream static pressure P_1
kPa
Perfect gas, isentropic and 1-D assumed. A physical expansion requires M_1 ≥ 1 and M_2 ≥ M_1.
Results
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Turning angle θ = ν_2 − ν_1
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ν(M_1)
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Downstream pressure P_2
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Temperature ratio T_2/T_1
Expansion Fan Schematic
Mach lines fan out from the convex corner. The flow accelerates from upstream (blue, lower Mach) to downstream (magenta, higher Mach) and turns by θ.
Prandtl-Meyer function ν(M)
Horizontal axis = Mach number M, vertical axis = ν(M) [°]. Yellow markers show the current M_1 and M_2; the arrow indicates θ = ν_2 − ν_1.
Theory & Key Formulas
A Prandtl-Meyer expansion is an isentropic process in which a supersonic flow turns around a convex corner through a fan of infinitely thin Mach waves. The Prandtl-Meyer function, defined for M ≥ 1, is:
For M_2 \gt M_1 the turning is positive (expansion, acceleration) and the stagnation quantities P_0, T_0 are preserved. With γ = 1.4 the function ν asymptotes to about 130.45° as M \to ∞.
What is the Prandtl-Meyer Expansion Simulator
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If a supersonic jet hits a corner where the surface bends outward, away from the flow, does a shock form there too?
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No — that convex corner does not give you a shock. It launches a Prandtl-Meyer expansion fan: an infinite set of Mach lines spread out like a fan, smoothly accelerating the flow and turning it. With M_1 = 2.0 and M_2 = 2.6 in the simulator the turning angle θ comes out to about 15.03°.
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Wait, the flow accelerates? Then what happens to the pressure?
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Yes, the Mach number rises and the static pressure and temperature drop. In our preset, P_2/P_1 ≈ 0.39 — the upstream 100 kPa falls to about 39.2 kPa downstream. The key point is that this is isentropic. A shock is irreversible and always destroys some stagnation pressure, but an expansion fan is reversible and keeps both P_0 and T_0 exactly. That is why expansion fans are so attractive on the lower surface of a supersonic airfoil.
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There is this mysterious function ν(M). What does it actually represent?
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Think of ν(M) as the cumulative wall-turning angle needed to expand a Mach 1 flow smoothly up to the current Mach M. So M_1 = 2.0 corresponds to a flow that has already turned by 26.38° from M = 1, and M_2 = 2.6 corresponds to 41.41°. The actual fan turning is the difference: θ = 41.41 − 26.38 ≈ 15.03°. The yellow vertical arrow on the ν(M) plot shows that gap visually.
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Does ν(M) keep growing forever as M increases?
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No — there is a hard ceiling. For γ = 1.4, ν approaches ν_max = (π/2)·(√6 − 1) ≈ 130.45° as M → ∞. The physical reading is that even an expansion into vacuum can only turn the flow by about 130°. Sharper convex corners cause the boundary layer to detach from the wall — exactly what limits how much a rocket plume can spread in vacuum.
Frequently Asked Questions
Because θ = ν(M_2) − ν(M_1), choosing M_2 \lt M_1 produces θ \lt 0 — formally a "compression" turning that decelerates the flow. In practice, smooth isentropic compression of a supersonic stream is unstable: any finite turning collapses into a discontinuous oblique shock instead of a continuous compression fan. Prandtl-Meyer expansion is therefore physically meaningful only for M_2 \gt M_1. The simulator still shows a numerical value for θ \lt 0, but it is a mathematical extrapolation and does not correspond to a real expansion fan.
The Mach angle μ = arcsin(1/M) is the angle a single Mach wave makes with the local flow direction at Mach M. Inside an expansion fan, the leading Mach line lies at angle μ_1 to the upstream flow and the trailing one at angle μ_2 to the downstream flow, so the wedge angle of the fan involves both μ values. The turning angle θ = ν(M_2) − ν(M_1), in contrast, measures how much the flow itself has changed direction. So μ describes the slope of a wave, while θ describes the rotation of the streamline. Going from M_1 = 2 to M_2 = 2.6 changes μ from about 30° to 22.6° — the waves lay down toward the flow as the gas accelerates.
For a thin diamond or symmetric double-wedge airfoil at zero angle of attack, oblique shocks form at the leading-edge concave corners on both upper and lower surfaces, and Prandtl-Meyer expansion fans form at the convex corners between the maximum-thickness station and the trailing edge. As the angle of attack is increased, the upper-surface leading-edge shock weakens and is eventually replaced by an expansion fan. The trailing edge restores the flow to ambient conditions through compression waves (shock waves). Lift and wave-drag analysis is built up directly from this combination of shocks and expansions.
For γ = 1.4 the limit is ν_max ≈ 130.45°. If you tried to turn an M_1 = 1 flow through a 130.45° convex corner, the math would give M_2 → ∞, P_2 → 0 and T_2 → 0 — clearly unphysical. Long before that limit, viscous effects and real-gas behavior (condensation, dissociation, vibrational relaxation) take over and the flow detaches from the wall, forming large-scale recirculation zones or a free-expansion jet. The way an under-expanded rocket plume continues to spread by more than 90° in vacuum is closely related to this same upper bound.
Real-World Applications
Supersonic and hypersonic airfoils: Diamond, double-wedge and other thin supersonic airfoils host Prandtl-Meyer expansion fans on their upper surface, trailing edge and at the hinge of any control surface. Because the expansion is isentropic, it generates lift without a stagnation-pressure penalty, but accurate pressure-coefficient predictions beyond linear (Ackeret) theory require the full ν(M) relation. The same combined shock-and-expansion analysis was used in designing the ogival wings of Concorde and SR-71, as well as the airfoils of hypersonic research vehicles such as the X-43.
Nozzle exhaust plumes (over- and under-expansion): When the exit pressure of a rocket or supersonic jet nozzle does not match ambient pressure, alternating expansion fans and oblique shocks form at the exit lip, producing the characteristic shock-diamond pattern (Mach disks). In an under-expanded jet, a Prandtl-Meyer expansion launches outward from the lip, accelerating and turning the flow before it is recompressed by reflected shocks. Predicting the impact of off-design pressure ratio requires accumulating the ν(M) turning angles along the jet boundary.
Supersonic intakes: External-compression intakes use wedges or cones to launch oblique shocks that decelerate the flow, but expansion fans are sometimes used just before the subsonic diffuser inlet to re-accelerate the flow and avoid boundary-layer separation at the foot of a shock. In SR-71 spike intakes and the variable-geometry intakes of ramjets and scramjets, maximizing total-pressure recovery at the intersection of expansion fans and shocks is a central design problem and the Prandtl-Meyer relations are a standard tool.
Supersonic wind tunnels and free-jet test facilities: When supersonic nozzles deliver Mach 2-5 flow to a test section, expansion fans and compression waves form along the free-jet boundary downstream of the nozzle. To preserve a uniform test rhombus, nozzle contours are designed using the method of characteristics, in which each Mach line behaves exactly according to the Prandtl-Meyer relations. The size of the usable test diamond is itself set by the geometry of these expansion fans and reflected waves.
Common Misconceptions and Cautions
The most common error is to assume that an expansion fan, like a shock, drops the stagnation pressure. Prandtl-Meyer expansion is built up from a continuous stack of infinitely thin Mach waves and is exactly isentropic, so the stagnation pressure P_0 and stagnation temperature T_0 are preserved across the fan. Only the static pressure P and static temperature T fall. The simulator may show P_2/P_1 ≈ 0.39, but that is a static-pressure ratio; the total-pressure loss is zero. This property is what makes expansion fans on the lower surface of a supersonic airfoil so attractive — they give a "free" pressure drop.
The next pitfall is to confuse the turning angle θ with the Mach angle μ. θ = ν(M_2) − ν(M_1) is how much the flow itself has rotated, while μ = arcsin(1/M) is the inclination of an individual Mach wave to the local flow direction — two different angles. The leading Mach line of an expansion fan lies at angle μ_1 to the upstream flow, the trailing line at μ_2 to the downstream flow, and the wedge between them reflects both quantities. From M = 2 (μ = 30°) to M = 2.6 (μ ≈ 22.6°), the waves lay down further toward the streamlines as the flow accelerates.
A third caution is the existence of an upper bound on ν(M). For γ = 1.4, ν_max = (π/2)·(√6 − 1) ≈ 130.45°, approached as M → ∞. This means that the total turning achievable by Prandtl-Meyer expansion of a finite supersonic stream is bounded; sharper convex corners than this cause the flow to separate and become a free-expansion jet. Pushing M_2 toward 8 in the simulator gives ν ≈ 95°, and lowering γ raises the ceiling further (closer to the high-temperature limit), as you can confirm directly from the curve.