Prandtl-Meyer Expansion Calculator — Supersonic Flow Around a Corner
Compute the downstream Mach number (M₂), pressure ratio, temperature ratio, and density ratio for a Prandtl-Meyer expansion fan. Enter the upstream supersonic Mach number, convex-wall turning angle, and ratio of specific heats γ.
°
Mach number after the expansion fan — always greater than M₁
How does this calculator work?
Given upstream M₁ > 1 and convex turning angle θ, compute ν₂ = ν₁ + θ then invert the Prandtl-Meyer function to get M₂. Property ratios follow from isentropic relations: p₂/p₁ decreases, T₂/T₁ decreases, M₂ > M₁. Uses 100-step bisection for inversion. Valid for calorically perfect gases.
Formula
How this is calculated
When supersonic flow turns around a convex corner (expanding flow), it does so isentropically through an expansion fan — a continuous array of Mach waves emanating from the corner. Unlike the abrupt discontinuity of an oblique shock (compressive corner), the Prandtl-Meyer expansion fan is smooth, conserves total pressure, and can be solved analytically.
The Prandtl-Meyer function ν(M) gives the angle by which the flow has turned from Mach 1 to Mach M under isentropic expansion. To find M₂ after a turning angle θ, compute ν₁ = ν(M₁), add the turning angle (ν₂ = ν₁ + θ), then invert the equation to find M₂ such that ν(M₂) = ν₂. Because ν(M) has no closed-form inverse, this calculator uses a 100-step bisection search which converges to machine precision in all cases.
Once M₂ is known, the downstream pressure, temperature, and density follow from the isentropic relations using the conservation of total (stagnation) conditions across the fan. The ratio p₂/p₁ = (p₀/p₁)/(p₀/p₂) where p₀ is total pressure. A maximum turning angle exists for each γ — for air (γ = 1.4) this is about 130.45°, corresponding to M₂ → ∞.
Frequently asked questions
The maximum Prandtl-Meyer angle is π/2 × (√((γ+1)/(γ-1)) − 1). For γ = 1.4 this evaluates to approximately 130.45°. Beyond this angle the flow cannot remain attached — the physical expansion would require M → ∞ and zero static pressure.
An oblique shock forms when supersonic flow turns toward the flow direction (compressive corner) and is always accompanied by entropy rise and total pressure loss. A Prandtl-Meyer expansion occurs when flow turns away from the flow direction (convex corner), is isentropic (no entropy rise, no total pressure loss), and always results in higher Mach number and lower static pressure and temperature.
Yes — change γ for your gas. Common values: monatomic ideal gases (He, Ar) use γ = 5/3 ≈ 1.667; diatomic gases (air, N₂, O₂) at room temperature use γ = 7/5 = 1.4; combustion gases are often around γ = 1.2–1.3. The isentropic assumption holds for calorically perfect gases at moderate temperatures.
Also known as
TG we-Calculate Editorial Team. (2026). Prandtl-Meyer Expansion Calculator — Supersonic Flow Around a Corner [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/prandtl-meyer-expansion-calculator
TG we-Calculate Editorial Team. "Prandtl-Meyer Expansion Calculator — Supersonic Flow Around a Corner." TG we-Calculate. 2026. https://we-calculate.com/calculator/prandtl-meyer-expansion-calculator.
TG we-Calculate Editorial Team, "Prandtl-Meyer Expansion Calculator — Supersonic Flow Around a Corner," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/prandtl-meyer-expansion-calculator
@misc{wecalculate_prandtl_meyer_expansion_calculator, title = {Prandtl-Meyer Expansion Calculator — Supersonic Flow Around a Corner}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/prandtl-meyer-expansion-calculator}}, year = {2026}, note = {TG we-Calculate} }
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