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Prandtl-Meyer Expansion Calculator

Expansion waves occur when supersonic flow encounters an outward-turning surface, accelerating as the flow expands into available space. This calculator applies Prandtl-Meyer expansion theory to determine the downstream Mach number, pressure, temperature, density, and characteristic angles from an upstream supersonic condition and a known deflection angle. Essential for compressible flow analysis in nozzle design, aerodynamic flow fields, and high-speed vehicle performance.

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Creators Steven Wooding, BSc Physics
7,733 people find this calculator helpful

Understanding Expansion Waves

When a supersonic flow encounters a surface that turns outward (convex), the flow expands into the increased space. Unlike shock waves, which compress flow abruptly across a discontinuity, expansion waves consist of an infinite family of weak Mach waves that gradually reorient the flow direction. The flow accelerates through the expansion fan, increasing in Mach number while pressure, temperature, and density decrease.

Expansion waves are isentropic—there is no entropy generation because the process involves no shock-induced discontinuity. This property makes them reversible and thermodynamically efficient. The continuous nature of the expansion preserves total pressure and stagnation properties, contrasting sharply with oblique shock behaviour.

Real-world examples include:

  • Rocket nozzles during flight, where internal expansion creates supersonic jets
  • Supersonic intakes on aircraft, where external expansion fans help match flow to the engine
  • Aerodynamic surfaces on hypersonic vehicles, where smooth turns trigger controlled expansion

Prandtl-Meyer Function and Downstream Conditions

The Prandtl-Meyer function, developed by Ludwig Prandtl and Theodor Meyer in the early 1900s, quantifies the turning angle available for a given Mach number. It allows us to compute downstream properties from the deflection angle and upstream state.

A = √[(γ + 1)/(γ − 1)]

B = √(M₁² − 1)

ν(M₁) = A × arctan(B/A) − arctan(B)

ν(M₂) = ν(M₁) + θ

μ₁ = arcsin(1/M₁)

μ₂ = arcsin(1/M₂)

T₂ = T₁ × [1 + (γ − 1)/2 × M₁²] / [1 + (γ − 1)/2 × M₂²]

P₂ = P₁ × {[1 + (γ − 1)/2 × M₁²] / [1 + (γ − 1)/2 × M₂²]}^[γ/(γ − 1)]

ρ₂ = ρ₁ × {[1 + (γ − 1)/2 × M₁²] / [1 + (γ − 1)/2 × M₂²]}^[1/(γ − 1)]

  • γ — Specific heat ratio of the gas (1.4 for air)
  • M₁ — Upstream Mach number (must exceed 1 for supersonic flow)
  • M₂ — Downstream Mach number after expansion
  • θ — Deflection angle in degrees (outward turn)
  • ν(M) — Prandtl-Meyer function value for a given Mach number
  • μ — Mach angle (angle of the Mach cone)
  • T₁, P₁, ρ₁ — Upstream temperature, pressure, and density
  • T₂, P₂, ρ₂ — Downstream temperature, pressure, and density

Using the Calculator

Select "Yes" if you need downstream flow properties (pressure, temperature, density) in addition to Mach numbers and Mach angles. For Mach number and angles alone, select "No".

Inputs required:

  • Gamma (γ): Specific heat ratio; use 1.4 for air, adjust for other gases
  • Upstream Mach number (M₁): Must be supersonic (M₁ > 1)
  • Deflection angle (θ): Outward turn angle in degrees
  • If properties selected: upstream pressure, temperature, and density

Outputs provided:

  • Downstream Mach number (M₂)
  • Prandtl-Meyer functions for both states
  • Mach angles (μ₁ and μ₂)
  • If properties selected: P₂, T₂, ρ₂

Practical Example

Consider air (γ = 1.4) flowing supersonic from a nozzle with:

  • Upstream Mach M₁ = 1.5
  • Deflection angle θ = 15°
  • P₁ = 1 atm, T₁ = 288 K, ρ₁ = 1.226 kg/m³

The calculator first computes the Prandtl-Meyer functions. For M₁ = 1.5, ν(M₁) ≈ 11.39°. Adding the deflection angle: ν(M₂) = 11.39° + 15° = 26.39°. Solving the Prandtl-Meyer equation for this function value yields M₂ ≈ 2.05.

Using isentropic relations, downstream pressure drops to approximately 0.383 atm, temperature to 242 K, and density to 0.553 kg/m³. The Mach angle decreases from 41.8° upstream to 29.7° downstream, reflecting the higher Mach number.

Common Pitfalls and Caveats

Avoid these typical mistakes when applying Prandtl-Meyer expansion theory.

  1. Upstream Mach must exceed 1 — Expansion waves only exist in supersonic flow. Subsonic flow cannot expand smoothly through a compression turn—it will shock instead. Always verify M₁ > 1 before using the calculator.
  2. Deflection angle direction matters — Outward turns (convex surfaces) cause expansion; inward turns cause compression. Expansion waves apply only to outward deflections. If your geometry turns inward, use oblique shock relations instead.
  3. Total pressure is conserved, static pressure is not — The calculator preserves stagnation pressure through the expansion because the process is isentropic. However, static pressure drops significantly as the flow accelerates. Do not confuse stagnation and static properties.
  4. Gamma varies with temperature and gas composition — Air at high altitude or with combustion products has a different γ than the standard 1.4. High-temperature flows may require γ ≈ 1.2 to 1.3. Consult gas tables for your specific conditions to avoid large errors in downstream properties.

Frequently Asked Questions

What is the difference between an expansion fan and an oblique shock?

Expansion waves are isentropic, continuous processes where flow accelerates, Mach number increases, and static properties (pressure, temperature, density) decrease. Oblique shocks are discontinuous, irreversible processes where flow decelerates, Mach number drops, and static pressure rises sharply. Expansion occurs at outward-turning surfaces; shocks occur at inward-turning or blunt surfaces in supersonic flow. Total pressure is conserved through expansion but decreases across a shock.

How do I find the downstream Mach number given deflection angle and upstream conditions?

Use the Prandtl-Meyer function approach: (1) Calculate ν(M₁) using the upstream Mach number and the formula shown in the calculator. (2) Add the deflection angle: ν(M₂) = ν(M₁) + θ. (3) Solve the Prandtl-Meyer equation for M₂ corresponding to ν(M₂). This typically requires iteration or a Mach-function table. The calculator performs this inversion automatically.

Why does temperature drop through an expansion wave?

As the flow expands and accelerates, kinetic energy increases. By energy conservation (stagnation enthalpy remains constant), static temperature must decrease. The isentropic relation T₂/T₁ depends on the ratio of total temperatures factored by the Mach number squared terms. Higher downstream Mach always yields lower static temperature, even though stagnation temperature is unchanged.

What is a Mach angle and why does it change across the expansion?

The Mach angle μ = arcsin(1/M) represents the half-angle of the cone formed by Mach waves at a given Mach number. It decreases as Mach increases because higher Mach numbers produce narrower wave cones. Upstream at M₁ = 1.5, μ₁ ≈ 41.8°. Downstream at higher M₂, the angle becomes sharper (e.g., μ₂ ≈ 29.7° at M₂ = 2.05). This change is crucial for sketching wave patterns and shock-expansion theory.

Can an expansion fan turn the flow through any angle?

No. For a given upstream Mach and γ, the maximum turning angle is limited by the Prandtl-Meyer function of the upstream state. As Mach approaches infinity, the Prandtl-Meyer function approaches a maximum value: ν_max = (√[(γ+1)/(γ−1)] − 1) × 90°. For air (γ = 1.4), ν_max ≈ 130°. Deflection angles larger than available expansion lead to supersonic separation or shock formation.

Does the calculator account for real gas effects or heat transfer?

No. The calculator assumes an ideal gas and isentropic (adiabatic, reversible) expansion. Real gases deviate from ideal behaviour at high pressures or low temperatures. Heat transfer and viscous losses violate isentropy and reduce the final Mach number. For high-precision engineering, use more sophisticated gas tables and account for wall friction if the expansion region is confined.

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