Prandtl-Meyer expansion waves

Centred Prandtl–Meyer expansion fans at convex corners: the isentropic ν(M) function, fan boundaries, νmax and isentropic compression.

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Why it matters

When a supersonic stream turns away from itself — over the shoulder of a diamond airfoil, at the lip of an under-expanded nozzle, behind a wedge — it expands through a fan of Mach waves and accelerates. The Prandtl–Meyer function makes this calculation exact and simple, and together with oblique shocks it gives the shock-expansion method for supersonic airfoils and the design of minimum-length nozzles.

Key ideas

What happens at a convex corner. A uniform supersonic stream meets a wall that turns away by angle θ. The flow cannot turn abruptly through a single wave (that would be an expansion shock, forbidden by the second law). Instead it turns gradually through a centred fan of infinitely many Mach waves radiating from the corner.

  • The first (forward) Mach line is at μ₁ = sin⁻¹(1/M₁) to the upstream flow.
  • The last (rearward) Mach line is at μ₂ = sin⁻¹(1/M₂) to the turned downstream flow.
  • Through the fan, Mach number increases; pressure, temperature and density fall; streamlines curve smoothly.
  • Each wave is infinitesimally weak, so the whole process is isentropic: p₀ and T₀ are constant through the fan.

Prandtl–Meyer function. Applying the geometry of a weak wave gives dθ = √(M² − 1)·dV/V. Integrating from M = 1 defines ν(M), the angle through which a sonic flow must be turned to reach M. For a turn of θ, simply ν(M₂) = ν(M₁) + θ.

  • ν(1) = 0; for air ν(2) = 26.38°, ν(2.5) = 39.12°, ν(3) = 49.76°.
  • As M → ∞, ν → νmax = (π/2)·(√((γ+1)/(γ−1)) − 1), 130.45° for air. A flow cannot be turned more than νmax − ν(M₁); beyond that the flow separates from the wall leaving a vacuum region (theoretical limit).
  • The same function describes isentropic compression by a gradual concave turn: ν(M₂) = ν(M₁) − θ, until Mach waves converge into a shock.

Simple-wave property. Because the fan is isentropic, once M₂ is known all static ratios follow from the isentropic relations using the constant p₀ and T₀.

Link with weak shocks. For small turns, a weak oblique compression and an isentropic compression give nearly the same pressure change (they differ at third order in θ). This is why linearised supersonic theory treats both compression and expansion with one formula.

Formulas

ν(M) = √((γ+1)/(γ−1)) · tan⁻¹√((γ−1)(M² − 1)/(γ+1)) − tan⁻¹√(M² − 1) — Prandtl–Meyer function (radians; convert to degrees for tables).

ν(M₂) = ν(M₁) + θ — expansion (convex turn). For an isentropic compression, ν(M₂) = ν(M₁) − θ.

νmax = (π/2)·(√((γ+1)/(γ−1)) − 1) — 130.45° for γ = 1.4.

μ = sin⁻¹(1/M) — Mach angle, fixes the fan boundaries.

p₂/p₁ = [(1 + (γ−1)/2·M₁²) / (1 + (γ−1)/2·M₂²)]^(γ/(γ−1)); T₂/T₁ = (1 + (γ−1)/2·M₁²)/(1 + (γ−1)/2·M₂²) — isentropic, same p₀, T₀.

Symbols: ν Prandtl–Meyer angle; θ turning angle of the wall (same units as ν); M₁, M₂ Mach numbers before and after the fan; μ Mach angle; γ ratio of specific heats; p pressure (Pa); T temperature (K). Valid for steady, 2-D, inviscid supersonic flow of a perfect gas (M ≥ 1 throughout).

Worked examples

Example 1 (standard). Air at M₁ = 2.0 turns 10° around a convex corner. Find M₂, p₂/p₁, T₂/T₁ and the fan boundaries. γ = 1.4.

  1. ν(2.0) = 2.4495 × tan⁻¹(0.7071) − tan⁻¹(1.7321) = 2.4495 × 35.26° − 60° = 26.38°.
  2. ν(M₂) = 26.38° + 10° = 36.38°; inverting (table or iteration) gives M₂ = 2.385.
  3. T₂/T₁ = (1 + 0.2 × 4)/(1 + 0.2 × 5.688) = 1.8/2.1376 = 0.842.
  4. p₂/p₁ = 0.842^3.5 = 0.548.
  5. Forward Mach line μ₁ = 30.0° to the upstream flow; rearward Mach line μ₂ = sin⁻¹(1/2.385) = 24.8° to the turned flow, i.e. 24.8° − 10° = 14.8° to the original direction.

Answer: M₂ ≈ 2.38, p₂/p₁ ≈ 0.548, T₂/T₁ ≈ 0.842; the fan spans 30.0° to 14.8° measured from the upstream direction.

Example 2 (GATE level). Air at M₁ = 1.5, p₁ = 100 kPa is to be expanded to p₂ = 40 kPa by turning around a corner. Find the turning angle.

  1. p₀/p₁ = (1 + 0.2 × 2.25)^3.5 = 1.45^3.5 = 3.671, so p₀ = 367.1 kPa (constant through the fan).
  2. p₀/p₂ = 367.1/40 = 9.178, so 1 + 0.2·M₂² = 9.178^(1/3.5) = 1.8839, M₂² = 4.419, M₂ = 2.102.
  3. ν(1.5) = 11.91°, ν(2.102) = 29.16°.
  4. θ = 29.16° − 11.91° = 17.25°.

Answer: θ ≈ 17.3° (with M₂ ≈ 2.10).

Common mistakes

  • Subtracting θ from ν for an expansion; for a convex turn ν increases.
  • Treating the expansion fan as a shock with entropy rise — it is isentropic, p₀ is unchanged.
  • Mixing radians and degrees: ν from a calculator in radian mode must be converted before adding a turning angle in degrees.
  • Measuring the rear Mach line from the upstream direction instead of from the turned flow.
  • Using the Prandtl–Meyer relation for subsonic flow (M < 1 has no Mach waves).

For GATE AE

Expect: find M₂ (and pressure ratio) after a given turn, find the turning angle needed for a given pressure or Mach number, Mach angles of the fan boundaries, νmax, and true/false statements on isentropy and wave direction. Practise evaluating ν(M) on a calculator in degrees and inverting it by two or three trials. Combined problems pair an expansion fan with an oblique shock (diamond or flat-plate airfoils, nozzle exit flows).

Quick check

  1. What is ν(2) for air?
  2. Does stagnation pressure change through an expansion fan?
  3. Air at M = 2.5 turns 15° (convex). Roughly what is M₂?
  4. What is the maximum turning angle from M = 1 for air?
  5. What is the angle of the forward Mach line for M₁ = 2?

Answers: 1. 26.38°; 2. no, the fan is isentropic; 3. about 3.24; 4. 130.45°; 5. 30°.

Interview questions

All Compressible Aerodynamics interview questions

Try answering each one aloud before you open it.

  1. 1.What is a Prandtl-Meyer expansion wave?Concept

    It is the centred fan of Mach waves through which a supersonic flow turns around a convex corner. It is not a shock: each wave is infinitesimally weak, so the flow accelerates smoothly and isentropically, with Mach number rising and pressure, temperature and density falling. The fan is bounded by the Mach line at μ₁ = sin⁻¹(1/M₁) to the incoming flow and at μ₂ = sin⁻¹(1/M₂) to the turned flow, and the downstream Mach number follows from ν(M₂) = ν(M₁) + θ.

  2. 2.Explain the difference between a Prandtl-Meyer expansion wave and a shock wave.Concept

    A Prandtl-Meyer expansion wave is an isentropic process where the flow expands and accelerates, resulting in a decrease in pressure and temperature. In contrast, a shock wave is a non-isentropic process where the flow decelerates, leading to an increase in pressure and temperature. Expansion waves are smooth and continuous, while shock waves are abrupt and discontinuous.

  3. 3.How does the Mach number change across a Prandtl-Meyer expansion wave?Concept

    Across a Prandtl-Meyer expansion wave, the Mach number increases. This is because the flow expands and accelerates as it turns around a convex corner, resulting in an increase in the velocity of the flow relative to the speed of sound.

  4. 4.Why are Prandtl-Meyer expansion waves important in aerospace engineering?Application

    Prandtl-Meyer expansion waves are important in aerospace engineering because they help in understanding and predicting the behavior of supersonic flows around objects. This knowledge is crucial for designing efficient supersonic aircraft and missiles, as it allows engineers to optimize the shape of the vehicle to minimize drag and maximize performance.

  5. 5.What happens if a supersonic flow encounters a concave corner instead of a convex corner?Application

    If a supersonic flow encounters a concave corner, a shock wave is generated instead of an expansion wave. This is because the flow is compressed and decelerated as it turns around the concave corner, leading to an increase in pressure and temperature.

  6. 6.Explain how the Prandtl-Meyer function is used to calculate the turning angle in an expansion wave.Application

    ν(M) is the angle through which a sonic flow must be turned isentropically to reach Mach number M, so turning angles are differences of ν. For an expansion from M₁ to M₂ the wall turning is θ = ν(M₂) − ν(M₁); conversely, for a given θ you add it to ν(M₁) and invert the function for M₂. If a pressure ratio is given instead, you first find M₂ from the isentropic relation with the constant p₀, then the required θ.

  7. 7.What is the effect of the specific heat ratio on the Prandtl-Meyer expansion wave?Application

    γ enters ν(M) through the factor √((γ+1)/(γ−1)). For a given turn from a given Mach number, a higher γ gives a larger rise in Mach number — from M = 2 a 10° turn gives M₂ ≈ 2.38 for γ = 1.4 but about 2.53 for γ = 5/3. The maximum possible turning angle νmax = (π/2)(√((γ+1)/(γ−1)) − 1) is however smaller for higher γ: 130.45° for air and 90° for a monatomic gas. The pressure ratio for a given Mach change also depends on γ through the exponent γ/(γ−1).

  8. 8.A supersonic flow with Mach number 2.5 expands around a convex corner. What is the final Mach number if the turning angle is 15 degrees and γ = 1.4?Numerical

    Turning angles add to the Prandtl–Meyer function, not to the Mach number. ν(2.5) = 39.12°, so ν(M₂) = 39.12° + 15° = 54.12°. Inverting with tables or iteration gives M₂ ≈ 3.24. The pressure then falls isentropically by p₂/p₁ = [(1 + 0.2 × 2.5²)/(1 + 0.2 × 3.24²)]^3.5 ≈ 0.33.

  9. 9.Describe a real-world application where Prandtl-Meyer expansion waves are utilized.Application

    Prandtl-Meyer expansion waves are utilized in the design of supersonic nozzles, such as those used in rocket engines. These nozzles are designed to expand the exhaust gases smoothly and efficiently to achieve high exit velocities, which is essential for maximizing thrust and performance in space vehicles.

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