Prandtl Meyer Expansion Calculator

How to use this Prandtl – Meyer expansion calculator

What this calculator does

This calculator models an ideal, two-dimensional Prandtl – Meyer expansion of a calorically perfect gas as a supersonic flow turns around a convex corner. From the upstream Mach number, the specific-heat ratio, and the outward turn angle, it solves the Prandtl – Meyer relation for the downstream Mach number. If you also provide upstream pressure, temperature, and density, it applies isentropic relations to estimate the corresponding downstream static properties. It is a gas-dynamics calculation, not a CFD solver: it does not model viscosity, boundary layers, shocks caused by an incorrect turn direction, chemistry, three-dimensional effects, or real-gas changes in γ. NASA Glenn's Prandtl – Meyer angle explanation provides useful physical context for the function used here.

When to use it

Use it when checking a supersonic nozzle or duct corner, estimating how an ideal external flow accelerates through an expansion fan, validating hand calculations in a compressible-flow course, or generating a quick engineering comparison before moving to a higher-fidelity analysis. The model is appropriate only when the incoming flow is supersonic and the turn represents an expansion rather than a compression.

How to calculate

1. The calculator opens with a complete demonstration: γ = 1.4, M₁ = 1.5, θ = 15°, P₁ = 1 atm, T₁ = 288 K, and ρ₁ = 1.22586 kg/m³. Results and a validated example XLSX are immediately available.
2. Choose Do you want flow properties? = Yes when you want pressure, temperature, and density as well as Mach and angular results. Choose No when only the Mach/angle solution matters.
3. Replace Gamma (γ), Upstream Mach (M₁), and Theta (θ) with your case. For a properties calculation, also enter Pressure (P₁), Temperature (T₁), and Density (ρ₁). Select the corresponding Pressure unit, Temperature unit, and Density unit.
4. Read Downstream Mach (M₂) first, then inspect ν₁, ν₂, μ₁, μ₂, the flow-deflection quantity, and – when enabled – the downstream pressure, temperature, and density. The comparison table puts both regions side by side.
5. Select Download Excel to export the current validated model as a real XLSX workbook. Reset clears the demonstration and all semantic input values; Download Excel is then disabled until a complete valid state is entered again.

Input guide

Do you want flow properties? is a required Yes/No control. Yes activates P₁, T₁, and ρ₁; No leaves the angular/Mach solution intact. Gamma (γ) is a required dimensionless decimal greater than 1 and at most 2; 1.4 is a common ideal-air approximation. Raising γ changes the shape and upper limit of the Prandtl – Meyer function, so do not substitute a heat-capacity ratio for a different gas or temperature regime without justification. Upstream Mach (M₁) is required and must exceed 1; 1.5 is the example. A value at or below 1 is outside this supersonic model. Theta (θ) is the required outward turn in degrees; 15 is the example. A larger positive θ demands a larger ν₂ and therefore a larger M₂, but it must remain below the finite maximum turn allowed by γ and M₁.

When properties are enabled, Pressure (P₁) is a required positive decimal, with the Pressure unit selectable as Pa, kPa, bar, atm, or psi; 1 atm is the example. Temperature (T₁) is a required temperature with the Temperature unit selectable as K, °C, °F, or °R; 288 K is the example. Celsius and Fahrenheit are converted to absolute temperature internally, and any entry at or below absolute zero is rejected. Density (ρ₁) is a required positive decimal with the Density unit selectable as kg/m³, g/L, or lb/ft³; 1.22586 kg/m³ is the example. Use a period as the decimal separator. Scientific notation and ambiguous decimal-comma forms are intentionally rejected rather than silently reinterpreted.

Output guide

Downstream Mach (M₂) is the solved supersonic Mach number after the expansion; it is an estimate from the ideal relation and should be at least M₁ for a nonnegative expansion turn. Prandtl – Meyer angle ν₁ and ν₂ are the function values in degrees, with ν₂ – ν₁ exactly equal to θ within numerical tolerance. Mach angle μ₁ and μ₂ are asin(1/M) in degrees; they normally decrease as Mach number increases. Flow deflection is reported as μ₂ – θ, matching the angular convention used by this calculator.

With properties enabled, Pressure (P₂), Temperature (T₂), and Density (ρ₂) are downstream static estimates in the selected input units. The Flow-state comparison table lists Mach number, Prandtl – Meyer angle, Mach angle, pressure, temperature, and density for Region 1 and Region 2.

Worked example

For the startup case γ = 1.4, M₁ = 1.5, and θ = 15°, the Prandtl – Meyer function gives ν₁ = 11.9052°. Adding the turn gives ν₂ = 26.9052°. Solving ν(M₂) = 26.9052° yields M₂ = 2.0191. The Mach angles are μ₁ = 41.8103° and μ₂ = 29.6874°, so the displayed flow-deflection quantity μ₂ – θ is 14.6874°. The isentropic temperature ratio is 0.798738; from P₁ = 1 atm, T₁ = 288 K, and ρ₁ = 1.22586 kg/m³, the model returns P₂ = 0.455424 atm, T₂ = 230.0367 K, and ρ₂ = 0.698960 kg/m³. These are the same canonical values used in the initial cards, table, and workbook checkpoints.