Prandtl Meyer Expansion Calculator
Expansion Fan Properties
What Is a Prandtl-Meyer Expansion Calculator?
A Prandtl-Meyer Expansion Calculator models an ideal supersonic expansion around a sharp convex corner. The tool starts with an upstream Mach number and gas-specific heat ratio. It calculates the initial Prandtl-Meyer angle, adds the entered turning angle, then numerically finds the downstream Mach number that corresponds to the new angle.
A Prandtl-Meyer expansion calculator finds the downstream Mach number after supersonic flow turns through a specified angle. This tool also calculates the initial and final Prandtl-Meyer angles, static pressure ratio, static temperature ratio, and stagnation pressure ratio for the modeled isentropic expansion.
The calculator includes preset values for air, carbon dioxide, helium, and methane. It also accepts a custom specific heat ratio. Its results apply to the ideal gas-dynamics assumptions built into the code, not to every real compressible-flow situation.
How the Prandtl-Meyer Expansion Formula Works
The calculation begins with the Prandtl-Meyer function. For an upstream Mach number greater than 1, the code calculates the initial angle ν₁ from Mach number and the specific heat ratio γ.
Here, M is Mach number and γ is the gas specific heat ratio. The function returns the angle in radians inside the calculation. The displayed Prandtl-Meyer angles are converted to degrees.
The final Prandtl-Meyer angle is found by adding the turning angle θ to the initial value:
The code then uses a bisection search to find the downstream Mach number M₂ that produces ν₂. It performs up to 100 iterations and stops early when the Prandtl-Meyer function is within 0.000000001 radians of the target.
Once M₂ is known, the calculator determines the static temperature and pressure ratios:
Worked example
For air, use γ = 1.400, an upstream Mach number of 2.0, and a turning angle of 20°. The initial Prandtl-Meyer angle is about 26.38°. Adding 20° gives a final angle of about 46.38°. The numerical search produces a downstream Mach number of about 2.830595.
The calculator therefore displays M₂ = 2.831, ν₁ = 26.38°, and ν₂ = 46.38°. It also displays p₂/p₁ = 0.2752, T₂/T₁ = 0.6917, and a stagnation pressure ratio of 1.0000.
If the turning angle is zero, the code skips the expansion calculation and returns the original Mach number with pressure, temperature, and stagnation pressure ratios of 1.0000.
How to Use the Prandtl-Meyer Expansion Calculator: Step by Step
- Select the Gas Type. Available presets are air or diatomic gas at γ = 1.400, carbon dioxide at 1.289, helium or monatomic gas at 1.667, and methane at 1.299.
- Select Custom γ if you want to enter your own specific heat ratio. The calculation requires that custom value to be strictly greater than 1 and strictly less than 2.
- Enter the Upstream Mach Number (M₁). The calculator requires a value strictly greater than 1.0 because the modeled Prandtl-Meyer expansion requires supersonic upstream flow.
- Enter the Turning Angle (θ) in Degrees. The value may be zero or positive.
- Select Calculate. The tool determines the downstream Mach number and the associated expansion properties.
- Review M₂, ν₁, ν₂, p₂/p₁, T₂/T₁, p₀₂/p₀₁, and the aerodynamic analysis message.
The downstream Mach number is displayed to three decimal places. The two Prandtl-Meyer angles are displayed to two decimal places with degree symbols. Pressure and temperature ratios are shown to four decimal places. Reset returns the gas selection to air and clears the custom γ, Mach number, turning angle, and displayed results.
How to Read Your Prandtl-Meyer Expansion Calculator Results
The outputs describe how the calculator’s ideal supersonic flow changes through the expansion. In a valid positive-angle calculation, the downstream Mach number rises while static pressure and static temperature fall. The stagnation pressure ratio remains fixed at 1.0000 because the code treats the expansion as isentropic.
| Output | What It Represents |
|---|---|
| Downstream Mach Number (M₂) | Calculated Mach number after the turn |
| Initial Prandtl-Meyer Angle (ν₁) | Expansion-function angle at M₁ |
| Final Prandtl-Meyer Angle (ν₂) | ν₁ plus the entered turning angle |
| Static Pressure Ratio (p₂/p₁) | Downstream static pressure divided by upstream static pressure |
| Static Temperature Ratio (T₂/T₁) | Downstream static temperature divided by upstream static temperature |
| Stagnation Pressure Ratio (p₀₂/p₀₁) | Displayed as 1.0000 in the modeled isentropic process |
Pressure-drop analysis
The calculator changes its explanatory message according to the pressure ratio. If p₂/p₁ is below 0.5, the code describes the static-pressure decrease as significant. At 0.5 or above, it describes the expansion as moderate. This wording affects only the explanation, not the numerical results.
Maximum expansion handling
The code estimates its upper Prandtl-Meyer limit by evaluating the function at Mach 1000. If ν₁ plus the requested turn exceeds that value, the calculator displays an infinite downstream Mach number, zero static pressure ratio, zero static temperature ratio, and stagnation pressure ratio of 1.0000. It also warns that the requested expansion is physically impossible under the model.
Model assumptions and limitations
The tool assumes a calorically perfect gas, steady inviscid flow, no external work, no heat transfer, and a sharp convex corner. It models an ideal isentropic expansion fan. It does not calculate viscous losses, shocks, boundary-layer effects, real-gas changes, geometry away from the corner, or other non-ideal effects.
Frequently Asked Questions
What is a Prandtl-Meyer expansion?
A Prandtl-Meyer expansion in this calculator is the modeled acceleration of supersonic flow around a convex turning corner. The calculation increases the Prandtl-Meyer angle by the entered turn, solves for a higher downstream Mach number, and determines the associated static pressure and temperature ratios under isentropic assumptions.
How do you calculate the downstream Mach number after an expansion?
The calculator first evaluates the Prandtl-Meyer function at the upstream Mach number. It adds the turning angle to obtain the target final angle. Because the code does not use a direct inverse formula, it applies a numerical bisection search to find the downstream Mach number corresponding to that target.
Can the upstream Mach number be exactly 1.0?
No. This calculator requires the upstream Mach number to be strictly greater than 1.0. If you enter 1.0 or a lower value, it displays an alert explaining that an expansion fan requires a supersonic upstream Mach number. The calculation does not continue until a valid value is entered.
What happens if the turning angle is zero?
A zero turning angle produces no expansion in this calculator. The downstream Mach number remains equal to the upstream Mach number, and the initial and final Prandtl-Meyer angles are identical. Static pressure, static temperature, and stagnation pressure ratios are all displayed as 1.0000.
What specific heat ratios can I use?
The calculator provides γ values of 1.400 for air or diatomic gas, 1.289 for carbon dioxide, 1.667 for helium or monatomic gas, and 1.299 for methane. The custom option accepts a numeric γ only when it is strictly between 1 and 2.
Why is the stagnation pressure ratio always 1.0000?
The calculator always displays p₀₂/p₀₁ as 1.0000 because its expansion model is isentropic. The code assumes no stagnation-pressure loss through the expansion fan. That result follows the idealized model used by the calculator and should not be interpreted as including real viscous or other non-ideal losses.
What happens if the turning angle is too large?
If the requested final Prandtl-Meyer angle exceeds the code’s maximum based on Mach 1000, the tool flags the turn as beyond the modeled physical limit. It displays “Infinite” for downstream Mach number, 0.0000 for static pressure and temperature ratios, and an explanatory warning about the impossible expansion condition.