Prandtl Meyer Expansion Calculator - Compressible Flow Function

The prandtl meyer expansion calculator solves the Prandtl-Meyer function nu(M), which gives the angle through which a supersonic flow can turn around a convex corner and still remain isentropic. The function is zero at Mach 1 and rises to about 130.45 degrees for air as the upstream Mach grows without bound, and it works for any constant specific-heat ratio gamma.

• • Aerodynamics homework: Get the textbook nu(M) value and the downstream isentropic ratios for an air flow at any upstream Mach.
• • Wind-tunnel planning: Estimate how much a supersonic nozzle wall can diverge before the flow separates, then check the new pressure and temperature.
• • Nozzle and inlet design: Predict the static pressure and temperature downstream of an expansion corner in a converging-diverging nozzle or external-compression inlet.
• • Propulsion back-of-envelope: Use the closed-form nu(M) and gamma = 1.4 default to size a corner in a ramjet or scramjet inlet during preliminary sketching.

The model is the same one used in Liepmann and Roshko's Elements of Gasdynamics and Anderson's Modern Compressible Flow. For air the function maps from Mach 1 up to the asymptotic ceiling, and the calculator exposes a custom gamma field so helium, argon, or any combustion gas with a tabulated cp/cv can be used.

Before working through the expansion function it is worth revisiting the dimensionless compressibility form of the ideal gas law in the Compressibility Calculator, since the same Mach and gamma show up here as input arguments.

The calculator evaluates the Prandtl-Meyer function nu(M) directly from its closed-form expression for a perfect gas, and uses bisection to invert it when the user supplies a deflection angle instead of an upstream Mach. The downstream state is then found from the standard isentropic relations, so the displayed pressure, temperature, and density ratios are deterministic outputs of the same Mach and gamma.

• M1: Upstream Mach number ahead of the convex corner; must be 1 or larger.
• M2: Downstream Mach number after the fan; equal to M1 in forward mode, found by bisection in reverse mode.
• theta: Wall deflection angle in radians (or degrees), equal to the change in flow direction across the fan.
• gamma: Specific-heat ratio cp/cv; 1.4 for air, 1.66 for helium or argon.
• p/p0, T/T0, rho/rho0: Isentropic ratios of static pressure, temperature, and density to their stagnation values.

The arctan term appears twice because the function comes from integrating the small-deflection Prandtl-Meyer equation along the reference process that brings the flow from Mach M back to Mach 1.

Bisection runs over the bracketed interval Mach 1 to Mach 80 with a 1e-9 tolerance. The downstream pressure, temperature, and density ratios are then computed from the same isentropic relations used for nozzle flows.

According to NASA Glenn Research Center - Prandtl-Meyer Angle, nu(M) = sqrt((gamma+1)/(gamma-1)) * arctan(sqrt((gamma-1)(M^2-1)/(gamma+1))) - arctan(sqrt(M^2-1)) for a perfect gas with specific-heat ratio gamma, with nu(1) = 0 by construction, and its physical interpretation is the angle a sonic flow must be expanded through to reach Mach M

The isentropic ratios come from the same energy-conservation family that the Bernoulli Equation Calculator handles for incompressible flow, just specialized here to the compressible limit.

Four ideas make the function predictable: why an expansion fan is isentropic, why Mach lines fan out at the Mach angle, what the 130.45 degree ceiling means in practice, and why the same isentropic ratios feed the downstream state.

These four ideas are the conceptual core of any Prandtl-Meyer expansion.

When the working fluid departs from a perfect gas, the Ideal Gas Calculator becomes the place to revisit the equation of state that the Prandtl-Meyer derivation assumes.

Pick the solve direction, enter the upstream Mach or a deflection angle, set gamma for the working gas, and read the four result cards.

1. 1 Choose the solve direction: Select Deflection angle from upstream Mach to return nu(M), or Downstream Mach from a deflection angle to invert nu.
2. 2 Enter the upstream Mach: Type the Mach number ahead of the convex corner. The default of 2 matches a typical wind-tunnel test section.
3. 3 Set the deflection angle in reverse mode: Enter the wall turn angle in degrees; the bisection then finds the matching downstream Mach.
4. 4 Set the specific-heat ratio: Use 1.4 for air, 1.66 for helium or argon, or any constant value your reference quotes.
5. 5 Pick the angle display unit: Switch between degrees and radians. The math is identical, only the display changes.
6. 6 Read the result cards: The first card shows the Prandtl-Meyer angle at the current Mach (nu(M1) forward, nu(M2)=nu(M1)+theta reverse), followed by the downstream Mach and the isentropic pressure, temperature, and density ratios.

For a 12 degree wall divergence downstream of a nozzle throat, enter inputMode = angle, M1 = 2.5, thetaDeg = 12, gamma = 1.4, angleUnit = deg. The calculator returns the downstream Prandtl-Meyer angle nu(M2) = nu(2.5) + 12 = 51.12 degrees, downstream Mach 3.072, p/p0 = 0.0245, T/T0 = 0.3464, rho/rho0 = 0.0706, so static pressure falls to 2.4 percent of stagnation.

When the expansion fan feeds a propulsion device rather than a wind tunnel, the Ideal Rocket Equation Calculator is a natural follow-up for estimating the resulting thrust.

The prandtl meyer expansion calculator gives textbook nu(M) values plus the downstream isentropic state in one panel, useful for homework, lab reports, and preliminary design.

• • Closed-form Prandtl-Meyer function: The angle is evaluated from the exact nu(M) expression, not a chart lookup, so the result matches what a textbook table would print for the chosen Mach and gamma.
• • Reverse solve with bisection: Supplying a wall deflection angle returns the matching downstream Mach using a bracketed bisection that converges fast.
• • Downstream isentropic state: Static pressure, temperature, and density ratios all come from the same stagnation-conserving relations used for nozzle flows.
• • Custom specific-heat ratio: The gamma field accepts 1.05 to 1.7, covering air, helium, argon, and any tabulated combustion product with a constant cp/cv.

The calculator is also a useful sanity check for any CFD or wind-tunnel result.

Three input factors and two physical assumptions determine the Prandtl-Meyer angle and the downstream state, and the same factors also bound where the closed-form model is valid.

• • The function is isentropic, so the corner must be convex and weak enough that the fan fully attaches to the wall.
• • Real high-Mach flows pick up vibrational non-equilibrium effects, so the constant-gamma assumption can drift by a few percent above Mach 7 for air.

For most homework and preliminary-design work the closed-form Prandtl-Meyer model is the right reference. The gamma input is the cleanest knob to turn when matching a real gas.

According to NASA Glenn Research Center - Centered Expansion Fan, an expansion fan is an isentropic process that raises Mach, lowers static pressure, and conserves total pressure as the flow turns away from itself around a convex corner

According to NIST Chemistry WebBook - Thermophysical Properties, the specific-heat ratio of dry air is 1.4 at standard conditions, while monatomic gases such as helium reach about 1.66, which is why the calculator exposes a custom gamma field for non-air working fluids

Beyond the inviscid corner, real expansion fans interact with the boundary layer; the Reynolds Number Calculator is a good way to estimate whether the boundary layer stays attached at the chosen deflection.