Prandtl-Meyer Expansion Calculator
Enter the upstream Mach number and deflection angle (or any two of the three main variables) to compute all downstream flow properties for a Prandtl-Meyer isentropic expansion fan. The calculator covers downstream Mach number, pressure, temperature and density ratios, Mach angles, and the Prandtl-Meyer function values. Choose a solve mode to work forward, backward, or in reverse.
Formula
Worked example
Air (gamma = 1.4) flows at M1 = 2.0 around a 15-degree convex corner. nu(M1) = 26.38 deg. nu(M2) = 26.38 + 15 = 41.38 deg. Inverting gives M2 = 2.583. Pressure ratio P2/P1 = (iso1/iso2)^3.5 = 0.4669. Temperature ratio T2/T1 = iso1/iso2 = 0.8519.
What is Prandtl-Meyer expansion?
When a supersonic flow encounters a convex corner (one that bends away from the flow), it cannot generate a compression shock. Instead, it fans out through a continuous set of Mach waves called a Prandtl-Meyer expansion fan. The fan is isentropic: no entropy is generated, total pressure is conserved, and the flow accelerates while static pressure and temperature both fall. This is the exact opposite of what happens at an oblique shock, where compression decelerates the flow and raises static pressure at the cost of total pressure. Prandtl-Meyer theory, developed independently by Ludwig Prandtl and Theodor Meyer around 1908, gives the exact analytical solution for this turning process in a calorically perfect gas.
How the Prandtl-Meyer function works
The Prandtl-Meyer function nu(M) is defined as the angle through which a sonic (M = 1) stream must be expanded to reach a given Mach number. Its formula is: nu(M) = sqrt[(gamma+1)/(gamma-1)] * arctan(sqrt[(gamma-1)(M^2-1)/(gamma+1)]) - arctan(sqrt[M^2-1]). Because expansion is isentropic, the total turning angle theta (the wall deflection) equals nu(M2) minus nu(M1). Given any two of the three quantities M1, M2, and theta, the third follows directly. Downstream isentropic property ratios (pressure, temperature, density) are then obtained from the standard isentropic flow relations using the isentropic factor 1 + (gamma-1)/2 * M^2.
Mach angles and the expansion fan boundaries
Each Mach wave in the fan makes an angle mu = arcsin(1/M) with the local flow direction. The leading edge of the fan lies at angle mu1 (relative to the incoming flow) and the trailing edge lies at mu2 - theta (relative to the turned downstream direction). Flow outside the fan is not affected; flow inside transitions smoothly from the upstream to the downstream state. For large deflection angles the fan can become very wide, and if theta exceeds the maximum Prandtl-Meyer angle (about 130.5 degrees for gamma = 1.4), a vacuum state would theoretically be reached, which is physically impossible for a real gas.
Applications in aerospace and engineering
Prandtl-Meyer expansion is fundamental to the aerodynamics of supersonic nozzles, rocket engine nozzles (where the diverging section accelerates flow through expansion waves), supersonic inlets, and the trailing edges of supersonic aerofoils. In the method of characteristics, the Prandtl-Meyer function is the primary tool for designing contoured nozzle walls that produce a uniform exit flow. On supersonic and hypersonic vehicles, pressure distributions over inclined surfaces are routinely estimated by combining oblique shock relations (for windward faces) and Prandtl-Meyer expansion (for leeward faces).
Prandtl-Meyer angle vs. Mach number (gamma = 1.4, air)
| Mach number (M) | nu (deg) | Pressure ratio (P/P0) | Temperature ratio (T/T0) | Mach angle, mu (deg) |
|---|---|---|---|---|
| 1 | 0 | 1.000000 | 1.000000 | 90 |
| 1.5 | 11.91 | 0.27240 | 0.82963 | 41.81 |
| 2 | 26.38 | 0.12780 | 0.72088 | 30 |
| 2.5 | 39.12 | 0.05853 | 0.64397 | 23.58 |
| 3 | 49.76 | 0.02722 | 0.57447 | 19.47 |
| 4 | 65.78 | 0.00659 | 0.46119 | 14.48 |
| 5 | 76.92 | 0.00189 | 0.37037 | 11.54 |
| 7 | 91.97 | 0.000242 | 0.24490 | 8.21 |
| 10 | 102.31 | 0.0000236 | 0.16667 | 5.74 |
Standard values of the Prandtl-Meyer function for air at common Mach numbers. These represent the total turning angle from M = 1.
Frequently asked questions
What is the Prandtl-Meyer expansion fan?
A Prandtl-Meyer expansion fan is a family of Mach waves that forms when supersonic flow turns around a convex corner. Unlike a shock wave, the fan involves no discontinuity in entropy: the flow accelerates continuously and isentropically, with static pressure and temperature decreasing while total pressure remains constant.
Why must the upstream Mach number be greater than 1?
Prandtl-Meyer expansion only occurs in supersonic flow. Subsonic flow cannot support Mach waves, and a subsonic stream encountering a concave wall would simply follow the wall without producing the characteristic fan structure. The minimum valid input for M1 is just above 1, corresponding to nearly sonic flow and a very narrow fan.
What is the maximum possible deflection angle?
For a perfect gas the maximum turning angle is achieved as M2 approaches infinity, giving nu_max = (pi/2) * (sqrt[(gamma+1)/(gamma-1)] - 1). For air (gamma = 1.4) this is about 130.45 degrees. Deflection angles beyond this value are not physically realizable for a perfect gas because the flow would need to reach a vacuum. Real-gas effects (dissociation, ionization) modify this limit at very high Mach numbers.
Is the expansion isentropic?
Yes. The entire Prandtl-Meyer expansion fan is isentropic: no shock waves form, entropy is unchanged, and total pressure is perfectly conserved across the fan. This makes it the compressible flow counterpart of ideal incompressible turning. By contrast, any shock wave, even an oblique one, generates entropy and reduces total pressure.
How does changing the specific heat ratio (gamma) affect the results?
A higher gamma (such as helium at 1.667) produces a smaller Prandtl-Meyer angle for the same Mach number and a lower maximum turning angle. A lower gamma (such as CO2 at 1.29) gives a larger nu(M) and a higher theoretical maximum turning angle. For air at standard conditions gamma = 1.4 is the conventional choice; above about Mach 5-7 real-gas effects reduce the effective gamma, which is why hypersonic computations require variable-gamma or equilibrium chemistry models.
How is this calculator different from an oblique shock calculator?
Oblique shock relations apply to concave corners where flow is compressed: the Mach number decreases, static pressure and temperature rise, and entropy increases (so total pressure drops). Prandtl-Meyer relations apply to convex corners where flow is expanded: Mach number increases, static pressure and temperature fall, and the process is isentropic. The two solutions together cover the full range of two-dimensional supersonic corner flows.