Prandtl-Meyer Expansion Calculator

A Prandtl-Meyer expansion fan forms when a supersonic flow turns around a convex corner and away from itself. Unlike a shock wave, which compresses and decelerates supersonic flow, an ideal expansion fan is continuous and isentropic. The flow accelerates to a higher Mach number while static pressure, temperature, and density decrease. This calculator relates the upstream Mach number and turning angle to the downstream state for a calorically perfect gas. The Prandtl-Meyer function assigns an angle to each supersonic Mach number for a selected ratio of specific heats. The downstream function value equals the upstream function value plus the expansion turning angle. Because the inverse does not have a simple elementary expression, the calculator solves for final Mach number numerically by bisection. The method is deterministic and converges tightly for valid angles below the theoretical maximum expansion. For air near ordinary temperatures, gamma is commonly approximated as 1.4. Other gases and high-temperature flows can require different values because molecular degrees of freedom and chemical effects change specific heat. Gamma must exceed one in this ideal-gas formulation. At very high temperatures, dissociation, vibration, and variable specific heats can make the constant-gamma model inaccurate. After finding downstream Mach number, the calculator applies isentropic stagnation relationships. Total temperature and total pressure remain constant through an ideal expansion, while static values change with Mach number. The temperature ratio is the upstream stagnation factor divided by the downstream factor. Pressure and density ratios follow by raising that factor to exponents based on gamma. The Mach angle is the inverse sine of one divided by Mach number. The model assumes steady, two-dimensional, inviscid, adiabatic, supersonic flow with no shocks embedded in the fan. Real corners have boundary layers, finite geometry, viscosity, and possible flow separation. An expansion followed by compression can also create wave interactions that this isolated calculation does not capture. Input turning angle is the magnitude of a favorable turn; a compression turn requires oblique-shock analysis instead. Prandtl-Meyer calculations are used in supersonic nozzle design, external aerodynamics, wind-tunnel analysis, aerospace coursework, and preliminary wave-pattern estimates. Use the ratios to scale known upstream static properties, not stagnation properties. Detailed aircraft, rocket, or safety-critical designs should be validated with appropriate gas models, computational fluid dynamics, experimental data, and qualified aerodynamic analysis.