Engineering
Prandtl-Meyer Supersonic Expansion Function nu(M) & Maximum Turn Angle
Calculate the isentropic Prandtl-Meyer expansion angle nu(M) (in degrees) from Mach 1 to supersonic Mach M and the vacuum limiting turn angle nu_max = (SQRT((gamma+1)/(gamma-1)) - 1) * 90.
Turn supersonic airflow around convex corners and rocket bell nozzles!
When supersonic flow turns away from itself around a convex corner of deflection angle theta, it accelerates isentropically through a centered Prandtl-Meyer Expansion Fan so that downstream Prandtl-Meyer angle satisfies nu(M_2) = nu(M_1) + theta! Measured from sonic M = 1 (where nu(1) = 0 deg), Ludwig Prandtl and Theodor Meyer's exact function is: nu(M) = SQRT((gamma + 1) / (gamma - 1)) * ATAN(SQRT(((gamma - 1) / (gamma + 1)) * (M^2 - 1))) - ATAN(SQRT(M^2 - 1)) (in radians, converted to degrees via DEGREES)!
In diatomic air (gamma = 1.40), accelerating from Mach 1.0 to Mach 2.0 requires a 26.38 deg turn, and Mach 3.0 corresponds to nu(3.0) = 49.76 deg (so turning a Mach 2.0 stream around a 23.38 deg convex corner accelerates it isentropically to Mach 3.0)! As M -> infinity into hard vacuum, ATAN approaches 90 deg and turning angle hits the theoretical ceiling nu_max = (SQRT(6) - 1) * 90 = 130.45 deg!
| Supersonic Mach Number M (Air gamma = 1.40) | Mach Factor SQRT(M^2 - 1) | Prandtl-Meyer nu(M) (Degrees) | Physical Interpretation |
|---|---|---|---|
| Sonic Throat (M = 1.00) | 0.0000 | 0.00 deg | Reference sonic state |
| Mach 2.00 Supersonic | 1.7321 | 26.38 deg | Turn from M=1 to M=2 |
| Mach 3.00 Supersonic | 2.8284 | 49.76 deg | 23.38 deg fan from M=2 to M=3 |
| Vacuum Limit (M -> Inf) | Infinity | 130.45 deg | Max turn before flow detachment |
Prandtl-Meyer Angle nu(M) in Degrees
Validates M >= 1 (B2) and gamma > 1 (B3), then evaluates nu(M) using DEGREES and ATAN.
=IF(B2 < 1, "Requires Supersonic Mach M >= 1", IF(B3 <= 1, "Invalid Gamma <= 1", ROUND(DEGREES(SQRT((B3 + 1) / (B3 - 1)) * ATAN(SQRT(((B3 - 1) / (B3 + 1)) * ((B2 ^ 2) - 1))) - ATAN(SQRT((B2 ^ 2) - 1))), 2)))Vacuum Limiting Turn Angle nu_max (Degrees as M -> Infinity)
Evaluates (SQRT((gamma + 1) / (gamma - 1)) - 1) * 90 in degrees.
=IF(B3 <= 1, "Invalid Gamma <= 1", ROUND((SQRT((B3 + 1) / (B3 - 1)) - 1) * 90, 2))Try the example
Use the original values or change them to test your understanding.