How this calculator works
A smooth family of expansion waves turns supersonic flow away from itself. The Prandtl-Meyer function measures the turn from sonic conditions to a chosen Mach number. Subtract upstream and downstream function values to obtain corner turn; inverse modes solve the monotonic function numerically. Stagnation pressure and stagnation temperature remain constant, while static pressure, temperature and density decrease. The ideal model assumes constant specific heats and excludes viscous effects and real-gas chemistry.
Formula
nu(M) = sqrt[(gamma+1)/(gamma-1)] atan(sqrt[(gamma-1)(M^2-1)/(gamma+1)]) - atan(sqrt[M^2-1]). theta = nu(M2)-nu(M1). Mach angle mu = asin(1/M). T2/T1 = [1+(gamma-1)M1^2/2]/[1+(gamma-1)M2^2/2]; p2/p1 = (T2/T1)^[gamma/(gamma-1)].
Worked example
For gamma = 1.4, expansion from M1 = 2 to M2 = 3 turns through about 23.37758593 degrees. Temperature ratio is 9/14, density ratio about 0.33134940 and static pressure ratio about 0.21301033.
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Frequently asked questions
Can this describe a compression turn?
No. A finite compressive corner generally produces a shock; use the oblique-shock calculator.
Why is there a maximum expansion angle?
The Prandtl-Meyer function approaches a finite limit as Mach number tends to infinity, leaving only a finite turn available from any upstream state.
Does total pressure decrease?
Not in this ideal isentropic expansion. Static pressure decreases while total pressure remains constant.