Prandtl Meyer

Prandtl-Meyer Expansion (aerodynamics/high-speed/prandtl-meyer)

Use when the task is the Prandtl-Meyer expansion of a supersonic flow: the expansion angle from the Mach number, the downstream Mach number after the flow turns away from itself, the total turning angle of the fan, and the static pressure ratio across it.

Domain quick reference

• Prandtl-Meyer function: nu(M) = sqrt((gamma+1)/(gamma-1)) * atan(sqrt((gamma-1)/(gamma+1) * (M^2 - 1))) - atan(sqrt(M^2 - 1)), in radians. nu(1.0) = 0.0 exactly: the expansion fan collapses to zero width at the sonic point and widens monotonically above it. The function is undefined for subsonic Mach numbers (M < 1 raises ValueError).
• Specific heat ratio: gamma (default 1.4 for air) must be > 1.
• Total turning angle of an expansion fan: delta = nu(M2) - nu(M1) in radians for a flow turning away from itself; positive for an expansion (M2 > M1), negative for a compression (not a Prandtl-Meyer fan).
• Downstream Mach after a turn: solve nu(M2) = nu(M1) + delta for M2, where delta is the turning angle in radians. Bisection on the bracket [1, 50] is deterministic and offline; a turning angle too large for the bracket raises ValueError.
• Static pressure ratio: p2/p1 = pr(M2) / pr(M1) with the isentropic relation pr(M) = (1 + (gamma-1)/2 * M^2)^(-gamma/(gamma-1)); always < 1 for M2 > M1 (the expansion drops static pressure as it accelerates the flow).
• Textbook anchor (Anderson, Modern Compressible Flow, Table A.5): nu(2.0) = 26.380 deg at gamma = 1.4; a 10 deg turn from M = 2.0 gives M2 = 2.385 and p2/p1 = 0.548.

Workflow

1. Collect the upstream Mach number M1 (must be >= 1) and the specific heat ratio gamma (default 1.4).
2. Compute the expansion angle with prandtl_meyer_function; verify the flow is supersonic before trusting the fan.
3. For a turn away from itself, find the downstream Mach number with mach_after_expansion from the turning angle in degrees.
4. Compute the total turning angle between two Mach numbers with flow_turn_angle.
5. Compute the static pressure drop with expansion_pressure_ratio.
6. Get the downstream state at once with expansion_properties for the airfoil, inlet, or nozzle analysis.

Pitfalls

• Feeding a subsonic Mach number: nu(M) is undefined below Mach 1; a subsonic input raises, never returns a negative angle.
• Using degrees where radians are expected: prandtl_meyer_function and flow_turn_angle return radians; mach_after_expansion takes the turning angle in degrees.
• Confusing expansion with compression: a flow turning toward itself is a shock, not a Prandtl-Meyer fan; flow_turn_angle is only the expansion (M2 > M1) case.
• Asking for an unreachable turning angle: nu(M) is bounded above, so a turn too large for the Mach bracket raises instead of converging to nonsense.
• Treating the fan as zero width above Mach 1: nu(M) = 0 only at M = 1; any M > 1 widens the fan strictly.

Behavior contract (gate 3)

The Prandtl-Meyer function, turning angle, downstream Mach, pressure ratio, and downstream state logic is exercised by the gate 3 contract test: scripts/test_prandtl_meyer.py against scripts/prandtl_meyer_logic.py (stdlib unittest, offline). Run: python3 scripts/test_prandtl_meyer.py

Compliance

• Standards referenced, not reproduced: NACA TR-824 anchors the compressible-flow tables (Anderson, Modern Compressible Flow, Table A.5 reproduces its values); the Prandtl-Meyer relations are standard compressible-flow methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.