ashfordeOU Oblique Shock Relations (aerodynamics/high-speed/oblique-shock)
Use when the task is an oblique shock in supersonic flow: the wave angle from the theta-beta-M relation, the weak and strong solutions, the deflection limit for an attached shock, and the downstream state.
Domain quick reference
- Geometry: a supersonic flow deflected into itself by the angle theta (wedge half-angle or compression-corner turn) forms an attached oblique shock inclined at the wave angle beta to the upstream flow, with the Mach angle mu = asin(1/M1) < beta <= 90 deg.
- Only the Mach component normal to the shock changes across it: M1n = M1 * sin(beta); the tangential component passes through unchanged. All downstream ratios come from the normal shock relations applied to M1n.
- theta-beta-M relation: tan(theta) = 2 * cot(beta) * (M1^2 * sin^2(beta) - 1) / (M1^2 * (gamma + cos(2*beta)) + 2). theta = 0 at both beta = mu (Mach wave, isentropic) and beta = 90 deg (normal shock).
- Two solutions for theta < theta_max: the weak solution (small beta, downstream flow usually still supersonic, the branch physically realized on a wedge) and the strong solution (large beta, downstream flow subsonic).
- Deflection limit theta_max: the apex of the shock polar, where the two branches merge. Above it no attached oblique shock exists and the shock detaches. theta_max grows with M1 toward about 45.6 deg (gamma = 1.4); at M1 = 2 it is 22.9735 deg.
- Downstream Mach: M2 = M2n / sin(beta - theta), with M2n the normal-shock downstream Mach at M1n.
- Ratios: p2/p1 = 1 + 2*gamma/(gamma+1) * (M1n^2 - 1), always > 1; rho2/rho1 and T2/T1 = (p2/p1)/(rho2/rho1) follow the normal shock relations; p02/p01 < 1 but far gentler than the normal shock at the same M1 (the weak oblique shock keeps almost all total pressure).
- Textbook anchor (Anderson, Modern Compressible Flow, Example 4.2) at M1 = 2.0, theta = 10 deg, gamma = 1.4: weak beta = 39.3139 deg, M2 = 1.6405, p2/p1 = 1.7066, p02/p01 = 0.9846; strong beta = 83.7001 deg, M2 = 0.6037, p2/p1 = 4.4438. At theta = 0 the strong branch is exactly the normal shock at M1 (p2/p1 = 4.5, M2 = 0.5773503).
Workflow
- Collect the upstream Mach number M1 (must be > 1), the deflection angle theta in degrees, and the specific heat ratio gamma (default 1.4 for air).
- Check the deflection limit with deflection_limit: theta above theta_max means a detached shock and no attached-shock answer.
- Find both wave angles with shock_angles; use the weak branch for the physically realized wedge flow unless the boundary conditions force the strong branch.
- Compute the downstream state with shock_properties (weak by default, strong=True for the strong branch): Mach number and the pressure, density, temperature, and stagnation pressure ratios.
- For the Mach wave limit, use mach_angle for mu at the given M1.
Pitfalls
- Feeding theta above theta_max: no attached oblique shock exists; the shock detaches. The functions raise instead of returning a fake wave angle.
- Using the normal shock relations on M1 instead of M1n: the oblique shock compresses only the Mach-normal component; using full M1 overstates every ratio.
- Picking the strong branch by default: on a wedge the weak solution is the one realized; the strong branch needs downstream pressure high enough to force it.
- Treating the Mach wave as a shock: at theta = 0 the weak branch gives beta = mu with p2/p1 = 1, an isentropic limit, not a shock.
- Confusing beta and theta: beta is the shock wave angle to the upstream flow, theta the deflection of the flow itself; M2 uses sin(beta - theta).
- Feeding a subsonic M1: the relations are only valid for M1 > 1.
Behavior contract (gate 3)
The oblique shock relation logic is exercised by the gate 3 contract test: scripts/test_oblique_shock.py against scripts/oblique_shock_logic.py (stdlib unittest, offline). Run: python3 scripts/test_oblique_shock.py
Compliance
- Standards referenced, not reproduced: oblique shock relations are standard compressible-flow methodology (public-domain textbook content, e.g. Anderson, Modern Compressible Flow, Example 4.2); NACA TR 824 is referenced as the pack's public-domain reference anchor, summary-only per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.