Bow Shock Standoff

Bow Shock Standoff Distance (aerodynamics/high-speed/bow-shock-standoff)

Use when the task is estimating the detached bow-shock standoff distance on the stagnation streamline ahead of a blunt nose in supersonic and hypersonic flow: the shock stands ahead of a sphere nose or a circular cylinder leading edge rather than attaching to a sharp point, and the shock-layer thickness between the nose and the shock sets the local environment for pressure and heating trades. This leaf implements the classical Billig-form standoff correlations for gamma = 1.4 in pure Python, stdlib only: the sphere form for an axisymmetric nose and the cylinder form for a two-dimensional leading edge, both as an exponential of 1 over the freestream Mach squared. It pairs with aerodynamics/high-speed/hypersonic-flow for the force coefficients of the blunt body behind the shock and with aerodynamics/high-speed/aerodynamic-heating, whose stagnation-point flux scales with the same nose radius. The method is the standard engineering estimate for the shock-layer geometry, not a CFD replacement.

Domain quick reference

• Sphere standoff ratio (axisymmetric nose): Delta / R = 0.143 * exp(3.24 / M^2), with M the freestream Mach number, R the nose radius and Delta the standoff distance of the detached shock ahead of the stagnation point.
• Cylinder standoff ratio (two-dimensional leading edge): Delta / R = 0.386 * exp(4.67 / M^2). The coefficient is about 2.7 times the sphere coefficient, so at the same Mach the cylinder shock layer is the thicker one.
• Physical distance: Delta = (Delta / R) * R, so the standoff distance scales linearly with the nose radius.
• Trends: the ratio is monotone decreasing in Mach (more compression pushes the shock closer at higher Mach) and the cylinder ratio exceeds the sphere ratio at every Mach above 1.
• High-Mach asymptote: as M grows, each ratio approaches its leading coefficient from above, exp(3.24 / M^2) and exp(4.67 / M^2) both tending to 1.
• Validity floor: the correlations are documented for freestream Mach above about 1.5; the ratio grows without bound as M approaches 1, where no detached bow shock exists.
• Units are SI throughout: M dimensionless, R and Delta in meters.
• NACA TR-824 frames the compressible-flow context; the standoff relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the flight point and the nose geometry: record the freestream Mach number M and the nose radius R in meters and pick the body, sphere for an axisymmetric nose or cylinder for a two-dimensional leading edge. Feed a Mach at or below 1 or a non-positive radius and the functions reject the point with ValueError, because no detached shock layer exists at or below Mach 1. (flight-point step)
2. Evaluate the standoff ratio on the stagnation streamline with standoff_ratio(mach, body), applying the Billig-form correlation: 0.143 * exp(3.24 / M^2) for the sphere or 0.386 * exp(4.67 / M^2) for the cylinder. The output is the geometry-free ratio Delta / R. (ratio-evaluation step)
3. Convert the ratio to the physical standoff distance with standoff_distance(mach, radius, body), Delta = (Delta / R) * R in meters; the distance scales linearly with the nose radius. (distance-conversion step)
4. Run the trend checks with standoff_report(mach, radius, body): decreasing_with_mach compares the ratio at M against the ratio at 1.1 * M and sphere_cylinder_order confirms that the cylinder ratio exceeds the sphere ratio at this Mach. Both sanity flags gate the blunt-body nose-radius trade and the shock-layer thickness estimate. (trend-check step)
5. Respect the validity floor: the correlations are documented for freestream Mach above about 1.5, so a ratio computed below the floor carries the floor caveat in the report and is never quoted as a converged shock-layer size. (validity-floor step)
6. Confirm the deterministic behavior and the non-physical input rejection with the contract test scripts/test_bow_shock_standoff.py. (contract-test step)

Worked example

A reentry or cruise nose at M = 8 and a blunt leading edge at M = 4:

• Sphere at M = 8: standoff_ratio(8.0) = 0.143 * exp(3.24 / 64) = 0.15043; with R = 0.5 m, standoff_distance(8.0, 0.5) = 0.07521 m.
• Sphere at M = 4: standoff_ratio(4.0) = 0.143 * exp(3.24 / 16) = 0.17510; with R = 0.5 m the standoff distance is 0.08755 m.
• Cylinder at M = 8: standoff_ratio(8.0, "cylinder") = 0.386 * exp(4.67 / 64) = 0.41522; with R = 0.5 m the standoff distance is 0.20761 m.
• Cylinder at M = 4: standoff_ratio(4.0, "cylinder") = 0.386 * exp(4.67 / 16) = 0.51683; with R = 0.5 m the standoff distance is 0.25841 m.
• Trend checks at M = 4, R = 0.5 m, sphere: standoff_report(4.0, 0.5) returns ratio 0.17510, distance 0.08755 m, sphere_cylinder_order True (cylinder ratio 0.51683 exceeds the sphere ratio 0.17510) and decreasing_with_mach True (the ratio at 4.4 is below the ratio at 4.0). The same flags hold at M = 8, whose ratio 0.15043 sits below the M = 4 sphere ratio: the standoff shrinks as Mach rises.

Verification

• Confirm standoff_ratio(8.0) returns 0.15043 within 1e-4 and standoff_ratio(4.0) returns 0.17510 within 1e-4.
• Confirm standoff_ratio(8.0, "cylinder") returns 0.41522 within 1e-4 and standoff_ratio(4.0, "cylinder") returns 0.51683 within 1e-4.
• Confirm standoff_distance(8.0, 0.5) returns 0.07521 m within 1e-4 and that distance at R = 1.0 m is exactly twice the distance at R = 0.5 m (linear scaling identity).
• Confirm the ratio is monotone decreasing: ratio at M = 6 below the ratio at M = 4 for both bodies, and the cylinder ratio exceeds the sphere ratio at M = 4 and M = 8.
• Confirm every non-physical input raises ValueError: mach 1.0, mach 0.8, radius 0, negative radius, and the body string "wedge".
• Confirm standoff_report returns exactly the keys ratio, distance, sphere_cylinder_order and decreasing_with_mach, and that two runs at the same point agree (determinism).
• Run the contract test offline: python3 scripts/test_bow_shock_standoff.py (deterministic).

Related leaves

• aerodynamics/high-speed/hypersonic-flow: the modified Newtonian force coefficients of the blunt body behind the shock.
• aerodynamics/high-speed/oblique-shock: attached shock turning on sharp-nosed bodies, the regime where no detached standoff exists.
• aerodynamics/high-speed/normal-shock: the stagnation streamline jump relations across the detached shock.
• aerodynamics/high-speed/aerodynamic-heating: stagnation-point convective flux with nose-radius scaling, the companion trade to the shock-layer thickness.
• aerodynamics/high-speed/flat-plate-skin-friction-heating: the thin-boundary-layer alternative on slender surfaces.

Pitfalls

• Quoting a ratio at Mach near 1: the exponential form grows without bound as M approaches 1 and the correlations are only documented above about Mach 1.5, so a low-Mach ratio needs the validity floor caveat (step 5 of the workflow).
• Feeding Mach at or below 1: no detached shock exists there, the correlation is not defined, and the functions raise ValueError rather than return a meaningless number.
• Using the sphere correlation for a two-dimensional leading edge or the cylinder correlation for an axisymmetric nose: the cylinder coefficient 0.386 is about 2.7 times the sphere coefficient 0.143, so swapping the body inflates or deflates the shock layer by that factor.
• Reporting the ratio as a distance: Delta / R is geometry-free and dimensionless; the physical distance is the ratio times the nose radius, so a 0.15043 ratio at R = 0.5 m is 0.07521 m, not 0.15043 m.
• Extending the gamma = 1.4 calibration: both exponentials use the gamma = 1.4 constants, so applying them to a real-gas flow with a different effective gamma needs a re-derived coefficient.
• Scaling the wrong radius: the distance is linear in the nose radius, so halving the radius halves the standoff; keep the ratio and the radius together in the report.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_bow_shock_standoff.py

The test covers the four worked-example anchors (sphere and cylinder ratios at M = 8 and M = 4 within 1e-4 of the module outputs), the standoff distance conversion at R = 0.5 m, the linear scaling identity in radius, the monotone decrease of the ratio with Mach for both bodies, the cylinder-over-sphere ordering at M = 4 and M = 8, the exact standoff_report key set and flag values, determinism, and ValueError rejection of Mach at or below 1, non-positive radii and unknown body strings.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is the compressible-flow reference framework; the Billig-form standoff correlations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.