Module blunt_tip 

A blunt or vertical nose tip faster than sound: modified Newtonian pressures on the cap, handed over to the second-order shock-expansion method (crate::shock_expansion) where the surface slope falls to the steepest wedge an attached shock can turn. After C. M. Jackson Jr., W. C. Sawyer and R. S. Smith, A Method for Determining Surface Pressures on Blunt Bodies of Revolution at Small Angles of Attack in Supersonic Flow, NASA TN D-4865 (1968) (J68).

Flown faster than sound since the milestone M1.8e7, for noses whose tip is vertical (power-series noses with n < 1, Haack series, elliptical noses) through crate::model::SupersonicBody. The guide’s Blunt tips explains it and how it was checked.

use hpr_aero::blunt_tip::{handover_angle_rad, newtonian_loading, pitot_pressure_ratio};

// Behind a normal shock at Mach 2 the pitot pressure is 5.640 times the free stream's
// (NACA Report 1135, Table II).
assert!((pitot_pressure_ratio(2.0)? - 5.6404).abs() < 1e-4);
// The cap hands over where its slope falls to the steepest wedge an attached shock turns
// at Mach 1.5, 12.1°.
assert!((handover_angle_rad(1.5)?.to_degrees() - 12.11).abs() < 0.01);
// Where the cap's slope is 45° its loading is C_p,max/2, 0.829 at Mach 2.
let loading = newtonian_loading(2.0, 45f64.to_radians())?;
assert!((loading - 0.8286).abs() < 1e-4);

The cap. Near a blunt tip the shock stands off the body and the flow behind it is subsonic, where the shock-expansion method, which marches supersonic flow from a pointed tip, can’t start. The report’s cap is modified Newtonian (eq. 1, p. 5):

p_s/p₀ = (p_t2/p₀ − 1) sin²δ + 1,

with δ the surface’s slope to the wind and p_t2 the pitot pressure behind a normal shock (the Rayleigh pitot formula, NACA Report 1135, 1953, eq. 100, p. 619):

p_t2/p₀ = [(γ + 1)M²/2]^(γ/(γ − 1)) · [(γ + 1)/(2γM² − (γ − 1))]^(1/(γ − 1)).

In coefficient form C_p = C_p,max sin²δ, C_p,max = (p_t2/p₀ − 1)/(γM²/2).

The handover. The shock-expansion method starts “at the point where the surface slope is the same as that required for shock attachment to a two-dimensional wedge at the free-stream Mach number”, chosen “simply because it gave the best agreement with the available data in the low supersonic-speed range” (p. 5): the wedge’s largest deflection δ_max, from the shock angle of NACA Report 1135 eq. 168 (p. 624),

sin²θ = [(γ + 1)M²/4 − 1 + √((γ + 1)((γ + 1)M⁴/16 + (γ − 1)M²/2 + 1))]/(γM²),

turned into a deflection by eq. 138 (p. 621), tan δ = 2 cot θ (M² sin²θ − 1)/(2 + M²(γ + 1 − 2 sin²θ)). It is 12.1° at Mach 1.5 and 22.97° at Mach 2. hpr hands over at the lesser of δ_max and 24° (MAX_HANDOVER_RAD), so from Mach 2.06 up the cap reaches further aft than the report’s. The cap is a cap because the method reads the tangent cone’s normal-force slope at the handover, and those tables (crate::shock_expansion::cone_normal_force_slope) once stopped at TN 3527 Fig. 2’s 24°. They now reach 30° (CONE_TABLE_CAP_RAD), and the handover doesn’t, because the march does not carry it there yet: measured over the whole sweep in ADR-043, a 30° handover reads nearer TN D-4865’s own sphere-cone at every row where a cap binds at all, and on the committed Arcas Robin nose above about Mach 4 it puts most of the march’s elements into η < 0 (issue #108), where the answer moves with the element count. No cap above 24° holds its answer to Mach 5, and the failure isn’t orderly in the cap: 28° is the worst of the four measured. handover_angle_capped_rad takes the cap as a parameter so both ends are measured rather than argued.

The flow behind it: hpr’s choice, not the report’s. hpr starts TN 3527’s march at the handover as the method starts at a pointed vertex: with the flow on the cone tangent to the body there (Taylor–Maccoll), that cone’s loading tan δ (dC_N/dα)_tc, and no pressure gradient (TN 3527 sketch (a), p. 6). The report starts it from the Newtonian pressure and Mach number instead (eq. 2 and p. 5). Read that way at α → 0, the march on the committed Arcas Robin nose reduces elements from Mach 2.96 (issue #81, the method’s open question there), so its answer changes as elements are added, and fails from Mach 3.96, where the Newtonian pressure at the handover lies below the tangent cone’s; the tangent cone’s start holds to Mach 5 and converges. crate::shock_expansion::HandoverStart keeps the report’s start to compare, and the decision record on it, ADR-038, gives both readings’ numbers.

The loading at α → 0. On the cap, TN 3527’s loading form (crate::shock_expansion), whose C_Nα = (2π/A_ref) ∫ Λ r dx makes Λ half the windward meridian’s ∂C_p/∂α, takes the wind’s slope δ + α cos φ in C_p = C_p,max sin²δ: Λ = C_p,max sin δ cos δ (newtonian_loading). A hemisphere then carries C_Nα = C_p,max/2, its Newtonian drag turned into the body’s axes, as it must. Behind the handover the loading is the method’s. The handover itself is held where it sits on the body, as TN 3527 holds every other point; the report’s equivalent bodies turn the body about the sphere’s center, sliding the handover along the surface, and that term is left out. Nothing here measures what it is worth.

What it covers. The report checked spherical caps only: a sphere-cone (a 0.175-diameter nose radius on an 11.5° cone) and a sphere on a flared body, from Mach 1.50 to 4.63 and up to 12°, calling its results “adequate engineering estimates … except where flow separation or detached secondary shock waves are present” (p. 13). A power-series, Haack or elliptical nose has no sphere at its tip; applying the slope rule to it is an extrapolation, stated. The wedge’s deflection, the pitot pressure and the Newtonian pressure are exact for a perfect gas with γ = 1.4; the loading is only as good as Newtonian theory on the cap. A cap that shrinks to nothing doesn’t reach the cone it sits on, because the march keeps its start cone’s total pressure (issue #101), and near the join’s start the cap can cover half a slender nose, far more than the report’s own.

Constants

CONE_TABLE_CAP_RAD

The steepest cap the method can be given, 30°: the steepest cone its normal-force slopes cover (crate::shock_expansion::cone_normal_force_slope), where NASA SP-3007’s tables stop (“cone angles from 2.5° to 30°”, Foreword, p. iii). A handover steeper than this has no tangent cone to start the march from.

MAX_HANDOVER_RAD

The steepest slope the cap hands over at, as hpr flies it: 24°. It was TN 3527 Fig. 2’s steepest tangent cone, the steepest whose normal-force slope the method could read; since the milestone M1.8e11, which took the cone slopes to 30°, the tables reach CONE_TABLE_CAP_RAD, and 24° is kept because that is as far as the march carries the handover, not as far as the tables do (ADR-043: what the handover’s cap is worth).

Functions

handover_angle_capped_rad

The handover slope at Mach mach under a cap of cap_rad rather than the flown MAX_HANDOVER_RAD, rad: the lesser of the wedge’s largest deflection and the cap. What ADR-043 sweeps; a body takes it through crate::shock_expansion::ShockExpansionBody::with_handover_cap_rad.

handover_angle_rad

The slope at which the cap hands over to the shock-expansion method at Mach mach, rad: the lesser of the wedge’s largest deflection (wedge_detachment_angle_rad, TN D-4865 p. 5) and MAX_HANDOVER_RAD, the cap hpr flies.

handover_loading

The loading just behind a handover that holds still in the wind, Λ = λ/(γM²), with λ = 2γp/sin 2μ at the handover’s pressure pressure_ratio (times the free stream’s) and surface Mach number surface_mach, and M the free stream’s: the Prandtl–Meyer flow’s ∂p/∂ν over the free stream’s γM²/2, halved as TN 3527’s loading is. It is TN D-4865’s equivalent bodies (eqs. 4a and 4b, the body turned about the sphere’s center) read at α → 0, hpr’s reading (crate::shock_expansion::HandoverStart::Newtonian).

newtonian_loading

The cap’s loading at α → 0 where its slope to the axis is slope_rad, Λ = C_p,max sin δ cos δ: half the windward ∂C_p/∂α of C_p = C_p,max sin²(δ + α cos φ), in the units of TN 3527’s loading (crate::shock_expansion).

newtonian_pressure_coefficient_max

Modified Newtonian C_p,max = (p_t2/p₀ − 1)/(γM²/2) at Mach mach.

newtonian_pressure_ratio

The cap’s surface pressure over the free stream’s where its slope to the wind is slope_rad (TN D-4865 eq. 1): (p_t2/p₀ − 1) sin²δ + 1.

newtonian_surface_mach

The cap’s surface Mach number where its pressure is pressure_ratio times the free stream’s, isentropic from the pitot pressure (TN D-4865 eq. 2); zero at the stagnation point. The report’s march starts from it (crate::shock_expansion::HandoverStart::Newtonian).

pitot_pressure_ratio

The pitot pressure behind a normal shock over the free stream’s, p_t2/p₀, at Mach mach (the Rayleigh pitot formula, NACA Report 1135 eq. 100, p. 619; TN D-4865 eq. 1).

wedge_detachment_angle_rad

The largest angle a two-dimensional wedge can turn the flow at Mach mach behind an attached shock, rad: the shock angle of NACA Report 1135 eq. 168 (p. 624) put into eq. 138 (p. 621).