Module afterbody 

The afterbody faster than sound: a boattail’s own pressure drag, and the base pressure behind it.

A boattail is a transition that narrows toward the tail. Below Mach 0.8 it keeps Niskanen’s rule, a share of the base drag on its decrease in area (Niskanen 2009 eq. 3.88, crate::drag::boattail_factor). Faster than sound the air expands around the boattail’s shoulder, its pressure falls below the free stream’s, and it pulls back on the boattail: a wave drag the rule doesn’t have. Behind the boattail the base’s pressure rises, which lowers the base drag. On the boattail’s fore (cylinder) area A₁ = π d₁²/4, for a boattail of length l from diameter d₁ to d₂, area ratio a = (d₂/d₁)² and half-angle θ = atan((d₁ − d₂)/(2l)):

• Attached flow, from Mach 1 (Boattail::attached_pressure_drag): MIL-HDBK-762’s chart for conical boattails (Fig. 5-122, printed p. 5-187), y = 4 C_D (l/d₁)² against x = √(M² − 1)/(2 l/d₁) for a from 0.25 to 0.80, read into conical_boattail_chart. The handbook cites no source for it; its values agree with Jack’s second-order theory (NACA TN 2972, 1953) within −10% to +8% for a up to 0.6 (validation/fixtures/aero/ measured-boattails.json). It is held to the 2D limit C_PM = −C_p,PM(M, θ)(1 − a): the pressure behind a Prandtl–Meyer expansion through θ (expansion_pressure_coefficient) over the whole annulus. On an axisymmetric boattail the pressure recovers aft of the shoulder, so the drag stays below that limit and approaches it as the boattail gets short against β d₁ (the quasi-cylinder solution’s recovery falls as 1/x). Past the chart’s end at x = 1.4, the drag closes the chart’s gap to the limit as 1/x: C_D = [1 − (1 − r) 1.4/x] C_PM(M), with r the chart’s share of the limit at x = 1.4 (at most 1).
• Separation (Boattail::separation_weight): steep boattails separate. Cubbage’s boattails (NACA RM L57B21, 1957, Mach 0.6–1.28) stay attached at 16° and separate completely by 30°, and a separated boattail sees about a cylinder’s base pressure. Between 16° and 30° the drag moves linearly in θ from the attached value to the base drag coefficient on the annulus, C_D,base(M)(1 − a).
• Through Mach 1: MIL-HDBK-762 finds no method for boattails at transonic speeds and advises extrapolating the supersonic drag “to peak value at a Mach number range of 1.0 ≤ M∞ ≤ 1.2, with a sharp reduction to a lower value at subsonic speeds” (p. 5-47). The chart’s near-sonic end is not used: from Mach 1 to 1.2 the attached drag is held at its Mach 1.2 value, below Mach 1 the drag falls on a straight line to the rule’s value at Mach 0.8, where the buildup’s other transonic terms start (Niskanen p. 47), and the rule holds below. The line is half-way up at Mach 0.9; measured boattails are half-way up by 0.89 (Compton, NASA TN D-6789) to 0.92 to 0.96 (Cubbage), peak at Mach 1.0 to 1.1, and at 1.2 are 0.83 to 0.90 of that peak (Cubbage), so holding the Mach 1.2 value reads under the peak.
• The base behind a boattail (boattail_base_pressure_ratio): MIL-HDBK-762 Fig. 5-141 (printed p. 5-210, after Rubin, Brazzel and Henderson, 1970) correlates a boattail’s base pressure with a cylinder’s at Mach 2.5 to 3.5 as p_cyl/p_bt = 0.442 + 0.558 a_b, with a_b the base’s area over the cylinder’s. hpr takes the cylinder’s pressure from Love’s correlation (Fig. 5-139, printed p. 5-208; NACA TN 3819) and scales its own base drag by the ratio of the two coefficients, k = C_p,bt/C_p,cyl. Below Mach 2.5, where the correlation over-predicts the relief of the measured bases, k is held at its Mach 2.5 value, which matches Cortright and Schroeder’s at Mach 1.91 and de Moraes and Nowitzky’s at 1.59; below Mach 1 it returns to 1 by Mach 0.8, where the base drag is Niskanen’s again. A separated boattail gives no relief (weight as above).

Narrowing parts of one smooth surface drag as one cone (crate::drag::BoattailTerm), and a lip behind a boattail sits in its wake (crate::drag::WakeTerm); every part between a boattail and what follows fades both, and the base’s relief, so each is continuous in the geometry.

Outside the data: the chart below a = 0.25 (Jack’s theory reaches 0.2), attached flow up to 16° where Jack’s theory stops at 11°, Cubbage’s separation angles (measured to Mach 1.28) at every Mach number, and Fig. 5-141 outside its bases’ area ratios of 0.25 to 0.67 and angles to 14°. How well each piece agrees with the measurements is in the guide.

Calisto’s boattail, 0.472 calibres long from a = 1 to 0.469 (18.4°), at Mach 1.5: the chart gives 0.219 and the 2D limit 0.211, so the attached drag is 0.211; separation takes it 17% of the way to the base’s 0.088, to 0.190.

use hpr_aero::afterbody::Boattail;

let calisto = Boattail::new(0.06, 0.127, 0.087)?;
assert!((calisto.attached_pressure_drag(1.5)? - 0.211).abs() < 5e-4);
assert!((calisto.pressure_drag_coefficient(1.5)? - 0.190).abs() < 5e-4);

See Boattails faster than sound in the guide.

Structs

Boattail

A boattail’s geometry and the terms of its pressure drag that don’t depend on the Mach number. Coefficients are on its fore area π d₁²/4.

Constants

BASE_RELIEF_MACH

The lowest Mach number of MIL-HDBK-762 Fig. 5-141’s correlation, 2.5; below it the base pressure ratio is held.

CHART_AREA_RATIOS

The chart’s curves, by area ratio (d₂/d₁)².

CHART_X

The chart’s abscissae x = √(M² − 1)/(2 l/d₁) at which CHART_Y was read.

CHART_Y

MIL-HDBK-762 Fig. 5-122 (printed p. 5-187, PDF p. 425), “Wave-Drag Coefficient of Conical Boattails at Supersonic Speeds”: y = 4 C_D (l/d₁)² on π d₁²/4, one row per CHART_AREA_RATIOS at CHART_X. Read from a 200-dpi render with the grid fitted for tilt, each curve fitted in ln y against ln x and checked on an overlay: ±(0.005 + 2%), the lines being about 0.017 thick. The 0.80 curve’s reading rises by 0.001 past x = 1.2, where the printed line doesn’t; it is held at 0.0332. The chart starts at x = 0.05, where the readings hook; they start at 0.06.

LOVE_BASE_PRESSURE

Love’s correlation of turbulent base pressure behind cylinders, −C_p,b against Mach number (MIL-HDBK-762 Fig. 5-139, printed p. 5-208, the solid line, after NACA TN 3819), read at Mach 2.5 to 5: ±0.001 (±0.0015 at Mach 3, where symbols hide the line).

MAX_TURNING_RAD

The largest turning angle of a Prandtl–Meyer expansion from Mach 1, to a vacuum: (π/2)(√((γ + 1)/(γ − 1)) − 1), 130.45° (NACA Report 1135, 1953, eq. 172, p. 626).

SEPARATION_COMPLETE_RAD

The boattail half-angle from which the flow is separated: 30°, Cubbage’s shallowest separated boattail (NACA RM L57B21).

SEPARATION_ONSET_RAD

The boattail half-angle up to which the flow stays attached: 16°, Cubbage’s steepest attached boattail (NACA RM L57B21).

SUPERSONIC_MACH

Where the transonic rise ends: from Mach 1 a boattail takes its supersonic drag (held to SUPERSONIC_MODEL_MACH’s value) and its base the supersonic relief.

SUPERSONIC_MODEL_MACH

The lowest Mach number at which the supersonic boattail drag is evaluated, 1.2, the top of MIL-HDBK-762’s “peak value” range (p. 5-47); from Mach 1 to 1.2 its value there is held.

TRANSONIC_ONSET_MACH

Where a boattail’s drag starts its transonic rise and its base its relief: Mach 0.8, where the rest of the buildup starts its transonic methods (crate::drag::SUBSONIC_MACH_LIMIT, Niskanen 2009 p. 47). An earlier draft used 0.9, chosen after seeing the Arcas Robin wind tunnel, a target (the decision record on the afterbody, ADR-030).

Functions

boattail_base_pressure_ratio

The base pressure behind a boattail over a cylinder’s, as a ratio of pressure coefficients k = C_p,bt/C_p,cyl that scales the base drag, for a base of area ratio a_b (its area over the boattail’s fore area) and an attached boattail (MIL-HDBK-762 Fig. 5-141, printed p. 5-210):

conical_boattail_chart

MIL-HDBK-762 Fig. 5-122’s y = 4 C_D (l/d₁)² for a conical boattail of area ratio a = (d₂/d₁)² at x = √(M² − 1)/(2 l/d₁) from 0 to 1.4 (CHART_Y): log-log in x between the readings, held below x = 0.06, and linear in a between the curves. Past the chart’s last curve, a > 0.8, it goes to 0 at a = 1 as (1 − √a)² (linear theory’s pressure is proportional to the surface slope, so at a fixed length the drag goes as the slope squared); below its first, a < 0.25, it continues the straight line through the 0.25 and 0.30 curves.

expansion_pressure_coefficient

The pressure coefficient behind a two-dimensional isentropic (Prandtl–Meyer) expansion of a flow at Mach mach ≥ 1 through turn_rad: M₂ from ν(M₂) = ν(M) + θ, then C_p = (p₂/p − 1)/(γ M²/2) with p₂/p = [(1 + (γ−1)M²/2)/(1 + (γ−1)M₂²/2)]^(γ/(γ−1)) (NACA Report 1135: the pressures from eq. 44, the dynamic pressure γ p M²/2 from eq. 31b, p. 616). Past the largest turning angle the flow reaches a vacuum, C_p = −2/(γ M²).

prandtl_meyer_angle

The Prandtl–Meyer function ν(M) = √((γ + 1)/(γ − 1)) atan √((γ − 1)(M² − 1)/(γ + 1)) − atan √(M² − 1), rad: the angle through which a flow at Mach 1 turns, expanding, to reach M (NACA Report 1135, 1953, eq. 171c, p. 626).