Module supersonic_boattail 

A conical boattail’s share of the normal force faster than sound, measured: W. D. Washington and W. Pettis Jr., Boattail Effects on Static Stability at Small Angles of Attack, U.S. Army Missile Command report RD-TM-68-5 (1968; DTIC AD-695658, approved for public release).

Washington and Pettis mounted a model’s aft section on its own balance and measured it with a boattail and as a plain cylinder of the same length, at Mach 1.75 to 4.5 (models T0 to T4), and a whole model with and without a boattail (No. 1) from Mach 0.8 to 4.5, Mach 0.8 to 1.5 of it in a second tunnel (pp. 1 to 2). The boattail’s increment, ΔC_Nα = C_Nα(with) − C_Nα(without) (eq. 1), correlates with

ΔC_Nα / [1 − (D_B/D)²] = F(√|M² − 1| / (L_B/D)) (Fig. 5, printed p. 8),

per degree on the cylinder’s area πD²/4, with D the cylinder’s diameter, D_B the boattail’s base diameter and L_B its length. Its center of pressure is “approximately 50 percent of its length” (abstract), from about 43% at Mach 2 to 64% at 4.5 (Fig. 6, printed p. 9). hpr reads both figures by hand from the page images (WP_PARAMETERS, WP_SLOPE_PER_DEG, WP_CENTER_MACHS, WP_CENTER_FRACTION).

What it covers. The models were conical boattails “with diameter ratios of 0.72 to 0.86 and angles from 4 to 10 degrees” (p. 1), 0.82 to 1.18 diameters long, behind a cylinder; the correlation’s points reach the peak near zero argument, with the next near 0.3 and the rest out to about 5.4; the curve is drawn to 6.1. Past 6.1 hpr holds its end, an extrapolation, as is any boattail steeper, shorter, longer or narrower than those tested: a long one reads the curve near zero argument, which comes from the report’s lowest supersonic runs. The data scatter about the curve by up to about 15%. Slender-body theory gives a boattail 2[(D_B/D)² − 1] per radian at any Mach number (Munk’s value, which the report plots for comparison at subsonic speeds, p. 3). The measured curve runs from 0.23 to 1.58 times it, crossing at an argument of 0.635; it is 0.24 to 0.47 of it at the Arcas Robin’s Mach numbers.

See docs/physics/aero.md (The body faster than sound in a flight).

Constants

WP_CENTER_FRACTION

The boattail’s center of pressure as a share of its length from its fore end, at WP_CENTER_MACHS: the middle of RD-TM-68-5 Fig. 6’s band (printed p. 9), read by tracing its two edges to about ±0.01; the band is about ±0.03 wide. Held outside Mach 1.5 to 4.8.

WP_CENTER_MACHS

The Mach numbers of WP_CENTER_FRACTION.

WP_PARAMETERS

The argument of Fig. 5’s correlation, √(M² − 1)/(L_B/D), at WP_SLOPE_PER_DEG’s rows.

WP_SLOPE_PER_DEG

ΔC_Nα / [1 − (D_B/D)²], per degree on the cylinder’s area, at WP_PARAMETERS: RD-TM-68-5 Fig. 5, the supersonic side of the faired curve, read by tracing it on a 300-dpi scan to about ±0.0003 per degree (±0.02 per radian). At 0 (Mach 1) the curve peaks near −0.055, past the axis’s last tick; from there to about 1.2 it follows model No. 1’s points, the report’s lowest supersonic runs. Their Mach numbers are not read off here: the report gives model No. 1 as Mach 0.8 to 4.5, 0.8 to 1.5 of it in the AEDC tunnel (p. 1).

Functions

wp_center_fraction

The boattail’s center of pressure at Mach mach, as a share of its length from its fore end (WP_CENTER_FRACTION).

wp_parameter

Fig. 5’s argument, √(M² − 1)/(L_B/D), for a boattail length_over_diameter long in its fore diameters at Mach mach (at least 1).

wp_slope

A conical boattail’s normal-force slope increment at α → 0, per radian on the area π fore_radius_m², at Mach mach (at least 1): negative, from Fig. 5’s correlation (WP_SLOPE_PER_DEG).