Module tube_fins 

Tube fins: a ring of short open tubes around the body, each flown as an annular wing (a ring airfoil).

Normal force. One tube of mean diameter d and length L, with λ = L/d, takes Weissinger’s approximation for a thin ring wing (Weissinger 1955, as quoted by Wagner 2021 eq. 15), on the area d L:

C_Lα = π² / (1 + πλ/2 + λ arctan(1.2 λ)) per radian.

Short rings tend to Ribner’s lifting-line result π² (Wagner eq. 13), and long ones to slender-body theory’s π/λ, which is Hoerner’s L = q d² π α for a ring of small aspect ratio (Hoerner 1965 p. 7-13): the ring deflects the air inside it as well as the air around it, so it lifts twice as much as a solid body of its diameter. Fletcher’s measured slopes on five rings (NACA TN 4117, 1957, Fig. 11, at Mach 0.13) lie within 3% of it, taken at his diameter (the rings’ inner one) and on his area; for a paper tube the inner and mean diameters differ by about 1%. Wagner gives the formula for λ < 5; past that it runs on to the slender-body limit, which is exact for a long ring.

Compressibility follows Göthert’s rule, as Barrowman’s fin slope does: the slope at Mach M is the incompressible slope of the ring stretched to λ/β, over β = √(1 − M²). It leaves the slender limit unchanged and turns the lifting-line one into Prandtl–Glauert’s. Nothing measures tube fins near the speed of sound, where the flow through a tube may choke, so the model refuses Mach TUBE_FIN_MACH_LIMIT and above.

Center of pressure. Against the ring’s aspect ratio A = d/L, at the stretched ring’s β A faster than Mach 0 (ring_center_fraction): Fletcher’s measured aerodynamic center from A = 2/3 to 3 (FLETCHER_AERODYNAMIC_CENTER, his Fig. 8); below A = 2/3, a straight line to the leading edge at A = 0. That end point is hpr’s derivation from slender-body theory, in which a section’s lift is the growth of its apparent mass along the body: a thin ring’s appears whole at its leading edge and stays, so all its lift is there. Hoerner and Borst (Fluid-Dynamic Lift, 1985, p. 19-16) assume the same of the air turned inside an open tube, that it turns “at or near the rim of the inlet”; they had no measurement of it. Fletcher’s fifth ring, at A = 1/3, is left out, a judgement: its center sits 0.11 of its chord ahead of its leading edge, which he puts down to its low aspect ratio making it act like a body of revolution (p. 4). hpr infers, beyond his text, that its thick section (a Clark Y 11.7% of a chord three bores long, outside a straight bore, so walls 0.35 of the bore thick) is what makes it so, that a paper tube’s center lies aft of it, and that his thinner-walled rings may carry the same forward bias in smaller measure. Holding his point below A = 1/3 instead put OpenRocket’s Tube fin rocket at a margin of 0.29 calibres rather than 0.79, in a one-off run not kept in the report. No thin tube’s center is measured. Rings shorter than a third of their diameter, past A = 3, are refused.

The set. N tubes add N times one tube’s slope, with no interference from the body or between the tubes: none is measured. The body’s own crossflow disturbance at the tubes goes as R²/s² e^{−2iφ} around it, which sums to zero over three or more tubes evenly spaced, so the model refuses fewer than three. That is a first-order derivation, not a measurement: it takes the body’s flow at each tube’s center and leaves out the images and the lift carried onto the body. Slender-body theory with the body included gives the set more lift than N isolated rings, by an unchecked estimate 1.13 to 1.96 times on three OpenRocket probes, at a gap of 0.005 radii and still rising as it closes (#234; docs/physics/aero.md, Tube fins).

Structs

TubeFinSetAero

A tube fin set’s precomputed terms.

Constants

FLETCHER_AERODYNAMIC_CENTER

Fletcher’s measured aerodynamic center of five annular airfoils, (A, x_ac/c): the aspect ratio A = d/c (diameter over chord) and the aerodynamic center’s distance aft of the leading edge as a fraction of the chord, from α = 0° to 10° at Mach 0.13 (NACA TN 4117, 1957, Fig. 8, p. 16). Read from the chart, two independent readings within 0.003 of the chord, and checked against the text: the center moves aft as A rises, and sits ahead of the leading edge at A = 1/3 (p. 4).

THIN_RING_CENTER

The points ring_center_fraction joins, (A, x_ac/c): slender-body theory’s leading edge at A = 0, then Fletcher’s four rings that act as wings (FLETCHER_AERODYNAMIC_CENTER from A = 2/3).

TUBE_FIN_MACH_LIMIT

The top of the tube-fin model’s range, Mach 0.8, where hpr’s fin model leaves its subsonic method (crate::fins::TRANSONIC_START_MACH). No source covers tube fins faster; a judgement.

Functions

ring_center_fraction

A thin ring’s aerodynamic center aft of its leading edge, as a fraction of its length, at an aspect ratio A = d/L: THIN_RING_CENTER interpolated linearly in A, and held at A = 3 beyond it (TubeFinSetAero::new refuses a ring that short).

ring_lift_slope

A thin ring wing’s normal-force slope per radian on the area d L, at a length-to-diameter ratio λ = L/d: Weissinger’s π² / (1 + πλ/2 + λ arctan(1.2 λ)) (Wagner 2021 eq. 15).