Module fins 

Fin sets: Barrowman’s subsonic normal-force slope and center of pressure, with the Prandtl–Glauert factor; supersonic linear theory and the transonic join between them (FinAero); the fin-count and roll terms, and fin–body interference.

• One fin (Diederich’s planform correlation as Barrowman applies it; Barrowman 1967 eq. 3-6, Niskanen 2009 eq. 3.40): (C_Nα)₁ = 2π (s²/A_ref) / (1 + √(1 + (β s² / (A_fin cos Γ_c))²)), β = √(1 − M²), with s the span from the body surface, A_fin one fin’s area and Γ_c the mid-chord sweep. At M = 0 it is Barrowman 1966 eq. 50 (eq. 57 for a trapezoid, where s²/(A_fin cos Γ_c) = 2ℓ/(c_r + c_t)). At M → 1 it tends to π s²/A_ref.
• Mean aerodynamic chord (Niskanen eq. 3.30–3.32): c̄ = (1/A)∫c² dy, y_MAC = (1/A)∫y c dy, x_MAC,LE = (1/A)∫x_LE c dy, and the center of pressure at the quarter chord X_f = x_MAC,LE + c̄/4, fixed through subsonic flow (Barrowman 1967 p. 6). For a trapezoid these give Barrowman 1966 eq. 76a (Niskanen eq. 3.34); for an ellipse on its root chord X_f = (½ − 2/(3π)) c_r.
• Freeform fins (Niskanen pp. 27–29): the chord runs from the leading edge to the trailing edge, so the gap of a jagged edge counts toward the center of pressure but not toward the area in (C_Nα)₁; Γ_c is the span average of the angle between the mid-chord points.
• N fins (Niskanen eq. 3.51–3.53, OpenRocket technical documentation 13.05 eq. 3.54): a fin at angle Λ to the lateral airflow adds (C_Nα)₁ sin² Λ in the plane of the flow, and Σ sin² Λ_k = N/2 for three or more even fins. Fin–fin interference scales 5, 6, 7 and 8 fins by 0.948, 0.913, 0.854 and 0.810: six and eight fins give 1.37 and 1.62 times four fins (MIL-HDBK-762(MI) p. 5-24), five and seven are interpolated. More than eight fins have no source and are refused.
• Fin–body interference (Barrowman 1966 eq. 77, Niskanen eq. 3.56): K_T(B) = 1 + r_t/(s + r_t), with r_t the body radius at the fins.
• Supersonic (FinOutline::supersonic; Barrowman 1967 appendix A, first order): the flat plate’s load 4α/β, β = √(M² − 1), halved inside the tip’s Mach cone, with the root a reflection plane: (C_Nα)₁ = (4/β)(A_fin − A_cone/2)/A_ref at the load’s centroid.
• Through Mach 1 (FinAero): the subsonic method to Mach 0.8, linear theory from M_s = max(1.2, 1/cos Γ_L, 1/cos Γ_T, √(1 + 1/A²), √(1 + (c_t/2s)²)), and slope and CP linear in M between (ADR-027, the normal force through Mach 1).

See docs/physics/aero.md.

Structs

FinAero

One fin’s normal force through the speed regimes: its geometry and outline, and the two ends of its transonic join, computed once.

FinGeometry

A fin’s aerodynamic geometry.

FinLoading

One fin’s normal-force slope and center of pressure at one Mach number.

FinOutline

A fin’s outline in its own plane, for supersonic linear theory: a simple polygon of [x, y] vertices, m, with x aft of the root leading edge and y out from the root, closed along the root.

FinRoll

One fin’s rolling moment at one Mach number, about the body axis, on the reference area A_ref and diameter d (FinAero::roll); the body’s interference is the fin set’s (roll_forcing_interference, roll_damping_interference).

FinRollTerms

One fin’s roll terms on one body that don’t change with Mach (FinAero::roll_terms): its span moments about the axis and the ends of the transonic join, built once per fin set.

Constants

SUPERSONIC_START_MACH

The lowest Mach number for supersonic linear theory: Mach 1.2, the bottom of the supersonic region (Niskanen 2009 Table 3.1, p. 19).

TRANSONIC_START_MACH

Where the fin slope and CP leave the subsonic method: the top of the subsonic region, Mach 0.8 (Niskanen 2009 Table 3.1, p. 19).

Functions

fin_count_factor

Fin–fin interference factor for count fins in one set (OpenRocket technical documentation 13.05 eq. 3.54, from MIL-HDBK-762(MI) p. 5-24): 1 up to four fins, then 0.948, 0.913, 0.854 and 0.810.

interference_factor

Fin–body interference factor K_T(B) = 1 + r_t/(s + r_t) (Barrowman 1966 eq. 77; Niskanen 2009 eq. 3.56), with s the span from the body surface and r_t the body radius at the fins.

roll_damping_interference

The body’s interference with the roll damping (Barrowman 1967 eq. 3-122 and 3-123), with τ = (s + r_t)/r_t and λ = c_t/c_r, for a chord falling linearly from root to tip:

roll_forcing_interference

The body’s interference with the roll forcing of canted fins (Barrowman 1967 eq. 3-95 and 3-105, from slender-body theory, his reference 23), with τ = (s + r_t)/r_t:

roll_sum

Σ sin² Λ_k over count evenly spaced fins, where Λ_k is the angle from the lateral airflow to fin k (Niskanen 2009 eq. 3.51–3.53). The first fin is at base_angle_rad and the airflow at flow_roll_rad, both from x_B toward y_B. Three or more fins give exactly N/2 at any roll.

side_sum

Σ sin(φ − θ_k) cos(φ − θ_k) over count evenly spaced fins at θ_k in a lateral airflow at φ: the side-force share, perpendicular to the flow’s plane. Each fin sees the local angle α sin Λ_k (Niskanen 2009 eq. 3.50) and pushes along its own normal; eq. 3.51 keeps the part of that push in the flow’s plane, sin² Λ_k, and this is the part across it. The sum vanishes for three or more fins; for one or two it doesn’t: two fins at 45° to the flow push along their common normal, √2 times their in-plane share. Derived here from eq. 3.50; Niskanen drops it.