Module crossflow 

Body lift: the viscous crossflow term of Jorgensen’s method for bodies of revolution at an angle of attack (L. H. Jorgensen, NASA TR R-474, 1977), and Galejs’s constant it replaces.

At an angle of attack α the air crosses the body sideways at V sin α, separates behind it as it would behind a cylinder in a cross-wind, and pushes it with the drag of that crossflow:

C_N = η C_dn (A_p/A_r) sin² α (TR R-474 eq. 2.12, printed p. 10),

with A_p the body’s planform (side-view) area, A_r the reference area, C_dn the crossflow drag coefficient of an infinitely long circular cylinder and η the ratio of a finite cylinder’s crossflow drag to an infinite one’s. Both depend on the crossflow Mach number M_n = M sin α (eq. 2.3, p. 8); η also on the body’s length over its diameter. The force acts at the planform’s centroid (eq. 2.21, p. 13). hpr takes each factor from Jorgensen’s figures, read by hand from the page images:

• C_dn (CROSSFLOW_DRAG, Fig. 1, printed p. 75) below the critical crossflow Reynolds number, where “C_dn = 1.2” at low M_n (p. 15). From M_n 0.6 to 1.2 it takes the filled points “extrapolated from data obtained in Ames 2’ × 2’ wind tunnel”, the values Fig. 6 was divided by (below); past 1.4, the faired curve through the experiments, to 4.8.
• η against length over diameter (ETA_BY_FINENESS, Fig. 4, printed p. 77): the circular cylinder at a crossflow Reynolds number of 88,000, measured “only at very low subsonic Mach numbers” (p. 17).
• η against M_n (ETA_BY_CROSSFLOW_MACH, Fig. 6, printed p. 78): Jorgensen’s η C_dn back-computed from the measured normal force of two bodies of fineness 10 and 12 at 45° to 60° (his Fig. 5), divided by Fig. 1’s C_dn, at the eleven crossflow Mach numbers from 0.4 to 1.6 he computed; below 0.4 it runs to Fig. 4’s value for those bodies. He uses Figs. 5 and 6 “in lieu of better information” (p. 18); past 1.6, η “probably can be assumed to be unity” (p. 17), and hpr holds the last point, 0.984.

Combining the two ηs, a judgement. Fig. 6 holds for bodies of fineness 10 to 12 only. For another fineness f, hpr scales Fig. 6’s η by how much longer or shorter Fig. 4 makes the body, and lets that scaling fade as the crossflow speeds up, by the share s Fig. 6’s own bodies have risen toward 1:

η(f, M_n) = η₆(M_n) [η₄(f) + (1 − η₄(f)) r] / [η₆(0) + (1 − η₆(0)) r],

s = [η₆(M_n) − η₆(0)] / [1 − η₆(0)] and r its running maximum over [0, M_n], with η₆(0) = 0.69, midway between Fig. 6’s starting points for fineness 10 and 12. r never falls back: Fig. 6 dips at M_n = 1 only because Jorgensen divided by Fig. 1’s peak there, not because the body’s length counts again. The rule gives Fig. 6 back for a body of fineness about 10.6 (where this reading of Fig. 4 gives 0.69), Fig. 4 at M_n = 0 for any fineness, and Fig. 5’s η C_dn for every fineness once M_n passes 0.8, where Fig. 6 reaches 0.99. Where r = s, below M_n 0.8, it equals η₄ + (1 − η₄) s.

Sampling, not smoothing. Fig. 1’s C_dn peaks at M_n ≈ 0.96 and Fig. 6’s η dips at 1.0; each is steep there. hpr samples both at Fig. 6’s points and interpolates each linearly between them, so their product is Jorgensen’s own η C_dn at those points (his Fig. 5, within the reading, test the_product_follows_figure_5) and moves smoothly between them, instead of multiplying two steep curves read separately.

Left out. Past the critical crossflow Reynolds number (about 2 × 10⁵, Fig. 2, p. 76) a cylinder’s C_dn falls to “between about 0.15 and 0.30” at low M_n (p. 15); Jorgensen computes that only for illustration, with nothing to check it against (p. 27), and hpr leaves it out. hpr’s potential-flow term stays its own (sin α, slender-body theory or TN 3527’s method), not Jorgensen’s sin 2α cos(α/2).

Galejs’s constant (BodyLift::Galejs): hpr’s body lift until the milestone that sized it (M1.8e6) was K (A_plan/A_ref) sin² α with K = 1.1 at every Mach number (R. Galejs, Wind Instability, after Hoerner; Niskanen 2009 eq. 3.26), kept to reproduce earlier results.

See docs/physics/aero.md (Body lift).

Enums

BodyLift

How a body’s crossflow lift is sized: its C_N = factor · (A_plan/A_ref) sin² α. In JSON, {"kind": "jorgensen"} or {"kind": "galejs", "k": 1.1}.

Constants

CROSSFLOW_DRAG

A circular cylinder’s crossflow drag coefficient C_dn at CROSSFLOW_DRAG_MACHS, below the critical crossflow Reynolds number: NASA TR R-474, Fig. 1 (printed p. 75), read by hand to about ±0.01. To 0.2, the “C_dn = 1.2” of p. 15; to 0.5, the curve through Lindsey’s points; from 0.6 to 1.2, the filled points extrapolated from the Ames 2’ × 2’ tunnel; from 1.4, the curve through the supersonic experiments. Held past 4.8.

CROSSFLOW_DRAG_MACHS

The crossflow Mach numbers M_n = M sin α of CROSSFLOW_DRAG.

ETA_BY_CROSSFLOW_MACH

Jorgensen’s η against the crossflow Mach number for bodies of fineness 10 and 12, at ETA_MACHS: NASA TR R-474, Fig. 6 (printed p. 78), the circles “computed from figures 1 and 5”, read by hand to about ±0.005. At 0, ETA_REFERENCE. Held past 1.6.

ETA_BY_FINENESS

A finite circular cylinder’s crossflow drag over an infinite one’s, η, at ETA_FINENESS, at very low crossflow Mach number: NASA TR R-474, Fig. 4 (printed p. 77), the curve for a circular cylinder at a crossflow Reynolds number of 88,000 (from Goldstein), read by hand to about ±0.005. Held outside 2 to 40.

ETA_FINENESS

The fineness ratios (length over diameter) of ETA_BY_FINENESS.

ETA_MACHS

The crossflow Mach numbers of ETA_BY_CROSSFLOW_MACH: M_n = 0 and Fig. 6’s eleven computed points.

ETA_REFERENCE

Fig. 6’s η at M_n = 0: 0.69, midway between its square and diamond there, about 0.68 and 0.70, which Jorgensen takes from Fig. 4 for its two bodies of fineness 10 and 12 (this module’s own reading of Fig. 4, ETA_BY_FINENESS, gives 0.685 and 0.701).

Functions

crossflow_drag

A circular cylinder’s crossflow drag coefficient C_dn at crossflow Mach number crossflow_mach, below the critical Reynolds number (CROSSFLOW_DRAG). A negative or NaN input reads as 0.

crossflow_eta

η for a body of fineness fineness at crossflow Mach number crossflow_mach: Fig. 6’s value scaled by Fig. 4’s for the body’s length, the scaling fading as Fig. 6 rises toward 1 (see the module’s Combining the two ηs).

crossflow_eta_low

Fig. 4’s η for a body of fineness fineness, at low crossflow Mach number (ETA_BY_FINENESS).

crossflow_factor

Jorgensen’s η C_dn for a body of fineness fineness at crossflow Mach number crossflow_mach: the factor on (A_plan/A_ref) sin² α in its body lift.

crossflow_factor_from_eta_low

crossflow_factor from the body’s Fig. 4 η (crossflow_eta_low), which a model computes once.