V Tail Sizing

V-Tail Sizing (vehicle-design/sizing/v-tail-sizing)

Use when the task is sizing a V-tail (butterfly or vee tail) empennage from equivalent horizontal and vertical tail volume requirements. The V-tail is the single canted pair of aft surfaces that replaces the conventional separate horizontal and vertical tails, combining the two tail volume requirements on one surface pair whose ruddervators blend the elevator and rudder functions. This leaf implements the planform-area projection convention in pure Python, stdlib only, deterministic. It pairs with vehicle-design/sizing/tail-sizing, which owns the conventional separate-surface volume coefficients and their typical ranges, and with vehicle-design/sizing/canard-sizing for the forward surface; control-surface-sizing owns elevator and rudder areas from moment requirements, and wing-planform-sizing owns the wing geometry the volume coefficients reference.

Domain quick reference

• Volume coefficient definition (inverse form): the required equivalent tail area for a target coefficient is S = V * S_ref * ref_len / tail_arm. The horizontal requirement uses the wing reference area S_ref, the wing reference chord c_bar and the tail arm l_h: S_h = V_h * S_ref * c_bar / l_h. The vertical requirement uses the wing span b and the tail arm l_v: S_v = V_v * S_ref * b / l_v.
• Projection convention (documented method, Raymer-style projected area for the volume coefficient formulas, name and paraphrase only): with the two panels canted up at the dihedral angle Gamma from the horizontal, the horizontal equivalent area entering the horizontal formula is the sum of the horizontal projections, S_h_eff = S_vt * cos(Gamma), and the vertical equivalent area is S_v_eff = S_vt * sin(Gamma). The cos^2/sin^2 loading convention is NOT used here.
• Vector-sum inversion: under that convention, S_vt = sqrt(S_h^2 + S_v^2) and Gamma = atan2(S_v, S_h), measured from the horizontal plane, are exact. The equal panel split makes the per-surface projections (S_vt / 2) * cos(Gamma) and (S_vt / 2) * sin(Gamma).
• Panel geometry: each panel carries area_per_surface = S_vt / 2 and is treated as a flat surface with its own aspect ratio (SURFACE_ASPECT_RATIO = 4.0 default), so span_per_surface = sqrt(aspect_ratio * area_per_surface) and chord_per_surface = area_per_surface / span_per_surface, the mean chord of one panel.
• Ruddervators: total movable control area = RUDDERVATOR_FRACTION * S_vt, with RUDDERVATOR_FRACTION = 0.35 (the documented engineering default), half on each panel. Area fraction only, no hinge-geometry output.
• Effective volume round trip: v_h_eff = S_vt * cos(Gamma) * l_h / (S_ref * c_bar) and v_v_eff = S_vt * sin(Gamma) * l_v / (S_ref * b). Met verdicts compare against the targets with VOLUME_TOL = 1e-9, which absorbs the cos/sin/atan2 round-trip error of order 1e-16.
• Units are SI throughout: m, m^2, rad, deg.
• FAR-25 frames the certification context for the empennage design requirements; the relations above are standard engineering methodology, summary-only.

Workflow

1. Gather the aircraft and requirement data: the wing reference area S_ref, the wing reference chord c_bar, the wing span b, the tail arms l_h and l_v (wing to V-tail root), and the target volume coefficients V_h and V_v. Pick targets inside the typical ranges quoted by the conventional tail-sizing leaf when starting from scratch.
2. Convert each target volume coefficient into the required equivalent area with tail_area_from_volume_coefficient: S_h = tail_area_from_volume_coefficient(V_h, c_bar, l_h, S_ref) for the horizontal requirement and S_v = tail_area_from_volume_coefficient(V_v, b, l_v, S_ref) for the vertical requirement.
3. Resolve the two equivalent areas onto the canted pair with vtail_geometry(S_h, S_v): read the total V-tail area S_vt, the dihedral angle gamma_rad and gamma_deg (from the horizontal), the equal panel split area_per_surface, and the per-surface span_per_surface and chord_per_surface at SURFACE_ASPECT_RATIO.
4. Size the ruddervators with ruddervator_sizing(S_vt, control_fraction): the default 0.35 fraction of the total V-tail area, split half per surface.
5. Verify the effective volume round trip with effective_volume_check(S_vt, gamma_rad, V_h, V_v, S_ref, c_bar, b, l_h, l_v): the projected S_h_eff and S_v_eff recover the targets within VOLUME_TOL, and read the met verdicts v_h_met and v_v_met. If either flag is False the tail is undersized for its requirement at the given arm.
6. Confirm the deterministic checks with the contract test python3 scripts/test_v_tail_sizing.py.

Worked example

Light aircraft: S_ref = 16 m2, c_bar = 1.5 m, b = 11 m, tail arms l_h = l_v = 4.5 m, targets V_h = 0.7 and V_v = 0.04 (both inside the typical ranges quoted by the tail-sizing leaf).

• Required equivalent areas (step 2): S_h = 0.7 * 16 * 1.5 / 4.5 = 3.73333 m2 and S_v = 0.04 * 16 * 11 / 4.5 = 1.56444 m2.
• V-tail resolution (step 3): S_vt = sqrt(3.73333^2 + 1.56444^2) = 4.04787 m2, gamma_rad = atan2(1.56444, 3.73333) = 0.396818 rad, gamma_deg = 22.7360 deg from the horizontal. The included vee angle between the two panels is 2 * Gamma = 45.472 deg.
• Per-surface geometry (step 3) at aspect ratio 4.0: each panel carries area_per_surface = 2.02394 m2, span_per_surface = sqrt(4.0 * 2.02394) = 2.84530 m, chord_per_surface = 2.02394 / 2.84530 = 0.711325 m.
• Ruddervators (step 4) at the 0.35 fraction: ruddervator_area_total = 0.35 * 4.04787 = 1.41676 m2, ruddervator_area_per_surface = 0.708378 m2.
• Effective volume round trip (step 5): S_h_eff = 3.73333 m2 and S_v_eff = 1.56444 m2 recover v_h_eff = 0.700 (0.6999999999999998) and v_v_eff = 0.04, so v_h_met and v_v_met are both True under VOLUME_TOL. A 10% smaller S_vt = 3.64308 m2 at the same Gamma gives v_h_eff = 0.63, below the tolerance band, and both met flags read False.

Pitfalls

• Using the cos^2/sin^2 loading convention: some V-tail treatments size the horizontal projection with cos^2(Gamma) and the vertical with sin^2(Gamma). This leaf uses the Raymer-style planform-area projection (cos and sin on the total), and its vector-sum inversion and round trip are exact only under that documented convention.
• Treating S_h and S_v as physical tail areas: they are equivalent areas entering the volume coefficient formulas. Building two separate physical surfaces from them, one horizontal and one vertical, is the conventional tail-sizing path, not the V-tail resolution onto the single canted pair.
• Swapping the reference lengths: the horizontal requirement runs on the wing reference chord c_bar (S_v = V_v * S_ref * b / l_v uses the span b). Mixing the two reference lengths corrupts both the dihedral angle and the round trip.
• Rating the met verdicts without the tolerance: the round-trip error is of order 1e-16, so an exact comparison would spuriously fail a correctly sized tail; VOLUME_TOL = 1e-9 absorbs it while a genuinely undersized tail (for example 10% smaller area) still fails both flags.
• Sizing the panel span on the total area: the aspect ratio convention applies to ONE panel of area S_vt / 2, so span_per_surface = sqrt(AR * S_vt / 2). Using S_vt in its place overstates the panel span by sqrt(2).
• Reading the dihedral as the included vee angle: Gamma is measured from the horizontal plane, so the angle between the two panels is 2 * Gamma (45.472 deg in the worked example, not 22.736 deg).

Verification

• Confirm tail_area_from_volume_coefficient(0.7, 1.5, 4.5, 16.0) returns 3.73333 m2 and (0.04, 11.0, 4.5, 16.0) returns 1.56444 m2.
• Confirm vtail_geometry on those areas returns S_vt = 4.04787 m2, gamma_deg = 22.7360, area_per_surface = 2.02394 m2, span_per_surface = 2.84530 m, chord_per_surface = 0.711325 m.
• Confirm the symmetric identity: vtail_geometry(1.0, 1.0) gives gamma_deg = 45.0 and S_vt = sqrt(2) * S_h.
• Confirm the arm identity: doubling the tail arm halves the required equivalent area.
• Confirm the projection identity: S_h_eff^2 + S_v_eff^2 equals S_vt^2 within 1e-12.
• Confirm the round trip recovers the target coefficients within 1e-9 with both met verdicts True, and that a 10% smaller S_vt at the same dihedral gives v_h_eff = 0.63 with both verdicts False.
• Confirm every non-positive area, arm, span, reference quantity, volume coefficient and every control fraction outside (0, 1) raises ValueError.

Related leaves

• skills/vehicle-design/sizing/tail-sizing: the conventional separate-surface horizontal and vertical tail volume coefficients, their required-area inverse and the typical transport ranges that seed the V-tail targets.
• skills/vehicle-design/sizing/canard-sizing: the forward surface alternative to the aft empennage pair.
• skills/vehicle-design/sizing/control-surface-sizing: elevator and rudder control areas sized from pitch and yaw moment requirements rather than from the volume-coefficient equivalence.
• skills/vehicle-design/sizing/wing-planform-sizing: the wing reference quantities (area, chord, span) the volume coefficients are built on.

Contract test

Run the deterministic stdlib unittest offline from the repo root:

python3 skills/vehicle-design/sizing/v-tail-sizing/scripts/test_v_tail_sizing.py

The 34-method suite covers the equivalent-area conversion for both targets with its arm and reference-area scalings and its ValueError rejections, the canted-pair resolution with the vector-sum total, the dihedral in radians and degrees, the equal panel split, the panel span and chord at the aspect ratio, the fixed result keys, the 45 degree symmetric identity and the quadrant bound, the ruddervator fraction sizing with its out-of-range rejections, the effective volume round trip with the met verdicts, the undersized-tail flip, the projection identity and determinism, and the ValueError rejection of every non-physical verification input.

Compliance

• Standards referenced, not reproduced: FAR-25 is the certification context (empennage design and control requirements); the sizing relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 skills/vehicle-design/sizing/v-tail-sizing/scripts/test_v_tail_sizing.py

The test exercises the full SKILL.md Workflow: the worked-example data gathering (step 1), the equivalent-area conversion from the volume targets (step 2), the canted-pair resolution and per-surface geometry (step 3), the ruddervator area fraction (step 4), and the effective volume round trip with its met verdicts (step 5), plus the projection and arm identities and ValueError rejection of every non-physical input. Exit code 0 with all tests passing is the gate.