Delta Wing Vortex Lift

Delta Wing Vortex Lift (aerodynamics/cfd/delta-wing-vortex-lift)

Use when the task is the separated-flow lift of a sharp-edged slender delta wing: the Polhamus leading-edge suction analogy (NASA TN D-3767, 1966) that adds a leading-edge-separation vortex term to the attached potential term. The leaf is the only separated-flow vortex-lift model in the library; it covers the regime the attached-flow siblings exclude. It pairs with aerodynamics/cfd/vortex-lattice-method, which owns the horseshoe-vortex attached-flow linear range, and with aerodynamics/high-lift/high-lift-systems for the mechanical-device alternatives. Valid for sharp leading edges, subsonic flow, aspect ratio about 0.5 to 2.0, alpha up to about 25 degrees. Does not model vortex breakdown onset (empirical charts only, not modeled), circulation control, blown flaps, or ice accretion.

Domain quick reference

• Aspect ratio of a full delta: AR = 4 / tan(Lambda_LE); 76 deg sweep gives about 1.0 and 45 deg gives 4.0.
• Total lift (TN eq. 15): CL = Kp sin(a) cos^2(a) + Kv cos(a) sin^2(a), the sum of the potential term and the vortex term.
• Potential term: CL_pot = Kp sin(a) cos^2(a), with the slender-wing small-angle slope Kp = pi * AR / 2.
• Vortex term: CL_vort = Kv cos(a) sin^2(a), where Kv is the leading edge suction force coefficient, linear from 3.14 at AR 0 to 3.45 at AR 4 (clamped beyond 4).
• Drag due to lift: CD_i = CL * tan(a), the product of the total lift coefficient and the tangent of the angle of attack.
• Crossing angle: tan(a) = Kp / Kv, where the vortex term equals the potential term; slender wings are vortex dominated earlier.
• All angles in degrees on input; CL(0) = 0 exactly.

Workflow

1. Get the aspect ratio: delta_aspect_ratio from the leading-edge sweep angle of the full delta, or supply the AR directly.
2. Compute the model factors: slender_delta_kp for Kp and delta_kv for Kv at that aspect ratio.
3. Split the lift at the angle of attack: polhamus_cl_potential and polhamus_cl_vortex, then polhamus_cl for the total.
4. Find the drag due to lift with cd_due_to_lift on the total lift.
5. Check which term dominates: vortex_potential_crossing_deg for the angle where the vortex term overtakes the potential term.
6. For a one-call breakdown use delta_lift_summary (aspect ratio by default, or pass sweep=True with the leading-edge sweep angle in degrees), returning the aspect ratio, kp, kv, alpha, the lift split, cd_due_to_lift, vortex_fraction and crossing_deg keys.
7. Confirm the deterministic checks with the contract test scripts/test_delta_wing_vortex_lift.py.

Worked example

AR 1.0 delta (Lambda_LE = 76 deg) at alpha = 15 deg, and an AR 0.5 delta (about 82.9 deg sweep).

• delta_aspect_ratio(76) = 0.9973, about 1.0; delta_aspect_ratio(45) = 4.0000.
• slender_delta_kp(1.0) = 1.5708 = pi / 2; delta_kv(1.0) = 3.2175, delta_kv(0) = 3.14, delta_kv(4) = 3.45.
• Alpha 15, AR 1.0: CL_potential = 0.3793, CL_vortex = 0.2082, CL_total = 0.5875, with the vortex term 35.4% of the lift.
• Sweep at AR 1.0: CL(5) = 0.1602, CL(10) = 0.3601, CL(15) = 0.5875, CL(20) = 0.8281, CL(25) = 1.0661, monotone increasing.
• cd_due_to_lift(15 deg) = 0.5875 * tan(15) = 0.1574.
• AR 0.5, alpha 20: CL_total = 0.5866 with the vortex term 59.6% of the lift, about 60% against 35% for the AR 1 case at 15 deg.
• Crossing: AR 1.0 at 26.0 deg, AR 0.5 at 13.9 deg, the slender wing vortex dominated much earlier.
• Identity: CL(0) = 0 exactly.

Verification

• Confirm delta_aspect_ratio(76) about 1.0 and delta_aspect_ratio(45) about 4.0, and that sweeps <= 0 or >= 90 deg raise ValueError.
• Confirm delta_kv anchors 3.14 at AR 0 and 3.45 at AR 4 with linear interpolation between.
• Confirm CL(0) = 0 exactly and CL positive and increasing over 5 to 25 deg at AR 1.0.
• Confirm the vortex fraction grows as AR shrinks (AR 0.5 alpha 20 about 60% against AR 1.0 alpha 15 about 35%).
• Confirm the crossing angle shrinks with AR, and that at the crossing angle the potential and vortex terms are equal.
• Confirm delta_lift_summary returns exactly its documented keys and identical floats run to run.
• Confirm ValueError rejection of non-physical inputs: sweep <= 0 or

  = 90, aspect ratio < 0, kv <= 0 for the crossing angle.

Related leaves

• aerodynamics/cfd/vortex-lattice-method: attached-flow linear sibling; this leaf covers the separated vortex regime it excludes.
• aerodynamics/high-lift/high-lift-systems: mechanical high-lift devices, the attached-flow alternative to vortex lift.
• aerodynamics/high-speed/swept-wing-aerodynamics: simple-sweep cosine corrections for attached subsonic swept wings.

Pitfalls

• Applying the suction analogy to a rounded or blunt leading edge: the model is valid for sharp leading edges only, where the flow separates at the edge and forms the leading-edge vortex; a rounded-edge wing stays in the attached-flow regime owned by vortex-lattice-method.
• Reading the model beyond its documented range: valid for subsonic flow, aspect ratio about 0.5 to 2.0 and alpha up to about 25 degrees; outside that range the Kv linear interpolation is clamped and the polynomial lift split loses its physical basis.
• Interpreting the potential term as the whole lift: at AR 1.0 and alpha 15 deg the vortex term is 35.4% of the total, and at AR 0.5 alpha 20 deg it reaches about 60% - the vortex contribution grows as the wing slims, so the attached-flow-only estimate is not a conservative low answer, it is wrong.
• Expecting the crossing angle to be constant across AR: it shrinks with aspect ratio (26.0 deg at AR 1.0 against 13.9 deg at AR 0.5), so a slender wing is vortex dominated much earlier than a moderate delta.
• Treating drag due to lift as the total drag: cd_due_to_lift = CL * tan(alpha) is the vortex-lift drag term only; parasite and wave drag come from the drag-polar leaves.
• Asking the model for vortex breakdown: breakdown onset is documented as not modeled (empirical charts only), so the CL growth beyond the valid alpha range is not a breakdown prediction - use the charts or a higher-fidelity method instead.

Behavior contract (gate 3)

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

python3 scripts/test_delta_wing_vortex_lift.py

The test covers the aspect ratio identity, the Kp and Kv anchors, the AR 1 alpha 15 potential/vortex/total split (35.4% vortex), the 5 to 25 deg sweep values and monotonicity, drag due to lift, the AR 0.5 alpha 20 high-vortex-fraction case, the crossing angles at AR 1 and AR 0.5 with the potential-equals-vortex identity, summary-dict keys, determinism, and ValueError rejection of non-physical inputs.

Compliance

• NASA TN D-3767 (Polhamus 1966) is the primary methodology source, named and paraphrased (public-domain NTRS, reference-only, no reproduction). NACA Report 824 anchors the section-data context per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.