# Rotorcraft Axial Descent Flow States > Use when you must classify the axial flow state of a rotor in vertical descent: hover at zero rate, the vortex-ring band from zero to twice the hover induced velocity, and the windmill-brake momentum state at and above that boundary. Computes the band limits, the windmill-brake induced velocity from the momentum-theory closed form, the signed rotor power and torque in descent (negative when the rotor absorbs power from the airstream), and the torque-reversal condition c = P_profile over k T versus v_h that decides whether the zero-shaft-power autorotative equilibrium is reachable on the momentum branch. Produces the flow-state verdict, the descent induced velocity, signed power and torque, and the momentum reachability verdict. Trigger: vortex-ring state, windmill-brake state, axial descent, descent induced velocity, torque reversal, rotor descent power. - Skill: `ashfordeou/rotorcraft-axial-descent-flow-states` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/rotorcraft-axial-descent-flow-states` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/rotorcraft-axial-descent-flow-states/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/rotorcraft-axial-descent-flow-states --- # Rotorcraft Axial Descent Flow States (flight-mechanics/performance/rotorcraft-axial-descent-flow-states) Use when you must categorize the axial flow state of a helicopter rotor in vertical descent from the descent-rate ratio w = Vd / v_h and evaluate the windmill-brake state with momentum theory: the vortex-ring / turbulent-wake band 0 < w < 2 versus the windmill-brake state w >= 2, the band induced velocity from the momentum closed form, the signed rotor power and torque in descent, and whether momentum theory can close to the zero-shaft-power autorotative equilibrium. It pairs with flight-mechanics/performance/rotorcraft-hover-performance (the hover state at zero rate) and with flight-mechanics/performance/rotorcraft-vertical-climb-performance (the climb-only momentum leaf). It implements the windmill-brake momentum model in pure Python, stdlib only, and never computes empirical inflow: inside the vortex-ring band the flow is momentum-invalid and the induced velocity is left None. Scope: this leaf owns the windmill-brake momentum-theory branch and the vortex-ring band boundary. Power-off descent estimation is owned by the empirical leaf flight-mechanics/performance/rotorcraft-autorotative-descent (which explicitly excludes momentum theory in descent, the vortex-ring state and vertical zero-airspeed descent); induced velocity at hover and in climb belongs to the hover and vertical-climb leaves; inflow in level flight belongs to the forward-flight leaf. Momentum theory is applied only where it is valid, Vd >= 2 v_h. ## Domain quick reference All quantities are SI (m/s, N, W, N m). Module constants are RHO_SL = 1.225 kg/m^3, G = 9.80665 m/s^2, K_INDUCED_DEFAULT = 1.15, PI = math.pi. - Hover induced velocity from thrust and disk area: v_h = sqrt(T / (2 rho A)), A = PI R^2. The reference rotor (R = 5.0 m, m = 2200 kg, T = m G) gives v_h = 10.5887 m/s. - Flow-state bands: Vd = 0 is hover, 0 < Vd < 2 v_h is the vortex-ring / turbulent-wake band (momentum invalid, empirical inflow, NASA TP-2005-213477 public-domain context), Vd >= 2 v_h is the windmill-brake state where momentum theory applies. - Band limits: vortex_ring_band_limits(v_h) returns (0, 2 v_h); the worked rotor gives (0, 21.18) m/s. - Windmill-brake induced velocity (physical branch): v_i = Vd/2 - sqrt((Vd/2)^2 - v_h^2), valid only for Vd >= 2 v_h. The boundary identity v_i(2 v_h) = v_h holds exactly, v_i never exceeds v_h, and v_i falls like v_h^2 / Vd as Vd grows (at Vd = 5 v_h, v_i / v_h = 0.2087, near the 1 / w asymptote). - Signed descent power: P = k T (-Vd + v_i) + P_profile. Negative P means the rotor absorbs power from the airstream (windmill-brake working state); positive P means the shaft drives the rotor. - Signed torque: Q = P / Omega. Negative torque opposes the engine drive while the rotor absorbs power. - Torque-reversal condition: c = P_profile / (k T) versus v_h. The zero-power condition P = 0 combined with the momentum quadratic v_i^2 - Vd v_i + v_h^2 = 0 gives the formal crossing at Vd = c + v_h^2 / c with v_i = v_h^2 / c. That root is physical only when v_i <= v_h, i.e. c >= v_h (then Vd >= 2 v_h by AM-GM). When c < v_h no momentum root exists and the equilibrium is momentum-unreachable. ## Workflow 1. Fix the operating point: descent rate Vd (m/s, positive downward), thrust T (N) and disk area A = PI R^2 (or the mass, radius, density that imply them). The sibling vertical-climb leaf owns climb, so Vd < 0 raises ValueError here. 2. Get the hover induced velocity from the thrust, density and area: v_h = sqrt(T / (2 rho A)); call axial_flow_state(Vd, v_h) for the verdict "hover", "vortex-ring-band" or "windmill-brake", and vortex_ring_band_limits(v_h) for the band in m/s. 3. In the windmill-brake state only, get the induced velocity with windmill_brake_induced_velocity(Vd, v_h); the function refuses Vd below 2 v_h because momentum theory does not apply in the band. 4. Compute the signed power with rotor_descent_power(T, Vd, v_i, P_profile, k) and, with the rotor speed Omega, the signed torque with rotor_descent_torque(P, Omega). 5. Decide zero-shaft-power reachability with torque_reversal_condition(P_profile, T, k, v_h): c >= v_h reports the momentum root Vd = c + v_h^2 / c on the windmill-brake branch; c < v_h returns momentum_root_Vd None with the momentum-unreachable verdict (the equilibrium lies in the empirical vortex-ring / turbulent-wake regime). 6. Bundle everything with descent_summary(T, R, P_profile, Vd, rho, k, rotor_speed_rad_s): it returns flow_state, v_h, band_limits, induced_velocity, power_W, torque_Nm and momentum_root_reachable, with the momentum fields None in the band and at hover. 7. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_axial_descent_flow_states.py. ## Worked example Reference rotor (shared with the hover and blade-element siblings): R = 5.0 m, m = 2200 kg (T = m G = 21574.63 N), rho = 1.225 kg/m^3, A = PI R^2 = 78.540 m^2, P_profile = 122935 W, k = 1.15. Real module outputs: - v_h = 10.5887 m/s, vortex_ring_band_limits -> (0, 21.177) m/s. - At Vd = 2 v_h (21.177 m/s): windmill_brake_induced_velocity returns v_i = 10.5887 m/s, the boundary identity with diff 0.0; the signed power is -139780.0 W (the rotor absorbs power from the airstream). - At Vd = 25 m/s: v_i = 5.8570 m/s (0.553 v_h), P = -352017.6 W (spec magnitude about -352019 W). - At Vd = 30 m/s: v_i = 4.3756 m/s (0.413 v_h), P = -512828.7 W (about -512829 W); with Omega = Vtip / R = 220 / 5 = 44 rad/s, Q = P / Omega = -11655.2 N m (about -11655 N m), torque opposing the engine drive. - At Vd = 40 m/s: P = -794246.6 W (about -794247 W); P stays negative across the whole windmill-brake band for this rotor. - torque_reversal_condition: c = P_profile / (k T) = 4.9549 m/s (about 4.955), c < v_h (4.955 < 10.589), verdict momentum-unreachable, momentum_root_Vd None. The formal crossing would sit at Vd = c + v_h^2 / c = 27.58 m/s only on the non-physical branch: it demands v_i = v_h^2 / c = 22.63 m/s, above v_h, impossible on the windmill-brake branch, so 27.6 m/s is never a momentum root of this rotor. ## Verification - Confirm axial_flow_state boundaries: Vd = 0 -> "hover", Vd just below 2 v_h -> "vortex-ring-band", Vd = 2 v_h and above -> "windmill-brake". - Confirm windmill_brake_induced_velocity(2 v_h, v_h) equals v_h to 1e-9, and that v_i satisfies the momentum quadratic identity v_i (Vd - v_i) = v_h^2 exactly on the physical branch. - Confirm the worked powers -139780.0, -352017.6, -512828.7 and -794246.6 W and the torque -11655.2 N m at Omega = 44 rad/s, all within the spec magnitude bounds, and that P is negative across the windmill-brake band. - Confirm the torque-reversal split: c >= v_h reports the momentum root Vd = c + v_h^2 / c (>= 2 v_h) as reachable; the worked rotor with c = 4.955 < v_h reports momentum-unreachable with no root, and the required v_i = v_h^2 / c = 22.63 m/s exceeding v_h is the reason the formal 27.58 m/s crossing is non-physical. - Confirm ValueError on Vd < 0 (climb), v_h <= 0, a windmill-brake call below 2 v_h, non-positive thrust, non-positive k, negative profile power or induced velocity, and Omega <= 0. - Confirm the descent_summary dict contains exactly the seven documented keys with momentum fields None in the band and at hover. - Confirm determinism: identical inputs give identical floats run to run (no RNG). - Run the contract test offline: python3 scripts/test_rotorcraft_axial_descent_flow_states.py (34 tests, deterministic). ## Pitfalls - Evaluating momentum theory inside the vortex-ring band: windmill_brake_induced_velocity refuses Vd below 2*v_h with ValueError because momentum theory is invalid there, and the band induced velocity is None; do not fill the band with the closed form. - Signing Vd wrong: descent rate is positive downward in this leaf and a negative Vd (climb) raises ValueError - vertical climb is owned by the sibling climb leaf. - Reading the signed power without its sign: negative P means the rotor absorbs power from the airstream (windmill-brake working state) and the torque opposes the engine drive; flipping the sign for a 'magnitude' erases the physical verdict. - Reporting the formal torque-reversal root as reachable: when c < v_h the momentum root Vd = c + v_h^2/c would demand v_i = v_h^2/c above v_h and is non-physical; the function returns momentum_root_Vd None with the momentum-unreachable verdict, as in the worked rotor (c = 4.955 < v_h = 10.589). - Reading descent_summary momentum fields at hover or in the band: the induced velocity, power and torque momentum fields are None outside the windmill-brake state by contract. - Non-positive thrust, k, or rotor speed and negative profile power raise ValueError; determinism is pinned (no RNG). ## Related leaves - flight-mechanics/performance/rotorcraft-autorotative-descent: the empirical power-off descent sibling; this leaf's torque-reversal verdict explains why momentum theory cannot close to the autorotative equilibrium. - flight-mechanics/performance/rotorcraft-vertical-climb-performance: the climb-only momentum leaf (Vc < 0 raises ValueError there). - flight-mechanics/performance/rotorcraft-hover-performance: the hover state at zero descent rate and its induced power terms. - flight-mechanics/performance/rotorcraft-blade-element-hover-performance: the coefficient-polar blade-element model sharing the reference rotor. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_rotorcraft_axial_descent_flow_states.py The test covers the reference-rotor worked example (v_h 10.5887 m/s, band (0, 21.18) m/s, v_i(2 v_h) = v_h identity, powers about -139780 / -352019 / -512829 / -794247 W, torque about -11655 N m at Omega 44), the flow-state boundaries at 0, just below, at and above 2 v_h, the momentum quadratic identity v_i (Vd - v_i) = v_h^2, the 1 / w asymptote at 5 v_h, the torque-reversal split (c >= v_h reachable with root at c + v_h^2 / c, c < v_h momentum-unreachable), the exact descent_summary keys with momentum fields None in the band, the pinned module constants, determinism run to run, and ValueError rejection of climb rates, non-positive v_h, windmill calls below 2 v_h, non-positive thrust, k, profile power and Omega. ## Compliance - Standards referenced, not reproduced: FAR-29 is named reference-only per standards-map.yaml. NASA TP-2005-213477 (public domain) is named in the body as the empirical-inflow context that marks the vortex-ring band momentum-invalid; the momentum relations above are summary-only standard engineering methodology. - compliance: STANDARDS-REF, gated: false.