Deep Stall Analysis

Deep Stall Analysis (flight-mechanics/stability-control/deep-stall-analysis)

Use when the task is to assess whether a T-tail or aft-fuselage-mounted tail airplane can enter a deep stall, a self-sustaining high-angle-of- attack trim beyond the stall sometimes called alpha lock, and whether the elevator retains enough pitch-down authority to recover from it. The model captures the loss of horizontal-tail effectiveness when the tail enters the wing or fuselage wake at high angle of attack through a tail blanking factor, adds a separated-flow wing-body pitch-up that rises after the stall and fades at very high angle of attack, solves for the post-stall trim point in the deep-stall band, and compares the elevator pitch-down moment available at that trim with the pitch-up moment hump the elevator must overcome to pitch back below the stall. It pairs with flight-mechanics/stability-control/spin-recovery as the departure and spinning neighbor and with flight-mechanics/stability-control/longitudinal-stability as the tail sizing and static stability neighbor. Deep stall here means the static pitch trim at high alpha with a wake-blanked tail, not a spin entry; developed spins and autorotation bands belong to spin-recovery, and stall and post-stall flight test programs belong to flight-test-operations stall-characteristics-testing and high-angle-of-attack-testing. All wake and separation coefficients in this model are documented typical engineering-model constants, not standard values.

Domain quick reference

• Stall angle: alpha_stall = (clmax / a_w) * R2D + VISC_STALL_SHIFT_DEG, with a viscous shift of 2.0 deg above the linear clmax/a slope.
• Tail blanking factor: 1.0 below the wake angle, linear ramp to eta_blank / eta_t0 across the next BLANK_DELTA_DEG = 10 deg, constant at eta_blank / eta_t0 above. The factor multiplies eta_t0, so the effective tail efficiency is eta = eta_t0 * factor. Setting eta_blank equal to eta_t0 gives factor 1.0 everywhere, a tail that is never blanked.
• Tail pitch contribution: Cm_t = -v_h * eta * a_t * (1 - d_eps_dalpha) * alpha (radians), from the tail volume coefficient v_h, tail lift slope a_t and downwash gradient d_eps_dalpha.
• Separated-flow pitch-up: rises linearly at slope sep_contrib over SEP_RISE_RAD = 0.6 rad beyond the stall, fades linearly back to zero over SEP_FADE_RAD = 0.4 rad, zero outside that band.
• Total moment: Cm = cm0_wb + cm_alpha_wb * alpha + Cm_sep + Cm_t. A positive Cm in the deep-stall band means the airplane pitches up into the post-stall region; a downward crossing of Cm through zero at high alpha is the stable deep-stall trim (alpha lock point).
• Recovery authority: max_down_moment = |cm_delta| * |delta_e_max|, compared with the maximum pitch-up moment hump between the stall and the locked trim. Positive recovery margin means full down elevator can pitch the nose back below the stall.
• FAR 25 and CS 25 frame the stall and controllability context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Gather the configuration: clmax, a_w, cm0_wb, cm_alpha_wb, v_h, a_t, eta_t0, eta_blank, alpha_wake_deg, d_eps_dalpha, sep_contrib, cm_delta and delta_e_max_deg into the inputs dict.
2. Compute the stall angle with stall_angle_deg(clmax, a_w) and the tail blanking factor across alpha with blanking_factor(alpha_deg, alpha_wake_deg, eta_blank / eta_t0).
3. Build the pitch moment curve with cm_total(alpha_deg, inputs) and check its sign at the stall with cm_at_stall(inputs). A negative Cm at the stall is the trimmed pre-stall state.
4. Solve for the post-stall trim with find_deep_stall_trim(inputs): it scans the band from stall + 5 deg up to hi_deg (default 60 deg) and returns the highest-angle Cm zero crossing, the stable high-alpha trim where the fading separated-flow pitch-up lets Cm fall back below zero. None means Cm never crosses in the band, so the tail keeps authority and no deep-stall trim exists.
5. Measure the lock with lock_depth_deg(alpha_lock_deg, alpha_stall_deg) and the blanking at the trim from blanking_factor at the lock angle.
6. Compute recovery_margin(inputs, alpha_lock_deg): the elevator pitch-down moment available over the pitch-up hump between the stall and the locked trim. Infinite when no lock exists.
7. Run analyze(inputs) for the verdict dict: deep_stall True when a post-stall trim sits at least 3 deg above the stall; alpha_lock True when the elevator cannot overcome the hump (negative recovery margin).
8. Confirm the deterministic checks with the contract test scripts/test_deep_stall_analysis.py.

Worked example

T-tail transport-like airplane: clmax 1.5, a_w 5.7, cm0_wb 0.02, cm_alpha_wb -0.6, v_h 0.9, a_t 4.2, eta_t0 0.9, eta_blank 0.25, alpha_wake_deg 20.0, d_eps_dalpha 0.35, sep_contrib 3.0, cm_delta -1.1, delta_e_max_deg -25.0.

• Stall angle: 1.5 / 5.7 * 57.29578 + 2.0 = 17.08 deg (module value 17.077837).
• Blanking factor: 1.0 at 20 deg, 0.63889 at 25 deg, 0.27778 at 30 deg and above (eta 0.25, tail almost fully in the wake).
• At 45 deg (0.7854 rad): the tail is fully blanked so Cm_t = -0.9 * 0.25 * 4.2 * 0.65 * 0.7854 = -0.482; the separated-flow pitch-up, x = 0.4873 rad past the stall, adds 3.0 * 0.4873 = 1.462; wing-body Cm = 0.02 - 0.471 = -0.451. Total Cm = +0.528, positive, the pitch-up deep-stall band.
• find_deep_stall_trim returns 58.77 deg, in the [55, 60] deg band, deterministically (two calls agree to 1e-9). Lock depth 58.77 - 17.08 = 41.69 deg.
• Recovery: max down moment = 1.1 * 25 * 0.0174533 = 0.480. The pitch-up hump peak between stall and lock is about 0.729, so the recovery margin is 0.480 - 0.729 = -0.249, negative. analyze verdict: "deep-stall alpha lock, elevator insufficient", with cm_at_stall -0.818 (pre-stall trim is stable) and blanking factor 0.27778 at the lock.
• With the same airplane at eta_blank 0.9 (tail never blanked, blank ratio 1.0) the tail keeps authority, Cm stays negative at high alpha, find_deep_stall_trim returns None and the verdict is "no deep-stall trim" with recovery margin infinite.
• With cm_delta -1.6 and delta_e_max_deg -30.0 the down moment rises to 0.838, the margin turns positive (+0.109) and the verdict becomes "deep-stall trim, elevator recovers".

Verification

• stall_angle_deg(1.5, 5.7) returns 17.077837 deg, within 1e-6 of the module contract value.
• blanking_factor(25.0, 20.0, 0.25 / 0.9) returns 0.638889 and blanking_factor(35.0, 20.0, 0.25 / 0.9) returns 0.277778.
• separation_pitch_up at 45 deg returns 1.4620 (rise branch); peak 1.8 at 0.6 rad past the stall; 0.9 at 0.8 rad; zero at and beyond 1.0 rad and below the stall.
• cm_total(45.0, inputs) returns +0.5283, positive in the deep-stall band; cm_at_stall(inputs) is -0.818, negative.
• find_deep_stall_trim returns 58.7701 deg deterministically; the eta_blank 0.9 sanity case returns None.
• analyze reports deep_stall True, alpha_lock True, lock depth 41.69 deg and the alpha-lock verdict for the worked example.
• ValueError rejection: clmax <= 0, a_w <= 0, v_h <= 0, eta_t0 <= 0 or

  1, eta_blank < 0 or > eta_t0, alpha_wake_deg <= 0, d_eps_dalpha outside [0, 1], sep_contrib <= 0, cm_delta >= 0, delta_e_max_deg = 0 and cm_alpha_wb >= 0 all raise ValueError.

• Run the contract test offline: python3 scripts/test_deep_stall_analysis.py (35 tests, deterministic).

Pitfalls

• Setting eta_blank above eta_t0: the blanking factor is eta_blank/eta_t0 and eta_blank is validated in [0, eta_t0]; setting eta_blank equal to eta_t0 models a tail that is never blanked (factor 1.0), not 'more blanking'.
• Expecting a deep-stall trim from every T-tail: find_deep_stall_trim returns None when Cm never crosses zero in the post-stall band (tail keeps authority), and recovery_margin is infinite then - None is the 'no alpha lock' answer, not a failure.
• Reading a positive Cm in the deep-stall band as recovery: positive Cm at high alpha means the airplane pitches up into the post-stall region; the downward crossing through zero is the stable lock point, and alpha_lock True means the elevator cannot overcome the hump.
• Sign conventions on the elevator: cm_delta must be negative and delta_e_max_deg negative (down elevator); cm_alpha_wb must be negative; violations raise ValueError and a positive cm_delta would report recovery authority with the wrong sign.
• Quoting the wake and separation coefficients as standard values: the blanking ramp, separation rise (0.6 rad) and fade (0.4 rad) and slopes are documented typical engineering-model constants, not standard values.

Related leaves

• flight-mechanics/stability-control/spin-recovery: the departure and spinning regime neighbor; developed spin modes and spin recovery control inputs are its claim, not this leaf.
• flight-mechanics/stability-control/longitudinal-stability: neutral point and static margin for the low-alpha stability context and the tail volume basis.
• flight-mechanics/stability-control/dynamic-stability: the short period and phugoid modes around the trimmed states.
• flight-test-operations/envelope/stall-characteristics-testing: the flight test program for stall characteristics, the testing boundary for this analysis.
• flight-test-operations/envelope/high-angle-of-attack-testing: the flight test program for the post-stall regime this model assesses.
• vehicle-design/sizing/tail-sizing: sizing the horizontal tail that sets v_h and the tail authority used here.

Behavior contract (gate 3)

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

python3 scripts/test_deep_stall_analysis.py

The test covers the stall angle value and scaling, the blanking factor ramp points, the separation pitch-up branches (rise, peak, fade, zero), the positive Cm deep-stall band at 45 deg, the negative Cm at the stall, the bisection root value in [55, 60] deg with determinism, the no-root sanity case, lock depth, recovery margins (negative, infinite, positive), the analyze verdict strings, the full output dict, and ValueError rejection of every non-physical input in the validation list.

Compliance

• Standards referenced, not reproduced: FAR 25 and CS 25 frame the stall and controllability context (25.103 stall, 25.143 controllability and 25.145 longitudinal control context, summary-only per standards-map.yaml). The wake blanking and separated-flow coefficients in this model are documented typical engineering-model constants, not values taken from the standards.
• compliance: STANDARDS-REF, gated: false.