Sears Function Gust Lift

Sears Function Gust Lift (aerodynamics/aeroelasticity/sears-function-gust-lift)

Use when the task is the frequency-domain unsteady lift response of a rigid thin airfoil to a convected sinusoidal vertical gust (the Sears problem, Sears 1941, JAS 8(3); Bisplinghoff, Ashley and Halfman, Aeroelasticity, the unsteady incompressible gust chapter; Fung, An Introduction to the Theory of Aeroelasticity): the complex sears function S(k) evaluated from the Bessel series with the Theodorsen lift-deficiency function C(k), the gust gain and the phase lag, and the unsteady gust-load amplitude against the quasi-steady 2pirhoVbw_g_amp reference. This leaf is the section aerodynamics transfer function of the rigid airfoil, the frequency-domain complement of the flexible-section time-domain discrete-gust response sibling aerodynamics/aeroelasticity/aeroelastic-gust-response (whose quasi-steady reference L_qs = 2pirhoVbw_g is exactly the k = 0 amplitude this leaf's gain is normalized against), and it evaluates the same published C(k) expression, built on the same Bessel machinery, that aerodynamics/aeroelasticity/flutter-speed-prediction documents for the V-g stability problem. It produces no stability content, no structural degrees of freedom and no vehicle load factor: the gust response of a flexible typical section, static divergence, added-mass coefficient estimation and the discrete-gust certification load-factor method all belong to their own leaves.

Domain quick reference

• Sears problem: rigid thin airfoil of semi-chord b = c/2 in incompressible flow at speed V encounters a frozen sinusoidal vertical gust with the gust vertical velocity field w_g(x, t) = w_g_amp * Re{e^(i*(omegat - kx/b))}, x measured downstream from the leading edge, so the gust front crosses the leading edge at t = 0. Time dependence e^(+iomegat); physical quantities are the real part. Reduced frequency k = omegab/V = 2pifb/V.
• Complex lift amplitude per unit span: L_hat = 2pirhoVbw_g_amp * S(k) with L(t) = Re{L_hate^(iomegat)}. The quasi-steady reference amplitude L_qs = 2pirhoVb*w_g_amp is the k = 0 value, identical to the reference the discrete-gust sibling documents.
• Bessel series (Abramowitz and Stegun 9.1.10 to 9.1.11, the same published series family the flutter sibling cites for C(k)): J0(x) = sum_m (-1)^m (x/2)^(2m)/(m!)^2, J1(x) = (x/2)*sum_m (-1)^m (x/2)^(2m)/(m!(m+1)!), and the log-harmonic Y0 and Y1 forms with the Euler-Mascheroni constant in the ln(x/2) + gamma terms. Each series runs until its term magnitude falls below SERIES_TOL times the running sum, with SERIES_TERMS_MAX as the safety cap.
• Hankel functions of the second kind: H0^(2)(k) = J0(k) - iY0(k), H1^(2)(k) = J1(k) - iY1(k). Theodorsen function C(k) = H1^(2)(k)/(H1^(2)(k) + i*H0^(2)(k)): C = 1 at k = 0 exactly (steady flow) and |C| falls toward the high-reduced-frequency limit 1/2.
• Sears function, leading-edge reference: S(k) = [C(k)(J0(k) - iJ1(k)) + iJ1(k)]e^(-ik), S(0) = 1 + 0i exactly. The classical mid-chord-referenced printed expression S_mid(k) = C(k)(J0(k) - iJ1(k)) + iJ1(k) is the exact identity S_mid(k) = S(k)e^(+ik), with |S_mid| = |S|; the module pins the leading-edge reference, the convention whose phase lag grows monotonically from zero.
• Gust gain and phase lag: gain |S(k)| is the unsteady-load amplitude ratio to the quasi-steady reference, monotone decreasing from |S(0)| = 1; the phase lag phi(k) = -arg S(k) in radians is the time by which the lift peak lags the gust peak at the leading edge, monotone increasing from zero. The half-amplitude reduced frequency where |S(k)| = 0.5 sits near k = 0.567.
• Wronskian gain identity (from the Bessel Wronskian J1Y0 - J0Y1 = 2/(pik)): |S(k)| = (2/(pik))/|H1^(2)(k) + i*H0^(2)(k)| for k > 0, exact to machine precision over the whole sweep of this leaf.
• Scope: single-frequency frozen sinusoidal gust, incompressible thin-airfoil theory, small gust amplitudes (linear), two-dimensional section per unit span, rigid airfoil with no motion degrees of freedom and no added-mass coefficient catalog. Loads are N/m per unit span. FAR 25 and CS 25 gust-load rules frame the certification context by name only (standards-map.yaml far-25 and cs-25, reference-only), never reproduced.

Workflow

1. Gather the gust encounter inputs: air density rho, flight speed V, semi-chord b = c/2, gust vertical velocity amplitude w_g_amp, and the reduced frequency k of the sinusoidal gust, either chosen directly for a target gust wavelength or converted from the gust frequency f with reduced_frequency (k = 2pif*b/V). The module rejects non-positive rho, V, b and f and negative or non-finite amplitudes with ValueError.
2. Confirm the Bessel machinery: bessel_j0, bessel_j1, bessel_y0 and bessel_y1 at x = 0.5, 1.0 and 2.0 reproduce the published Abramowitz and Stegun constants to about 1e-15, so the series are accurate far beyond the 0.1 to 2.0 reduced-frequency sweep of this leaf.
3. Evaluate the complex sears function: theodorsen_c builds C(k) from the Hankel ratio of the Bessel functions, then sears_function combines C(k) with the J0, J1 terms into S(k) on the leading-edge gust reference. S(0) returns exactly 1 + 0j, the quasi-steady limit |S| = 1 that the gain is normalized against.
4. Sweep the gust gain and the phase lag: sears_gain and sears_phase_lag over the reduced-frequency sweep show the monotone gain roll-off from 1 and the monotone phase-lag growth, the signature of the convected sinusoidal gust response.
5. Compute the gust-load amplitudes: quasi_steady_gust_lift gives the reference L_qs = 2pirhoVbw_g_amp (the k = 0 amplitude) and unsteady_gust_load gives the rigid-section load estimate L_hat = L_qs|S(k)| in N/m per unit span.
6. Locate the half-amplitude reduced frequency: half_gain_reduced_frequency bisects deterministically on |S(k)| = 0.5 over [BISECT_LO, BISECT_HI] = [0.1, 2.0], the reduced frequency where the unsteady gust-load amplitude falls to half the quasi-steady value. A reversed or non-straddling bracket raises ValueError.
7. Verify with the Wronskian gain identity |S(k)| = (2/(pik))/|H1^(2)(k) + iH0^(2)(k)| and run the contract test scripts/test_sears_function_gust_lift.py under python3.

Worked example

Rigid thin section of chord c = 2 m (semi-chord b = 1 m) at V = 80 m/s in standard sea-level air rho = 1.225 kg/m^3, encountering a sinusoidal vertical gust of amplitude w_g_amp = 5 m/s at reduced frequency k = 0.5. Gust frequency f = kV/(2pi*b) = 6.366197723675814 Hz, gust wavelength lambda = V/f = 12.56637061435917 m = 6.283185307179586 chord lengths. All values are the module's real outputs, matching the wave-45 spec anchors.

• Quasi-steady reference: L_qs = 2pirhoVb*w_g_amp = 3078.760800517998 N/m, the amplitude the same gust would produce at k = 0.
• Complex sears function: S(0.5) = 0.4392999993899336 - 0.2901613576384412i (leading-edge gust reference). Gust gain |S(0.5)| = 0.5264770678107253, so the unsteady gust load retains 52.65 percent of the quasi-steady amplitude at this reduced frequency.
• Phase lag: phi(0.5) = 0.5837270902719581 rad = 33.44509866003521 deg, the lift peak following the gust peak at the leading edge by phi/omega = 0.014593 s (omega = k*V/b = 40 rad/s at the worked point).
• Unsteady gust-load amplitude: L_hat = L_qs*|S(0.5)| = 1620.896958747317 N/m per unit span, the amplitude a rigid-section sinusoidal-gust load estimate would carry into a structural check.
• Half-amplitude reduced frequency: |S(k)| = 0.5 at k_half = 0.5672427076304547, where L_hat = 1539.380400258999 N/m = 0.5000000000000001*L_qs exactly: a gust only about half a semi-chord shorter than the k = 0.5 case halves the rigid-section load.
• Gain table read-off: |S(0.1)| = 0.8373543986997829, |S(0.5)| = 0.5264770678107253, |S(1.0)| = 0.3895689126581746, |S(2.0)| = 0.2801153775512997 shows the monotone roll-off; even a gust a full 6.28 chords long (k = 0.5) loads the section at about half the quasi-steady amplitude, and the phase lag grows from 0 to 41.52 deg (0.7247005314150557 rad at k = 2.0) across the sweep, the signature of the convected-gust response.

Verification

• Confirm the Bessel machinery: bessel_j0/j1/y0/y1 at x = 0.5, 1.0 and 2.0 match the published constants within 1e-12 (real anchor worst error 9.44e-16).
• Confirm the Theodorsen values C(0.5) = 0.597936064250132 - 0.1507095031626353i, C(1.0) = 0.5394348710777939 - 0.1002729028641078i and C(2.0) = 0.5129548124291317 - 0.05769128342167992i within 1e-9 on each part (the classic published values the flutter sibling's own machinery reproduces), with C(0) exactly 1 + 0j and |C(2.0)| above the 0.5 high-k limit.
• Confirm the sears function values at k = 0.1, 0.25, 0.5, 1.0, 2.0, the gain read-off values, the phase-lag values in radians and degrees, and S(0) exactly 1 + 0j with gain exactly 1.0 (the quasi-steady limit).
• Confirm the Wronskian gain identity |S(k)| = (2/(pik))/|H1^(2)(k) + iH0^(2)(k)| at every 0.1 step of k in [0.1, 2.0] (real anchor max residual 2.22e-16, test bound 1e-12).
• Confirm the monotonicity: gain strictly decreasing and phase lag strictly increasing at every 0.05 step of the sweep, so the half-amplitude crossing is unique.
• Confirm the worked-example loads and geometry (L_qs, L_hat, their ratio, the reduced-frequency round trip, the gust wavelength in meters and chord lengths, and the half-amplitude load).
• Confirm ValueError rejection of every non-physical input class: zero or negative Bessel arguments, negative or non-finite k, non-positive rho/V/b/freq, zero frequency, negative or non-finite gust amplitude, and reversed or non-straddling half-gain brackets.
• Confirm determinism: two identical full sweeps return identical bits; the logic module imports only math and has no RNG.
• Run the contract test offline under both interpreters: python3 scripts/test_sears_function_gust_lift.py and the pyenv 3.13.12 python3 (32 tests, deterministic, well under 1 s).

Related leaves

• aerodynamics/aeroelasticity/flutter-speed-prediction: the V-g flutter stability owner of the same typical section; its published C(k) convention, built on the same Bessel series family, is evaluated inside S(k) here with no stability content.
• aerodynamics/aeroelasticity/aeroelastic-gust-response: the flexible-section time-domain discrete-gust sibling whose quasi-steady peak reference L_qs = 2pirhoVb*w_g is the k = 0 limit this leaf's gain is normalized against.
• aerodynamics/aeroelasticity/divergence-speed: the static torsional instability of the same section geometry.
• aerodynamics/aeroelasticity/added-mass-coefficients-potential-flow: the added-mass coefficient catalog of accelerating bodies; this rigid airfoil has no acceleration degree of freedom.
• aerodynamics/high-speed/wave-drag-area-rule: the Sears-Haack minimum-drag body and area-rule leaf, the owner of the transonic sears-haack token, distinct from the sears-function gust response.
• structures/loads/gust-maneuver-loads: the rigid-aircraft discrete-gust certification load-factor method (FAR 25.341 context); this leaf is the section aerodynamics transfer function, no vehicle inertia or load factor.
• structures/loads/random-vibration-analysis: the PSD machinery home of continuous-turbulence spectral gust content; this leaf is single-frequency, not spectral.

Pitfalls

• Confusing the gust-load amplitude with the quasi-steady value: at the worked k = 0.5 point L_hat = 1620.896958747317 N/m is only 0.5264770678107253 of L_qs = 3078.760800517998 N/m because the unsteady gain rolls off with reduced frequency; run the gain, do not substitute the quasi-steady anchor for a finite-frequency gust.
• Mixing the leading-edge and mid-chord gust references: S_mid(k) = S(k)e^(+ik) differ by the propagation phase over the semi-chord, so the phase lag only grows monotonically from zero on the leading-edge reference this module pins; |S| is identical in either reference.
• Reading the phase lag with the wrong sign convention: with the e^(+iomegat) convention the lag is phi(k) = -arg S(k) = atan2(-Im S, Re S), positive for all k > 0 here; flipping the time convention flips the sign.
• Pushing the Bessel series past their sweep: the series forms are double-precision accurate across the 0.1 to 2.0 reduced-frequency range documented here; the flutter sibling's C(k) machinery covers the same family for its own k range.
• Treating the Sears response as a structural or spectral result: the rigid-airfoil load of this leaf has no inertia weighting, no load factor and no PSD content; the discrete-gust certification method and the continuous-turbulence spectra live in the structures loads leaves.
• Forgetting the certification context is reference-only: FAR 25 and CS 25 gust-load rules are named as the context, never reproduced; the Sears relations are standard engineering methodology, summary-only.

Behavior contract (gate 3)

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

python3 scripts/test_sears_function_gust_lift.py

The test covers the Bessel reference constants and the worst series error bound, the Theodorsen values and limits, the exact S(0) quasi-steady limit, the complex sears function values, the gain and phase-lag read-off values, the monotone gain roll-off and phase-lag growth over the sweep, the Wronskian gain identity, the half-amplitude reduced frequency and its bracket sanity, the worked-example loads, frequencies, wavelengths and the half-amplitude load, the zero-amplitude gust case, ValueError rejection of every non-physical input class, sweep determinism, and the stdlib-only purity of the logic module. It passes under both /usr/bin/ python3 (3.9.6) and the pyenv 3.13.12 interpreter.

Compliance

• The Sears gust response is public-domain textbook methodology (Sears 1941; Bisplinghoff, Ashley and Halfman, Aeroelasticity; Fung, An Introduction to the Theory of Aeroelasticity); the airworthiness context is FAR 25 and CS 25 gust-load rules, referenced by name only, summary-only per standards-map.yaml (both reference-only).
• compliance: STANDARDS-REF, gated: false.