Ackeret Linearized Supersonic

Ackeret Linearized Supersonic (aerodynamics/high-speed/ackeret-linearized-supersonic)

Use when you must compute the section coefficients of a thin airfoil at supersonic Mach by ackeret linearized supersonic theory (Liepmann and Roshko ch. 10, Anderson sec. 9): the ackeret parameter, the linearized surface pressure law, and the section lift, wave drag and leading edge moment of the flat plate, the thin biconvex circular-arc section and a cambered thin plate. The leaf integrates the linearized pressure coefficient over the surface slopes by deterministic Simpson quadrature and exposes the closed forms of the canonical thin sections, in pure stdlib Python. It pairs with the exact-march sibling aerodynamics/high-speed/shock-expansion-airfoil, whose diamond double-wedge march is the exact answer this leaf's linear values cross-check within 10%, and with the area-Mach duct relations of aerodynamics/high-speed/isentropic-flow-relations.

Domain quick reference

• Ackeret parameter: beta = sqrt(M^2 - 1), the divisor of every linear supersonic coefficient. beta -> 0 as M -> 1 (the coefficients diverge, hence the M <= 1 guard) and beta -> M at large M. All angles are RADIANS; positive alpha means the freestream sits above the chord, so the lower surface is windward.
• Linearized surface pressure law: on a surface deflected by theta from the freestream (theta positive compresses, Cp positive windward), Cp = +-2theta/beta. Upper surface cp_u = 2(phi_u - alpha)/beta, lower surface cp_l = 2*(alpha - phi_l)/beta, with phi = dy/dx the surface slope relative to the chord.
• Section lift: cl = (1/c) int_0^c (cp_l - cp_u) dx = 4*alpha/beta for every thin CLOSED section: thickness and camber integrate out (integral (phi_u + phi_l) dx = 0 around a closed section). Lift-curve slope cl_alpha = 4/beta per radian.
• Leading edge moment (nose-up positive): cm_le = -(1/c^2) int_0^c x (cp_l - cp_u) dx = -2alpha/beta - (2/(beta c^2)) int (y_u + y_l) dx. Flat and symmetric sections sit at cm_le = -2alpha/beta with the center of pressure at mid-chord, x_cp/c = 0.5 (the supersonic linear-theory position, versus quarter-chord subsonically).
• Wave drag by the linearized pressure resolution (drag direction at +alpha to the chord): cd = alphacl + (2/(beta c)) int_0^c [phi_u^2 + phi_l^2 - alpha(phi_u + phi_l)] dx.
• Closed forms (xhat = x/c): flat plate cd = 4alpha^2/beta; biconvex parabolic-arc of thickness ratio tau, phi_u = 2tau*(1 - 2xhat), cd = (4alpha^2 + (16/3) tau^2)/beta; circular-arc cambered plate of camber ratio h/c, phi = 4*(h/c)(1 - 2xhat), cd = (4alpha^2 + (64/3) (h/c)^2)/beta and cm_le = -2alpha/beta - (8/3)(h/c)/beta.
• Pressure reconstruction (the only gamma-dependent relation): p/p_inf = 1 + (gamma/2) M^2 Cp, gamma default 1.4.
• NACA TR-824 frames the thin-section supersonic aerodynamics context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the supersonic flight condition: freestream Mach M > 1 and angle of attack alpha in radians. Traverse the ackeret parameter with ackeret_parameter(mach) to confirm beta = sqrt(M^2 - 1) and that the regime is linear supersonic, not M = 1 or below.
2. Evaluate the ackeret surface pressure law with cp_linear(theta_rad, mach) for each local surface deflection theta, and reconstruct the surface pressure with surface_pressure_ratio(cp, mach, gamma) when p/p_inf is needed.
3. Compute the section lift with lift_coefficient(alpha_rad, mach) and the supersonic lift curve slope with lift_curve_slope(mach): cl = 4*alpha/beta holds for every thin closed section, independent of thickness and camber.
4. Take the flat plate reference set with flat_plate(alpha_rad, mach): cl, cd_wave = 4*alpha^2/beta, cm_le and the mid-chord center of pressure x_cp_over_c = 0.5.
5. Size the wave drag and moment of the closed thin sections with biconvex_section(alpha_rad, thickness_ratio, mach) and cambered_plate(alpha_rad, camber_ratio, mach), using the additive thickness and camber drags on the flat-plate alpha drag.
6. Integrate a general thin section with section_coefficients(alpha_rad, mach, slope_upper, slope_lower, chord, panels): pass the surface slope callables phi_u(x) and phi_l(x) over the chord and read cl, cd_wave and cm_le from the linearized pressure integral over the surface slopes by deterministic composite Simpson quadrature (SIMPSON_PANELS = 2000 even panels, identical inputs give identical bits). Closed-form agreement with the three canonical families is an identity of the engine.
7. Cross-check the linear values against the exact shock-expansion march result of the sibling leaf (cl ratio ~1.015 at M = 2, 3 deg, the sibling's quoted 1.5% handover) and close with the deterministic contract test scripts/test_ackeret_linearized_supersonic.py.

Worked example

Representative point: M = 2.0, alpha = 3 deg = 0.052359877560 rad, with the biconvex tau = 0.06 (6% thick) and the cambered plate h/c = 0.02 (2% circular-arc camber), gamma = 1.4 where used. All values are real module outputs.

• Ackeret parameter and pressure law: beta(2.0) = 1.732050807569. A surface deflection of theta = +0.05 rad (2.865 deg, compression) carries Cp = 0.057735026919, the mirror deflection Cp = -0.057735026919. The flat plate's windward lower surface at alpha = 3 deg: cp_l = 2alpha/beta = 0.060459978808, so p/p_inf = 1 + 0.5gammaM^2cp = 1.169287940662 on the lower surface and 0.830712059338 on the suction upper surface.
• Lift and slope: cl = 4*alpha/beta = 0.120919957616 for the flat plate, the biconvex and the cambered plate alike (thickness and camber drop out of the lift integral); the supersonic lift-curve slope is 4/beta = 2.309401076759 per radian at M = 2.
• Wave drag: flat plate cd_wave = 4*alpha^2/beta = 0.006331354175. Biconvex tau = 0.06: zero-lift part (16/3) tau^2/beta = 0.011085125168, total cd_wave = 0.017416479344 at alpha = 3 deg (the additive thickness drag, 2.75 times the flat plate's alpha drag); at M = 3, alpha = 0 it falls to 0.006788225099, the M^-2 fall. Cambered plate h/c = 0.02: zero-lift part (64/3)(h/c)^2/beta = 0.004926722297, total cd_wave = 0.011258076472.
• Moments: cm_le = -2*alpha/beta = -0.060459978808 (nose-up positive) for the flat plate and the symmetric biconvex, with the center of pressure at mid-chord, x_cp_over_c = 0.500000000000. The cambered plate adds the nose-down couple (8/3)(h/c)/beta = 0.030792014357, giving cm_le = -0.091251993165: camber produces no lift in the linear theory but pitches the section nose-down.
• Engine identity: section_coefficients over the slope callables of the three families reproduces every closed form above with residuals below 4.1e-15 at the worked point (the engine and the closed forms are one theory).
• Sibling cross-check: the exact-march diamond of eps = 5 deg (10 deg included angle, thickness ratio tau = tan(5 deg) = 0.087488663526) at M = 2 has linear zero-lift cd_wave = 4*tau^2/beta = 0.017676770709, matching the shock-expansion-airfoil worked-example cd_wave = 0.0177 inside 0.2%, and cl(exact)/cl(linear) = 1.014720832024 at alpha = 3 deg, the sibling's quoted 1.5% above handover.

Verification

• Confirm ackeret_parameter(2.0) = 1.732050807569, beta(1.5) = 1.118033988750, beta(3.0) = 2.828427124746, beta(sqrt(2)) = 1.0.
• Confirm cp_linear(0.05, 2.0) = 0.057735026919 with the exact antisymmetry cp(-theta) = -cp(theta), and lift_coefficient(3 deg, 2.0) = 0.120919957616 with the lift-curve slope 2.309401076759 per radian.
• Confirm flat_plate(3 deg, 2.0) returns cl = 0.120919957616, cd_wave = 0.006331354175, cm_le = -0.060459978808 and x_cp_over_c = 0.5, and that cd_wave = alpha*cl at linear order.
• Confirm biconvex_section(3 deg, 0.06, 2.0) cd_wave = 0.017416479344 = 0.006331354175 + 0.011085125168 (drag additive) and cambered_plate cm_le = -0.091251993165 = -0.060459978808 - 0.030792014357 (camber couple).
• Confirm section_coefficients reproduces every closed form within 1e-9 and that cl is identical across the flat plate, biconvex and cambered plate within 1e-12 (thickness and camber independence).
• Confirm every mach <= 1.0 (1.0, 0.9, -2.0, non-finite), thickness or camber ratio <= 0.0, chord <= 0.0, odd or sub-2 panel count, and gamma <= 1.0 raises ValueError.
• Run the deterministic contract test offline: python3 scripts/test_ackeret_linearized_supersonic.py (35 tests, exit 0, under 20 s; also passes under the pyenv 3.13.12 hook interpreter).

Related leaves

• aerodynamics/high-speed/shock-expansion-airfoil: the exact shock-expansion march over the diamond double-wedge section; this leaf's linear values are its 10% cross-check.
• aerodynamics/high-speed/isentropic-flow-relations: area-Mach relations and total-to-static ratios of the same supersonic flow regime, no surface pressure law.
• aerodynamics/high-speed/hypersonic-piston-theory: the M >> 1 surface pressure law of unsteady hypersonic surfaces, beyond the moderate supersonic regime of this leaf.
• aerodynamics/airfoil/thin-airfoil-section-theory: the subsonic Glauert camber-line theory of the M ~ 0 regime, the incompressible cousin.

Pitfalls

• Applying the linear values where the exact march is required: the ackeret coefficients are the first-order answer; at M = 2, alpha = 3 deg the exact shock-expansion march cl sits 1.47% above the linear value (ratio 1.014720832024) and the wave drag differs by the same order, so quote the linear set as the cross-check the shock-expansion-airfoil sibling assigns it, not as the exact result.
• Using degrees in the ackeret formulas: cl = 4*alpha/beta is a radian identity, so alpha = 3 deg must enter as 0.052359877560 rad; feeding 3.0 directly returns a cl 57 times too large.
• Running the module at M <= 1: beta -> 0 as M -> 1 and every linear coefficient diverges, so the module raises ValueError at M <= 1; the transonic-similarity leaf owns the near-Mach-1 regime the ackeret formulas diverge away from.
• Expecting camber to add lift: in the linear theory the camber terms integrate out of the closed-section lift integral, so the cambered plate carries the same cl as the flat plate; camber shows up only in the nose-down moment couple and the camber drag.
• Summing thickness drag by hand with the wrong slope law: the biconvex parabolic-arc slopes run from +2tau at the leading edge to -2tau at the trailing edge, and only that law gives the (16/3) tau^2/beta zero-lift drag; a wedge or double-wedge section instead carries 4*tau^2/beta (see the sibling diamond cross-check).
• Treating the engine callables as values: section_coefficients takes slope callables phi_u(x) and phi_l(x), not fixed slope numbers; the upper slope carries the negative-lift suction side and must be passed with its own sign.

Behavior contract (gate 3)

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

python3 scripts/test_ackeret_linearized_supersonic.py

The test covers the ackeret parameter table and its M <= 1 rejections, the linearized surface pressure law table with exact antisymmetry and the p/p_inf reconstruction with the gamma default, the section lift coefficient and the supersonic lift curve slope, the flat plate reference set with its mid-chord center of pressure, the biconvex section with additive thickness drag and its M^-2 Mach fall, the cambered plate with the nose-down couple split, the shape independence of cl across the three canonical families, the section_coefficients Simpson engine identities against every closed form, engine zero-lift and determinism checks, the sibling diamond cross-validation, and ValueError rejection of non-physical mach, thickness, camber, chord, panel count and gamma inputs.

Compliance

• Standards referenced, not reproduced: NACA TR-824 frames the thin airfoil supersonic aerodynamics context; the ackeret relations above are standard engineering methodology (Liepmann and Roshko, Anderson), summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.