Aerodynamics: incompressible limits and compressible conservation

Density is kg/m^3, speed m/s, area m^2, lift/drag N, pressure Pa, angles rad, and temperature K. Lift is signed; drag is nonnegative. Thin-airfoil Cl=2*pi*(alpha-alpha_l0) assumes small angle and attached inviscid incompressible flow. No stall prediction is included. Finite-wing slopes use an idealized lifting-line correction; 0<e<=1. The empirical Oswald helper is restricted to aspect ratio 2..10 to avoid nonsensical extrapolation.

Compressible models

A calorically perfect gas with constant gamma>1 and gas constant R [J/(kg K)] is assumed. Stagnation relations are isentropic and do not conserve total pressure through a shock. Prandtl-Glauert requires 0<=M<1; it is unreliable near transonic flow despite the mathematical bound allowing values close to one. Normal-shock routines require upstream M>=1 (the sonic case is a limit). Area-Mach returns A/A* for M>0 and does not choose a sub/supersonic inverse branch.

from engineeringkit.aerodynamics import normal_shock_mach2, normal_shock_pressure_ratio
M2 = normal_shock_mach2(2)  # sqrt(1/3)
pressure_ratio = normal_shock_pressure_ratio(2)  # 4.5

The Mach-2 benchmark independently reconstructs density, temperature and speed, and checks mass flux, momentum flux and total enthalpy across the shock. Tests also check sonic limits, the infinite-aspect-ratio lift slope, and the stationary best-glide point of the parabolic drag polar.

Oblique shocks and expansion fans

oblique_shock_angle solves the theta-beta-M cubic in closed form (Thompson, 1950) and returns the weak branch unless strong=True. It requires M1>1 and 0<theta<=theta_max, where oblique_shock_max_deflection gives theta_max; a steeper wedge raises ValueError because the real shock is detached and curved, which no algebraic relation describes. oblique_shock_deflection is the forward relation, defined for beta from the Mach angle to 90 degrees. Expansions are isentropic: prandtl_meyer_expansion solves nu(M2)=nu(M1)+theta with a bracketing root search, and turning angles beyond the vacuum limit (130.45 degrees for gamma=1.4) are rejected. Benchmarks check mass, normal momentum, tangential velocity and total enthalpy across an oblique shock, and the Prandtl-Meyer function against quadrature of its ODE.

import numpy as np
from engineeringkit.aerodynamics import oblique_shock, prandtl_meyer_expansion
shock = oblique_shock(2.0, np.radians(20.0))  # beta 53.42 deg, M2 1.210
M2 = prandtl_meyer_expansion(1.5, np.radians(20.0))  # 2.207

Fanno and Rayleigh duct flow

fanno_flow and rayleigh_flow return a DuctFlowState of ratios to the sonic state on the same line, matching the tables in Anderson’s Modern Compressible Flow. The Fanno friction parameter uses the Darcy friction factor, fL*/D. The Darcy factor is four times the Fanning factor, so this is the same number as the 4fL*/D of compressible-flow tables, which use the Fanning factor; pass a Darcy factor (for example from fluid_engineering.colebrook_friction_factor) to fanno_choking_length. Fanno flow is adiabatic with constant friction factor; Rayleigh flow is frictionless, with the heat given per unit mass and constant cp. Neither predicts what happens beyond choking (a shock in the duct or a changed inlet state). Benchmarks integrate the friction ODE and check mass, momentum and energy along both lines.

Lifting line and panel method

lifting_line solves Prandtl’s equation with Glauert’s Fourier series at n_terms collocation stations, keeping every harmonic so asymmetric twist is allowed. Chord, angle of attack and zero-lift angle may be constants or functions of the spanwise station y in metres. It assumes a straight, unswept wing of moderate to high aspect ratio in attached flow, so it does not predict stall, and a rectangular wing converges slowly near the tips. The elliptic wing is reproduced exactly (e=1). panel_method is the Hess-Smith constant-source, uniform-vortex method. Points run clockwise from the lower trailing edge, and an Airfoil is ordered for you. It is inviscid and incompressible, so it gives no drag, and with a blunt (open) NACA trailing edge the circulation lift and pressure-integrated lift differ by up to about 1%. The circular cylinder is matched to round-off.

Pitching moment and stability

naca4_moment_coefficient is the thin-airfoil moment about the quarter chord (nose up positive), independent of angle of attack; aerodynamic_center gives x_ac/c = 1/4 - m0/a0 from measured slopes. Stability helpers follow Anderson’s Introduction to Flight, ch. 7, with positions h as fractions of the mean aerodynamic chord: neutral_point uses the wing-body lift slope a_wb, tail efficiency eta in (0, 1] and downwash gradient in [0, 1). pitching_moment_slope is a_wb*(h - h_n), and static_margin is positive when stable. downwash_gradient is the far-wake elliptic estimate 2a/(pi AR), which overstates the downwash at a tail close behind the wing. Only stick-fixed static stability is modelled.

Performance

rate_of_climb, climb_angle, load_factor, turn_radius and turn_rate are point-mass force balances for steady climbs and level coordinated turns, with load factor n>1 for a turn. takeoff_ground_roll takes lift-off at 1.2 times the stall speed with a constant mean net force T - [D + mu(W - L)], evaluated by the caller at about 0.7 V_LO. breguet_range_propeller uses power-specific fuel consumption in kg/(W s) (0.45 lb/(hp h) is about 7.6e-8) and is independent of speed. The benchmarks integrate Newton’s law over the ground roll and time-step the cruise.

Flat-plate skin friction and boundary-layer thickness assume a smooth plate with zero pressure gradient. Re must be positive; transition location, roughness and flow separation are not detected by these algebraic helpers.

Textbook reference problems

tests/test_reference_problems.py works ten examples from Anderson’s Fundamentals of Aerodynamics (3rd ed.) and Modern Compressible Flow (3rd ed.) and checks the printed answers. Where they differ, the cause is the method, not the theory: Anderson reads four-figure tables and charts, and the functions here evaluate the closed forms exactly. In Fundamentals Example 9.2 (M1 = 2, 20 degree wedge) he rounds the normal Mach number 1.606 to the table entry 1.6 and prints p2 = 2.82 atm, where oblique_shock gives 2.843 atm. In Example 9.8 (M1 = 1.5, 15 degree corner) he takes the nearest table entry M2 = 2.0, where prandtl_meyer_expansion returns the root 2.019, so p2 is 0.455 atm, not 0.469 atm. Modern Compressible Flow uses the Fanning friction factor: f = 0.005 in Examples 3.11 to 3.13 is a Darcy factor of 0.02 here.

Reference: NASA Glenn normal-shock equations.