Airfoil and Wing Lift: Method

Airfoil and Wing Lift: Thin-Airfoil Theory and Prandtl's Lifting Line

The tool answers two questions in sequence. What lift does a section (an airfoil of infinite span) produce at a given angle of attack, and where is its zero-lift direction? And what lift does a wing of finite span made of that section produce, how is it spread along the span, what induced drag does it pay for it, and where does it begin to stall? The first is thin-airfoil theory (Munk, Glauert); the second is Prandtl's lifting-line theory in the Fourier-series form Glauert gave it in 1926, solved here as a linear system at Multhopp's stations. Both are linear, inviscid, incompressible theories; their limits are listed at the end.

Thin-airfoil theory of the section

A thin cambered section is replaced by its camber line z(x) carrying a vortex sheet of strength γ(θ), with x = (c/2)(1 − cos θ) so that θ runs from 0 at the leading edge to π at the trailing edge. The condition that the flow follows the camber line gives γ as a Fourier series whose coefficients are integrals of the camber-line slope,

A₀ = α − (1/π) ∫₀^π (dz/dx) dθ,    A_n = (2/π) ∫₀^π (dz/dx) cos nθ dθ,

and the section coefficients follow: c_l = π(2A₀ + A₁) = 2π(α − α_L0), so every thin section has the lift-curve slope a₀ = 2π per radian (0.1097 per degree) and the zero-lift angle

α_L0 = −(1/π) ∫₀^π (dz/dx)(cos θ − 1) dθ,

negative for positive camber. The moment about the quarter chord, c_m,c/4 = (π/4)(A₂ − A₁), does not change with angle of attack: the quarter chord is the aerodynamic centre of every thin section. The ideal or design angle of attack α_i is the one at which A₀ = 0 and the flow meets the leading edge without turning round it (the lowest drag of the real section); the lift there is the design lift coefficient c_li = π A₁.

NACA four-digit camber line. The mean line of the NACA 2412 is two parabolic arcs, z/c = (m/p²)(2px − x²) ahead of the point of maximum camber and (m/(1 − p)²)((1 − 2p) + 2px − x²) behind it (x in fractions of the chord, m = 0.02 the maximum camber, p = 0.4 its position: the first two digits). Its slope is (2m/p²)(p − x) and (2m/(1 − p)²)(p − x), linear in cos θ on each arc, so the integrals above are elementary and the tool evaluates them in closed form, splitting them at θ_p = arccos(1 − 2p). NACA Report 824 tabulates α_i, c_li and c_m,c/4 for the NACA 62 to 67 mean lines (6 % camber at 20 to 70 % chord) and notes that the values scale linearly with the camber; the closed form reproduces them within the handbook's own rounding (its ideal angles were integrated from tabulated slopes and are up to 0.2° below the exact values). The report also records that measured zero-lift angles of four-digit sections are about 0.93 of the thin-airfoil value and that measured slopes of smooth sections are 0.100 to 0.110 per degree: the tool prints the 0.93 value beside the theoretical one, and a measured section can be entered directly as its a₀ and α_L0.

Prandtl's lifting line

A wing of span b is replaced by a bound vortex of strength Γ(y) along its quarter-chord line (the lifting line) and the sheet of trailing vortices it sheds, of strength −dΓ/dy per unit span. The trailing sheet induces a downwash w(y) at the lifting line, which turns the local flow down by the induced angle α_i = w/V, so the section at y works at the effective angle α_eff = α + ε(y) − α_L0 − α_i(y), where α is the geometric angle of attack of the root chord and ε(y) the geometric twist of the local chord relative to the root (washout negative). The lift of the section is both a₀ α_eff (its own lift curve) and 2Γ/(V c) (Kutta–Joukowski), and the downwash is the Biot–Savart integral of the sheet: eliminating w gives Prandtl's integro-differential equation for Γ(y).

Glauert's series and the monoplane equation. With y = −(b/2) cos θ (θ = 0 at the left tip, π/2 at the root, π at the right tip) the circulation is written

Γ(θ) = 2 b V Σ A_n sin nθ,

which vanishes at both tips term by term; the downwash integral then has the closed form α_i(θ) = Σ n A_n sin nθ / sin θ, and Prandtl's equation becomes the monoplane equation

Σ_n A_n sin nθ [ 4b/(a₀ c(θ)) + n/sin θ ] = α − α_L0 + ε(θ),

an algebraic equation in θ. For a wing symmetric about the root only the odd terms survive. The tool keeps N odd terms and satisfies the equation exactly at N stations on the half wing, θ_k = kπ/(2N), k = 1 … N: the cosine stations of Multhopp (1938), which put θ_N = π/2 at the root and crowd the others toward the tip, where the loading falls most steeply. This gives an N × N linear system, solved by Gaussian elimination. Glauert used N = 4 (θ = 22.5°, 45°, 67.5°, 90°) and the tool reproduces his tables for rectangular aerofoils to their printed digits at that setting; 20 terms are converged to 0.1 % in C_L and 0.2 % in C_Di for plain tapered wings, and a pointed tip or a step in the chord needs 40 or more.

The right-hand side is linear in the angle of attack, so the system is solved twice: once with a unit right-hand side (the coefficients A_n⁽¹⁾ per radian of absolute angle) and once with the twist alone (A_n^(ε)). The solution for any angle is A_n = (α − α_L0) A_n⁽¹⁾ + A_n^(ε), and one solve covers every angle of attack.

What the coefficients give

• Lift. Integrating ρ V Γ over the span, C_L = π A A₁: only the first coefficient carries lift (A = b²/S is the aspect ratio).
• Induced drag. Integrating ρ V Γ α_i, C_Di = π A Σ n A_n² = C_L² (1 + δ)/(π A) with δ = Σ_n>1 n (A_n/A₁)² ≥ 0. The elliptical loading (A₁ alone, constant downwash C_L/(πA)) has the least induced drag for a given lift and span; δ measures the excess of any other loading and e = 1/(1 + δ) is the span efficiency factor. Glauert's rectangular wing of aspect ratio 6 has δ = 0.046; a taper ratio of 0.3 to 0.5 brings a straight-tapered wing within 1 or 2 % of the ellipse. The tool reports δ and e of the actual loading (twist included) and of the additional loading alone (the untwisted wing, the value the tables give), and the induced drag of the basic loading alone, which is the price of the twist at zero lift.
• Lift-curve slope. From the unit-angle solution, a = π A A₁⁽¹⁾, which Glauert writes a = a₀/[1 + a₀(1 + τ)/(πA)]; τ = 0 for the elliptic wing and 0.163 for a rectangular wing of aspect ratio 6 (his Table 11 against A/a₀). Anderson's factor f of Report 572 is the same slope divided by the elliptic value.
• Zero-lift angle of the wing. The twist shifts it: α_L0,w = α_L0 − A₁^(ε)/A₁⁽¹⁾, measured from the root chord (Anderson's α_L0 + J ε, J about −0.4 for a linear twist). Washout makes it less negative.
• Span loading. c c_l = 2Γ/V = 4b Σ A_n sin nθ is the running lift per unit dynamic pressure (a length), the quantity the Wing Span Loads tool integrates; c_l = c c_l/c is the section lift coefficient. Following Anderson (Report 572), the loading is split into the additional part, proportional to C_L and independent of the twist (from A_n⁽¹⁾, normalised to unit C_L: Anderson's L_a = c_la1 c b/S), and the basic part, the twist's loading with its lift removed, A_n^(b) = A_n^(ε) − (A₁^(ε)/A₁⁽¹⁾) A_n⁽¹⁾, which integrates to zero lift and does not change with angle of attack (Anderson's L_b). Washout gives a basic loading positive at the root and negative at the tip.
• Stall onset. With a section maximum lift coefficient c_l,max (taken constant along the span) the first station to stall is where c_l(y) = c_l,b(y) + C_L c_l,a1(y) reaches it first, i.e. the wing C_L,stall is the minimum over the span of (c_l,max − c_l,b)/c_l,a1; the angle follows from the lift curve. This is Anderson's criterion for the beginning of the stall of a tapered wing; the actual maximum lift is somewhat higher because the stalled sections still carry lift. A stall that begins outboard of about 60 % of the semi-span (tip stall, typical of strong taper without washout) loses the ailerons first; washout, a fuller tip or a tip section of higher c_l,max moves it inboard.

Schrenk's approximation beside the exact loading

The Wing Span Loads tool uses Schrenk's method (NACA TM 948, 1940): the additional loading is the mean of the planform-proportional curve and the semi-ellipse of equal area, c c_l = ½ (c + c_ell) C_L, and the basic loading of a twist is half the chord-proportional loading about the chord-weighted mean twist, ½ a₀ (ε − ε̄) c. The present tool evaluates both lines for the same wing at the same C_L and prints them beside the lifting-line values with the percentage difference, so that the user of the loads tool knows what the approximation costs for the planform at hand: for a straight taper of 0.5 the additional loading agrees within about 2 % over the inner 90 % of the span and is fuller than the exact curve near the tip; the basic loading of a linear twist is 10 to 15 % high at the root. Schrenk's own slope correction, a₀/[1 + a₀(1 + κ)/(πA)] with κ from his Figure 1, is printed too; it is Glauert's formula with τ replaced by a function of the taper ratio.

Wing-body factor

With the box checked the tool applies Raymer's conceptual-design estimate of the lift-curve slope of the wing-body combination, C_Lα,WB = C_Lα F S_exp/S with F = 1.07 (1 + d/b)², where d is the fuselage width at the wing and S_exp the wing area outside the fuselage: the body carries lift across its width, the exposed wing is smaller than the reference area. It is information for a performance or stability estimate; the span loading remains the wing's.

Conventions and inputs

• Angles in degrees on the form, radians inside: the angle of attack is that of the root chord; the geometric twist ε(y) is the angle of the local chord relative to the root chord, washout negative; with one section along the span the aerodynamic twist equals the geometric twist, so the absolute angle of a section is α + ε − α_L0.
• The planform is a straight taper (span, root and tip chord, linear twist to the tip) or a typed table of chord and twist against span station from the root to the tip, linear between the rows; rounded tips or a constant-chord centre section are typed (the chords of Report 572's rounded-tip wing, for example). The span is twice the last station.
• The quarter-chord sweep is drawn and reported but not used: the lifting line is a straight bound vortex. The usual correction for moderate sweep is a slope reduced by about cos Λ (Helmbold–Diederich form a = a₀ cos Λ/[√(1 + (a₀ cos Λ/(πA))²) + a₀ cos Λ/(πA)]) and a loading shifted toward the tips of a swept-back wing; a swept wing needs lifting-surface or vortex-lattice theory for its distribution.
• The number of Fourier terms is also the number of stations; the table has the stations the user asks for, interpolated from a 41-point grid in θ; the charts use the grid.

Limits

The theory is linear (small angles, attached flow), inviscid (no profile drag, no maximum lift: the stall-onset station is where the theory stops being valid), incompressible (apply the Prandtl–Glauert factor to a₀ for subsonic Mach numbers), and for straight wings of aspect ratio above about 4 (lifting-surface theory for lower aspect ratios, where the lifting line over-predicts the slope). A single section is used along the span; part-span flaps and ailerons (a step in ε) are resolved only slowly by the series; dihedral, unsymmetric loading and the fuselage interference on the loading are not included.

Checks

The library behind the page is checked without the framework (test_lifting_line.php, 279 checks): Glauert's Tables 11, 12 and 13 for rectangular aerofoils and his aspect-ratio-6 values (τ 0.163, δ 0.046, 3.55° per C_L, 0.0555 C_L²) to their printed digits at four terms; NACA Report 572's L_a and L_b tables for aspect ratio 6, taper 0.5 with rounded tips and the factors of its illustrative example (u 0.995, J −0.408, w 0.0039, C_Di 0.0786 at C_L 1.2) at Anderson's four points, with the converged values recorded; the Report 824 mean-line data (NACA 62 to 67, the NACA 2415 and 4415 examples); the elliptical wing (e = 1, constant downwash, Report 572 Table III); Schrenk's two lines; and the default case against an independent numpy solution at 20 and 200 terms.