Prandtl lifting-line theory

Lesson 8 quoted results for elliptical wings and gave numbers for tapered ones. This lesson derives them. Prandtl's lifting-line theory, published between 1918 and 1919, was the first practical theory of the finite wing, and in the form Hermann Glauert gave it a few years later it is still used every day in early design. It predicts how lift is distributed along the span, the induced drag and the span efficiency, and, crucially for the Block 2 wing, which part of the wing will stall first. Arjun uses it to choose the taper ratio and the twist.

The lifting line

Prandtl replaced the wing by a single bound vortex along the quarter-chord line, the lifting line, whose strength \(\Gamma(y)\) varies along the span. Wherever the strength changes, Helmholtz's theorems require vorticity to leave the line and trail downstream, so a vortex sheet of strength \(-d\Gamma/dy\) per unit span streams back from the wing. Each element of that sheet induces downwash at every point on the lifting line, and the total downwash at a station \(y_0\) is an integral over the whole span.

Each section is then assumed to behave like a two-dimensional airfoil at its effective angle of attack, which is its geometric angle less the induced angle \(w/V_\infty\). The section's lift from two-dimensional data must equal the lift implied by the local circulation through the Kutta-Joukowski theorem. Equating the two gives Prandtl's lifting-line equation, an integro-differential equation for the circulation.

Here \(c(y_0)\) is the local chord, \(\alpha(y_0)\) the local geometric angle of attack including any twist, and \(\alpha_{L0}(y_0)\) the local zero-lift angle. All the geometry of the wing, its planform, twist and section choice, enters through these three functions. The theory assumes a straight, unswept wing of moderate to high aspect ratio, attached flow and incompressible conditions. For the Tern, with aspect ratio 10.3 and almost no sweep, it is an excellent model below the stall.

Glauert's Fourier solution

Glauert made the equation solvable by hand. Change the spanwise variable to an angle with \(y = -\tfrac{b}{2}\cos\theta\), so \(\theta\) runs from 0 at one tip to \(\pi\) at the other, and write the circulation as a sine series. Each term automatically vanishes at both tips, as the circulation must.

Substituting into the lifting-line equation, the downwash integral can be done exactly for each sine term, and the equation becomes a simple algebraic condition at each angle \(\theta\). Writing \(\mu = c\,a_0/(4b)\), it reads as follows.

Choosing \(N\) stations along the span gives \(N\) linear equations for \(N\) coefficients. For a symmetrical wing at symmetrical flight conditions, only the odd terms appear, so stations on one half of the span are enough. The program uses 12 odd terms and 12 stations from the tip to the root, which is more than enough: the results do not change in the third decimal place with more terms.

Once the coefficients are known, the wing's lift and induced drag follow from remarkably simple formulas. Only the first coefficient contributes to lift; the higher terms change how the lift is distributed and add induced drag.

The elliptical distribution is the case in which only \(A_1\) is nonzero, so \(\delta = 0\) and \(e = 1\). Any other distribution adds higher harmonics, each of which adds induced drag in proportion to its square. That is the formal proof that elliptical loading minimizes induced drag.

# Prandtl lifting-line theory solved by Glauert's Fourier method for a tapered, twisted wing
import math
def solve(A, b):
    n = len(b); M = [r[:] + [b[i]] for i, r in enumerate(A)]
    for k in range(n):
        p = max(range(k, n), key=lambda r: abs(M[r][k])); M[k], M[p] = M[p], M[k]
        for r in range(k + 1, n):
            f = M[r][k] / M[k][k]
            for c in range(k, n + 1): M[r][c] -= f * M[k][c]
    x = [0.0] * n
    for k in range(n - 1, -1, -1): x[k] = (M[k][n] - sum(M[k][c] * x[c] for c in range(k + 1, n))) / M[k][k]
    return x
def wing(alpha_deg, taper, washout_deg, b=17.0, S=28.0, a0=2 * math.pi * 0.96, aL0_deg=-4.0, N=12):
    cr = 2 * S / (b * (1 + taper))
    odd = [2 * k + 1 for k in range(N)]                 # symmetric loading: odd Fourier terms only
    th = [math.pi / 2 * (i + 1) / N for i in range(N)]  # stations from tip (small theta) to root (pi/2)
    A, rhs = [], []
    for t in th:
        y = -b / 2 * math.cos(t); eta = abs(2 * y / b)
        c = cr * (1 - (1 - taper) * eta)
        twist = -washout_deg * eta                      # linear washout: tip pitched down
        mu = c * a0 / (4 * b)
        A.append([math.sin(n * t) * (mu * n + math.sin(t)) for n in odd])
        rhs.append(mu * math.sin(t) * math.radians(alpha_deg + twist - aL0_deg))
    An = solve(A, rhs)
    AR = b * b / S
    CL = math.pi * AR * An[0]
    delta = sum(n * (An[i] / An[0]) ** 2 for i, n in enumerate(odd) if i > 0)
    e = 1 / (1 + delta)
    def cl_local(eta):                                  # section lift coefficient at spanwise position eta = 2y/b
        t = math.acos(-eta) if eta < 1 else 1e-9
        c = cr * (1 - (1 - taper) * abs(eta))
        G = 2 * b * sum(An[i] * math.sin(n * t) for i, n in enumerate(odd))   # circulation / V
        return 2 * G / c
    return CL, e, cl_local
for taper, wash in ((1.0, 0.0), (0.45, 0.0), (0.45, 2.5), (0.3, 0.0)):
    CL, e, clf = wing(4.0, taper, wash)
    cls = [clf(-eta) for eta in (0.0, 0.4, 0.7, 0.9)]
    peak = max(range(19), key=lambda i: clf(-i / 20) / CL) / 20
    print(f"taper {taper:4.2f}, washout {wash:3.1f} deg: CL = {CL:.3f}, e = {e:.4f}, cl/CL at eta 0/0.4/0.7/0.9 = "
          + " ".join(f"{c / CL:.2f}" for c in cls) + f", peak cl/CL near eta = {peak:.2f}")

taper 1.00, washout 0.0 deg: CL = 0.685, e = 0.9154, cl/CL at eta 0/0.4/0.7/0.9 = 1.11 1.08 1.00 0.79, peak cl/CL near eta = 0.00
taper 0.45, washout 0.0 deg: CL = 0.703, e = 0.9792, cl/CL at eta 0/0.4/0.7/0.9 = 0.95 1.04 1.04 0.92, peak cl/CL near eta = 0.55
taper 0.45, washout 2.5 deg: CL = 0.609, e = 0.9715, cl/CL at eta 0/0.4/0.7/0.9 = 1.05 1.06 0.96 0.79, peak cl/CL near eta = 0.15
taper 0.30, washout 0.0 deg: CL = 0.705, e = 0.9803, cl/CL at eta 0/0.4/0.7/0.9 = 0.91 1.03 1.07 0.99, peak cl/CL near eta = 0.70

Choosing the Block 2 planform and twist

The program compares four candidate wings, all with the Block 2 span of 17 m, area of 28 m² and NACA 44-series sections, at 4 degrees angle of attack. A rectangular wing, with constant chord, has a span efficiency of only 0.915 and its local lift coefficient is highest at the root. A wing tapered to 0.45 at the tip is almost elliptical, with \(e = 0.979\), and produces slightly more lift at the same angle. But its local lift coefficient, relative to the wing's, peaks at about 55 percent of the semispan and is still 0.92 at 90 percent. A taper of 0.3 is slightly more efficient again, \(e = 0.980\), but pushes the peak out to 70 percent of the semispan.

Why does the location of the peak matter so much? A wing begins to stall where its sections first reach their maximum lift coefficient. If that happens near the tip, the stall starts outboard, where the ailerons are, and if it spreads unevenly the aircraft can roll sharply and lose aileron control just when the pilot needs it. If the stall starts near the root, the outer wing keeps flying, the ailerons keep working, the turbulent wake from the stalled root buffets the tail as a natural warning, and the nose drops as the root lift is lost behind the centre of gravity. Certification rules in both countries require that the aircraft can be controlled in roll and that it gives clear warning before and during the stall, and aircraft designers almost always aim for root-first stall.

Tapering a wing is good for induced drag and for structure, since it moves the lift inboard and reduces the root bending moment, but it moves the section lift peak outboard, which is bad for stall. The tip sections also have a smaller chord and so a lower Reynolds number, which lowers their maximum lift. The standard cure is washout: twisting the wing so the tip is at a lower angle of attack than the root. With 2.5 degrees of linear washout, the program shows the peak moving all the way in to about 15 percent of the semispan, with the local lift coefficient at 90 percent of the semispan falling to 0.79 of the wing's. The price is a small loss of span efficiency, from 0.979 to 0.972, and a lower wing lift coefficient at the same angle, because the tips now carry less load. Tomasz's team chose the taper of 0.45 with 2.5 degrees of washout. The extra induced drag in cruise is about 0.7 percent of the induced drag, or roughly 7 N, a price everyone agreed was worth paying.

Beyond the classical lifting line

Lifting-line theory does not apply well to swept wings, low-aspect-ratio wings or wings with strong flow variations along the chord, because one line cannot represent how the load varies in the chordwise direction. The vortex lattice method extends the idea by covering the wing's planform with a grid of horseshoe vortices, several along the chord as well as along the span, and is widely used in conceptual design and for stability derivatives in AERO-330. Nonlinear versions of lifting-line theory use measured section lift curves instead of the straight-line slope, which extends the method up to the stall and lets designers estimate the wing's maximum lift, as Lesson 11 does in a simple way.