Elliptical planform

Elliptical Planforms: A Capsule History

Frederick Lanchester, an Englishman born in 1868, discovered high efficiency elliptical planforms. In 1897 Lanchester presented a paper on the origin and nature of lift generated by an aerofoil (a term he coined). The Physical Society of London immediately rejected his idea.

Not accepting defeat Lanchester continued to work and 10 years later published Aerodynamics. In his book Lanchester presented the concept of the elliptical distribution of lift, and the vortex theory of the finite aerofoil. The vortex theory predicted the existence of the tip vortex. Lanchester predicted all of this with no testing or evidence to guide him; he didn’t use models or wind tunnels.

Lanchester also secured a patent for bent up wing tips to control tip vortices, in 1897. Lanchester devised these theories about the same time the Wright brothers were learning to fly. 20 years later other aerodynamicists proved his theories correct.

During WWII the English exploited elliptical lift distribution in the design of the Spitfire airplane wing. This arrangement was chosen because it gives the most lift for the least amount of drag.

Racing yacht designers began to use the elliptical planform in their mainsails during the 1930’s and now designers use elliptical sails that can sometimes be almost rectangular.

Planform- the shape of an object when viewed from the top-down perspective (plan view).

As a note, this is a brief history of this shape. I am by no means an expert.

Sources:
Aerohydrodynamics of Sailing, C. A. Marchaj
Eric Sponberg

Proof that an elliptic planform makes the most aerodynamically efficient wing (assuming low speeds)

We begin by stating, without proof, that the equation for induced drag C_Di over a wing with circulation Γ(y) in a flow with density ρ_∞ and a downwash velocity of w(y). y represents position on the wing measured from the root, varying from -b/2 to +b/2, where b is wingspan.

D_i = -∫_-b/2^+b/2ρ_∞w(y)Γ(y)dy

Next, we take the elliptical distribution of circulation around a wing, which depends on y:

Γ(y) = Γ₀√(1 - (y/(b/2))²)

Finally, the downwash velocity for this planform simplifies to:

w(y) = -Γ₀/(2b)

We may now substitute into the integral expression:

D_i = -∫_-b/2^+b/2ρ_∞(-Γ₀/(2b))(Γ₀√(1 - (y/(b/2))²))

We introduce the coordinate transfomation y = -s cos φ, and simplify the expression to:

D_i = (ρ_∞Γ₀²/(2b))∫₀^π(b/2) sin φ√(1 - cos² φ) dφ
D_i = (π/8)ρ_∞Γ₀²

We now apply the formula for the drag coefficient:

C_Di = D_i/(0.5*ρ_∞V_∞²S)
C_Di = πΓ₀²/(4V_∞²S)

This is all well and good, but the circulation at the wing root is not a practical design variable. We then state, without proof, that based on the expression for the lift coefficient of the same wing:

Γ₀ = 2C_LV_∞S/(πb)

Thus, we can substitute into our expression for the drag coefficient:

C_Di = {π/(4V_∞²S)}{2C_LV_∞S/(πb)}²
C_Di = C_L² / (π*AR)

We have now derived the coefficient of induced drag for a wing with an elliptical circulation, and thus, elliptical lift distribution and planform (these follow directly from the Kutta-Joukowski lift theorem and the proportionality of chord to circulation).

Now, *poof* I present the formula for the coefficient of induced drag in general:

C_Di = C_L² / (π*AR*e)

where the airplane efficiency factor e ≤ 1. Note then that for the elliptical planform, e = 1. Therefore, a wing with an elliptical planform is the most efficient (provided aspect ratio and lift coefficient are fixed), because it has the least possible value for its coefficient of induced drag.