Elliptical wing planform and elliptical lift distribution

Question

I know that there are similar topics already here, but after reading them I still do not understand everything. From what I know elliptical lift distribution gives the least amount of induced drag. That is because there is the least amount of lift at the wingtips compared to other parts of the span - is it true what I wrote? And know question with elliptical wing planform. Is this planform generating the least amount of induced drag? And I read somewhere that this wing planform is most efficient among other planforms in subsonic flight. Is it true and if yes why? And does elliptical wing has elliptical wing distribution? Because I heard lot of versions.

Answer 1

Lift force over the wing is said to be of elliptical distribution if the lift per span ($L'$) along the wing span is of the following:

$$L'(y) = \rho_\infty V_\infty \Gamma_0 \sqrt{1-(\frac{2y}{b})^2}$$

where $\rho_\infty$ is density of the airflow, $V_\infty$ is the free-stream airspeed, $b$ is length of wing span, $y$ is lateral coordinate along the wing (0 being wing root), and $\Gamma_0$ is the circulation at wing root.

The above is essentially an equation of an ellipse, and hence the name elliptical lift distribution.

In low Mach flight, it can be theoretically demonstrated that an elliptical lift distribution produces the least induced drag for a given flat span. Induced drag of a lifting surface in incompressible flow can be expressed as (Ref Anderson, Fundamentals of Aerodynamics):

$$C_{D_i}=\frac{C_L}{\pi e A}$$

where $C_L$ is the lift coefficient, $A$ is aspect ratio and $e$ is the span efficiency factor.

For a flat span, $e$ must be smaller or equal to 1, and it's only equal to 1 when the lifting surface has an elliptical lift distribution.

When the wing has zero twist, an elliptical planform produces an elliptical lift distribution. However, elliptical lift distribution can also be coaxed by judiciously twisting the wing for a planform that is not geometrically elliptical.

Addendum on non-flat span:

In the case where the span is not flat, either because it has dihedral or is curved, it has been theoretically demonstrated that a non-flat span can achieve lower induced drag than a flat wing of equal projected span. This forms the theoretical basis behind winglets, end plates, spiroids, etc.

However, if we compare the non-flat wing to a flat wing whose length is that of the unfurled non-flat wing, the flat wing always has lower induced drag. That is, if there is no limit on the wing span length, elongating it is always better from an induced drag perspective.

Below is a graph comparing the induced drag of an optimal non-flat (circularly cambered wing) wing to that of an elliptically loaded flat wing. A value of 1 indicates no change. A value greater than 1 indicates more induced drag and a value less than 1 indicates less. $\beta$ is the span camber factor.

Data and figure compiled from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19630006412.pdf

Answer 2

… elliptical lift distribution gives the least amount of induced drag.

Not quite. The elliptical lift distribution (read JZYL's answer for an explanation what it is) will only give the least amount of induced drag for a given span. That means that wings with more span and a more triangular lift distribution will be lighter, thus needing less lift and, consequently, producing even less induced drag than that elliptical wing. This was shown by Ludwig Prandtl in 1932 and R.T. Jones in 1950. For his elegant proof, Prandtl used the moment of inertia of the lift distribution which is smaller when more of the lift is concentrated near the wing root as a proxy for wing mass.

That is because there is the least amount of lift at the wingtips compared to other parts of the span

This is true for all realistic lift distributions. Even a triangular wing (one with zero tip chord) which has its highest lift coefficient near the tip will have its lift drop to zero at the tip. The reason for the low induced drag can be found in the downwash, which is constant over span for an elliptical lift distribution.

Is this planform generating the least amount of induced drag?

The least amount of induced drag is only possible with the least amount of lift. A more practical figure of merit is how much lift on top of what is needed to keep the wing aloft can be produced for the least amount of induced drag. This will be driven foremost by span: The more wingspan is available to produce a given amount of lift, the lower the induced drag will be. By using flaps or twist, the lift distribution can be optimized further, again (almost) independent of planform.

… this wing planform is most efficient among other planforms in subsonic flight.

No, and on top of that it has unpleasant stall characteristics. Next, scaling laws have also a say what planform gives best efficiency. Large aircraft are better off with a smaller taper ratio.

does elliptical wing has elliptical wing distribution? (I read this as lift distribution)

Yes, this is the one undisputed advantage of an unswept elliptical planform without washout: It will have an elliptical lift distribution over the full angle of attack range with fully attached airflow. As soon as separation starts, however, that is no longer true. But all other planforms need to be tweaked to produce an elliptical lift distribution and then do it only at one angle of attack.

Because I heard lot of versions.

Oh yes, there is a lot of mythology to be found on elliptical wings. Mostly because of oversimplification. I prefer to keep things as simple as possible, but no simpler. That is why this answer has become so long.