Why is induced drag less on a high span wing?

Question

I have a basic question and sorry if it is a well known fact.

I understand that the induced drag is due to the tip vortices changing the effective angle of attack (downwash).

I searched some website and they mentioned that for high span wings the disturbance caused by tip vortices is less and hence, less induced drag.

Now the confusion arises.

As per the lifting line theory, the tip vortices strength will be as same as the bound vortex.

Does this mean that the induced drag is reduced only for a high span wing when having same bound vortex strength as a low span wing?

[For simplicity we can assume a elliptical lift distribution]

Answer 1

You start from wrong assumptions, which explains your doubts. The line

the induced drag is due to the tip vortices

is as true as saying that wet streets cause rain. Also, the opinion that the

tip vortices strength will be as same as the bound vortex

is wrong. Unfortunately, many authors don't understand the topic themselves and copy what others have written before without thinking the issue through. Ideally, you would forget about all what you have heard about vortices and lifting lines, but since you ask I will try to explain potential flow theory a little.

In potential flow theory, lift is caused by vortices which are caused by the movement of a wing through air. These vortices run along a closed line: Within the wing they form the bound vortex, then they leave the wing backwards as trailing vortices and are connected at the point where the movement started by the starting vortex.

Now comes the important part which most authors conveniently leave out: There is no single vortex; instead, potential flow assumes an infinite number of infinitesimally small vortices which form out of nowhere when lift is increased or speed is reduced. Consequently, no single vortex leaves the wing at the tips, instead, a sheet of vortices leaves the wing at the trailing edge. The change in strength of the bound vortices over span is equivalent to the strength of the vortices leaving the wing, so the vortices fade out towards the tips.

My advice is: If you do not want to operate or to write a potential flow code, do yourself a favor and forget all that. It is much better to interpret lift as the consequence of a pressure field around a wing which accelerates the air flowing around this wing downwards. Induced drag is simply the component of the resulting pressure forces parallel to the direction of movement, while the perpendicular component is lift. Please make sure to follow at least the last link; it gives a very good explanation what induced drag really is.

Tip vortices are the consequence of air filling the void above the downward moving air behind the wing. They are not originating from the wingtips, but the consequence of the vortex sheet rolling up (if you want to stay in that picture). Note that the distance between the cores of the vortices is much smaller than the wingspan. For an elliptic wing of span $b$, it is actually only $\frac{\pi}{4}\cdot b$

A higher wingspan allows to capture more air for lift creation, so less downward acceleration is needed. Lower downwash speed also causes a less powerful trailing vortex. Note that the mass of air affected by the wing grows with the square of the wingspan!

Answer 2

A simple way to understand this which I've never seen anywhere is to consider the air pushed down by the wing as the aircraft passes. If aircraft 1 has a wider wingspan than aircraft 2 (but the same wing area), then aircraft 1 accelerates a wider swath of air downwards than aircraft 2. Lift is proportional to the momentum transferred to the air being pushed downwards, while the energy required to do the downwards pushing is proportional to the square of the downward velocity. Pushing down on the wider swath of air means the air doesn't have to move as fast as the smaller swath to give the same lift. Since momentum is the same for both the large and small swath, the energy imparted to the larger (more massive) swath of air is smaller because of the velocity-squared term.

This offers an intuitive explanation for why induced drag drops as speed rises: at low speeds the mass of air pushed downward is proportional to the wingspan times the distance covered per second. At higher speeds the distance covered per second rises, giving the aircraft more mass to push against. Double the air mass by doubling the speed and the downward air velocity drops by half, which cuts the energy required by a factor of root two.

This extends to explain why an infinitely long wing produces zero induced drag: the accelerated air mass is infinite, so the air velocity imparted by downwards acceleration is zero, which costs no energy.

In a real aircraft the air doesn't go straight down, so you get vortices. But the momentum/energy relationship holds, as does the main principle: the larger the mass of air, the less you have to accelerate it downward to get the upward lift you want.