### Lift, Induced Drag, and Span Efficiency

Where Lift and Drag Come From
Work On More Air
How To Find More Air
Span Efficiency

#### Where Lift and Drag Come From

Flying is like running on soft ground. The ground gives way under you, so it takes more energy to run since you are providing the energy to move not only yourself but the ground under your feet.

So it is with flight: the airplane is “running” on the air, which, like soft ground, “gives way.” More accurately, the pressures on the airplane’s surfaces accelerate the airmass, creating a total aerodynamic force on the airplane equal and opposite to the rate of change of momentum (mass x velocity) induced in the accelerated airmass. The energy required to accelerate the air comes from the airplane and shows up as drag. The downward component of the accelerated airmass corresponds to the lifting force on the airplane, and the energy required to accelerate the airmass downward shows up as “lift-induced” or “vortex” drag.

#### Work 0n More Air

The key to understanding flight lies in these two equations from high school physics:

LIFT is equal to the rate of change of downward momentum induced in the airmass, where:

Momentum = mass x downward velocity

While DRAG is proportional to the energy added to the accelerated airmass, where:

Energy = 1/2 x mass x velocity squared

The more weight you have to lift, the more downward momentum you have to impart to the atmosphere. So, to increase lift, one must either increase the mass of air worked upon or increase its induced velocity (assuming a constant atmosphere). Because lifting forces increase proportionally with the induced velocity of the airmass, while drag increases proportionally with the square of the induced velocity, to produce lift more efficiently we must work on a larger mass of air and accelerate it less.

#### How To Find More Air

There are three ways to work on a larger airmass, equating to the three spacial dimensions:

1. Fly faster to increase the airmass in the longitudinal (forward/aft) dimension (jet fighter).
2. Increase wingspan to increase the airmass on the lateral (left/right) dimension (sailplane).
3. Create a non-planar lifting system to increase airmass on the vertical (up/down) dimension (biplane, winglets, annular wings).

#### Span Efficiency

SPAN EFFICIENCY is a number used to measure the relative lifting efficiency of a wing of specific span, and is inversely proportional to induced drag. Thus, all else being equal, an aircraft with span efficiency of 1 will have twice as much induced drag as an aircraft with span efficiency of 2.

A planar wing with a perfectly elliptical lift distribution (span-wise) induces minimum energy into the airmass for a given span and lift (thus minimizing induced drag) and has a theoretical span efficiency of 1. Since aircraft rarely exhibit an elliptical lift distribution, monoplanes generally have a span efficiency of less than 1.

By comparison, biplanes commonly demonstrate span efficiencies of 1.2 to 1.4, winglets often yield span efficiencies of around 1.2, and annular wings have a theoretical span efficiency of 2.0

This is the very straightforward physics of lift and induced drag, often overlooked and misunderstood even by practicing engineers.

The same physical principles apply to propulsion: To increase the ideal efficiency (thrust/power input) of a propeller at a given airspeed, work on more air by increasing diameter (disk area), or by using a “non-planar” propulsion system (duct or shroud).