Why Airplanes fly

There are three different correct ways to explain how an airplane wing produces lift.

1. Bernoulli, Use the Bernoulli effect to calculate the pressure around the wing and so the force on the wing..
2. Newtonian action and reaction. The wing deflects air downward and so in turn is pushed upward.
3. Circulation, The flow of air by the wing is modelled as a circulation and linear flow, the circulation can be used to calculate the pressure distribution on the wing and so the force.

Many people make mistakes when aplying these techniques, other people attack those mistakes and then say that the tecnique is incorrect. Here are some helpful ways to use these explanations correctly.

Using Bernoulli's law

The simplest correct staement of Bernoulli's law is:

This statement speaks about the change in speed of one parcel of air versus the change in pressure within that parcel.

To use Bernoulli's law correctly as a beginner you must point to the parcel of air you are going to follow, label its speed and pressure at one point and use its speed at a second point to calculate its different pressure at that second point.

Bernoulli only allows you to calculate a change in pressure from one point to another along the path of a parcel of air, along a streamline.

Go to the Mathematics of Bernoulli.

As Robert Heinlein said," if it can't be expressed in figures it's not science it's opinion."

Once you know the pressure at every point of the wing you can integrate over area to find the force on the wing. To do this you need to know the velocity of the air at every point on the wing. This is hard to do.

We'll return to use Bernoulli's law correctly and to show how it is often used incorrectly.

Using Newton's Law

Newton's law of action and reaction says that when a wing forces air down the air forces the wing up. All wings producing lift deflect air.

The amazing thing about wings is that because they are flying through air which is a fluid, the top of the wing deflects air down as well as the bottom of the wing.

Wind tunnel photographs show that air is deflected downward by wings that ae producing lift.

Newton's law is almost never used in this form to calculate the lift produced by wings.

Go to the Mathematics of Newton

Circulation

The way that aerodynamicists actualy calculate the lift due to a wing is by calculating the circulation of the air as it flows over the wing. The calculations are complicated and involve conformal mapping, which is beyond high school mathematics.

Bernolli math root

Start with the work energy theorem and ignore any changes in height (gravitational potential energy is ignored.)

1/2 mv₁² + W = 1/2 mv₂²

The kinetic energy of a parcel of air of mass, m, and velocity, v, at point 1 changes to a different kinetic energy at point 2 when work, W, is done on the particle.

Work is force times distance, W = f d.

To create Bernolli's law divide each erm by the volume, V, of the parcel of air, assume the parcel is incompressible and so does not change from point 1 to 2, amazingly for subsonic flows this is a good approximation.

1/2 m/Vv₁² + W/V = 1/2 m/Vv₂²

Mass per unit volume is density, r
Work per unit volume is force per unit area or pressure difference, DP = P₂-P₁

1/2 r v₁² -P₁ = 1/2 r v₂² - P₂

This relates the pressure and airspeed at one point to the pressure and airspeed at another.

Incease the airspeed from point 1 to point 2 and the pressure at point 2 must decrease from that at point 1.

This is Bernoulli's law, accept no substitutes. Always require someone who claims to be using the law to indicate point 1 then point 2.

Return

Newton Math Root

One of Newton's laws is F = ma where F is force, m is mass and a is acceleration.

This is a simplified form of the larger law that force is produced by a change in momentum, p = m v, where v is velocity

F = dp/dt = dmv/dt so if m is constant = m dv/dt = ma

So if a wing deflects downward a mass of air m in every time interval Dt and changes the vertical component of the velocity of the air from zero to an amount f v then the force on the air will be.

F = m fv/ Dt

Expressed in terms of density of the air

F = r A L v / Dt = r A v Dt fv / Dt = r A f v²

So what matters is:

the density of the air, there is less lift when the air is less dense due to altitude, temperature or humidity.

The cross sectional area of air that the wing deflects downward.

The ffraction of the velocity of the air that is turned into the downward direction.

and the speed at which the airplane flies, squared.

It is difficult to determine both the cross sectional area and the fraction of the air speed deflected downward, so this equation is not used to calculate the lift of a wing based on its shape.

Return

Many people particularly in the U.S. say that airplanes fly because of the bernoulli effect.
Other people say that airplanes fly because of Newton's laws of action and reaction, i.e. the wings push air down and the reaction force pushes the wing up.

Usually each of these groups says that they are followers of the truth and the others are wrong.

Actually, both explanations are explain airplane flight if used correctly.
Followers of one theory usually attack the other based on examples which use the other theory incorrectly.

The funny thing is that aerodynamicists actually use a third theory called "circulation" to calculate lift produced by a wing.

Here is an example of the incorrect use of Bernoulli:

Compare the pressure exerted on flat ground by stationary air above the ground and by air blowing over the ground at a high speed. Incorrect use of Bernoulli's law would say that the high speed air exerts less pressure on the ground. Correct use says that Bernoulli's theorem cannot be used in this case since we are not talking about the same parcel of air. In fact, the still air and the moving air could both exert atmospheric pressure on the ground.

The glancing blow fallacy.

Some people say that the pressure of air moving over the surface is reduced because the high speed air molecules exert glancing blows on the ground. This is incorrect. The speed with which molecules hit the ground is purely the result of their thermal motion the temperature can remain the same in the stationary air case and the moving air case so the molecules it the ground with the same speed, the wind adds speed only parallel to the ground this does not change the speed with which the molecules hit the ground. (The definition of temperature does not count the motion of the molecules due to the wind.) The density of molecules in both cases is also the same. If the same number of molecules hit the same area of ground each second with the same average speed perpendicular to the surface then they will exert the same force on the surface.

Correct use of Bernoulli's theorem, air flowing through a constricted pipe called a venturi tube starts out at atmospheric pressure the tube constricts and the parcel of air speeds up. Therefore the pressure in the narrow part of the venturi tube is less than atmospheric pressure.

Here is another common incorrect question.
What fraction of the lift is due to Bernoulli effect and what fraction is due to Newton's laws?

Bernoulli vs Newton

Prelude Conservation of Energy vs Newton's Laws
First consider the following example:

I drop my keyring 2 meters to the ground.
Trick question: Is the falling of my keyring described by:

1. The law of conservation of energy? or
2. Newton's laws, F = ma

The answer of course is that both of these describe the fall of the keys.
If I wanted to know the speed with which the keys hit the ground it is easier to use the law of conservation of energy whereas if I wanted to know the time it took to hit the ground I'd use Newton's laws. See Math Root 1 for details.

The important thing is that both conswrvation of energy and newton's laws are correct it is just easier to use one over the other depending on the question for which you need to calculate the answer.

For Flight

Bernoulli's theorem is a statement of the law of conservation of energy, the force on a wing from the deflection of air is calculated from Newton's laws.

Both are correct if used appropriately.

In particular, you cannot say that half or any other fraction of the lift is due to Bernoulli's theorem and the rest is due to Newton's laws.

All of the lift is due to Bernouli's Theorem and all of it is due to Newton's laws too.

Math Root

Using the Law of Conservation of Energy to find the final velocity
For a falling object where I can ignore friction forces the law of conservation of energy says that:

KE + PE = 0

where KE is kinetic energy in joules
KE = 1/2 mv²

where m is the mass in kg and v is the speed in m/s

and PE is potential energy PE = mgh

where g is the acceleration of gravity, 10 m/s² and h is the height of the drop for the keys, in m, h = -2m.

Solving for final velocity use

1/2 mv² = - mgh
v = (2gh)^0.5

Notice that the mass doesn't matter. The final velocity is about 6.3 m/s down.

Using Newtons laws and kinematics to find the fall time

F = ma where F is force in Newtons for gravity F = mg

a = F/m = mg/m = g

from kinematics

h = 1/2 a t² with a = g

so t = (2h/g)^0.5

so t = 0.63 s

Scientific Explorations with Paul Doherty

© 2000

20 October 2000