Bernoulli vs. Newton

Some teachers are adamant that airplanes fly because the pressure above the wing is reduced due to the Bernoulli effect.

Others are equally adamant that airplanes fly because wings deflect air downward so that in reaction the plane is forced upward.

Many of the supporters of one of these points of view believe that the other point of view is wrong.

Yet both points of view are correct.

Bernoulli's theorem is just a statement of the law of conservation of energy and so it is true for airplane wings.

Newton's law of action and reaction is also true in every case including airplane wings.

It is interesting that while both of these theories is true, neither is used by airplane wing designers to compute the lift of a wing. They use a theory in which lift is due to circulation of air about the wing.

**Bernoulli**

This is Bernoulli's theorem in its simplest form::

When the speed of a fluid increases the pressure decreases.

Notice that this is a statement about the relationship between the change in speed of a fluid parcel and the change in its pressure.

Measurements made by attaching pressure sensors to the top and bottom of a wing show that the pressure is reduced at the top of the wing and that the pressure either remains at atmospheric pressure, or is above atmospheric pressure, on the bottom of the wing depending on the angle of the wing with respect to the direction of motion of the wing through the air, the angle of attack.

See the Bernoulli Bottle exploration. (Coming Soon.)

Here is a case study using Bernoulli's equation.

Air flows over an airplane wing that is convex on the top and flat below.

The flat bottom of the wing is parallel to the velocity of the airplane.

The air a great distance in front of the airplane is at atmospheric pressure.

The air speeds up as it flows over the top of the wing.

Why? The same amount of air must flow through every cross sectional area so when the air flows through a smaller area over the top of the wing it must speed up. (This is due to the conservation of mass flow, no air is created or destroyed.)

When the air in front of the wing speeds up as it passes over the wing its pressure must drop. Thus the air flowing over the wing has lower than atmospheric pressure.

The air flowing along the bottom of the wing travels at the same speed and so remains at atmospheric pressure.

The combination of atmospheric pressure below the wing and lower pressure above leads to a net upward force called lift.

**Bernoulli Math Root**

Start with the law of conservation of energy:The sum of the kinetic and potential energies of a mass at one place is equal to the sum at another place plus the work done on the mass between the two points.

KE1 +PE1 = W +KE2 +PE2

1/2 mv

_{2}^{2}+ mgh_{1}= integral of F dx +1/2 mv_{2}^{2}+ mgh_{2}Where m is mass, h is height, v is velocity, g is the acceleration of gravity, F is the external net force on the mass, and dx is the distance moved.

Let's keep everything at the same height so that h

_{1}= h_{2}and so we can drop the potential energy terms.We'll also talk about a parcel of air with volume V and divide every term by V.

1/2 m/V v

_{2}^{2}= (F2-F1) Dx/V +1/2 m/V v_{2}^{2}Now m/V is density r

and the Volume V is the cross sectional area, A, of the air parcel times its length dx

(F2-F1) Dx/V is P

_{2}- P_{1}where P is the pressure F/A

so we have

1/2 r v

_{2}^{2}= (P_{2}-P_{1}) +1/2 r v_{2}^{2}or

P_{1 }+ 1/2 r
v_{2}^{2} = P_{2} +1/2
r
v_{2}^{2}

**Newton**

Photographs show that when a flying wing collides with air the air is defected downward. The downward force on the air is associated with an upward reaction force on the wing called lift.

A flat plate wing like that on a paper airplane will produce lift if it has an angle of attack. That is, if it collides with the air so that the air hits the bottom of the wing. The collision of the air with the bottom of the wing deflects the air down and produces lift.

A wing with an airfoil shape curve on top and flat surface on the bottom will produce more lift than a flat wing. Indeed, more air is deflected down by the top of the wing than by the bottom.

You can see that more air is deflected by the top of the wing than by the bottom particularly well when the wing has zero angle of attack. In this case the bottom of the wing deflects no air, and yet the wing still produces lift because the top is deflecting air downward.

The downward force on the air is equal and opposite to the upward force on the wing.

**Newton Math Root**

The downward force on the air is given by Newton's law F = ma

which is also F = dp/dt

where p is momentum, mv, and t is time.

From the point of view of the wing the air approaches the wing parallel to the ground, with a momentum parallel to the ground. After colliding with the wing the air travels downward, with a momentum toward the ground. The change in momentum of the air is directed downward. The force on the air is downward.

The force is thus:

F = dp/dt = dmv/dt

where the mass of air is the density, r, of air times the volume, V, deflected

m = r*V

The volume of air deflected is the cross sectional area of the air deflected times the length of the air parcel deflected in the time interval dt. This is:

m = r*A*v*dt

so F = r*A*v*v
= r*A*v^{2}

In both cases the lifting force increases as the speed of the air squared.

**How to use the Bernoulli
equation.**

To use the Bernoulli equation you
**must**:

Point to two different positions along a streamline.

A streamline is the path followed by a parcel of air.

Once you have pointed to the two positions you can comment on the change in the pressure between the two positions if you know the differences in the speed of the parcel at the two positions.

You can alternatively learn the change in speed if you know the change in pressure.

**Pitot Tube**

Consider a pitot-static tube otherwise known as a pitot tube. This is used to measure the speed of an airplane through the air. It is a tube that sticks out in front of the airplane with a hole in its end and several holes along its sides.

In the frame of reference of the airplane, the airplane is stationary and the air is moving past it.

Far ahead of the airplane the air is at atmospheric pressure.

The air collides with the front of the pitot tube and comes to rest. Since the air has slowed down its pressure increases. The increase in air pressure at the front of the pitot tube is proportional to the speed of the air relative to the plane squared!

The air flows past the sides of the pitot tube at the same speed as the distant air. It has no change in speed and so no change in pressure. It is at atmospheric pressure.

The difference in these two pressures can be used to calculate the speed of the plane.

A common mistake: The air rushing by the sides of the pitot tube does not have less pressure because of its speed relative to the surface of the tube! The Bernoulli equation gives the change in pressure due to changes in velocity along a streamline not due to relative velocity with respect to a surface.

**The Curve Ball that Curves the Wrong
Way!**

Consider a spinning baseball moving from the pitcher to the batter. We have a right handed pitcher and a right handed batter. Viewed from above the ball is rotating counterclockwise.

This means that as the ball crosses the plate the side of the ball near the batter has a higher airspeed than the side opposite the batter.

Yet the pressure on the baseball is lower on the side away from the batter and the ball is curving away from the batter. This is because the lowering of pressure due to Bernoulli's equation is not due to the relative velocity of the air and the surface of the ball!

It is due to the change in the speed of the air along a streamline.

TRAC streamlines!

NASA has a good site about the lift of airplane wings. (Site)