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Usual "explanation" of lift using Bernoulli principle

This theory is:

  • Air over the wing is accelerated, its dynamic pressure increases with velocity.
  • The static pressure over the wing decreases (due to Bernoulli principle, see below).
  • The opposite occurs below the wing, static pressure increases.
  • This results in an imbalance in static pressure between sides, and explains lift origin.
    enter image description here
    Source

Bernoulli's principle

Bernoulli's principle explains "an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy":

$$p + \frac{1}{2}\rho V^2 + \rho gh = k$$

where $k$ is a constant. In the case of a wing, $\rho gh$ is so small it can be omitted, the principle can be stated:

$$p_0 = p + \frac{1}{2}\rho V^2$$

where $p_0$ is a constant known as total pressure, the first term is the static pressure, and the second one the dynamic pressure.

The explanation of lift ignores the imbalance of dynamic pressure

The previous theory ignores a similar, but opposite, imbalance of the dynamic pressure (as the total pressure is constant, the arithmetic sum of the two imbalances is also constant):

enter image description here

Why is the force created by static pressure put ahead in the explanation of lift, and the force created by dynamic pressure is not mentioned? If this force is ignored because not significant, why and how much is this force smaller?

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In short, "dynamic pressure is not pressure at all" (A Princeton paper on dynamic pressure and Bernoulli's theorem). "Dynamic pressure" is a term that designates the decrease in pressure, as a consequence of the fluid's / medium's flow velocity.

I recommend that you also examine what NASA has to say:

Integrating this differential equation:

ps + .5 * r * u^2 = constant = pt

This equation looks exactly like the incompressible form of Bernoulli's equation. Each term in this equation has the dimensions of a pressure (force/area); ps is the static pressure, the constant pt is called the total pressure, and

.5 * r * u^2

is called the dynamic pressure because it is a pressure term associated with the velocity u of the flow. Dynamic pressure is often assigned the letter q in aerodynamics:

q = .5 * r * u^2

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The dynamic pressure is not a pressure as normally perceived.

From the wikipedia article:

Dynamic pressure $q$ is the kinetic energy per unit volume of a fluid particle.

$$q = \frac{1}{2} \rho u^2$$

I.e., it is the increase in static pressure that you would have if you would slow down the fluid to a complete halt, but until that moment it is kinetic energy that is not "pressing" on anything, the same way the kinetic energy of a car is not pressing on the road it is travelling (but would press on a wall that would try to suddenly stop said car).

It is well visualized in the image included in the wikipedia article as well

enter image description here

The flow on the left is travelling faster, and has increased dynamic pressure, but as the static component decreased, it is pressing less against the walls of the glass tube, leading to a pressure differential.

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  • $\begingroup$ Why does dynamic pressure reduce static pressure? $\endgroup$ – Koyovis Jun 27 '17 at 14:06
  • $\begingroup$ @Koyovis that's not how it works. as stated in the question, total energy is constant, hence if one component increases, the other decreases. $\endgroup$ – Federico Jun 27 '17 at 14:12
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I do find the question fascinating, and I like Federico's answer. What follows started as a comment regarding the energy [down] arrow in the diagram, but it's too big for a comment now.


1. Energy and pressure

Federico established that dynamic pressure is the kinetic energy per unit volume.

The pressure differential above and below the airfoil can also be thought of in terms of energy. From Wikipedia:

Since a system under pressure has the potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa).

$$P=\frac {\text{force $\times$ distance}}{\text{area $\times$ distance}}=\frac {\text{work (energy)}}{\text{volume}}$$


2. Energy transfer = pressure difference

enter image description here
(Source)

Let's imagine a tied-down taildragger on a windy day.

Wind is rushing towards the nose. It hits the underside of the airfoil, and transfers kinetic energy (loses speed, loses dynamic pressure) to the plane trying to push it. Above the wing the air leaps over the leading-edge and finds a drop, which it needs to fill so it speeds up (gains kinetic energy).

Here's the cool bit, it doesn't gain KE [per unit volume] from the plane, it gains KE by trying to fill a void, and in the process it loses potential energy per unit volume (pressure).

3. Direction of strength

So we have a group of atoms that are rushing for reasons beyond themselves, they meet an obstacle, jump over it, and accelerate. Now they have less time to push against the airfoil (they've become weaker in the vertical). However, the atoms below the airfoil are the opposite, and the easiest direction for them to push is now up.

There's no down arrow (except for weight). The decrease in dynamic pressure / kinetic energy below (and increase above) causes the system to do work upwards. Which the plane pays for in fuel by recreating that very windy day.

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Adding my own answer to summarize some elements.

Pressure is not always pressure

This wording dynamic pressure is indeed very confusing, as this is not a pressure which can be measured with a barometer or that kind of instrument. Similarly total pressure itself is not a pressure either. In Bernoulli's principle which is summarized in...

$\text {Total pressure} = \text {static pressure} + \text {dynamic pressure} = \text {constant}$

... the only pressure is the static pressure. Instead total pressure and dynamic pressure are energy. So to answer the question: Lift is a matter of forces exerted on the wing, and the only force involved is the one from the pressure (i.e. the static pressure).

Stagnation point

An interesting way to look at this equation is to start from the situation where the flow is interrupted by a body. The flow must take another path, for example:

Total, static and dynamic pressure

The flow is deflected in all directions around a central point. At this point air is blocked and accumulates until some point, preventing more air to come from the direction of the flow. This point where air is stagnant is the stagnation point. As air is not moving here there is no "dynamic pressure". Per Bernoulli's principle the total energy is converted into pressure, and this point is the location where pressure, known as stagnation pressure, is maximal.

At any other point, energy is split between flow (static) pressure and kinetic energy (dynamic pressure).

Pitot

On the other hand, if the wall is moving and air is static, at the stagnation point one side of the wall is subject to total pressure, and the other to static pressure. The difference is therefore the dynamic pressure which is known to be proportional to the velocity of the wall.

This is the principle used in the Pitot sensor to measure airspeed:

Pitot principle

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