# What is the relation between pressure and airflow speed above an airfoil?

Does decreased pressure on the top surface of an aerofoil cause high velocity airflow or does the high speed airflow result in decreased pressure?

## 2 Answers

Both pressure and velocity are related: The total energy of an air molecule outside of the boundary layer is constant and the sum of its pressure and its velocity component. Mathematically, the energy per unit of volume is $$\frac{\rho}{2}\cdot v^2 + p = const$$ which is actually the simplest form of Bernoulli's equation which neglects changes in altitude and temperature.

In the end, it's not this causes that, but both components fluctuate in sync and combine to a constant total.

Nomenclature:
$$\rho\:\:$$ density
$$v\:\:$$ speed
$$p\:\:$$ pressure

Reducing the cross-section area

When air encounters the airfoil, the streamlines over the top surface are compressed, because the surface is an obstacle pushing them vertically upwards, and the rest of the atmosphere prevents them from moving freely in block upwards. Note this reduction of area is widely accepted as real, but is not well explained.

This would work the same way for the other side of the cylinder, as atmosphere pressure exerts in any direction, normal to the surface.

Increasing speed

If we assume a simplified case of little change in density (nearly incompressible fluid), the same quantity of air must go through a smaller area in the same time, and to do that it "must" accelerate. This is similar to what happens to water in a garden hose: When squeezing the extremity water accelerates at this location relative to the rest of the hose.

Decreasing pressure

If we still assume a nearly incompressible fluid, and neglect the effect of viscosity then, per Euler's equation, an infinitesimal variation of velocity $dV$ leads to a variation of pressure $dP$ equal to $-\rho VdV$ ($\rho$ the air density, $V$ the velocity).

We see pressure varies inversely to the velocity. Therefore a cross-section area reduction leads to a velocity increase, which in turn leads to a pressure decrease.

If you need to put a name on this effect, then this is Bernoulli's principle! More on Euler and Bernoulli equations: Fluid Mechanics, Euler And Bernoulli Equations

Which one change first, pressure or speed?

Pressure and velocity in a fluid carry some energy (pressure energy, which is a kind of potential energy, and kinetic energy). The total energy is constant.

The obstacle changes the ratio between potential energy (pressure) and kinetic energy (speed). None change first. It's like action and reaction, their existence is linked and simultaneous.

• Me again, hi, what about an airfoil at a +ve none flat angle of attack? I don't see how the ball relates.
– user14897
Commented Jul 2, 2016 at 21:57
• To compare: With a full cylinder, pressure will be the same top and bottom. With a usual airfoil with a water droplet section, the principle will not change. When adding the AoA (which is not possible with the cylinder indeed), the pressure will be different on each side, and the accelerated airflow will be deviated down, creating aerodynamic lift. Using the steady flow assumptions (incompressible, inviscid and irrotational) is no more possible (e.g. I believe the flow on bottom side will be significantly subject to viscosity, air will be slowed down, and pressure will increase).
– mins
Commented Jul 2, 2016 at 23:05
• The cross-sectional area is not (necessarily) reduced!. Remeber, that a thin, flat plate, at slight angle to the air stream, produces lift, and does so using the same mechanism. So while what you say is true, it does not describe what happens around the wing well. Commented Jul 4, 2016 at 9:09
• @JanHudec: The question is not about lift, only pressure/velocity relationship. With all due respect for your expertise, I believe the flat plate also reduces the cross section of the airflow at appropriate Reynolds numbers and AoA,
– mins
Commented Jul 4, 2016 at 19:33
• @mins, well, after all the Bernoulli's equation still does hold, so the faster flow must be using less space. There is, however, no obvious argument why it should be the case, so it is not usable as explanation (speaking of cause in physics not always makes sense and this is such case). Commented Jul 4, 2016 at 19:40