# What does air becoming compressible change? [duplicate]

You always hear that around Mach 0.3, air technically becomes compressible. What actually changes when this happens, in subsonic and supersonic flow?

Does it affect the formation of shocks?

• Every macroscopic substance is compressible. Perhaps you are asking if the compression is relevant to aerodynamics? Feb 24 at 21:36

## 1 Answer

Air does not suddenly become compressible. It is a gradual change that can be delayed or accelerated by a number of factors. Mach 0.3 to 0.6 is a good range for most airplane related things.

We usually treat air as a continuum -- it is of course actually made up of particles bouncing off of one another. We can derive the equation of state $$P=\rho\,R\,T$$ by considering particles bouncing around in a box. The velocity of the particles is related to the temperature. The density is related to the mass of the particles and how many are in the box. The pressure is the average force caused by the particles hitting the walls of the box.

From this same argument, we can derive the equation for the speed of sound $$a=\sqrt{\gamma\,R\,T}$$. The speed of sound is the speed of a pressure wave in a gas. It is related to how fast those colliding particles are running into one another -- those collisions propagate the wave at the molecular level. We care about it at the continuum level.

Pressure waves are how disturbances are communicated in a fluid. If you put a blockage in a flow -- how is the existence of that blockage communicated upstream and downstream? How much time does that communication take?

The Mach number is the ratio of the flow's characteristic speed (usually the freestream) to the speed of sound. I.e. how fast is the flow moving vs. how fast does information propagate.

In the limit of incompressible flow (M=0), information propagates infinitely faster than the flow. I.e. information travels everywhere instantly. A change in a flow is felt everywhere instantly.

Incompressible flow is the condition where the time for information flow is not important to the flow (because the flow is slow).

Compressible flow is the condition where the time for information flow is important to the flow.

This is actually represented by a change in the differential equations for the flow. Incompressible flow is modeled with an elliptical differential equation -- everything is felt everywhere instantly. Compressible flow is modeled with a hyperbolic differential equation -- propagation time matters.

We call this incompressible vs. compressible because thinking back to our box of particles -- if the flow is slow compared to the collisions, then the particles can always spread themselves out equally -- the density will be uniform, and the flow will not compress. However, if the flow is fast compared to the collisions, then the particles can bunch up. They can compress in certain areas relative to others, the density of the flow can be non-constant.

If a disturbance is small (say a very thin airfoil), then the effects of compressibility (how much information travel time matters) may not become important until M=0.8.

If a disturbance is large (a cylinder in the flow), then the effects of compressibility will become important earlier.

Shocks happen because the flow is faster than the disturbance -- so the flow can't gradually react to a disturbance. Instead, it must find some way to abruptly react. This abrupt reaction is a shock.

Let us change gears for a moment.... If you put your thumb across the opening to a garden hose, the flow speeds up and the spray out of the hose goes further. Your thumb is the blockage. The water is incompressible. It must speed up to get by your thumb.

Imagine you are in heavy traffic on a four-lane highway. Up ahead, the highway narrows down to two lanes. Traffic slows down and everyone goes through the blockage very slowly.

How are these two situations different?

In traffic, information about the existence of the blockage travels slowly. You notice the car in front of you slowing down (brake lights). You have some reaction time before you press your brakes. This human reaction time dictates the speed that information propagates through traffic. As you get close to the blockage, the space between cars is reduced (compressibility), and everything slows down to some fixed rate (choked flow). It should not surprise you to learn that traffic flow is modeled with a hyperbolic differential equation.

What if traffic were modeled with an elliptical differential equation? This would require communication of information between cars to be instant. Let's say they were all robot cars communicating through a coordinated computer network (no humans and brake lights and reaction times). If a highway full of instant communicating and reacting cars approached a blockage, they would (instead of slowing down to a crawl), accelerate to 2x their speed and merge perfectly together to make it through the blockage -- just like water through the garden hose.