Compressibility effects

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A gas can be compressed, this means that when we exert an increasing pressure on a cilinder filled with gas, the volume of that unit of air will decrease. Liquids, like water, are as good as incompressible. Take a cilinder filled with water, put pressure on the top, and the water volume will hardly change with increasing pressure.

What is the result of this compressibility? Density changes. With increasing pressure a gas will have a density increase as well. With increasing pressure a liquid will not have a density increase, with a liquid, density will not change.

Now look at the formula's used in basic aerodynamics (Bernouilli, total pressure = static + dynamic pressure). As you should know, dynamic pressure depends on density. And since air is a gas, hence compressible, density will change. It will increase with increasing pressure.

But we never make this correction, most of the time we use a density value for air in "not moving" conditions. On the ground ie, it's 1.225kg/m**3. But when air is moving and being brought to a sudden stop, the density of the air will slightly increase because it will be compressed. So we make an error. We use a value for density to calculate dynamic pressure, but this value depends on the value of dynamic pressure itself. So I hope you understand something is wrong here. Dynamic pressure depends on density and density depends on dynamic pressure... so we can't really use the formula's anymore.

The higher the speed of the air, the more the real density (the density of "air under pressure" or "moving air being brought to a stop") will be higher than the density we use, the larger the error.

Now there have been clever engineers who calculated the errors because of compressibility: at what speeds are the errors in density between "not moving air" and "air under pressure being stopped" large enough to create a noticeable error? That's when we start to use compressibility correction factors. These factors allow us to use the "static" density of air, and depend on the speed of the air of course.

Hope this helps?

PS: Formula of Bernouilli can only be used for incompressible fluids, that's a basic requirement. However, at low speeds and low altitudes, the errors created are so small we can neglect them. It's far more easier to use Bernouilli than the alternatives available (thank god...). But at high speeds/high altitude we have to make corrections... but still easier than the alternatives.