download the script: Relation between velocity and density in gas dynamics

Consider a steady, one dimensional and frictionless flow, we can apply the Euler equation expressed as:

Here

- p= pressure
- ρ= density
- V= flow velocity
- z= height relative to reference level

For ideal gas flow, the potential energy difference term dz is small enough to be negligible. Thus we can now rewrite the Euler equation as:

This expression indicates that the increase (decrease) of flow velocity leads to the decrease (increase) of the pressure in the direction of flow.

Now we try to let the Mach number appear in this equation:

We notice that the relative change in velocity is proportional to relative velocity change.

For subsonic (Ma<<1):

That means, the relative change in density is very small in subsonic case.

For supersonic (Ma>>1):

That means, the relative change in density is very high in supersonic case.