15. Compressible Fluid Mechanics#
15.1. Compressible Inviscid Fluid Mechanics#
15.1.1. Shocks#
15.1.2. Quasi-1d flows#
If no shock occurs in the flow, Euler equations in differential form governs the dynamics of the flow
Quasi-1d model for steady flows is a simple model that provides good-enough results for flows delimited by streamlines that varies gently in the streamwise direction (or by solid walls, in the limit of the high-Reynolds flow without separations, where viscous effects are confined to a thin region - boundary layer - close to the walls).
This model is derived integrating over the sections of the stream tube, so that the physical quantities are functions of the streamwise coordinate
where the last equation comes from the contribution of the lateral surfaces, that has non zero contribution in the streamwise component of momentum equation if sections is not constant, and thus the unit normal vector
Thus the equations become
having used the mass equation to simplify the first term in the momentum equation, since
Now, from momentum equation a relation from changes in velocity and pressure holds
If the flow is isentropic (i.e. negligible viscous effects, no heat conduction, no shocks),
can be used to write a relation between changes in pressure and density
and thus, introducing Mach number
From this equation, it’s immediate to realize that:
for subsonic flows
, if section of the stream tube increases, , thus velocity descreases , and pressure increases , and viceversafor supersonic flows
, if section of the stream tube increases, , thus velocity increases , and pressure decreases