Intuitive explanation of supersonic flow behavior?

It is well known that once the flow of a gas becomes supersonic, it behaves very differently to subsonic flow:

• Velocity increases as flow area increases.
• Velocity decreases along a pipe with friction.
• Velocity decreases if a flow is heated externally.

The opposite is true for all of these in the case of subsonic flow.

I am familiar with the mathematical explanations that go through several pages of 1D compressible flow theory; however, I struggle to understand intuitively why a supersonic flow behaves so differently to a subsonic flow, in the sense that it practically reverses in behavior like this.

Does anyone know of a good, intuitive explanation as to why supersonic flows behave like this (beyond just: 'it does, because the math says so')?

• – honeste_vivere Aug 5 '18 at 18:29
• Does not velocity decrease along a pipe with friction in the subsonic case, too? – Mathews24 Oct 20 '18 at 5:47
• @Mathews24 no, subsonic flow accelerates along a pipe with friction. The pressure drops, which means the density drops, which means velocity has to increase, to maintain mass continuity. – Time4Tea Oct 20 '18 at 11:31

I think I have an explanation for at least one of them with partial answers for the other two. This is a great question that has bugged me since the first day I read it.

Velocity increases as flow area increases.

In subsonic flow, the flow is mediated by particle-particle collisions and those can occur faster than the flow can move, since the speed of sound, $$C_{s}$$, is larger than the bulk flow speed. Therefore, the gas acts like an incompressible fluid because any changes in flow are communicated through particle-particle interactions faster than the bulk flow of the fluid. For example, the fluid can flow around an obstacle without causing significant changes in pressure. The closer the flow speed to $$C_{s}$$, the more the incompressible flow assumption breaks down.

In supersonic flow, the bulk speed exceeds the sound speed and so the tube volume becomes more important since the particles can collide with the wall faster than particle-particle interactions. It's similar to why shock waves are unstable in converging geometries (e.g., see https://physics.stackexchange.com/a/137842/59023). The high flow rate can also change the compressibility of the fluid since the tube walls can create back-pressures on the flow. It's kind of like a pileup of traffic in a section of highway with a lane reduction and then the acceleration after the return to normal number of lanes. The change occurs faster than the constituent parts can respond (compressible) and a bunching occurs.

Velocity decreases along a pipe with friction.

First for reference, pipes with friction are dealt with using things like the Darcy–Weisbach equation or the Fanning friction factor. Both of these are used to determine pressure loss along a length of pipe due to friction with the walls of the pipe. In general, the pressure loss along a length of pipe, $$L$$, with radius, $$R$$, is given by: $$\Delta P \propto \frac{ L \ \rho \ U^{2} }{ 4 \ R }$$ where $$\rho$$ is the fluid mass density, $$U$$ is the bulk fluid flow speed, and the proportionality is one of the two friction factors mentioned above. Thus, the pressure loss per unit length of pipe increases with increasing flow speed. Note, however, these relationships are only valid in the incompressible flow regimes, i.e., subsonic flow.

Regarding the difference, I am guessing that subsonic flow sets up a boundary layer near the pipe walls with things like a Kelvin–Helmholtz instability or KHI arising between the two layers. I see why the presence of friction on the walls would cause a fluid to locally accelerate in the bulk flow (i.e., not near the boundary layer) due to a pressure gradient that maintains mass flow continuity. However, the formation of a boundary layer with KHI should reduce the drag on the bulk of the fluid flowing in the tube not near the boundary layer, though it does introduce viscosity near the shear layer.

In supersonic flow, the boundary layer might not form as the speed of the particles reflecting off the walls could exceed $$C_{s}$$. It's another way of saying that the transit time of a particle across a boundary layer must be larger than the growth time scale of the instability. In subsonic flow, the transit time is dominated by $$C_{s}$$ while in supersonic flow it's dominated by the bulk flow. The thickness of the shear layer will depend upon the transit speed of the particles, so for bulk flow speeds exceeding $$C_{s}$$ the shear layer would increase. That is, for supersonic flows the issue becomes whether a stable shear layer can even form because the boundary layer thickness would become comparable to the pipe size preventing laminar flow and increasing drag/loss of pressure.

Velocity decreases if a flow is heated externally.

I will start by saying my comments for this are only true for specific limits, which is true of all physics but I am qualifying this as I am not sure about the generality of the above statement.

Per the above explanation regarding flow in a pipe with friction, an external heat source would heat the fluid and can, thus, increase the thermal speed of the particles. As the thermal speed increases, the difference between it and the bulk flow speed will decrease, reducing the Mach number of the flow and the compressibility of the flow.

As I laid out in the first part above, a supersonic flow accelerates in an expanding cross-section due to a pressure gradient caused by the compressibility of the fluid induced by the walls of the tube. The speed decreases in a constriction for similar reasons. That is, if we ignore friction, then the decrease would be akin to reducing the area of the pipe as the transverse (to the bulk flow) transit time would decrease.

Conversely, the flow rate in incompressible flow is limited by $$C_{s}$$ because as one approaches $$C_{s}$$, the flow starts to become compressible. Therefore, if one increases $$C_{s}$$, it would imply that the flow rate in constricted channels could increase to maintain mass flow continuity.