I'll avoid energy conservation, but cannot set aside mass conservation obviously.
Both speed and pressure settle in a way that abides the laws of physics. It's not like one settles first, then the second follows. Don't let my wording fool you.
I won't consider temperature in addition to speed, density and pressure. It's too much for an explanation.
There are many, many ways of seing this.
The wrong but easy answer
People who 'simply' say:
Velocity goes down and pressure goes up.
Are, often, wrong. This only holds in perfect, incompressible, irrotational flows and is known as Bernouilli's principle. If my (French!) aerodynamics teacher heard anyone mention it, that person would probably lose points! These conditions are often far from reality so the consequences don't hold every time. However it can be used as a guideline to get a gut feeling about a low-Mach flow.
(Note in locally supersonic flows, the exact opposite might apply, so this warning is legit!)
The easiest answer is energy conservation. You mentioned you wanted another:
I understand the energy conservation
So I'll pass.
Your nozzle is a convergent manifold, your diffuser is divergent. One acts exactly opposite to the other (just reverse the time in steady flow) so I'll address convergent axisymmetric shallow nozzles. Shallowness means speed is quasi-axial everywhere.
Furthermore, in aviation we are concerned with planes, so I'll assume the intake velocity is constrained and cannot go down.
Let's get to equilibrium first.
Now try to cram a still fluid through a restricting pipe. It won't like it, and will resist (inertia, plus the surface pressure of the nozzle's axial component, etc. see? Forces!), thereby increasing its pressure. This pressure increase will propagate to the whole intake and raise that pressure. And now the intake has everything it needs to push the whole fluid forward:
- high intake pressure P0
- constrained intake speed V0
So the fluid effectively moves through the pipe.
How to describe the equilibrium
First off, any radial section of fluid has quasi-uniform pressure, otherwise the fluid would pick up a large radial speed, so we would not be at equilibrium yet.
Next, apply conservation of mass. It forces conservation flow throughout any section of fluid with
A×V×rho=Cte. If frontal area is restricted, then either or both speed and density must increase. The amount by which each increases depend on the fluid's properties (compressibility etc). So here you go: speed must increase in the convergent nozzle. Also as a by-product, density may increase.
How does that happen?
Simply, a force must push the fluid. Remember the intake has very high pressure necessary to maintain steady-flow? Well that pressure is propelling the fluid (and there must be a low pressure at the tighter end (exit) as well).
If we do a static analysis of a thin section of fluid, it has low speed behind, high speed in front, so each molecule is accelerating. The section behind provides high pressure, that pressure translates into an axial force, the molecules accelerate. The section in front has lower pressure, and doesn't push back as much. That pushing-forward-hard/not-pushing-back-as-much is the pressure differential.
So along the flow, we should see the pressure decreases.
There you go, I've used:
- steady-flow assumption
- conservation of mass
- second law of Newton
Look Ma, no energy!
P.S sorry if I sound like I talk to a 5 y.o. !