Rocket engine fog

I would need to create the effect of fog that flows quickly and chaotically from the rocket engine in all directions. It’s from 0:16 to 0:20 in this video.

Any ideas how to do this?

Assuming you are using a particle system for the fog…
Add a Turbulence Force Field and keyframe its strength. Perhaps use several of them.
You can see two streams and just after the third one appears the distortion wave hits. Add a keyframe on the turbulence Force Field (with low value) a frame or two before it hits, then raise the strength at the right moment and keyframe again.

A mix of different Force fields may give the best effect.