How does air travel after leaving air nozzle?

Without more details, we can't find a specific case that will match what you are doing. However, I did find a decent example to show you what to look for to answer your question.

Data for a nearly-sonic round jet can be found in this paper. If you look at Figure 5(a), you'll see how the normalized centerline velocity from several experiments collapses together and has essentially a hyperbolic tangent shape. You could find experimental data for conditions near your operating point (speed, Reynolds number) and create a tanh function that approximates the data and go from there.

You can find similar figures and/or correlations for the width of the jet as a function of downstream coordinate also. Standard textbooks on turbulence should have the subsonic jet correlations, but for a sonic jet I can't think of any off the top of my head that will definitely have it. But you can search through the literature using keywords learned in the paper I linked to and you should be able to find what you need.