Well most texts treat the steady behavior of jets, so that will comprise the bulk of information you would find. The actual startup process is likely to be rather complex and will involve essentially a slug of fluid leaving the orifice and pushing the still fluid out of the way. I'd be willing to bet that it would typically also involve the formation of a vortex ring near that leading slug much like what you can observe with a mushroom cloud, as the shear would be rather large as it penetrates into the surrounding stagnant air.
Other than that, I am not sure what you are seeking here. Jet's aren't exactly something I deal with a lot, though, so if you want to get too much more detailed, I suspect we will quickly reach a point where your guess is as good as mine.