Turbulent Jets

Immediately upon exit from the nozzle, the flow will be relatively uniform across the jet (notwithstanding whatever upstream effects may exist, and the existence of boundary layers inside the nozzle). If we have a very high contraction ratio nozzle such that the boundary layer is very thin, we might describe the velocity at exit as having a “top hat” profile. Mixing between the jet and the ambient fluid results in momentum exchange between the two; slowing the jet while entraining ambient fluid. The region where this mixing is taking place is known as the “shear layer”. The initial shear layer is the thickness of the boundary layer at the nozzle exit, but grows with increasing downstream distance. Enclosed by the boundary layer is the region of jet flow that has not yet exchanged fluid or momentum with the ambient fluid; this region is called the “potential core”. As we move downstream, the shear layer grows, and the potential core shrinks. Eventually, we reach the end of the potential core, where shear layers from each side meet in the centre; a lot of dynamics relevant to sound production take place at the end of the potential core.


Where does this potential core end? It depends on a number of things, like the Mach number of the jet, the initial state of the boundary layer, whether there is any kind of forcing, and so on. In a simple subsonic turbulent jet, you might expect the end of the potential core to occur about four nozzle diameters from the nozzle exit plane. In a supersonic jet, it could be ten diameters away or more; compressibility effects slow down mixing. Beyond the potential core we have a transitional region; the jet is spatially developing in a way that defies a simple description. But at some distance further downstream, the jet reaches a “fully developed” or “self-similar” state. What does this mean? First, let’s have a look at what some velocity profiles across the jet might look like. (To be inserted)


So, we can see that the jet is changing pretty drastically as we move downstream. So what does it mean to call it “self-similar”?

Well, we need to introduce a couple of new parameters to explain it. First we’ll take the time-averaged velocity along the centreline, which we’ll call U0, which is a function only of axial distance x for a given jet, and then we’re going to define something called the “jet half-width”, which is the radial distance from the centreline at which the time-averaged velocity reaches half the centreline value. This is also a function only of x.


Let’s now plot our velocity profiles, but instead of normalizing by the jet exit velocity and jet exit radius, let’s normalize by the local centreline velocity and jet half-width. We see all the profiles in the self-similar region collapse on top of each other.


We can dig into this further, or at least take advantage of the fact that other people have done it for us, and see that once we are in this self-similar region, the jet spreads linearly, and the inverse of the centreline velocity also increases linearly. This final plot gives us the last definition needed for classical turbulent jet theory; the “virtual origin” is the location at which the jet would originate if the entire flow behaved as the self-similar region did, with no potential core of transitional region.