The potential core is where the velocity is still pretty much the exit velocity.

In the mixing layer, there is a shear, which results in "vorticity".

Obviously, the potential core is considered to be a region where there is no vorticity. In regions where there is shear, there is vorticity. This can be seen by expanding the cross-product definition above:

Where there are gradients of velocity perpendicular to
the velocity direction, there is shear. The mixing layer thickness is approximately
0.2x to 0.25x, where x is the distance downstream of the nozzle exit.

A plot of the velocity component parallel to the axis, along the axis, would look as follows. In the potential core, the velocity can chance if the external pressure changes, as for example, for a jet inside a duct of changing cross-section. Otherwise, there would be no change inside the potential core. Beyond the potential core, the jet slows down through mixing and momentum exchange with the outside flow: the influence of the jet spreads outwards, while the influence of the outside is felt at the axis.

Now these large-scale structures achieve transport, but this is not enough to complete the mixing. Inside the large-scale structures, there are rolled-up shear layers. These again form smaller vortical structures, and so on down to much smaller scales. Thus, many levels of shear layers occur. At the smallest levels, there are tiny vortical structures, with relatively large contact area between fluid that originated in the two streams. At these levels, thorough mixing is occurring: the two fluids are getting to be indistinguishable. Thus, the transport is achieved through the large-scale coherent structures, but the mixing actually occurs through the fine-scale structures.

As the Reynolds number increases, the variety of size
scales increases, and probably the smallest scales get smaller, so mixing
is promoted.

The concept of mixing and transport can be applied not
only to chemical composition ("species transport"), but also to mass and
momentum. The thickening of a boundary layer is due to "momentum transport:
think of it as the outer lanes of traffic slowing down because the inner
lanes are slowing down. A fast car from the outer lane zipping into the
inner, slow-moving lane, is "mass transport".

If we are able to control the jet, we should be able to change the
length of the potential core, and the extent of the mixing layer.

These structures are also referrred to as "Kelvin-Helmholtz instability". Consider what happens when the jet is axisymmetric: these structures would extend all round the jet. So they can be considered to be like: "Vortex Rings", similar to what we see when someone is trying to impress everyone by blowing smoke rings. Smoke rings are vortical structures developing in the shear layer of a suddenly-accelerated jet flow, and they remain somewhat stable because the jet continues to accelerate upwards due to buoyancy, so there is continuous shear.

Wave number: Tell us about changes with distance.

Wave # k = 2p/l = w/c

where c is propagation speed of the wave (or structure)

l is wavelength. w is 2pf , and f is frequency (# per unit time, cycles per second)

Wavelength changes due to vortex pairing, among other things.

Extending this, we can try to describe complicated-looking variations as sums of several simple periodic functions, with different amplitudes. We can describe these as a series of terms, involving sine and cosine functions of a fundamental frequency and its multiples. This is the concept of a Fourier Series. A few assumptions are implicit here. One is that "linear superposition is valid".

We will see that operations involving "convolution integrals" can be reduced to simple multiplication (of complex numbers) in the frequency domain. There are several such concepts, which help us visualize relations between variables.

*Example: Autospectrum of velocity.*

The autospectrum is really the "auto-spectral density
function, integrated over finite intervals of frequency". Its units are
"velocity-squared per interval of frequency". In other words, it is a measure
of fluctuation energy (per unit mass) of the property which is being described:
the velocity. The word "auto" refers to the fact that it is a spectral
density function obtained as the product of the property with itself, i.e,,
velocity times velocity. In contrast, there can be "cross-spectral density
functions", for example, the product of velocity with pressure, or pressure
with density, etc.

Consider a velocity measurement at a point, as a function
of time.

Below, we see a spectrum where there are several frequencies of instability waves seen: the fundamental and several harmonics.

Below is the broad-band spectrum of turbulent flow: there is energy at every frequency, because there just so many different kinds of fluctuations occurring.

Receptivity is usually highest at the jet exit plane edge.

Excitation frequency for a jet is a function of the size
of the shear layer, which is proportional to the jet diameter.

Strouhal number fD/U is around 0.3 to 0.5 for jets, and
0.42 to 0.48 for laminar jets.

Example: D = 2", approx. 50 mm.

U = 100 fps = 30.48 m/s

f = 293 Hz.

References:

Laurence, J.C. (NASA TR 1292, 1956)

Davies, Fisher, Barrett (Journal of Fluid Mechanics, Vol. 15, pt. 3, 1963, p. 337-367.)

Bradshaw, Ferris, Johnson, Journal of Fluid Mechanics.