Chapter 3: Shear Layer Control
Free Shear Layers: jets
Letís begin by looking at the structure of a subsonic jet
issuing from a carefully-designed nozzle into the atmosphere.
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
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.
Why does mixing between streams occur when there is shear?
Due to instabilities of the shear layer, some disturbances
will amplify and form vortical structures, where the shear layer "rolls
up": the inner portion being composed of fluid moving faster than the outer
portions. These vortical structures merge and form large coherent structures,
which are big enough to go far out into each stream. Thus, fluid which
originated in each stream gets "transported" far out into the other stream.
This is called "convective transport", to distinguish it from the much
slower process of molecular diffusion across boundaries. When convective
transport occur, large "packets" of fluid, containing zillions of molecules,
get flung far out into each stream from the other.
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
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".
Control of Jets
Velocity components can be written as the sum of the time-averaged
(or mean), the random fluctuation about the mean, and the fluctuation due
to coherent structures (which are not quite random).
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.
vs. Subsonic Jets
Shown below is an underexpanded supersonic jet, coming out
of a convergent-divergent nozzle. As the jet comes out of the exit, it
expands because the outside pressure is less than the nozzle exit pressure.
The flow at the edge turns, sending out Prandtl-Meyer expansion waves.
Beyond this, the "slip line" bordering the jet flow is a line across which
the velocity changes sharply, but the static pressure is the same. This
"slip line" is where the shear layer develops. At the axis (or plane of
symmetry if its a 2-D planar jet), more P-M expansions go outward as the
flow expands further. When these expansion waves reach the outer boundary,
the pressure becomes lower than the outside, forcing the flow to turn back
in so that the pressure stays unchanged across the slip line. This inward
turn and pressure rise occurs across an oblique shock. This process continues
until a normal shock occurs and the flow becomes completely subsonic, spreads
out and slows down.
Supersonic jets contain shocks and expansions. We all
know that in supersonic flow, information cannot propagate upstream, so
what you do downstream cannot affect what happens upstream. But
note that the mixing layer contains subsonic flow. So here, information
can go back upstream! This causes phenomena which can be explained
using "feedback". It also opens up opportunities for controlling such jets.
Now lets look closely at the shear layer.
A disturbance which amplifies in a shear layer can be described
as an instability wave. If you monitor velocity fluctuations at a few fixed
points in space, what you see will be a change in velocity which moves
downstream. This similar to having 3 police cars stationed at different
point aclong the Interstate: each feels a pressure wave and a gust of wind
as something red flashes by, the ones further down the road feel this a
few seconds after the ones stationed up the road. On the other hand,
if you change your coordinates to a system which moves at the speed of
the convecting structures, you will see what appears to be a vortex: something
going round and round. Thus, the police car which moves at the speed of
the red flash sees a Corvette, whose shape remains constant in time. In
the case of fluid structures, note that the structures move at a speed
different from that of the fluid itself: thus its never the same fluid
molecules that are spinning around inside the coherent structure. Instead,
fluid goes in and out of the structure continuously. This is very important
to remember when one looks at, for example, smoke flow visualization of
vortices: the smoke moves at a speed different from that of the vortex.
Now you see why we refer to these structures as "waves": they are disturbances
propagating like waves through the fluid. We can associate a "wavelength",
"wavenumber" "frequency" etc. with these structures.
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
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.
Many things which look hopelessly complicated in the time
domain can be expressed with beautiful simplicity in the frequency domain.
To see this, consider the problem of describing something which goes up
and down, in a sinusoidal fashion, several times a second. Once we
recognize that it does repeat itself periodically, it becomes simple to
describe it. We describe it by one frequency, the amplitude, and maybe
a phase reference. These are fixed.
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
"Spectrum" (plural: spectra)
Refers to the distribution of the energy of fluctuations
as a function of frequency.
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
This variation can be supposed to be built up of various
sine and cosine waves, of different frequencies and amplitudes. This is
the idea of a Fourier Series. Conversely, we can "sample" this variation
for some time, and then break it up into the various frequencies, and compute
the amplitudes at each frequency. This is done by taking the "Fourier Transform"
of the sample of the signal. If we do this a number of times, and
average the result, it will probably all cancel out, because we have no
control about the phase of the sines and cosine waves as we begin each
sample. That is, some samples will have velocity going up while other will
have velocity going down, at the same parts of the sample. Instead,
however, if we square the fluctuation, we can avoid this problem. We can
find the average "energy" (i.e., something proportional to the square of
the amplitude) in each frequency interval. This is the process of
constructing a spectrum. Below we see how these "spectra" are used to discover
the various interesting features buried in what looks like a random fluctuating
Example of a Periodic or Narrow-Band fluctuation occurring in a turbulent
The spectrum shows a sharp peak in a narrow band of frequencies:
this means that there is something varying almost like a sine wave (nearly
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.
A spatial instability is a disturbance which grows as it
goes downstream. Below is shown a jet, in which instabilities grow as they
go downstream, eventually resulting in total turbulence. If you place a
hot-film probe in the shear layer just outside the potential core, you
might see the narrow-band peaks of the instability waves, superposed on
the turbulent spectrum. If your sensor is shifted downstream, you will
see that the frequency of this instability wave has decreased, while the
amplitude has increased. Out in the fully-turbulent zone, it will be hard
to distinguish any spectral peak on top of the broad-band turbulence.
Excitation of coherent structures
Control or excitation of a shear layer can be described using
the frequency, amplitude and "receptivity". Receptivity
refers to how a flow "receives" the excitation. This concept applies
to flows over airfoils (wall-bounded shear layers) as well as flows coming
out of jet nozzles (free shear layers). The figures below show speakers
blasting out sound at carefully selected frequencies, right close to the
points of maximum receptivity. In the case of the jet exit, the receptivity
is maximum in the immediate vicinity of the exit edge. In the case of an
airfoil, its harder to decide such a point: if the flow is at the point
of separating from the surface and forming a free shear layer, then receptivity
would be maximum at the beginning of the shear layer.
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.
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.