00:00:28
for
00:00:47
though turbulence is not particularly
00:00:49
easy to Define it's not hard to find
00:00:58
examples
00:01:10
in these we can find certain common
00:01:14
characteristics one of the most apparent
00:01:16
is disorder as can be seen in this
00:01:18
channel
00:01:22
flow the disorder is of such a
00:01:24
fundamental nature that the flow never
00:01:26
is reproducible in detail no matter how
00:01:30
one attempts to reproduce all the
00:01:31
boundary
00:01:34
conditions although the details are not
00:01:37
reproducible averages over suitably
00:01:39
large intervals of space or time may be
00:01:42
very well defined and
00:01:47
stable disorder then is a necessary
00:01:50
factor in any definition of turbulence
00:01:53
it is not however
00:01:58
sufficient
00:02:01
here is a pretty disordered fluid motion
00:02:03
but it would be unwise to include it in
00:02:05
turbulence a wave field like this does
00:02:07
very little mixing and mixing is an
00:02:10
essential feature of turbulent
00:02:13
motion the mixing action of turbulence
00:02:15
can lead to complete blending if the
00:02:17
volume is confined or to the dilution
00:02:21
which is the only thing that makes
00:02:22
pollution like this or this barely
00:02:27
tolerable another characteristic of
00:02:29
turbulence is the presence of
00:02:31
vorticity in a turbulent field the
00:02:34
vorticity is distributed continuously
00:02:36
but irregularly and in all three
00:02:40
dimensions so turbulent flow has more
00:02:43
than one distinguishing characteristic
00:02:45
or symptom perhaps we can borrow the
00:02:47
word Syndrome from pathology and say
00:02:50
that we have a defining syndrome or set
00:02:53
of symptoms for turbulence these are
00:02:56
disorder irreproducible in detail
00:03:00
efficient mixing and vorticity
00:03:02
irregularly distributed in three
00:03:05
dimensions this definition effectively
00:03:08
isolates turbulence from various kinds
00:03:10
of wave
00:03:11
motion it also eliminates all
00:03:13
two-dimensional
00:03:15
flows something roughly like turbulent
00:03:17
motion can exist in two
00:03:19
Dimensions weather systems on a large
00:03:21
scale represent nearly two-dimensional
00:03:25
flows however the characteristics of
00:03:27
such flows are in many ways so different
00:03:29
that it is perhaps unwise to include
00:03:31
them in
00:03:33
turbulence some flows like this jet are
00:03:37
clearly
00:03:40
turbulent others are equally clearly
00:03:44
not what is it that determines whether
00:03:46
or not a flow is
00:03:52
turbulent to help answer this question
00:03:54
we'll use this
00:03:56
apparatus which is a version of that
00:03:58
used by Haan over 100 25 years ago in a
00:04:01
study of flow through pipes in these two
00:04:04
tanks we have two different glycerin
00:04:06
water
00:04:08
mixtures one mixture has viscosity about
00:04:10
three times out of the
00:04:13
other the valve permits us to obtain any
00:04:16
combination of the two
00:04:21
mixtures the fluid is pumped by this
00:04:23
constant displacement pump so that the
00:04:25
flow rate Remains the Same no matter how
00:04:27
the mixing valve is set
00:04:34
our pipe begins
00:04:41
here Midway down the pipe we start a
00:04:43
monometer tube which we carry
00:04:47
Downstream so that we can see the
00:04:49
Upstream pressure simultaneously with
00:04:51
the end of the tube where the pressure
00:04:52
is
00:04:54
atmospheric the monometer reading will
00:04:56
give us a measure of the average
00:04:57
pressure gradient along the tube
00:05:02
now we start the
00:05:05
pump the monometer reads about 10
00:05:10
units now I change the valve setting to
00:05:12
reduce the
00:05:16
viscosity as the low viscosity fluid
00:05:18
flows into the pipe the monometer
00:05:20
reading
00:05:24
decreases once more I Chang the valve
00:05:27
setting to still lower viscosity
00:05:35
but look
00:05:36
here we have increased the pressure
00:05:39
difference which means we have increased
00:05:41
the drag let's look at it again I
00:05:45
increased the
00:05:50
viscosity and down goes the pressure
00:05:58
difference
00:06:01
I decrease it
00:06:07
again up it
00:06:13
goes now let us look at the fluid coming
00:06:16
out of the end of this
00:06:18
pipe notice that the fluid comes out in
00:06:20
quite a smooth
00:06:23
stream we have the intermediate
00:06:25
viscosity with its corresponding low
00:06:27
pressure difference now let's go to the
00:06:30
lower viscosity with a high pressure
00:06:38
gradient the edges of the stream coming
00:06:41
out of the tube have become
00:06:43
blurred if we look at this stream in
00:06:45
slow motion we can see that the blurring
00:06:47
is resolved into an irregular motion of
00:06:50
the
00:06:51
surface the flow has become turbulent
00:06:54
and the onset of turbulence in the pipe
00:06:56
has revealed itself both in The
00:06:57
Irregular motion of the fluid stream
00:07:00
and in the greatly increased drag in the
00:07:07
tube in the early 1880s osbor Reynolds
00:07:10
did a series of experiments on flow
00:07:12
through tubes and came to the conclusion
00:07:15
that the Criterion for the onset of
00:07:18
turbulence was a dimensionless function
00:07:20
of the flow parameters which has since
00:07:22
been called the Reynolds
00:07:25
number there is usually a certain amount
00:07:28
of arbitrariness in the definition of a
00:07:30
Reynolds number but for pipe flow let's
00:07:33
take it as the diameter multiplied by
00:07:36
the average speed divided by the
00:07:38
kinematic
00:07:41
viscosity although the question is still
00:07:43
under investigation it seems that if the
00:07:45
Reynolds number so defined is
00:07:48
appreciably less than 2,000 the fluid is
00:07:51
not
00:07:52
turbulent in this experiment we have
00:07:54
deliberately made the input condition
00:07:56
somewhat abrupt turbulence occurs at
00:07:58
aryal's number not much over
00:08:01
2,000 however if we improve the entrance
00:08:04
conditions by putting a nicely flared
00:08:06
funnel at the Upstream end we can delay
00:08:10
the transition to turbulence to a much
00:08:12
higher rentals
00:08:14
number now we have increased the flow
00:08:16
rate and further decreased the viscosity
00:08:19
with very great care it is possible to
00:08:21
push the rentals number up as high as
00:08:23
nearly 100,000 without
00:08:26
turbulence however here we can reach
00:08:28
only about 8,000
00:08:38
the onset of turbulence depends upon the
00:08:40
growth of perturbations because of an
00:08:42
instability for example at this rental's
00:08:45
number the flow alternates between
00:08:47
laminer and turbulent modes apparently
00:08:51
randomly depending presumably on some
00:08:53
random variations of the
00:08:58
perturbations
00:09:00
prominent in our defining syndrome was
00:09:02
the word
00:09:04
mixing what about
00:09:06
mixing let's modify our experiment by
00:09:09
introducing a thin streamer of dye into
00:09:11
the
00:09:13
flow notice that the Dy forms a thin
00:09:16
filament which maintains its identity
00:09:18
with very little change right down the
00:09:21
tube the only mixing is molecular and
00:09:24
the concentration gradients are so small
00:09:26
that this process is very slow
00:09:29
this is laminer
00:09:33
flow now we increase the rentals
00:09:41
number at the onset of turbulence our
00:09:43
die filament seems to explode and the Dy
00:09:46
is rapidly mixed across the
00:09:50
tube if we want to we can regard the
00:09:53
increase in pressure difference as a
00:09:54
manifestation of mixing
00:09:56
two mixing of momentum
00:10:05
in laminer pipe flow the velocity
00:10:07
profile is parabolic near the walls
00:10:10
there is a considerable quantity of
00:10:11
relatively slowly moving
00:10:14
fluid Dy injected near the wall marks
00:10:18
some slowly moving fluid when it leaves
00:10:20
the pipe it falls with a steep
00:10:27
trajectory notice that the slowly moving
00:10:29
fluid near the top of the pipe also
00:10:31
falls
00:10:38
steeply the rapidly moving fluid from
00:10:41
the middle of the tube Falls with a flat
00:10:50
trajectory compared with laminer flow in
00:10:52
turbulent flow the mixing of momentum
00:10:54
causes the velocity to be much more
00:10:56
nearly
00:10:58
uniform the fastest fluid is not quite
00:11:00
so fast and there is very little slowly
00:11:03
moving
00:11:04
fluid what little there is can be
00:11:06
dragged along with the rest when it
00:11:07
leaves the
00:11:15
pipe nevertheless the velocity must
00:11:18
vanish at the wall so we can regard the
00:11:20
wall as a sink for
00:11:23
momentum turbulence increases the rate
00:11:25
at which momentum is transferred toward
00:11:27
the wall thus with turbulence we need a
00:11:30
larger pressure gradient to replace the
00:11:32
momentum lost to the
00:11:37
wall although the principal motion of
00:11:39
these bubble patches is Downstream there
00:11:42
is a very appreciable cross-stream
00:11:45
component fluid moving across the stream
00:11:47
in this way tends to carry its
00:11:50
properties with
00:11:53
it it is these cross stream velocities
00:11:56
which do the
00:11:57
mixing
00:12:00
for example the momentum close to the
00:12:03
wall is appreciably less than that in
00:12:04
the center of the
00:12:06
stream the cross stream motion carries
00:12:08
low momentum fluid into the center of
00:12:10
the stream and high momentum fluid
00:12:12
toward the
00:12:13
wall thus the turbulence transports
00:12:19
momentum in this shot of the Fraser
00:12:21
River at Hell's Gate we can see great
00:12:23
volumes of slow moving fluid which come
00:12:25
up from near the bottom and very much
00:12:27
reduce the average speed of the surface
00:12:28
floow
00:12:32
turbulence can transport more than
00:12:34
momentum with scalers such as D and heat
00:12:38
mixing may be primarily a matter of
00:12:39
stirring a fluid which is already
00:12:41
grossly
00:12:43
homogeneous on the other hand if we have
00:12:45
a mean property gradient the most
00:12:47
important effect of the mixing may be a
00:12:49
transport of some property as dye is
00:12:51
transported across this
00:12:55
channel the wall here has been
00:12:57
deliberately roughened in order to
00:12:58
increase increase the ratio of turbulent
00:12:59
to mean flow
00:13:03
speeds the die also helps us examine the
00:13:05
velocity
00:13:06
field the blue dye is injected at the
00:13:09
center of the channel where the flow is
00:13:11
fastest the red dye is injected near the
00:13:14
wall into the slowly moving
00:13:18
fluid notice how even at the same
00:13:20
distance from the wall the filaments of
00:13:22
blue dye move more rapidly than do the
00:13:24
red dye
00:13:26
filaments the forward momentum of fluid
00:13:29
moving towards the wall is on the
00:13:30
average greater than that of fluid
00:13:32
moving away from the
00:13:35
wall the region near the wall then
00:13:38
continuously gains momentum at the
00:13:40
expense of the region near the center of
00:13:42
the
00:13:43
flow since rate of change of momentum is
00:13:46
force and force per unit area is stress
00:13:50
we see that the presence of the
00:13:51
turbulence produces a stress the rental
00:13:54
stress within the
00:13:56
fluid the rental stress is additional
00:13:58
the ordinary viscous shearing stress and
00:14:01
produces the increase in drag which we
00:14:03
found in our pipe flow when it became
00:14:10
turbulent now let's have a closer look
00:14:12
at the mixing of a scaler in this vessel
00:14:15
we have two missible liquids one
00:14:17
floating on top of the
00:14:19
other if we leave them for a week or so
00:14:21
molecular diffusion will do a fair job
00:14:23
of mixing
00:14:27
them
00:14:32
much more thorough mixing can be
00:14:34
accomplished in less than a minute if we
00:14:36
make the fluid
00:14:37
turbulent in this case too the end
00:14:40
result is intimate mingling on a
00:14:42
molecular scale although the turbulent
00:14:44
motions themselves are not much smaller
00:14:46
than about a
00:14:52
millimeter the role of the turbulence is
00:14:55
to make the inhomogeneities more
00:14:57
vulnerable to the effects of molecular
00:14:59
diffusion perhaps this can be clarified
00:15:02
by
00:15:03
animation let us suppose we have a blob
00:15:05
of something which we plan to mix into
00:15:07
the surrounding
00:15:08
fluid if the fluid is turbulent The
00:15:11
Irregular motion will result in a strain
00:15:14
which will pull out the blob into a
00:15:16
greatly elongated form like
00:15:18
this in turbulence there is a great
00:15:21
range of different scales of motion at
00:15:24
the same time as the blob is being
00:15:25
pulled out smaller scales are distorting
00:15:27
it
00:15:31
and smaller scales still produce an even
00:15:33
finer grain
00:15:35
structure eventually the interfacial
00:15:38
area becomes so large and the property
00:15:40
gradient so steep that molecular
00:15:42
diffusion is able to act quickly and
00:15:44
produce efficient
00:15:47
mixing in reality of course all the
00:15:50
events we saw sequentially occur
00:15:56
simultaneously one of the curious
00:15:58
properties of turbulence is the fact
00:16:00
that although the rentals number is very
00:16:02
important in determining whether or not
00:16:04
a particular flow will be turbulent once
00:16:07
it is
00:16:08
turbulent the value of the rentals
00:16:10
number is of very little importance as
00:16:13
far as a large scale motion is
00:16:15
concerned these two jets look pretty
00:16:17
much the
00:16:19
same on the large scale a turbulent jet
00:16:22
is a turbulent jet
00:16:24
period however if we turn our attention
00:16:27
to the small scale motion
00:16:29
as seen in these shadow graphs the
00:16:32
effect of the rentals number is pretty
00:16:35
clear notice how much finer grained is
00:16:38
the structure in the higher Ryals number
00:16:41
jet the reason for this can be
00:16:43
understood if we consider the energy
00:16:45
dissipated in these two Jets the rentals
00:16:47
number difference is produced by having
00:16:49
different
00:16:50
viscosities all other conditions are the
00:16:52
same including the energy input into the
00:16:55
jet therefore the two Jets disappear
00:16:58
energy at the same
00:17:00
rate now energy dissipation in a
00:17:02
Newtonian fluid is given by the
00:17:04
viscosity multiplied by the mean square
00:17:06
of the strain rate dimensionally
00:17:09
viscosity multiplied by the speed
00:17:11
squared divided by some characteristic
00:17:13
length
00:17:14
squared this can be
00:17:20
written in the two Jets we have the same
00:17:22
energy dissipation and the same
00:17:24
characteristic speed but have different
00:17:27
viscosities therefore the length scales
00:17:29
must also
00:17:31
differ the high Ral number jet with low
00:17:34
viscosity must correspondingly have a
00:17:36
smaller characteristic length
00:17:41
scale this leads us to one of the most
00:17:43
important Concepts in the study of
00:17:45
turbulence the idea of the energy
00:17:51
Cascade as we have seen under certain
00:17:54
circumstances a large scale motion can
00:17:56
become turbulent
00:18:03
some of the energy in the large scale
00:18:04
motion is converted into turbulent
00:18:08
energy the largest scales of the
00:18:10
turbulence are usually smaller than
00:18:12
although comparable with the scale of
00:18:14
the basic mean
00:18:18
flow however usually these large scale
00:18:21
motions are themselves unstable and
00:18:24
break into smaller scale motions which
00:18:26
take energy from them
00:18:29
finally the energy passes down to scales
00:18:32
like those revealed by the shadow graph
00:18:34
which are so small that the rentals
00:18:36
number is too low for
00:18:38
instability the energy is dissipated by
00:18:40
the action of
00:18:43
viscosity the analogy with a Cascade of
00:18:45
water is a useful one here the only
00:18:48
property of the flow at the top which
00:18:50
matters at the bottom is the rate at
00:18:53
which water passes down the
00:18:56
Cascade similarly in the turbulent
00:18:58
energy
00:18:59
Cascade at the smaller scales of motion
00:19:02
it is only the rate of energy
00:19:03
dissipation which is of any
00:19:05
consequence this rate together with the
00:19:08
viscosity determines the size of the
00:19:10
smallest scales of
00:19:12
motion at high enough rentals number the
00:19:15
small scale turbulence loses all
00:19:17
directional
00:19:19
orientation it becomes
00:19:22
isotropic figuratively it doesn't know
00:19:25
which way is
00:19:27
up
00:19:44
further at small scales the turbulent
00:19:47
structure ceases to depend upon the
00:19:48
nature of the large scale
00:19:52
flow macroscopically the difference
00:19:54
between a jet and a channel flow is
00:19:56
marked but on a small enough scale as
00:19:59
revealed by the shadow graphs the
00:20:01
difference in structure
00:20:03
disappears because of the size
00:20:05
difference the similarity between the
00:20:06
small scale structures may be somewhat
00:20:10
obscured let's change the enlargement of
00:20:12
the channel
00:20:13
flow now they're nearly
00:20:16
indistinguishable this is what we mean
00:20:18
by similarity similar structure despite
00:20:21
differences in
00:20:23
scale thus we find that there is a
00:20:25
locally isotropic regime at the small
00:20:28
small scale end of the energy Cascade
00:20:30
which is similar for all kinds of
00:20:33
turbulence we have already seen that the
00:20:35
large scale motion does not depend upon
00:20:37
the Reynolds
00:20:38
number what the Reynolds number does is
00:20:41
to determine the size ratio of the
00:20:43
largest scales to the smallest scales of
00:20:45
the turbulent
00:20:49
motion knowledge of this behavior of
00:20:51
turbulence provides a useful exercise in
00:20:53
trying to outwit the movie studios
00:20:56
frequently they prefer to burn down a
00:20:58
rather than a full scale set now as
00:21:01
we've seen this deception is fairly
00:21:03
effective because the difference in
00:21:04
rentals number which is the only major
00:21:06
difference between model and full scale
00:21:08
is not apparent in the large scale
00:21:11
motion the difference lies in the small
00:21:18
scales one of these scenes is
00:21:23
phony just look at the small scale
00:21:25
motions and then make up your own mind
00:21:27
which is which
00:21:49
in decaying turbulence energy appears to
00:21:52
pass from small scales to large in fact
00:21:55
the energy transfer is still mostly from
00:21:57
large scales to small the rate of
00:22:00
dissipation at these small scales is so
00:22:02
great that as the turbulent field decays
00:22:05
it is the large scale motions which are
00:22:07
the last to
00:22:08
die similarly in cumulus clouds one can
00:22:12
differentiate between vigorously
00:22:13
convecting clouds with their abundance
00:22:16
of small scale
00:22:18
structure and those which have consumed
00:22:20
most of their
00:22:27
energy
00:22:33
it would be unwise to consider that in a
00:22:35
turbulent flow it is merely the rentals
00:22:38
number which is of importance and that
00:22:39
nothing else
00:22:40
counts in some cases the rentals number
00:22:43
may be enormous many millions and still
00:22:46
no turbulence will exist because of the
00:22:49
presence of some other influence like
00:22:52
rotation or density
00:22:54
stratification or for conducting fluids
00:22:57
mag IC
00:22:59
fields of these buoyancy effects are the
00:23:02
easiest
00:23:05
understood here we have a water Channel
00:23:07
which partway is divided by a horizontal
00:23:10
partition the flows are identical except
00:23:12
for
00:23:15
color in this case the two turbulent
00:23:17
flows mingle fairly quickly and produce
00:23:19
a single turbulent Channel
00:23:26
flow
00:23:30
suppose we put hot water through the
00:23:31
upper section and cold water through the
00:23:44
lower buoyancy forces tend to resist the
00:23:47
motion of fluid from the
00:23:48
upper in doing the work against buoyancy
00:23:51
forces require to raise the center of
00:23:52
gravity the turbulence loses
00:23:56
energy stable straic of density that is
00:23:59
light fluid above heavier fluid thus
00:24:01
acts to inhibit
00:24:09
turbulence on the other hand if we
00:24:11
invert the situation and put the light
00:24:14
water through the lower Channel and
00:24:15
heavier water through the upper one we
00:24:18
greatly increase the turbulent
00:24:26
activity
00:24:31
let's look at that
00:24:32
again
00:24:47
stable
00:24:53
unstable in the atmosphere both stable
00:24:56
and unstable buoyancy effects are Ur
00:24:59
frequently here the air close to the
00:25:01
ground is colder and heavier than the
00:25:03
air above it this stable situation is
00:25:06
called an inversion by
00:25:08
meteorologists vertical turbulent
00:25:10
motions are strongly inhibited and any
00:25:13
motion which occurs tends to be almost
00:25:15
purely
00:25:17
horizontal smog can accumulate when an
00:25:19
inversion at some height above a city
00:25:21
prevents pollution from mixing
00:25:24
upwards on the other hand it is not
00:25:26
uncommon for the air close to the ground
00:25:28
to be heated this produces instability
00:25:31
and vigorous
00:25:33
convection convective effects also occur
00:25:36
in some Stars including the
00:25:40
sun these cells are called
00:25:43
granulations although most of them are
00:25:44
more than a th000 kilm in
00:25:47
diameter they are thought to indicate
00:25:49
convective
00:25:51
turbulence convective turbulence can
00:25:53
easily be seen in a porridge
00:25:56
pot
00:26:00
in our defining syndrome of turbulence
00:26:03
we did not use the word random although
00:26:05
it would seem to have been
00:26:07
AO the reason that this word was avoided
00:26:10
was because that at least to some people
00:26:13
it carries with it the connotation of a
00:26:15
gaussian
00:26:16
process turbulent distributions are more
00:26:19
complicated than
00:26:20
that one of the ways of studying
00:26:22
turbulent distributions is with a hot
00:26:24
wire
00:26:26
anemometer
00:26:29
the output of the hot wire is
00:26:31
proportional to the air
00:26:33
speed in this record obtained in an
00:26:36
atmospheric boundary layer the large
00:26:38
scale motion is closely
00:26:40
gaussian however if we differentiate
00:26:43
this signal or if we examine any other
00:26:45
property that is strongly dependent upon
00:26:47
the small scale motions we find that the
00:26:50
property seems to be distributed in
00:26:51
concentrated bursts separated by regions
00:26:54
of comparative
00:26:56
inactivity
00:26:58
a stationary gaussian process could not
00:27:00
behave in this
00:27:01
way there the derivatives would look
00:27:04
much like the original signal except for
00:27:06
a change in
00:27:09
scale the higher the Ryals number of
00:27:11
turbulence the more marked this
00:27:13
intermittency becomes and it is
00:27:16
particularly noticeable in geophysical
00:27:19
flows This Record is of temperature
00:27:21
fluctuations in the
00:27:23
atmosphere and this one of velocity
00:27:26
fluctuations of a tidal flow in the
00:27:31
ocean as with many other aspects of
00:27:34
turbulent Behavior we do not have a
00:27:36
fully satisfactory theoretical
00:27:38
explanation for this kind of
00:27:39
intermittency it is another
00:27:41
manifestation of the baffling but
00:27:43
fascinating complexity of
00:27:56
turbulence
00:28:26
for
00:28:56
e
00:29:26
e
00:29:56
e
00:30:26
e
00:30:56
e
00:31:26
e
