Swirling
flow
Introduction
I have tried all sorts of ways to introduce and
to structure this chapter and have found it to be most difficult. The central
problem is that steady swirling flow, that is, flow in circles, does not
occur in nature but transient swirling flow seems to be everywhere.
Normally when we try to analyse some device through which fluid flows we start
with a flow pattern for steady flow, make some simplifying decisions and use
the steady flow energy equation to find out what we can. Then we go back and
take other effects into account. When you try to sort out the dynamics of
swirling flow you can draw what looks like a flow pattern but is not an
ordinary flow pattern in that every flow line is a circle and then you find
that there is no obvious way to idealise the flow for analysis as there is for
say, flow in pipes or flow over a weir because steady flow does not normally
come back to flow through the same point. Yet swirling flow is used in separators
and centrifugal casting and crops up in rotodynamic machines and causes no end
of unwanted problems in unforeseen rotating flows. Mostly it is not a facility
to make predictive calculations that is needed by the engineer but a
familiarity with the behaviour of swirling fluids. This has found its form in
the idea of a vortex and most people regard the word “vortex” as synonymous
with swirling flow. However it seems to me that we need to reserve the word
“vortex” for our models of swirling flow in the three forms of the forced
vortex, the free vortex and the free spiral vortex.
I came to the conclusion that the most easily
understood way to handle this is to recognise just two types of swirling flow,
the forced vortex and the free vortex and to consider the mechanics of radial
flow because it can co-exist with a free vortex to give the free spiral vortex.
Then I can look at other types of swirling flow.
The stability of swirling flow
Suppose that somehow we could
have an extensive mass of fluid that is rotating steadily about a
vertical axis. We could draw a flow pattern for some plane horizontal surface
in the flow. It would be just a set of concentric flow lines such as those in
figure 15-1. I have spaced these flow lines at equal increments of radius but,
to fit in with the common flow patterns, the flow lines might better be pitched
so that the lines have equal flows between any two adjacent lines although it
is difficult to see how this might be achieved.
This is a flow pattern for steady flow but it
is not like other flow patterns. It is for a flow where the properties of the
fluid, eg pressure, speed, density and so on do not change along a given flow
line. This immediately rules out the use of the steady flow energy equation
because, even though the properties are constant in time and the flow is
steady, the very fact that the properties do not change along the flow line
makes it pointless to even attempt to use the steady flow energy equation. It
means that, as the properties do not vary along the flow lines, its only
value to be derived from considering this model comes from the fact that the
properties vary across the flow lines, that is, with radius.
For this flow pattern to exist the flow must in
some way be stable and we can look to see what the conditions are for this to
be the case.
Stability comes from there being
a balance between the pressure forces and inertia forces that are applied to
the fluid that is flowing in circles. I have a flow pattern in which
gravitation does not affect the flow but it has no thickness. If we are to
consider inertia forces we must have mass somewhere. With this in mind we have
to consider not a line but a ring like the one shown in figure 15-2. The ring
shown is really just one of many rings that make up a thin disk of fluid.The
disk has a small thickness and the ring has a radial
thickness . As the outcome of this analysis will
be a differential equation these dimensions are really 
infinitesimals
that I have had to exaggerate in order to make the drawing clear. So the
drawing is of a ring of very small cross section that is rotating at uniform
speed. There is nothing in this ring of fluid to hold it together other than
the forces exerted by the other rings that surround this elemental ring. These
act through pressures and ultimately these pressures acting on the ring must
create the inward radial forces that gives the ring its centripetal
acceleration.
We can look at the forces on a
small element of the ring contained between two radial surfaces subtending a
small angle of at the centre and shown in figure
15-3. Having let the ring of fluid rotate in a horizontal plane the small
element of this elemental ring is in equilibrium under pressure forces acting
on the four vertical surfaces, two radial and two tangential, and the inertial
force on the mass of fluid in the element.
I shall suppose that the
tangential force on one side of the element is equal to the area of the element
multiplied by the mean pressure on it and that the resultant of the two forces
is equal to one force multiplied by . Then the inward radial force on the
element is given by :-
The mass of the element is where is the density of the fluid. The
centripetal acceleration is where is the tangential velocity of the
fluid in the ring.
It follows that :-
This can be simplified if it is
expanded and second order quantities are ignored and reduced still more to :-
and finally to
This is the fundamental
expression for swirling flow to be stable. It is is how the tangential velocity
varies with radius that determines whether the vortex is “forced” or “free”. I
will start with the forced vortex.
The forced vortex
A real fluid in a suitable
container can be made to rotate like a wheel. When it rotates steadily like a
wheel the relationship between tangential speed and radius is simple. It is where is the angular velocity and is the tangential velocity.
If
it can be re-written . This can be where is the pressure at radius .
This is not very suitable for use
by engineers who are more accustomed to work in terms of head so it can be
rewritten:-
.
This can be plotted for some
values of , and .
I have plotted graph 15-1 for a
maximum radius of 250 mm and a speed of 250 rpm. The relationship between
pressure head and radius is parabolic. At a radius of 250 mm the pressure head
is about 2.2 m of fluid.
I think that a reader will find it to be
valuable to explore this graph for other values of speed and radius.
It is worth looking at a worked example of a
real system that could be made to flow with a forced vortex.
Figure 15-4 represents a cylindrical
tank that is fitted with a top and is mounted so that it can be rotated about a
vertical axis at a steady speed. The tank is completely filled with water. The
top has a small hole in its centre so that the pressure of the water adjacent
to the hole is atmospheric.
When the tank is rotated at a steady speed of
120 rpm the water will ultimately rotate with the tank as if the tank and the
water formed a solid. The water then rotates as a forced vortex. One tends to
associate swirling flow with free surface effects but the presence of the top
on the tank has prevented free surface effects. However we can predict the
pressure distribution through the water and the obvious thing to do is to
predict the pressure distribution over the lower face of the top so that we can
see what happens when the top is removed. We can also predict the pressure
distribution over the upper face of the bottom.
We have so the pressure head can be
calculated using, say, Mathcad. It is shown in graph 15-2 where the scale of
the pressure head is the same as the linear scale of the tank. The red line is
clearly a parabola. The pressure head acts vertically upwards on the lower face
of the top and will produce a force on the top tending to lift it.
This force can be calculated. It is:-
.
The distribution of pressure head over the
bottom of the tank is that for the top plus the depth of the tank, that is, 0.1
metres. The force exerted on the bottom is equal to the upward force on the top
plus the weight of water. Therefore:-
force = .
This tank has one more thing to tell us.
Suppose that the top were to be removed and the tank extended to a depth of 200
mm. If the tank is once again rotated at 120 rpm the water could only be in
equilibrium if the surface took on the shape of this parabola and formed a
paraboloid of revolution. Necessarily the middle would drop and the sides rise
and, if water were to be added, the apex of the paraboloid could be brought
back to its original position at a depth of 0.1 metres.
I have shown the result in
figure 15-5 where the red lines show the surface as a paraboloid of revolution.
It is most unlikely that anyone would see such
a surface under normal day-to-day activity because it clearly requires some
mechanical device in which to produce it. I wondered about this surface as I
had seen one created in a rig filled with water made by someone else and it was
very fragile in that it was very easily disturbed. Thumping the bench on which
it stood produced a fascinating array of surface waves on the paraboloidal
surface.
It seemed to me, as an engineer, that the whole
process of bringing such a tank to a state of equilibrium is not as simple as
it sounds and there is some value in looking more closely at this process.
Generating a forced
vortex
Suppose that the open-topped tank
described above were to be brought up to speed very quickly. Shearing of the
water would take place at the sides and over the bottom. The water has
viscosity and this shearing would cause a shearing force to act on the water
and this force would progressively make the water rotate, first slowly and then
faster and eventually all the water would rotate like a wheel. The viscous
shearing force is acting on a mass of fluid to make it accelerate. The time
taken would depend on the coefficient of viscosity for the water and on the
density of water. If the liquid in the tank were to be changed to, say, a
lubricating oil with viscosity perhaps 2000 times that of water or to mercury
with a high density and low viscosity the process of coming to equilibrium
would be very different.
I thought that I might make a very simple rig
to find out what does happen. On the face of it, all that is needed is a
cylindrical drum with its axis vertical, partly filled with liquid, and rotated
about its vertical axis. I made a very simple rig from odds and ends in my
workshop. It is shown in figure 15-8. I had a nicely-made, plastic dome that
came with a free, crudely-made clock. It was a short job to make a wooden
fitting to attach the dome to my wheel brace and mount the wheel brace in the
vice. I found that I needed the white plastic disk to reduce coloured
reflections from the red bench-top. The dome turns 3.75 times for each turn of
the handle and a typical speed of rotation of the dome is 240 rpm. It is easy
to turn the handle at these speeds and there was a need to limit the speed to
stop the fluid overtopping the dome. As it is turned by hand there will
inevitably be fluctuations in speed.
[I
think that it turned out to be more interesting that I expected. I started by
wanting to take some pictures of the free surface of a forced vortex but ended
up with a device that is very suited to the illustration of the concept of
Reynolds number. In the end I tried this little rig with water, lubricating oil
for internal combustion engines, with steam oil for steam engines and with a small
quantity of mercury.]
At this point I will just illustrate
the shape of the free surface of water rotating as a forced vortex. I used
water and I found that the rig was very sensitive to having the axis of
rotation vertical. The surface became lopsided very easily. There were waves on
the surface and I thought that I might get rid of these by adding some
detergent to the water to reduce surface tension. This worked and led to the
pictures in figures 15-7a to e. But they are not pictures for untreated water.
It is
inevitable that there will be refraction of light in passing through the
plastic dome and through the interface between the plastic and the water and
this distorts the shape of the surface as seen by the camera. The black
“lozenge” is a reflection of the black circle of adhesive foam used to attach
the dome to the fitting in the chuck.
Figures
15-7a to d show the shapes at progressively increasing speed. Figures 15-7d and
e show the way in which the surface can become lop-sided very easily.
The rig
gave me an opportunity to watch the water as the water accelerated to its final
state of equilibrium. The water quickly rose up the sides with a fairly flat
centre but the rotation spread towards the centre and in less than half a
minute the shape was as steady as it ever would be with a hand turned device.
When turning was just stopped the water went on rotating and the surface
quickly became flat with a rotating pattern of small surface waves like the
arms of a spiral galaxy. These died away but the rotation continued until it,
too, died away.
One might
argue for having a mechanical drive to make the water rotate. One might fit
radial blades inside the tank so that the volume is divided into several
compartments. This would speed up the process of coming to a steady state but I
am not sure whether,
In that
case, the original flow pattern of concentric circles is still appropriate.
Applications
of the forced vortex.
I cannot
think of an application of a forced vortex per se.
Steel
pipes are cast by running molten steel into a mould that rotates on a
horizontal axis. This seems to be a process that is much dependent on empirical
methods for its design.
There are
separators that depend on rotation to achieve their function. Oil, in an
emulsion of oil and water, can be separated by rotating the emulsion in a
cylindrical vessel that rotates about a vertical axis. If the emulsion were to
be just left in a vessel for a long time the two fluids would separate under
the action of gravity but rotation at high speed imposes a centripetal
acceleration that is much greater that the acceleration due to gravity and the
separation takes place in a much shorter time. As the centripetal acceleration
at any point in a forced vortex is given by where is the angular velocity the
separator will work with the emulsion occupying an annulus of the vessel and
not full so that there is an inside radius to require a centripetal
acceleration. A separator could also separate suspended solids from oil.
There are
all sorts of cyclone separators. A wide range of solids can be carried in pipes
in a flow of air. Typical examples are dust extractors and pneumatic conveying.
In both cases particulate solids are entrained in the flow and must be
separated at discharge. The cyclone has an upright conical shape with a
cylindrical section at the top. The air and particulates enter the
cylindrical upper section tangentially. The solids slow down and follow a
spiral path to the bottom of the cone and fall out into a collection system.
The air is discharged to the atmosphere from the centre of the top at low
speed. It is moot point whether this process depends on the forced vortex.
One might
see the centrifugal pump as an application of the forced vortex. Certainly,
when a centrifugal pump runs against a closed delivery valve there is a forced
vortex in the pump and the net head on the valve can be calculated from the
dimensions of the impeller and its speed but, as soon as the valve is opened,
there is a flow and we need a better explanation for the action of the pump
than the forced vortex.
It seems
to me that an understanding of the mechanics of the forced vortex can lead to
ideas for the design of devices to achieve some objective but mostly, once the
ideas have come into existence, the realisation of a practical device in
hardware results from trial.
A forced
vortex appears to occur in nature at the centre of rotating storms.
The
free vortex.
This is for a flow pattern of
concentric circles that is not driven mechanically but left to find its own
relationships between radius and pressure and radius and velocity and still be
stable. It is a flow pattern that cannot have a steady state because at every
circle there is a velocity gradiant and therefore a shearing force owing to the
effects of viscosity with no way to drive it.
We need some condition comparable to that for the forced
vortex that must be fulfilled as well as the condition for stable swirling
flow.
Condition for a free vortex
Suppose that in some way
we could produce a free vortex in an extensive mass of fluid and rotating about
a vertical axis. If the fluid were to be real, viscosity would soon alter the
flow pattern so let us suppose that the fluid is frictionless so that we can
see what might happen. The important feature would be that the flow had no
torque exerted on it anywhere because the presence of a torque is not
consistent with steady flow. As torque equals the rate of change of angular
momentum the necessary condition for zero torque is that the rate of change of
angular momentum with radius is the same everywhere in the vortex and is zero.
If we consider the flow to be made up of many particles of fluid each with the
same mass every particle must have the same angular momentum about the centre
of rotationof the ho vortex.
For any particle the angular
momentum is given by :-
where is the mass of the particle.
It follows that the condition for
a free vortex is that where is a constant or of course that . So where is a new constant.
Then :- .
This can be integrated between
two radii, both of which must be finite because the condition of uniform
angular momentum leads to infinite tangential velocity at zero radius and zero
tangential velocity at infinite radius.
. Then:-
and this can be put in Newton
metre/Newton
:-
This result shows that not only
is the angular momentum of the particles of fluid uniform throughout the vortex
but the sum of the pressure energy and the tangential kinetic energy, , is the same everywhere in the vortex
as well[1].
It is now possible to draw a
graph of pressure head aganst radius for a free vortex with, say, the pressure
at some suitably large radius being atmospheric. Then the pressure throughout
the vortex will be less than atmospheric. We can put:-
and this can be evaluated as a
graph.
I have plotted a graph of
pressure head against radius to suitable values in graph 15-3. It is the red
line. It tells us the character of the free vortex. All the interesting things
happen in the middle where the tangential velocity tends to infinity as the
radius tends to zero.
Any real free vortex will occur
in a real fluid and real fluids have viscosity. This means that energy will be
lost in the fluid wherever there is a velocity gradient with the inevitable
shearing. It is pertinent to plot a graph of the velocity gradient versus
radius to see how the rate of change of shearing strain varies with radius.
We have for a free vortex :- and . This can be plotted for the same value
of the constant as was used in graph 15-3 and I
have added it in blue to graph 15-3 to give 15-4. Note the dividing factor of
100 to get the two graphs on the same pair of axes. In the middle the velocity
gradient is very large indeed.
So a free vortex in a real fluid
would lose energy to viscous friction very quickly in the centre but very
slowly in the rest of the vortex. It might exist for a short time in a fluid of
low viscosity but is unlikely ever to come into existence for fluids with high
viscosity. A real fluid, even if it has low viscosity, cannot flow with a free
vortex in the centre. Some change in the flow pattertn must occur. It looks to
an observer with a knowledge of the forced vortex as though a forced vortex
forms in the centre of the free vortex and this is a very plausible
interpretation of this mode of flow.
The size of the forced vortex
will depend on both the viscocity and density of the fluid and may be quite
small for mercury and very large for thick oil. But there is nothing practical
about it and in truth we are only interested in the character of this notional
flow so that we can understand it when the free vortex is combined with inward
radial flow and the flow pattern ceases to be imaginary and becomes real in the
sense that the energy equation can be applied toit.
Uniform inward radial flow
This is important because, in
nature, rotating flow often occurs where there is an inward radial flow. This
inward radial flow provides the energy to sustain the rotating flow with its
continuous loss of energy to friction. Inward radial flow can be produced quite
easily in engineering so we deal with it like any other flow using the steady
flow energy equation and the continuity equation because this is a normal flow in
which energy can be lost steadily and made up externally whereas, as we have
seen, the circulating flow lacks any mechanism by which energy can be imparted
to the flow.
Figure
15-8 represents a simple real system in which some form of radial flow must
exist. It is an open-topped tank with a radiused nozzle fitted centrally in its
base and with a circular plate set symmetrically above the nozzle at a fixed
distance above the base. When water is
supplied steadily to the tank, in a way that minimises the disturbance caused
by the filling, it will flow between the base and the plate and out of the
nozzle. Eventually the level in the tank will become constant. What has been
made is really a convergent nozzle that converts potential energy to kinetic
energy and like all other convergent flows it will be stable and involve little
loss of energy. Such a system might produce rotation as well as flowing
radially but, if we treat the flow between the base and the plate as wholly
radial, we can use our normal methods of ignoring friction and regarding the
flow as one-dimensional and relate pressure, velocity and radius.
is a point in the free surface and its total
energy relative to the level midway between the plate and the base is and and and both are equal to zero. The
total energy at = .
I have drawn a single flow line
in blue that starts somewhere in the free surface and goes through the gap
between the plate and the base to emerge in the jet. I have shown another
identical flow line in the the plan view to show that it is radial. The
flow line passes through point at radius between the plates. Having chosen
to ignore friction we can say that :- where is the pressure and is the velocity of the water at and is the radial direction.
The same equation applies to
every other point between the plate and the base.
We need a relationship between
the velocity and the radius and that will come from the continuity equation.
The area of crossection of the flow at radius is and so the inward velocity at radius is where is the volume flow between the
plates. It follows that:-
and this is of the
same form as the expression for pressure for the free vortex. It means that
inward radial flow and the free vortex can co-exist.
The free spiral vortex.
In
describing the system shown in figure 15-8 I said that the tank was filled
using some arrangement that minimises the disturbance in the water and this produces
radial inward flow. Now suppose that a ring of guide blades were to be fitted
as shown in figure 15-9 on the edge of the plate and fitting between the plate
and the bottom of the tank.. These blades will impart angular momentum to the
water and the flow between the bottom and the plate will not only be inwards
but it will also rotate. A free vortex and steady inward radial flow will
co-exist. This arrangement is used to create a swirling flow in water turbines
and the runner would fit inside the ring of guide blades.
For the arrangement shown the
flow pattern will change from the simple radial lines and there will be flow
lines that curve from entry to the plate and exit to the nozzle. We need to
know what these flow lines will look like.
If this is a combination of
inward radial flow and a free vortex we can say for the inward radial flow that
where is the radial velocity at radius and is the volume flow. For the free
vortex the tangential velocity at radius is given by . If the flow at any point is a
combination of inward radial flow and a free vortex the absolute velocity at that point is the vector sum of
and .
In figure 15-10 I have drawn a
flow line in red and the radial velocity and the tangential velocity at radius are shown. The vector sum of these
velocities is , the absolute velocity, and this makes
an angle with the tangent to the circle at
radius .
We know that both and are inversely proportional to , so must be the same at every radius.
This is the characteristic feature of a logarithmic spiral that, for a polar
plot, has the equation where are constants and is the independent variable.
It will be clear that constant simply increases the radius for
any combination of values of and . Constant changes the angle and when =1 the angle is 45°.
Graphs 15-5 shows the effect of
changing the value of which is to change the angle and
graph 15-6 shows the effect of changing .
If the guide blades are all
mounted so that they can be rotated together they will alter angle and change the free spiral vortex.
I have drawn a set of flowlines
for a combination of inward flow and a free vortex for the case of = 45°.
It is figure 15-10. Patterns like this occur in nature in the atmosphere and in
spiral galaxies on a large scale, and in eddies on a small scale. I think that
all that this tells us is that the logarithmic spiral occurs in many guises not
necessarily that we are looking at fre spiral vortices.
We
see immediately that this is a pattern for steady flow and the each flow line
starts at one point and finishes at another unlike the flow lines of a vortex
that are constrained to circles. The shearing of the free vortex has gone and
this free spiral vortex has flow lines that have no velocity change across the
lines.
The rotation reduces the rate of
flow because whilst we can still write for point B in figure 15-8 the relevant velocity is the
absolute velocity and not the radial one. It follows that the radial velocity
at B is smaller and this means that the flow is reduced.
Application of the spiral vortex
Early
water turbines were designed to replace water mills that were used for
grinding, fulling, crushing etc. They used the small differences in level in
dammed rivers. Figure 15-11 shows the arrangement. The water entered the
forebay and flowed into the ring of guide blades where a spiral vortex was
formed. In this vortex pressure energy was changed to kinetic energy. The water
leaving the guide blades then flowed through the rotating runner where the
kinetic energy and the remaining pressure energy was transferred to the shaft
to drive its load. The water then left through the draft tube. All the
subsequent pressure turbines developed from this simple design. Clearly the
spiral vortex, even if it only exists in the small space between the guides and
the runner, is an essential feature of water turbines and we have sufficient
information to size the components of the guide system. I shall come back to
this when I deal with water turbines.
Real swirling flow.
In this chapter I have set out
the mechanics of swirling flow for special cases. In doing so I have
deliberately ignored some of the inconvenient features of this type of flow. I
rather glibly ignored losses when I applied the energy equation. I now want to
examine that decision. It implies that the fluid had no viscosity and that is
difficult to handle because if
a
fluid had no viscosity it would be impossible to make a small quantity
of fluid rotate if it were to be at rest and impossible to stop rotation if
rotation could be established. This follows from the fact that there is no
way to apply a tangential force to the small quantity of fluid in the absence
of viscosity. We can see what this means from figure 15-12.
Here I have drawn the plan view of a disk mounted on a
vertical axis so that it can rotate in the horizontal plane. At some point on
the disk a small wheel is mounted on a short axle so that the wheel can rotate
freely. There is nothing notional about this mechanism, it could be made quite
easily. Suppose that the disk is made to rotate at a slow steady speed. The
arrowhead drawn on the disk would appear to turn once per revolution of the
disk in the clockwise direction. I have marked the small wheel with a black dot
and, if there were to be no friction between the small wheel and the axle on
which it is mounted, the small wheel would not rotate and, in my diagram, the
dot will always be at the lowest point on the small wheel. The wheel appears to
be rotating anti-clockwise relative to the disk at the speed of rotation of the
disk. This, for obvious reasons, is called counter-rotation. In reality there
will be some friction and the small wheel will gradually speed up until it has
no rotation relative to the disk. Then the disk and the wheel would appear to
rotate as a solid.
This has obvious consequences for
our notional rotating fluid with the flow pattern in figure 15-4 that we have
called a vortex. Every molecule of the fluid will tend to counter-rotate and
this may lead to groups of molecules counter-rotating[2]. It is effectively impossible to allow
for this complication. All that the engineer can do is ignore counter rotation
and then try to take the effect of viscosity into account at a later stage[3]. On top of this, in real fluids there
will be a shearing process going on at every radius and we cannot deal with
this either so, in order to get some idea of what happens, we ignore viscosity
as well. This puts us in the contradictory position of choosing to ignore
viscosity knowing that the flow is only possible if there is internal friction.
That decision does lead to an insight to the behaviour of fluids when they
rotate but leaves us with the need to consider the affect of viscosity.
The
effects of the viscosity and density of real fluids on swirling flow
Let me
return to my simple hand driven device for creating a forced vortex. In this
device fluid friction is used to create the motion. This is not how we normally
meet fluid friction. Usually viscosity opposes some flow that we want to create
and we really want to minimise its effects. If it is viscosity that drives the
fluid in this small rig the magnitude of the driving force will depend on the
viscosity of the fluid being driven and, as the object of the driving is to
make the fluid rotate, the effect produced will depend on the density of the
fluid as well. This means that what happens depends on viscosity and density
but it will also depend on the speed of rotation and the size of the rig. These
are the quantities that appear in Reynolds number. However, this time viscosity
has an active role and not the passive role that it normally plays. As a result
the operator who turns the handle can feel and see the effects of viscosity and
density. I thought that it was most instructive.
I took the pictures in figure 15-7 and then
thought that I might find out what happens if I changed to oil. I used 10-w40,
semi-synthetic, motor oil. The result is shown in figure 15-13.
The
process of making the oil rotate was much faster than for water and the shapes
were not sensitive to speed or want of alignment. The process started with the
oil rising up the wall of the dome leaving the centre only slightly dished as
in figure 15-13a. As the rotary motion progressed through the oil, the centre
became more depressed but the rise at the wall was still evident as in 15-13b.
Eventually the surface took on the parabolic shape of 15-13c. It was still
liable to go askew as is shown in 15-13d. Overall it was much less sensitive
than water. It was attractive to look at.
Then I
changed to steam oil. This is thick oil that flows very reluctantly at room
temperature. It is effectively opaque so I could take no photographs. The first
surprise was the short time needed to reach its final shape. The second was
that when I just released the handle of the wheel brace the dome went on
rotating for about 1 and a half turns of the handle. The oil acted like a
flywheel. I was then interested to see that I could not turn the handle at a
steady enough speed to stop the parabolic surface from rising and falling in
time with the rotation of the handle. This suggested that there might be
something to be learnt from using mercury with its low viscosity and very high
density.
I had a
small quantity of mercury that has been in my workshop for about 40 years going
back to the days before it was declared to be so toxic that it had to be banned.
I cleaned it and put it in the dome.
The
photographs in figures 15-14a to f give an idea of how the mercury behaved but
the rig has to be used to really appreciate it. Pictures 14a, b and c are for
the same speed. At the start the mercury goes up the sides leaving the centre
flat and not rotating. Later more mercury rotates and the flat area diminishes
as in 14b to take on the paraboloidal shape in 14c. In 14d the speed has been
increased. Picture 14e shows the surface after the turning has been suddenly
stopped. It looks remarkably like a free vortex. This rotation went on as the
surface flattened but the high density, low viscosity and high surface tension
led to a “skin” of mercury with rotation going on under it and visible to the
eye but not the camera. In the final picture the mercury split. It started as a
break in the surface tension and ended with two separate “flows”. It is the
result of the shape of the dome and the limited quantity of mercury that I had
available. But this rig cost me nothing and I learnt a lot.
I
think that the most important feature of this testing was that it was the first
time that I had met a system in which the viscous drag is used to drive the
flow. When I first learnt about non-dimensional groups the question students
asked was “What is Reynolds number?” There were all sorts of answers given to
this but they all seem to start at the non-dimensional group and try to read
something into it as a ratio of this to that. Here we have a very simple system
where the ratio of the viscosity to the density is dominant if the speed and
shape are fixed and only a little thought shows that a change in the speed
and/or the size will change things again. It is so obvious that some
combination of these quantities is significant. That combination proved to be
Reynolds number.
The
mercury tests showed the effect of surface tension that prevented the mercury
rising as high as it would have done in the absence of surface tension. Figure
15-14f shows the surface tension providing the centripetal force needed to hold
the mercury at the bottom of the dome together and on centre. Hence the Weber
number.
The real spiral vortex
This
is the flow that we see most often. It is the swirling flow that we see every
day in wash basins and baths as they empty. The swirling motion that we see is
in part determined by the geometry of the wash basin or bath. We need to get
rid of this undefinable geometry in favour of something that anyone can
visualise.
I found a yoghurt pot that was
about 110 mm in diameter and drilled a hole of 6.3 mm diameter in it to use as
a vortex tank. Figure 15-15 shows a free spiral vortex in this very small
tank. It was easy to generate. This picture was taken after the inside of the
plastic pot has been sprayed with black paint to give a contrasting background
to show the powdered aluminium floating on the surface. This powder showed the
flow of the water on the surface and that the powder was being carried to the
eye and down through the vortex so that the powder had all gone before the tank
was empty. Unfortunately the still camera just stops the powder and gives this
speckled appearance but the shape of the surface looks to be very like that in
figure 15-8 although there is no certainty that is identical. I made a larger
tank of 200 mm diameter and about 180 mm deep also with a 6.3 mm hole. It too
gave a free spiral vortex as shown in figure 15-16 where the shape of the free
surface is quite obvious. The powder in figure 15-15 shows the water at the
surface following a path that could be a logarithmic spiral going round faster
and faster as it approaches the small hole that extends through the flow and on
into the jet. It does not show us what happens in the main body of the water. I
injected ink into the flow and it is evident that all the water takes part in
the flow moving steadily more or less horizontally in spirals towards the eye
and then being swept down towards the hole. But ink did not prove to be a good
marker unless the rate at which it is injected could be controlled.
In
order to explain what goes on in water in a tank such as this I must go back to
the tank that I have already mentioned in chapter 4. When I was working I
had a clear perspex tank that was large by laboratory standards. It is shown
again in figure 15-17. In this photograph dye is being injected to water that
is not rotating but flowing freely through a sharp-edged orifice. It was
possible to create a free spiral vortex in the water in the tank and such a
vortex is visible in figure 15-18 at the top of the picture. It was taken when
the level in the tank was low to get the attractively shaped jet below the
orifice. Figure 15-19 shows the jet at much greater depths and the jet bursts
open and surface tension is not sufficient;y strong to pull it back together.
In figure 15-19 the hollow core is visible but not the trumpet shaped section.
In order to produce the free
spiral vortex the tank was filled through a pipe at the top that was near to
the outer edge and was set up to supply water tangentially and horizontally.
I think that there were two
important observations to be made about this free spiral vortex that formed in
this tank. The first is that no special procedure was adopted in filling the
tank or timing the opening of the orifice yet the resulting vortex was always
the same. The second is that the creation of this free spiral vortex did not
affect the coefficient of discharge of the orifice. The time taken to drain the
tank between two set limits of depth was the same for flow with no discernable
rotation and for a fully developed free spiral vortex. This is cannot be
accidental, somehow the free spiral vortex in this tank can increase or
decrease its angular momentum to come to some quasi-steady state that, for this
tank, is always the same.
When I had access to this tank I
was intrigued by this vortex but had too little time to stop and think about it
in depth. I did use the dye injection system that was fitted to show the flow
lines on non-rotating flow to look at this vortex. The following is based on
what I recorded at the time.
Dye
could be injected using the several pipes and then the dye showed what happens
in the tank. In chapter 4 picture 4-1 I showed a perspex tank that I used to
have in the laboratory. Its primary purpose was for a laboratory exercise on
the time taken for the level to fall between set values. It had several
secondary functions. The tank was fitted with a filling pipe that admitted
water more or less horizontally and tangentially to the tank. If the orifice
was closed and the tank filled as quickly as possible and the supply of water
stopped, the water in the tank would rotate in some random way and looked to be
totally unco-ordinated except for the fact that it was all going round in the
same direction. In a short time, perhaps 30 seconds, the surface began to lose
its random motion and it was clear that the flow was changing to something that
was not random. In a further short time the water in the central region could
be seen to be moving much faster than that at the outer edge and then a dimple
appeared in the middle which grew to be 100 mm or so in diameter with a flat
part in the middle. The random energy had become organised so that there was a
forced vortex in the middle surrounded by a free vortex. This visible vortex
inevitably died away if left alone[4]
but, if the orifice in the bottom of the tank was opened, the flow changed
again. The flat middle disappeared and with it went the forced vortex and the
dimple was re-established to be come deeper and deeper until a column of air
was created right through the water from surface to orifice and beyond.
Figure
15-8 is of the tank taken from the underside and the hole at the centre of the
rotation can be seen at the middle upper part of the picture. The hole extends
through the orifice and permits the jet to burst open.
If dye
is injected into the main body of the water it will be seen to be moving in
horizontal circles with no inward or downward components other than that due to
the emptying. The dye lines are so well defined that, had it been part of
Reynolds’ experiment we would conclude that the flow was laminar. Certainly the
flow is two-dimensional in the form of a vortex that is not a forced vortex and
so is presumably a free vortex.
Clearly
this flow is being sustained in some way and we have to look for the source of
the energy and a mechanism by which it is given to the rotating flow. The
source of the energy is obviously the original potential energy of the water
that is falling steadily in the tank. To find the mechanism by which this is
transferred we shall have to look more closely at the flow.
If dye
is now injected into the water in the vicinity of the orifice it is be evident
that this is a region in which pressure energy and a little potential energy is
being exchanged for kinetic energy. This is a real flow and not to be confused
with the notional flows that are the forced and free vortices. The kinetic energy
that is being created near to the orifice has a tangential component and
therefore has angular monentum. This angular momentum is carried upwards
through the main body of the water in viscous forces and sustains what appears
to be a free vortex throughout the tank. It is finally dissipated in friction
between the water and the tank. Provided that there is sufficient initial
angular monentum to establish the rotating flow in the vicinity of the orifice
the flow will ultimately settle down to a state of quasi-equilibrium in which
the angular kinetic energy being created near the orifice is lost in friction
at the solid wetted surfaces of the tank.
I do not know why the coefficient
of discharge is not affected by the rotation but it is easy to think of possible
reasons.
The fact is that this orifice
tank is not important in engineering practice. Any value that it may have lies
in being able to study it so easily. I think that one should recognise that
none of this would happen in the same tank filled with engine oil. I tried
engine oil in my 200 mm tank and it simply would not rotate for more than a few
seconds after stirring was stopped. Small bubbles in the oil showed the flow to
be without rotation. All the interest in vortices stems from cyclones and tornadoes
in the air and swirling flow in bath tubs and they are both possible because
air and water have low viscosities.
What is important is that, in a
real flow that is caused to rotate, as, say, that in a centrifugal pump, this
same viscosity can affect the upstream flow by a steady transfer of momentum
upstream.
Eddies
Eddies appear in all sorts of
flows where there is no mechanical constraint on the flow. I am thinking
of the flow that occurs in wakes. in a tidal river that flows through a rocky estuary
bed one might see deep quickly rotating eddies in the wake produced by water
flowing over rocky outcrops. One can see the same where a river flows under a
bridge as shown in figure 15-20
Figure 15-20 shows the river Stour
flowing under a bridge at Sandwich in
Another form of eddy is very common.
Figure 15-21 shows a paint picture of the flow over a stalled aerofoil. It is
due to Prandtl. I have seen a virtually identical photograph of a model sail in
a water channel. What is different is first that these eddies are
position-fixed relative to the aerofoil for a given wind speed and for a given
angle of attack and secondly that the eddies are not circular. Eddies have to
fit into a space between the solid boundary and the separated free stream.
For
another article I converted Prandtl’s picture to a negative and drew lines to
show other details of the flow. There are eddies rotating in both directions
and in fact a flow path between the main one and the rest of the wake. These
eddied are not random even if they do wobble about somewhat; they are a proper
flow pattern.
Endnote
I think
that there is something worrying in my problems with this chapter. During my
teaching career I did not spot the inherent contradiction in deciding to treat
the fluid as frictionless and to ignore counter rotation[5]. In some ways I feel deceived because it
is normal for the simplifications that we make to facilitate analysis to be
comprehensible whereas this decision leads us to an impossible position. It
makes the treatment of swirling flow very difficult.
I used to
lecture it quite happily in what I now recognise to be a simplified form. When
I had time to stop and think about it I realised that swirling flow does not
fit in with most other topics in engineering. We do not have important
applications for swirling flow yet it occurs so frequently that it cannot be
ignored. I think that I learnt a great deal more about this type of flow and I
think that it has convinced me that wakes containing eddies are not just to be
called an eddying wake and forgotten but seen to be an integral part of the
flow pattern.
I have
added some worked examples but they are trivial. The very nature of swirling
flow makes the design of suitable examination questions very difficult. They
either ignore the fundamental conflict of it being necessary to have fluid
friction for the flow to exist at all and needing to pretend that the fluid is
frictionless in order to analyse it or to produce exercises in mathematics.
Worked
examples on vortices etc.
Q1 An open-topped, cylindrical tank is mounted
with its axis vertical and arrangements are made for it to be rotated at
constant speed. The tank has a diameter of 400 mm and a height of 600 mm. It is
filled to a depth of 500 mm with water. Calculate the speed at which the water
just starts to spill over the top edge.
Q2 Find the speed at which the bottom of the tank in Q1 is exposed.
Q3 Suppose that the tank in Q1 was fitted with a lid with a small hole in
the middle of it and the tank completely filled with oil of density 900 kg/m3. Calculate the forces that would be
exerted on the top and bottom of the tank when:-
a) when the tank is stationary,
b) when the tank and the oil rotate at 100 rpm.
Q4 Calculate the total amount of mechanical energy imparted to the oil in
Q3 during the process of raising its rotational speed to 100 rpm.
Q5 Water flows radially inwards between two plane, horizontal, circular
plates. At a radius of 300 mm the pressure is 0.5 bar absolute and the radial
speed is 3 m/s. Calculate the pressure at a radius of 100 mm.
Q6
Suppose
that, in the system in Q5, the water formed a free vortex as well as the inward
radial flow and the volume flow remained unchanged. If the radial and
tangential velocities are equal at every point in the flow calculate the new
velocities at 300 mm and 100 mm and, if the pressure at 300 mm is now1 bar
absolute, calculate the pressure at the radius of 100 mm.
Q7 An open-topped tank of 700 mm diameter is set up with
its axis vertical and filled with water to a depth of 800 mm. A stirrer made up
of two crossed blades as shown in the diagram is set up coaxially with the tank
and rotated at 60 rpm. This rotation of the stirrer will impart angular
momentum to the water and ultimately the water will rotate steadily with the
water inside the containing cylinder of the stirrer approximating to a forced
vortex and the rest of the water rotating as a free vortex. The stirrer has a
diameter of 75 mm.
The water that rotates as a free vortex will
have a depression that may be too small to detect by the unaided eye. Ignoring
the meniscus at the wall estimate the difference in level between the wall and
the free surface at the radius of the stirrer. Could you draw this profile
to scale?
Q8
We
have seen that a fee spiral vortex can be formed in cylindrical tank by giving
the water an initial rotation and then allowing the water to discharge through
a central hole in the bottom of the tank.
Suppose that such a system was rotating with a
fully developed, air entraining, vortex and the outflow was stopped. The
central core cannot now be maintained and the flow pattern will change to
produce a so-called compound vortex. Suppose that this is made up of a central
forced vortex surrounded by a free vortex. If the radius at which the two
vortices meet is show that the depth of the central
depression below the outer level of the surface is given by where is the angular velocity at radius .
Solutions
Q1 As no water is spilt the volume
of water in the can is unchanged and therefore the volume of air is unchanged.
All that has happened to the air is that its shape has changed from a cylinder
to a paraboloid of revolution. We shall need to be able to relate the volume of
the paraboloid to its height and diameter.
I have marked points 1 at the lowest
point of the paraboloid and on the axis, 2 at the upper edge of the tank
and 3 on the inside surface of the tank at the same level as 1. A
plane surface is shown dotted and a free vortex will exist in this surface.
For that vortex . In this case and . So :-
.
We now have an expression for the height of the
paraboloid of air and its diameter. We need an expression for the volume in
order to find .
Volume .
.
We have . Therefore:-
Volume or the volume equals one-half of
the containing cylinder. This means that = 0.2 m. Hence:- from which rpm.
Q2 In this case . Also from which . Speed of revolution =
Q3a Force on top is zero.
Force on bottom
3b When the tank and the oil are rotating at 100
rpm there will a forced vortex distribution of pressure over both the inside of
the lid and over the bottom and the pressure at the middle of the lid will be
zero. If we find the force on the lid the force on the bottom will greater by
666 N.
This requires an integration of the force on an
elemental ring of area forming part of the lid. If the elemental ring has a
radius its area is . For a forced vortex with zero pressure
at its centre the pressure head at radius . Therefore:-
Pressure at radius . radians/sec.
Pressure at radius .
From this the force on the elemental ring is and the total force on the inside
of the lid .
The force on the bottom is 790 N
Q4 During the starting phase an unknown
amount of energy is imparted to the internal energy of the oil but, once start
up is complete, the oil contains pressure energy and kinetic energy that it has
acquired. Two integrations will be required to calculate these two amounts.
If the volume of oil is taken to be made up of
many cylindrical shells the weight of a shell of radius and thickness where is the length of the shell. The
weight = which is Newton.
Kinetic energy/unit weight at radius J/N.
But the pressure energy per unit weight, , also equals = J/N and the total energy imparted
to the shell is = .
So the total energy stored is
Note The question implies that the oil is made to rotate by the
action of viscosity that starts at the solid surfaces of the tank and gradually
reaches a steady state. Energy will be lost in this process. However the oil
could be made to rotate by the fitting of radial blades inside the tank and
this would make the start-up much shorter and waste less energy. But suppose
that the tank were to be open topped as in Q1 and oil could be fed into the
bottom. Then the oil could be continuously lifted by this tank and its blades
and it would be the basis of a crude centrifugal pump. Its efficiency would be
less than 50%.
Q5 This is really a
convergence to be dealt with using the energy equation and continuity of flow.
The total energy at 300 mm radius is given by :-
At 100 mm the velocity will be and the kinetic energy per unit
weight will be . This is converging flow and if losses
are ignored, the total head is unchanged and the pressure head is . The pressure will be .
At 300 mm the radial velocity is
still 3 m/s and, as the tangential velocity equals the radial velocity, the
absolute velocuty will be . So the total head at the 300 mm radius
is
At 100 mm radius the radial
velocity of the water will again be 9 m/s so the absolute velocity will be . Then the pressure head will be . This corresponds to 0.28 bar.
Q7 This complicated
question is teling us the tangential velocity of the water at a radius of 37.5
mm. The tangential velocity is . (This is very small because 60 rpm is
very low. Wind powered generators turn at about 12 rpm.)
As this is a free vortex the
product of the radius and the tangential speed is constant. Therefore the
tangential speed of the water near to the wall is .
The difference in level will be or 2.7 mm. This is the difference
in level over a distance of 312.5 mm!
Q8 If the diameter of the
tank is large compared with the radius the kinetic energy head at will be given fairly accurately by
. This means that the surface level at will be lower that the level at
the wall by . The further drop in level to the
centre of the forced vortex will also be given by so the total depression will be .
The case of the flow
between paralel circular plates is clearly controlled flow that an engineer
might put to practical use. Most people will only ever see the uncontrolled
swirl that occurs in eddies and in
[1] This is consistent with real eddies where
there is always a dip towards the centre.
[2] I have a vehicle with an
unobstructed space in it that is about 3 metres long, 1.7 metres wide and about
0.8 metres high. Often after I have travelled only for 3 or 4 minutes I have to
turn left then right and the left again to get on to a main road and, on cold
mornings, I can feel the air in the car turning relative to the car. It
counter-rotates.
[3] It is normal to ignore
counter-rotation and proceed but this tendency to counter-rotation does not
disappear and for water, with its relatively low viscosity, it causes trouble
in centrifugal pumps that are designed for high efficiency and low running costs
rather than low first cost and high running costs.
[4] I used to have the tank filled
using a sprinkler system 24 hours before the lab session to give time for all
the internal motion to die away but the water still rotated from time to time.
[5] The mathematicians call it inviscid,
irrotational flow.