Flow at Different Reynolds Numbers Video Lecture | Fluid Mechanics for Mechanical Engineering

in one of the experiments at Cambridge
the students dropped spheres with
different densities and diameters into
tanks of oil with different viscosities
they then measure the terminal velocity
of the spheres and this allows the drag
coefficient of the spheres to be
calculated one of these spheres drops
very slowly
and one of these oils is very viscous in
this situation the flow creeps around
the sphere and the Reynolds number which
is the ratio of inertial to viscous
forces is very very small this scenario
in which the viscous forces dominate is
known as creeping flow this picture
which is taken from Van Dyke's album of
fluid motion shows the path lines in
creeping flow now it looks a little bit
like those that you get for inviscid
flow but it's important to point out
that the velocity profile is very
different in creeping flow the flow is
like one big boundary layer for example
at this point and at this point around
the sphere the azimuthal velocity V
theta as a function of the radius R
measured from the center of the sphere
looks like this there is zero velocity
on the surface of the sphere and then
the flow tends towards the free stream
velocity V as R tends to infinity at
very low Reynolds number the inertial
forces are negligible compared with the
viscous forces if we take the
navier-stokes equation which is F equals
MA written for a viscous fluid I'll
write as a equals 1 upon M times F the
acceleration is the material derivative
of the velocity field DV by DT that can
be written as partial DV by DT plus V
dot grad V and then the forces per unit
volume come in two forms one is the
pressure forces and the other is this
the viscous forces if the inertial
forces are very small then this equation
can be arranged into grad P is
approximately equal to MU del squared V
and intriguingly this equation can be
solved analytically for a sphere this is
known as Stokes flow and the radial
velocity is given by this expression and
the azimuthal velocity is given by this
expression one can then integrate the
shear stress around the surface of the
sphere to find the viscous drag on the
sphere and it has a very simple
expression 6 PI mu R V where R is
radius of the sphere if we then work out
the drag coefficient using PI R squared
the cross-sectional area of the sphere
at his widest point as the
characteristic area then we find that
the drag coefficient is simply 24
divided by the Reynolds number where the
Reynolds number is defined as Rho V D
the diameter of the sphere divided by mu
the viscosity and flows at low Reynolds
number seem very strange to us because
we very rarely see them from day-to-day
a particularly curious effect is known
as kinematic reversibility which is that
if an object does a symmetric motion
such as the flapping of the tail of a
fish it will simply return to its
original position once it has completed
one cycle in order to move in a very
viscous fluid one has to break some
symmetry for example the rotating
corkscrew can rotate its way through a
viscous fluid and the symmetry that's
broken is that the corkscrew has to wind
either one way or the other
but that's not the topic of this chapter
what I want to do now is consider flows
at a slightly higher Reynolds number
than creeping flow I shall call this
fairly low Reynolds number by which I
mean the flow around the sphere at a
Reynolds number of around 100 at this
Reynolds number the boundary layers
separate on the picture you can see flow
separation here and here and on my
diagram there shown here now the
pressure along the back of this object
is roughly the same as the pressure at
the separation point and this is lower
than the corresponding pressure at the
front of the sphere and this gives a net
force on the body from left to right in
the direction of the flow this is known
as form drag and if we plot the drag
coefficient as a function of the
Reynolds number for the flow around a
sphere and then we plot the line 24
divided by the Reynolds number which is
the contribution of viscous drag then we
say that the total drag is greater than
24 divided by Reynolds number and the
difference between the two is the
contribution of the form drag and note
that the vertical axis is a log axis on
the Left we see therefore that viscous
drag dominates at low Reynolds number
while at higher Reynolds number we see
that form drag dominates because the
viscous drag becomes very small as
Reynolds number increases the form drag
increases relative to the skin friction
beyond Reynolds number of around 1,000
the skin friction is negligible between
a Reynolds number of one thousand and
two hundred thousand the point of
separation remains very close to the
equator we can see this on the
instantaneous image on the left at a
Reynolds number of 15,000 but perhaps
it's more player in the time averaged
image on the right we're here we can see
the separation point here and the
recirculation zone behind the sphere and
because the position of the separation
point doesn't change much with Reynolds
number and because the pressure
difference between one side and the
other Delta P scales with a half Rho V
squared where V is the speed the V
incoming flow the drag coefficient in
this range of Reynolds number stays
approximately constant at around naught
point four and if we go back a page you
can see this region around here but you
can also see that something rather
curious happens around here the drag
coefficient suddenly drops for a smooth
sphere this is at a Reynolds number of
around 200 thousand and what's happening
here is that the boundary layer becomes
turbulent upstream of the equator this
increases the rate of stream wise
momentum transfer from the free stream
and this causes the boundary layer to
remain attached for longer such that the
separation point is quite far back
towards the rear of a sphere and this is
seen perhaps more easily on the time
averaged image on the right in which one
can see the separation point and the
much smaller region of recirculating
flow and this is beneficial in two ways
firstly there is some pressure recovery
in this region here by which I mean that
the pressure at the separation point is
greater than the pressure at the equator
so the pressure along the back face of
the sphere is larger than it was when
the separation point was at the equator
and secondly this lower pressure acts
over a smaller region of the sphere so
not only is the pressure drop between
the front and back of the sphere not as
big but it also acts over a smaller area
and these two effects can
to greatly reduce the form drag if we go
back to the diagram of drag coefficient
versus Reynolds number you see that the
drop in form drag just here is quite
large and it's worth noting on this
figure that the drop in CD due to the
form drag is due to the turbulent
boundaries layers remaining attached for
longer but what happens later at high
Reynolds number while one finds that
this drag goes back to around not point
four as even the extra momentum transfer
due to the turbulent boundary layers is
no longer able to keep the flow attached
around the back of the sphere and one
returns to the case in which the
separation point is at the equator of
the sphere