Re: [Phys-L] Buoyancy Questions

[John Denker <jsd@av8n.com>]

On 07/01/2013 08:21 PM, Michael Barr wrote:

> I was at my sister's pool a few days ago and I noticed that when I
> took a ball (a pool volleyball) and submersed it just below the
> surface and released it, the ball would jump around 3 or 4 feel above
> the surface of the water.  When I pushed the ball deeper, about 2 or
> 3 feet below the surface and let go, the ball would rise and then
> sort of sputter at the surface and not rise above the surface at all.
> Does anyone know how to explain this.  I thought it would be a nice
> challenge for my students but the problem is, I don't think I know
> what's going on either.

This is a "challenge" all right.

Strictly speaking it is not a "buoyancy" question.  Buoyancy is a 
hydrostatic phenomenon, whereas here we are dealing with hydrodynamics.
This is relevant because of Denker's Law of Fluid Dynamics:

   Everything having to do with fluid dynamics turns 
   out to be N times trickier than you expected.

For most people, N is at least 10, if not more.  For some really smart
people who specialize in the field, N might be as small as 2, meaning
they get fooled only about half the time.

Let the aforementioned volleyball be experiment #1.  Just now I did 
some further experiments. 

#2:  Tennis ball.  Compared to a volleyball, this has a smaller size
 and a larger average density.  It will not jump several feet out of
 the water.  It will jump out, but just barely.  The top barely gets
 one diameter above the ambient water level.

#3:  Three liter pickle bottle.  This is very buoyant, and about the 
 same size as a volleyball, but more squared off, almost cubical.  I
 could not get it to jump out of the water at all.

#4:  Half liter water bottle.  This is very blunt on one end, with a
 rather long conical taper on the other end.
   4a) With the tapered end going first, it does not jump out at all.
    The top doesn't even get above the surface by half the overall
    length of the bottle.
   4b) With the blunt end going first, it jumps like crazy.  Starting
    it deeper (within reason) works better (very unlike the volleyball,
    for which deeper is worse).

Any theory that explains the volleyball had better explain these other
results.  See also #5 below.

At this point you may be wondering, how did I know in advance it would
be important to do these additional experiments?  Why did I not attempt
to answer the original question without additional data?  The answer
can be found in another of my favorite proverbs:

   Eight times burned, ninth time shy.

I've been fooled enough times to know that fluid dynamics is seriously 
tricky.

I don't know the answer to the original question, but I can say a few 
things that may serve as a starting point, to give you some feel for 
what's going on, and for where to look for the rest of the answer.

Start by thinking about the density.  The volleyball is much less dense
than the water.  This has the obvious static effect (buoyancy) but it
also affects the dynamics.  When the ball moves through the water, we
cannot ignore the water;  the water has to get out of the way somehow.
This means there is some kinetic energy /in the water/.  We are
accustomed to ignoring this kinetic energy when dealing with baseballs
or aircraft moving through the air ... but water is very much denser
than air, and we absolutely cannot ignore this dynamic effect.

One way of dealing with this involves some sophisticated physics.  It
involves changing the definition of what we mean by /particle/.  There
are two choices:
  a) For some purposes, we want to treat the volleyball as "the"
   particle.  We call this the _bare_ particle.
  b) For some purposes, it will be convenient to treat the volleyball
   /and the water it stirs up as it moves along/ as "the" particle.
   We call this the _dressed_ particle.

The idea of dressed states shows up all over the place in optical, atomic,
and solid-state physics.  It also shows up in elementary particle physics.
Note that nobody has the slightest idea what is the bare mass of the
electron.  The only thing we've ever been able to observe is the dressed
electron, including the electric field it stirs up as it moves along.  The 
term "renormalization" is another word for basically the same idea.

You can easily observe this by building an underwater upside-down pendulum
clock, with a buoyant pendulum-bob.  You can measure the force-versus angle
relationship, statically.  Then the period of the pendulum depends on the
mass.  If you don't use the dressed mass, you'll get a wildly wrong answer.

Next, we evaluate the Reynolds number.  That tells you the ratio of
inertial contributions to viscous contributions in the equation of motion.
For small Reynolds numbers, everything is overdamped.  Things are relatively 
simple.  Viscous drag dominates over everything else.  However, for the
volleyball in water, the Reynolds number is on the order of 10^5.  That's
big enough to guarantee that there will be turbulence and all sorts of
other stuff going on.

Next, we pull a rabbit out of a hat.  There is no good reason why this 
should work, but it does:  To a first approximation, we are going to
ignore both viscosity and turbulence.  Logic says that in one extreme
or the other, you should be able to ignore one or the other ... but
never both at the same time.  However, for streamlined objects, we
can get away with this, to a first approximation.  Feynman called this
"the flow of dry water".  The more conventional name is /potential/
flow.  There is also something called "inviscid" flow which may or may
not mean the same thing.

In this approximation, you can use everything you know about electric
potential contours and electric field lines to help visualize the
streamlines of the flow.  The flow around a sphere is isomorphic to
an electric dipole.

The history of fluid dynamics goes way back, before the Maxwell equations, 
and many decades before the electron was discovered or airplanes were 
invented or vectors were invented.  It's hard to imagine how these old 
guys like Stokes and Bessel could be so smart, but they were.

The Stokes flow around a sphere is symmetrical:  There is a stagnation
point at the front ("north pole") as you might expect, and also a
stagnation point at the back ("south pole").  In accordance with 
Bernoulli's principle, there is high pressure at these locations.
The relative pressure is +1 Q.  Meanwhile, around the equator of the
sphere, there is high velocity and low pressure.  The velocity field
and pressure field extends for quite a ways in every direction before
gradually dwindling to insignificance.

You can calculate this analytically for a sphere, but for almost any
other shape, such as a practical boat or torpedo or whatever, you 
would need CFD i.e. computational fluid dynamics.  Generally this is
based on FEA or FEM (finite-element analysis or finite-element modeling).
with a few additional tricks thrown in.

The fact that a streamlined object has high pressure in back (not just
in front) is the reason why streamlining is advantageous if you want
low drag.  A streamlined object has a lot of pressure on the front,
but this is counteracted by pressure on the back.  This is called
/pressure recovery/.  In contrast, if you're trying to build an
old-fashioned round parachute, you don't want pressure recovery,
so you choose something exceedingly non-streamlined such as a cup
with the open end forward.  A flat plate has tremendous pressure
drag i.e. almost no pressure recovery, and a forward-facing cup has 
even more drag.  The pressure in back is much closer to ambient
than to +1 Q.

This has to do with /laminar/ flow transitioning to /separated/
flow.  I don't pretend to understand the transition in fundamental
physics terms, and I strongly suspect nobody else does either.  On 
the other hand, an experienced fluid dynamicist will have a feel 
for what shapes are well streamlined and which are not.

  This stands in contrast to non-experts, who generally have no 
  clue whatsoever.  Most people think a car "looks fast" if it
  is pointy in front.  They think a minivan that is tapered in
  front and blunt on the back is reasonably well streamlined.
  This is pretty much nonsense.  Rockets and rifle-bullets and
  front-line jet fighters are pointy in front, but that's because
  they go /supersonic/.  In the subsonic regime, it works much
  better to be gently rounded in front and tapered in the back,
  like a minnow.

So now you know why I expected the half-liter water bottle to
jump better with the tapered end in the rear:  It was much more
streamlined.  I was cultivating pressure recovery.

A sphere is not very streamlined.  The quasi-taper on the back
is much to abrupt.  There is lots of separation, and therefore
lousy pressure recovery.  It's quite a bit better than a flat
plate or a parachute, but it's tenfold worse than a streamlined
body.
  http://www.engineeringtoolbox.com/drag-coefficient-d_627.html

=========

Returning to the idea of dressed states:  As the object crosses
the water/air boundary, it loses its dressing.

You may be wondering, does the object retain its momentum as it
crosses the boundary?  Does it retain its kinetic energy?

Simple high-school physics suggests that if we ignore the medium,
the particle should retain both its momentum and its energy.
However, this is obviously not possible.  You cannot change the
mass in such a way as to preserve both momentum and KE.  If
p is the same, then p^2/(2m) cannot be the same ... and vice
versa.  This should be enough to convince you that the simple
answer is not right, and the right answer is not simple.

The fact is, when the object pops out of the water, it makes a
mess.  It sheds loads of vorticity into the water.

One way you can kinda maybe sorta get some appreciation for what
is going on is as follows:  Let's suppose, hypothetically, that
we think the surface doesn't move very much as the object approaches
from below.  Subject to that hypothesis, we conclude that no water
molecules cross the plane of the surface.  This is a boundary
condition.  The easiest way to enforce that boundary condition
is by the method of images.  That is to say, the flow field 
below the water/air boundary is the same as we would get if we
replaced the air with water containing a _mirror image_ object,
mirrored in the plane of the erstwhile surface.

So, even subject to two approximations (Stokes flow and stationary
surface) things are getting complicated;  we have the Stokes flow
of /two/ objects, not just one.

There's probably a clever way to analyze this, perhaps as mutual
annihilation of two dipoles, but that requires more cleverness
than I've got right now.

Still, we can make some qualitative predictions.  There must be 
high pressure in the water, directly above the center of the 
rising object.  Therefore we are motivated to do one more
experiment:

#5:  Drill a 2cm hole in a modest-size piece of plywood.  Float the
 plywood on the surface of the water.  Let the object rise up and
 hit the wood, centered on the hole.  A geyser is produced.  This
 is more of a geyser than you could produce just holding the object
 in your hand and swatting the hole.  That's because there's more
 fluid involved in the steady-state flow field.

This illustrates one more thing that you really need to know if
you're going to think about such things:
 ++ The water is a /fluid/ with pressure and velocity everywhere.
 ++ Pressure is defined everywhere, not just at the surfaces 
  where the fluid hits something.
 ++ The water is a /fluid/, not a bunch of separate, non-interacting
  particles.  The fluid interacts with itself, quite strongly.  The
  pressure-field and velocity-field surrounding a moving object
  extends a long ways in every direction.

The technical term for the opposite of a fluid is /dust/.  If you
disturb one dust particle, the nearby dust particles are not affected.

=======================

Last but not least, almost everything I've said up to this point
is restricted to steady flow, or to slowly-changing almost-steady
flow.

In reality, if the object changes velocity, it takes some time for
the flow pattern to get established.  So the actual flow field
depends on history.  For this you need something like the 
Boussinesq–Basset equation
  http://en.wikipedia.org/wiki/Basset_force

Without this history-dependence, we would have no hope of ever
explaining the observed result that starting the volleyball deeper 
does not necessarily make it jump better.