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Re: Bernoulli and viscosity



At 02:45 PM 12/11/02, you wrote:
The 2000-2001 Physics Olympiad preparation exam (mechanics) has a
question in which water comes out of a bullet hole in a water tower.

<http://helios.physics.utoronto.ca/~poptor/problems00/ps2_mechanics.p=
df>

The solution...

<http://helios.physics.utoronto.ca/~poptor/solutions00/pss2_mechanics=
.pdf>

is a little overly complicated but essentially it calculates the
difference in pressure (inside the tank at the level of the hole vs.
outside the tank at the level of the hole) is rho*g*h (where h is
the depth of the water; the difference in air pressure over height
h is ignored) and sets this equal to the change in kinetic energy
per volume, 1/2 rho*v^2. From this one gets the speed of the water
out the hole.

The problem is that this doesn't seem to work in real life. When
I set up my little demo, my speed is significantly less (i.e.,
my stream doesn't go as far as one would expect). I assume the
problem is due to viscous friction but I'm struggling to get an
estimate on what the actually v should be. I've checked
the PHYS-L archives and the web and could not find anything.

Any help would be appreciated.

____________________________________________
Robert Cohen; rcohen@po-box.esu.edu; 570-422-3428; http://www.esu.edu=
/~bbq
Physics, East Stroudsburg Univ., E. Stroudsburg, PA 18301


I took a look at one or two standard engineering references. Marks' asserts
quite confidently that the exit velocity in the narrowest part of the vena
contracta for a sharp edged round hole is 0.98 to 0.99 sqrt(2gh). This
occurs at a downstream point about one half hole diameter away from the hole.
This result is apparently rather old. It is sometimes called the Torricelli
value, as representing the free fall velocity to the center line of the
hole from the liquid surface (a distance h).
Engineers are usually more interested in flow rate, and given that the
waisting at the vena amounts to an area of about 0.62 of the exit hole area
A, they use a flow rate value of
Q = 0.61A.sqrt(2.g.h) This 0.61 factor is labeled the discharge
coefficient, the product of
of the contraction coeff and the velocity coeff for this kind of hole.
Hamilton Smith (-1885) using his own results, and those of Poncelet &
Lesbros (1827-1835) found a relation between the discharge coefficient for
water at ordinary low temperatures for a sharp edged round (and square)
hole which varied with water head and hole diameter. This varied between
0.65 for 1/4 inch holes at low heads, 8 inch up to 0.60 for 3 inch diameter
holes at heads of 1.5 ft up.
However, flow rate depends quite strongly on details of the hole shape,
smoothness, profile, wall thickness, and details such as greasiness (!) of
the exit.

This much does not appear particularly argumentative. However, deducing the
horizontal range given the exit velocity is probably the issue.
I suggest that knowing a reasonable value for the average exit velocity of
the liquid
(as offered above) is insufficent to give a complete measure of the
horizontal range, for given vertical displacements, where viscosity &
surface tension effects may play a part.
The URLs given in Bob's note no longer point to this question and answer,
by the way.





Brian Whatcott
Altus OK Eureka!