Dynamic Similarity
Representation of the motion of any object falling in any fluid
We will show here using the concept of dynamic
similarity that the motion of any object falling in any fluid is
governed by a simple equation in which only one parameter, the
terminal velocity, , is needed to
distinguish one physical situation from another.
The bob in the experiment has the shape:
Based on the comparison of drag coefficients found here, the drag
coefficient, , is related to the
dimensionless drag coefficient by the equation
where |
= dimensionless drag coefficient
of the bob based on the area,
=
,
(m2)
= fluid
density, (kg/m3)
|
so that
The drag coefficient,
, measured in a wind tunnel using
air at 20 degrees Celsius is 1.277. Thus
is
A dimensionless (or similarity) plot of all geometrically similar bobs
falling in ANY fluid would plot as
vs. where
similar to
the plot for spheres shown
here.
Next we want to link the fall of the bob and the fall of the
skydiver or any other object. To do this we return to eqn. (9.4) and rewrite it as applies to the bob:
Replacing in terms of
we have
The motion of the bob and the skydiver both satisfy eqn. (9.12)
provided that the terminal velocity of each object is used. More
generally, the motion of any object falling in any fluid will satisfy
eqn. (9.12) if the terminal velocity of the particular object is
used.
To summarize, the motion of the bob, and the
skydiver with and without the parachute are all similar if the
terminal velocity is used as the parameter for dynamic
similarity. Differences in fluid density, viscosity, mass of the
object as well as the size and shape of the object are all taken into
account properly by one parameter:
Copyright
1998-1999
Rensselaer Polytechnic Institute. All Rights Reserved.