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:

bob

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

    (FL. 3.3)

The drag coefficient, , measured in a wind tunnel using air at 20 degrees Celsius is 1.277. Thus is
=
= Ns2/m2
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:

    (9.11)
Replacing in terms of we have
    (9.12)

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:


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