3.2. REQUIREMENTS OF SIMILITUDE
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Many scale ratios cannot be chosen independently, but are instead a
derived result of other selected scales. For instance, the scale ratio for area
in a model directly depends on the length scale because area has units of
length squared. In terms of scale ratios this is represented as
Na =
T 2
LP
Lm2
-t-'m
Likewise, flow velocity has dimensions of length divided by time, so the
scale for velocity can be derived as
This example illustrates that scale ratios can be expressed as products of
other scale ratios as determined from the dimensional units of the variable
in question.
Example 3.1. Similarity by Differential Equation
where
The partial differential equation for a uniform vibrating string under constant
tension is given as:
d2ri _ (_R_\ d^L
dt2
y p A J dx2
where
7]
-
string displacement [77 = (z, t)]
t
- time
x
— coordinate parallel to the motionless string
R
- string tension
p
- material density
A
- string cross-sectional area
We can nondimensionalize the vibrating string equation by defining the nondimensional variables
- _ V
- _ 1 . ; _ A
T,~ Z’
X’
T