2.64.
CHAPTER 2, PROBLEM 64
235
2.64
Chapter 2, Problem 64
Problem:
Experimental studies are planned for a rotational axisymmetric flowfield.
The vorticity of the flow,
ω
, has dimensions
1
/T
and depends on initial circulation,
Γ
o
(dimensions
L
2
/T
), radius,
r
, time,
t
, and kinematic viscosity of the fluid,
ν
. How
many dimensionless groupings are there? What are the dimensionless groupings?
Solution:
The dimensional quantities and their dimensions are
[
ω
] =
1
T
,
[
Γ
o
] =
L
2
T
,
[
r
] =
L,
[
t
] =
T,
[
ν
] =
L
2
T
There are 5 dimensional quantities and 2 independent dimensions
(
L, T
)
, so that the
number of dimensionless groupings is 3.
The appropriate dimensional equation is
[
ω
] = [
Γ
o
]
a
1
[
r
]
a
2
[
t
]
a
3
[
ν
]
a
4
Substituting the dimensions for each quantity yields
T
−
1
=
L
2
a
1
T
−
a
1
L
a
2
T
a
3
L
2
a
4
T
−
a
4
=
L
2
a
1
+
a
2
+2
a
4
T
−
a
1
+
a
3
−
a
4
Thus, equating exponents, we arrive at the following two equations.
0
= 2
a
1
+
a
2
+ 2
a
4
−
1
=
−
a
1
+
a
3
−
a
4
We can solve immediately for
a
1
from the second equation, viz.,
a
1
= 1 +
a
3
−
a
4
Substituting into the first equation yields
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 Spring '06
 Phares
 Zagreb, Zagreb bypass, E. S. Taylor

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