ENG 103
Fall 2011
P5.11
During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional
analysis to estimate the wave speed of an atomic bomb explosion.
He assumed that the blast
wave radius
R
was a function of energy released
E
, air density
ρ
, and time
t
.
Use dimensional
analysis to show how wave radius must vary with time.
Solution
:
The proposed function is
R
=
f
(
E,
ρ
,
t
).
There are four variables (
n
= 4) and three
primary dimensions (MLT, or
j
= 3), thus we expect
nj
= 43 = 1 pi group.
List the dimensions:
Assume arbitrary exponents and make the group dimensionless:
The single pi group is
P5.17
If you disturb a tank of length
L
and water depth
h
, the surface will oscillate back
and forth at frequency
Ω
, assumed here to depend also upon water density
ρ
and the acceleration
of gravity
g
.
(
a
) Rewrite this as a dimensionless function.
(
b
) If a tank of water sloshes at 2.0
Hz on earth, how fast would it oscillate on Mars (
g
≈
3.7 m/s
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 Fall '08
 Chattot
 Fluid Dynamics, atomic bomb explosion, Sir Geoffrey Taylor, appropriate dimensionless variables, British fluid dynamicist

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