ES330 Assignment 8 Solutions
Chapter 7
Due Date: Thursday October 28, 2010
II
Problem 2
A student team is to design a humanpowered submarine for a design competition.
The overall
length of the prototype submarine is
L
p
, and its student designers hope that it can travel fully
submerged through water at
V
p
. The design team builds a onetenth scale model to test in their
university’s wind tunnel. A shield surrounds the drag balance strut so that the aerodynamic drag
of the strut itself does not influence the measured drag. Write an expression for the air speed,
V
,
they need to run the wind tunnel in order to achieve similarity.
Solution
To achieve similarity, the Reynolds numbers from the wind tunnel (model) and prototype
must match. Therefore:
Re
m
=
Re
p
(1)
ρ
m
V
m
L
m
μ
m
=
ρ
p
V
p
L
p
μ
p
Solving for
V
m
(or
V
):
V
=
V
m
=
ρ
p
ρ
m
μ
m
μ
p
L
p
L
m
V
p
The length of the model tested in the wind tunnel is 1
/
10 the length of the prototype, so
L
m
= (1
/
10)
L
p
. Therefore,
L
p
/L
m
= 10 in the above equation.
V
= 10
V
p
ρ
p
ρ
m
μ
m
μ
p
We could also further reduce the problem by using values of
ρ
and
μ
for air and water.
Assuming that the air in the tunnel and the water used for the prototype are both at about
20
o
C
and atmospheric pressure, we can find the density and viscosity for both fluids.
ρ
air
=
ρ
m
= 1
.
204
kg/m
3
ρ
water
=
ρ
p
= 998
kg/m
3
μ
air
=
μ
m
= 1
.
825x10

5
kg/
(
m
*
s
)
μ
water
=
μ
p
= 1
.
002x10

3
kg/
(
m
*
s
)
1
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We can now solve for
V
:
V
= (10
V
P
)
998
kg
m
3
1
.
204
kg
m
3
1
.
825x10

5
kg
m
*
s
1
.
002x10

3
kg
m
*
s
V
= 151
V
p
III
Problem 3
The students from the previous problem measure the aerodynamic drag on their model submarine
in the wind tunnel. They are careful to run the wind tunnel at conditions that ensure similarity
with the prototype submarine. Their measured drag force is
D
m
. Write an expression for the drag
force on the prototype submarine from Problem 2.
Solution
To solve the problem, we can match:
F
d,m
ρ
m
V
2
m
L
2
m
=
F
d,p
ρ
p
V
2
p
L
2
p
(2)
Solving for the drag force on the prototype, or
F
D,p
:
F
D,p
=
ρ
p
ρ
m
L
p
L
m
2
V
p
V
m
2
F
D,m
We know
V
m
and
L
p
/L
m
= 10 from Problem 2, and
F
D,m
is given as
D
m
in the problem
statement, so:
F
D,p
=
ρ
p
ρ
m
(10)
2
V
p
10
V
p
ρ
p
ρ
m
μ
m
μ
p
!
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 Fall '10
 Bohl
 Aerodynamics, Wind Tunnel

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