M&AE 3050
Problem Set 1
Fa 2009
Solution
1. Redo the dimensional analysis of the force acting on an object of length
`
in a uniform
steady ﬂow of a gas of density
ρ
,v
iscos
i
ty
μ
and free stream velocity
V
taking
com
pressibility efects
into consideration. Assume the force now depends upon
ρ
,
μ
,
`
and
a
, the speed of sound.
Solution
The development is similar to what was done in class, however we now include the
speed of sound,
a
, in the list of variables
f
(
F, ρ, μ, v, `, a
)=0
.
In this case
m
=6
,
k
=3and
n
−
3 = 3. Selecting
ρ
,
v
and
`
as the working
variables (as before), we obtain the same Π
1
and Π
2
groups. The last group will be
Π
3
=
aρ
α
v
β
`
γ
The units are
±
L
T
²±
M
L
3
²
α
±
L
T
²
β
[
L
]
γ
Forcing Π
3
to be dimensionless implies
M:
α
=0
,
T:
−
1
−
β
,
L: 1
−
3
α
+
β
+
γ
.
From this we see
∴
α
,
∴
β
=
−
1and
∴
γ
. WeseethatΠ
3
=
a/v
=
M

1
is
the inverse Mach number. Putting the result in a more historic form we have
F
ρv
2
`
2
=
f
³
ρv`
μ
,
v
a
´
.
2. A model of a harbor is made on the length ratio of 360:1. Storm waves of 2 m am
plitude and 8 m/s velocity occur on the breakwater of the full scale prototype harbor.
Signi±cant variables are the length scale, velocity and
g
, the acceleration of gravity.
The scaling of time can be made with the aid of the length scale and velocity scaling
factors.
(a) Neglecting friction (ﬂuid viscosity), what should be the size and speed of the
waves in the model?
(b) If the time between tides in the prototype is 12 hrs, what should be the tidal
period in the model?