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ES2B0 ‐ Fluid Mechanics
Model Answers to Example Questions (Set II)
Question 1: Dimensional Analysis
Use dimensional analysis to determine the period t for small oscillations of a simple
pendulum (Fig. 1) of length l . Assume that the period depends on the length of the
pendulum, the mass of the oscillating body and the gravitational acceleration of the Earth.
(N.B.: You will find that the correct answer to the question will imply that the period must in
fact be independent of the mass of the oscillating body.)
Figure 1: Sketch of Pendulum Geometry
SOLUTION
Quantities involved in the problem are:
QUANTITY UNITS DIMENSIONS
t: period of oscillation [s] T
l: length of pendulum [m] L
m: mass of oscillating body [kg] M
g: gravitational m = [ m s 2 ] L T 2
acceleration s 2
Assume:
t l m g
thus
, 2
T 1 L M L T 2
such that
L0 M 0 T 1 L M L T 2
and hence
L0 M 0 T 1 L M T 2
compare exponents to obtain:
From ‘ M ‘one gets : 0
1
From ’ T ‘ one gets : 2 1 and thus
2
1
From ’ L ’ one gets : 0 and thus
2
The final result is thus
1 1
t l 2 m0 g 2
such that
l
t const
g
The (non‐dimensional !) constant can in principle be determined from one single experiment
or one can obtain it from some other theoretical considerations. It turns out that it has a
value of const 2 6.28
N.B.: Dimensional analysis yields the result that the period of oscillation must be
independent of the mass!
Question 2: Dynamic Similarity
The flow around an airship with a diameter d 3 m and a length l 20 m needs to be
studied in a wind tunnel. The airspeed range to be investigated is at the docking end of its
range, a maximum of v p 2 ms‐1. Calculate the mean model wind tunnel speed if the model
is made to 1/10 scale. Assume the same air pressure and temperature for model and
prototype.
SOLUTION
Dynamic similarity requires that the Reynolds number of model and prototype need to be
the same. This gives
ES2B0 ‐ Fluid Mechanics
Model Answers to Example Questions (Set II)
Question 1: Dimensional Analysis
Use dimensional analysis to determine the period t for small oscillations of a simple
pendulum (Fig. 1) of length l . Assume that the period depends on the length of the
pendulum, the mass of the oscillating body and the gravitational acceleration of the Earth.
(N.B.: You will find that the correct answer to the question will imply that the period must in
fact be independent of the mass of the oscillating body.)
Figure 1: Sketch of Pendulum Geometry
SOLUTION
Quantities involved in the problem are:
QUANTITY UNITS DIMENSIONS
t: period of oscillation [s] T
l: length of pendulum [m] L
m: mass of oscillating body [kg] M
g: gravitational m = [ m s 2 ] L T 2
acceleration s 2
Assume:
t l m g
thus
, 2
T 1 L M L T 2
such that
L0 M 0 T 1 L M L T 2
and hence
L0 M 0 T 1 L M T 2
compare exponents to obtain:
From ‘ M ‘one gets : 0
1
From ’ T ‘ one gets : 2 1 and thus
2
1
From ’ L ’ one gets : 0 and thus
2
The final result is thus
1 1
t l 2 m0 g 2
such that
l
t const
g
The (non‐dimensional !) constant can in principle be determined from one single experiment
or one can obtain it from some other theoretical considerations. It turns out that it has a
value of const 2 6.28
N.B.: Dimensional analysis yields the result that the period of oscillation must be
independent of the mass!
Question 2: Dynamic Similarity
The flow around an airship with a diameter d 3 m and a length l 20 m needs to be
studied in a wind tunnel. The airspeed range to be investigated is at the docking end of its
range, a maximum of v p 2 ms‐1. Calculate the mean model wind tunnel speed if the model
is made to 1/10 scale. Assume the same air pressure and temperature for model and
prototype.
SOLUTION
Dynamic similarity requires that the Reynolds number of model and prototype need to be
the same. This gives
