6.a Describe what is meant by the term dimensional analysis. Your explanation should include the meanings and relevance of the terms geometric similarity, dynamic similarity and kinematic similarity as well as identifying some uses form the technique.
(8 marks)
6.b Assuming the drag force, F, exerted on a body is a function of the following
fluid density r
fluid viscosity m
diameter d
velocity u
Show that the drag force can be expressed as
where f is some unknown function and Re is the Reynolds number.
(10 marks)
6.c A prototype boat propeller has a diameter of 1.0m. It is necessary to determine the force it will experience when water flows past at 5 m/s. A model propeller is available of diameter 0.1m and can be placed in a wind tunnel. To obtain the dynamically similar conditions at what velocity would the air need to flow in the wind tunnel?
(7 marks)
m water = 1.0 ´ 106 kg/ms m air = 1.7 ´ 105 kg/ms
r water = 1000 kg/m3 r air = 12.5 kg/m3
6.a.
Dimensional analysis is used when constructing physical models of prototype structures. Physical models are used when the fluid flow is particularly complex and difficult to analyse by other means. It enables physical measurements - forces, velocities etc. - taken from the scale models to be converted to the equivalent measurement which would be found on a prototype.
The term similarity relates to physical a scale models.
Geometric similarity - all dimensions are in the in the same ratio.
Dynamic similarity - all velocities are in the same ratio - requires geometric similarity
Kinematic similarity - all forces are in the same ration - requires dynamic similarity.
6.b.
Assume the governing variables r , u, d
According to Buckingham's p theorem there are n-m groups
where
n = number of variables (5) and
m = number of dimensions (i.e. MLT, giving 3)
n-m = 5-3 = 2 groups
Dimensions of the variables are:
r = density (kg/m3) = ML-3
m = viscosity (kg/m/s) = ML-1 T-1
u = velocity (m/s) = ML-1
d = length (m) = L
F = newtons (kg m /s2) = MLT-2
For p 1
For p 2
Inverting p 1 gives 
Rearranging this gives 
6.c
For dynamic similarity the Reynolds numbers are equal for both the water and air situation
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