Dimensional Analysis
• A tool to help one to get maximum information from
a minimum number of experiments
• facilitates a correlation of data with minimum number
of plots.
• Can establish the scaling laws between models and
prototype in testing.
Parameter Dimensions
Consider experimental studies of drag on a cylinder
Drag (F) depends upon
Flow Speed V, diameter d, viscosity m, density of fluid r
Just imagine how many experiments are needed to study
this phenomenon completely,
It may run into hundreds
A dimensional analysis indicates that Cd and Reynolds number,
Re or the Mach number M can determine the Cd behaviour
thus making it necessary to perform only a limited number of
experiments.
Buckingham Pi Theorem
Consider a phenomenon described by an equation like
g = g(q1, q2, q3, ………..,qn)
where q1, q2, q3, ………..,qn are the independent variables.
If m is the number of independent dimensions required to
specify the dimensions of all q1, q2, q3, ………..,qn then one can
come up with a relation like,
G(P1, P2, P3, P n-m) = 0
where P1, P2, P3, P n-m are non-dimensional parameters.
In other words the phenomenon can be described by
n-m number of non-dimensional parameters.
Important non-dimensional numbers in Fluid
Dynamics
•Reynolds Number
•Euler Number
•
or Pressure Coefficient
•Froude Number
•Mach Number
•Cavitation Number
•Weber Number
•Knudsen Number
Re
Cp
Fr
M
Ca
We
Kn
Reynolds Number, Re
Ratio of Inertial forces to Viscous forces.
Re 
r V 2 L2
m VL

r VL
m
Flow at low Reynolds numbers are laminar
Flows at large Reynolds numbers are usually turbulent
At low Reynolds numbers viscous effects are important
in a large region around a body.
At higher Reynolds numbers viscous effects are confined
to a thin region around the body.
Euler Number or Pressure Coefficient, Cp
Ratio of Pressure forces to Inertial Force
Cp 
p
1
2
rV 2
An important parameter in Aerodynamics
Cavitation Number
In cavitation studies, p(see formula for Cp) is taken
as p - pv where
p is the liquid pressure and pv is the liquid vapour pressure,
The Cavitation number is given by
Ca 
p  pv
1
2
rV 2
Froude Number
Square of Froude Number related to the ratio of
Inertial to Gravity forces.
Fr 
V
gL
Important when free surfaces effects are significant
Fr < 1
Fr > 1
Subcritical Flow
Supercritical Flow
Weber Number
Ratio of Inertia to Surface Tension forces.
We 
r V 2L

Where  is surface tension
Mach Number
Could be interpreted as the ratio of
Inertial to Compressibility forces
M
V
c
M 
2
r V 2 L2
2
Ev L
Where c is the local sonic speed, Ev is the
Bulk Modulus of Elasticity.
A significant parameter in Aerodynamics.
NOTE: For incompressible Flows, c =  and M = 0
Similitude and Model Studies
For a study on a model to relate to that on a prototype
it is required that there be
Geometrical Similarity
Kinematic Similarity
Dynamic Similarity
Geometrical Similarity
Physical dimensions of model and prototype be similar
Hp
Hm
Lm
Lp
Lp
Hp

Lm
Hm
Kinematic Similarity
Velocity vectors at corresponding locations on
the model and prototype are similar
up
vp
vm
up
vp

um
vm
um
Dynamic Similarity
Forces at corresponding locations on
model and prototype are similar
Ftp
Fnp
Fnm
Ft p
Fn p

Ftm
Fnm
Ftm
Problem in Wind Tunnel testing
While testing models in wind tunnels it is required that following
non-dimensional parameters be preserved.
Reynolds Number
Mach Number
Cd = f (Re, M)
But the available wind tunnels do not permit
both these numbers to be preserved.
Solution for Wind Tunnel testing
Remedy is offered by nature itself
At low speeds viscous effects are more important than
the compressibility effects. So only Reynolds number be
preserved.
Cd = f (Re)
At higher speeds compressibility effects are dominating.
So only Mach number need be preserved.
Cd = f (M)