Dimensional Analysis

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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.

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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.
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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.
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Important non-dimensional numbers in Fluid
Dynamics

• Reynolds Number Re
• Euler Number
• or Pressure Coefficient Cp
• Froude Number Fr
• Mach Number M
• Cavitation Number Ca
• Weber Number We
• Knudsen Number Kn

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Reynolds Number, Re
Ratio of Inertial forces to Viscous forces.
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.
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Euler Number or Pressure Coefficient, Cp
Ratio of Pressure forces to Inertial Force
An important parameter in Aerodynamics
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Cavitation Number
In cavitation studies, Dp(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
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Froude Number
Square of Froude Number related to the ratio of
Inertial to Gravity forces.
Important when free surfaces effects are
significant
Fr lt 1 Subcritical Flow Fr gt 1 Supercritical Flow
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Weber Number
Ratio of Inertia to Surface Tension forces.
Where s is surface tension
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Mach Number
Could be interpreted as the ratio of Inertial to
Compressibility forces
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
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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
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Geometrical Similarity
Physical dimensions of model and prototype be
similar
Hp
Hm
Lp
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Kinematic Similarity
Velocity vectors at corresponding locations on
the model and prototype are similar
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Dynamic Similarity
Forces at corresponding locations on model and
prototype are similar
Ftm
Fnm
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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.
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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)