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 , density of fluid

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 likeg = g(q1, q2, q3, ………..,qn) where q1, q2, q3, ………..,qn are the independent variables.

If m is the number of independent dimensions

required tospecify the dimensions of all q1, q2, q3, ………..,qn then one can come up with a relation like,

G(1, 2, 3, n-m) = 0

where 1, 2, 3, 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 Re •Euler Number • or Pressure Coefficient Cp•Froude Number Fr•Mach Number M•Cavitation Number Ca•Weber Number We•Knudsen Number Kn

Reynolds Number, Re

Ratio of Inertial forces to Viscous forces.

VL

VL

LV

22

Re

Flow at low Reynolds numbers are laminarFlows at large Reynolds numbers are usually turbulentAt 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

221 V

pCp

An important parameter in Aerodynamics

Cavitation Number

221 V

ppCa v

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

Froude Number

Square of Froude Number related to the ratio of Inertial to Gravity forces.

gL

VFr

Important when free surfaces effects are significant

Fr < 1 Subcritical FlowFr > 1 Supercritical Flow

Weber Number

Ratio of Inertia to Surface Tension forces.

LV

We2

Where is surface tension

Mach Number

Could be interpreted as the ratio of Inertial to Compressibility forces

c

VM

Where c is the local sonic speed, Ev is the Bulk Modulus of Elasticity.

2

222

LE

LVM

v

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 prototypeit is required that there be

Geometrical SimilarityKinematic SimilarityDynamic Similarity

Geometrical Similarity

Physical dimensions of model and prototype be similar

Lp

Hp Hm

Lm

Hm

Lm

Hp

Lp

Kinematic Similarity

Velocity vectors at corresponding locations on the model and prototype are similar

vp

up vm

um

m

m

p

p

v

u

v

u

Dynamic Similarity

Forces at corresponding locations on model and prototype are similar

Fnp

Ftp Fnm

Ftm

m

m

p

p

Fn

Ft

Fn

Ft

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

But the available wind tunnels do not permit both these numbers to be preserved.

Cd = f (Re, M)

Solution for Wind Tunnel testing

Cd = f (M)

Remedy is offered by nature itself

At low speeds viscous effects are more important thanthe 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.