1) The document discusses dimensional analysis and similitude, which are qualitative techniques used in many fields of science and engineering. 2) Dimensional analysis involves identifying the fundamental dimensions involved in a physical phenomenon and representing the phenomenon in terms of dimensionless parameters. This allows experiments to determine the nature of relationships. 3) An example of dimensional analysis of flow velocity V in terms of gravitational acceleration g, height h, and kinematic viscosity ν is shown. Dimensionless groups like the Froude and Reynolds numbers are derived.

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ESO204A/IITK/mkdas@iitk.ac.in 1
Dimensional Analysis and
Similitude
ESO204A, Fluid Mechanics and rate Processes
Chapter 5 of F M White
Chapter 7 of Fox McDonald
Simple and powerful qualitative technique applicable to
many fields of science and engineering
c
a b

A
B C
and c
Right‐angled triangle ABC is
uniquely determined by
Area of triangle
ABC:  1 ,cA f c
Rearranging:
 2 ,cf A c 
dimensionless
L2
L
3 2
cA
f
c

 
   
 
 2
cA
f
c
 
 2
cA c f  
2 2 2
c a b 

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c
a b

2 2 2
c a b 

:smaller of two acute angles
A
B CD
Area of triangle
ABC:  2
cA c f 
Similarly area
of triangle ABD:
 2
aA a f 
Similarly area
of triangle ACD:
 2
bA b f  c a bA A A 
     2 2 2
c f a f b f    2 2 2
c a b  
The theorem is proved using dimension‐related
arguments only
For quasi‐steady,
incompressible,
frictionless flow, we have
derived (Bernoulli Eq.)
2V gh
Alternate approach
Experiment (or thought experiment) suggests
 1 ,V f g h  2V f u 
where is made of and
and has the same dimension as that of
u g h
V
m n
u g h
u gh

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 2V f gh
m n
u g h    1 2 m n
LT LT L 

L
LT-2
2m n m
L T 

LT-1
1; 2 1m n m    
1 2m n u gh
Think about the nature of f2
; is a constantV c gh c
The constant can be
evaluated from experiments
; is nondiemsnional
V
c
gh
  
The solution may
also be written as
Summary
  0; is nondimensional  
Where V, g, and h are
dimensional, as shown before
The technique described above is known as
Dimensional Analysis
 , , 0f V g h Given
The above system is equivalent to
V
gh
 

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Dimensional Analysis
If certain physical phenomenon is governed by
 1 2, ,.... 0nf x x x 
 
where some/all of the
variables are dimensionalx
Then the above phenomena can be represented as
 1 2, ,.... 0m      where all the variables
are non-dimensional

m n
The nature of f and ψ may be obtained from
experiments
Dimensional Analysis: Buckingham Pi Theorem
 1 2, ,.... 0nf x x x 
where some/all
are dimensional
x
 1 2, ,.... 0m    
where all are
non-dimensional

where ,m n m n k  
Minimum number of fundamental dimensions
involved: k
 , , 0f V g h Example:
3n  2k  1m n k  

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Experiment shows
Above Equation suggests
 , , 0f V g h 
a b c
V g h 
  0  
where is made of , and and
is dimensionless
V g h

     0 0 1 2a b c
L T LT LT L 

2a b c a b
L T   

2 0a b c a b      2b c a   
a
V
gh

 
   
 
Any arbitrary value of a
should be ok 0
V
gh

 
  
 
Pi Theorem: Repeating and non‐repeating variables
 1 2, ,.... nx x x  1 2 1 2, ,.... ; , ,....r r rk nr nr nrmx x x x x x
       11 12 13 1
1 1 1 2 3 ... ka a a a
nr r r r rkx x x x x 
Construction of Pi‐terms
       21 22 23 2
2 2 1 2 3 ... ka a a a
nr r r r rkx x x x x 
       1 2 3
1 2 3 ...m m m mka a a a
m nrm r r r rkx x x x x 
.....
.....
Selection of repeating
variables:
o They must be
dimensionally
independent
o Together, they
must include all
fundamental
dimensions

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Experiment shows, for
viscous flow  , , , 0f V g h  
2k 
We have to select two
(02) repeating variables
Let’s take the repeating variables: ,g h
2m 4n 
Non‐repeating variables: ,V 
M L T
V 0 1 ‐1
g 0 1 ‐2
h 0 1 0
ν 0 2 ‐1
 , , , 0f V g h  L T
V 1 ‐1
g 1 ‐2
h 1 0
ν 2 ‐1
2k 
Repeating variables: ,g h
2m 4n 
Non‐repeating variables: ,V 
       11 12 13 1
1 1 1 2 3 ... ka a a a
nr r r r rkx x x x x 
   1
a b
V g h     0 0 1 2 a b
L T LT LT L 
  1 1 2a b a
L T   

2 0 1 2a b a     
1 2a b    1
V
gh
 
   2similarly
a b
g h     0 0 2 1 2 a b
L T L T LT L 
 
1 2, 3 2a b     2 3
gh

 

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 , , , 0f V g h   1 3
, 0
V
f
gh gh
 
  
 
 
Fr
V
gh

Fr
Re
Froude number 3
V
Vhghgh
 

We may also write  2 Fr,Fr Re 0f   Fr Fr Re
Frictionless flow: Fr constant
Viscous flow: Experiments are
necessary to find the
nature of function
 Fr Fr Re