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9/19/2016
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 
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 2
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
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 3
 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
 
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 4
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  
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 5
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
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 6
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

 
9/19/2016
ESO204A/IITK/mkdas@iitk.ac.in 7
 , , , 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

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Lec20 dimension1

  • 1. 9/19/2016 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 
  • 2. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 2 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
  • 3. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 3  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  
  • 4. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 4 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  
  • 5. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 5 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
  • 6. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 6 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   
  • 7. 9/19/2016 ESO204A/IITK/mkdas@iitk.ac.in 7  , , , 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