1. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Dimensional analysis
1
IIT Madras
Dr. Dhiman Chatterjee
Department of Mechanical Engineering
Indian Institute of Technology Madras

2. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Need to Non-dimensionalize variables
IIT Madras 2
It helps us to conduct experiments at laboratory scale (often called, model scale) to gather
information about the performance of a prototype before making a prototype.
It enables comparison of results obtained from experiments in one facility with that gathered
from a different facility. e.g. drag force (FD
) calculated for flow past an object
It reduces the number of experiments needed to gather sufficient information about any
phenomenon.
Non-dimensional form of governing equation helps in identifying variables/terms which are
more (or, less) important compared to other terms in the equation.

3. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 3
Principle of similarity
Geometric similarity:
√ X
Kinematic similarity:
U1m
U1p
V1m
V1p
W1m
W1p
Angles remain same

4. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 4
Principle of similarity (contd)
Dynamic similarity:
Types of forces Origin Expression
Force
Viscous force Fluid viscosity
Pressure force Pressure difference
Inertia force Fluid inertia
Capillary force Surface Tension
Gravity force Acceleration due to
gravity
Elastic force Compressibility
μ: dynamic viscosity
ρ: fluid density
σ: surface tension
K: bulk modulus of elasticity

5. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 5
Principle of similarity (contd)
Ratio of forces:

6. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 6
Principle of dimensional homogeneity
If an equation truly expresses a proper relationship between variables in a physical process,
it must be dimensionally homogeneous.
Buckingham’s Pi Theorem
Let us consider a physical process that satisfies the principle of dimensional homogeneity
and involves m dimensional variables. Then we can express this phenomenon/relationship
as:
Let n be the number of fundamental dimensions (like, mass, length, time, temperature)
involved in these m variables.
Then Buckingham’s Pi theorem states that the phenomenon can be described in terms of
(m-n) non-dimensional groups:

7. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 7
Buckingham’s Pi Theorem: example 1
Let us consider fully developed flow inside a pipe. Diameter of pipe is d and its length is Lp
.
Roughness of the pipe is εp
. Average velocity inside the pipe is V. Fluid properties required
are density (ρ) and viscosity (μ). We need to find out pressure drop (Δp).
Number of variables (m): 7 Number of fundamental dimensions (n): 3
Number of non-dimensional groups (m-n): 4
Let us choose ρ, V and d as repeating variables.

8. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 8
Buckingham’s Pi Theorem: example 1 (contd)

9. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 9
Buckingham’s Pi Theorem: example 1 (contd)
From Darcy-Weisbach relationship,

10. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 10
Buckingham’s Pi Theorem: example 2
Let us consider flow past an aerofoil at different angles of attack (α). Characteristic length
for an aerofoil is its chord length (Lc
). Freestream air velocity is V. Fluid properties required
are density (ρ) and viscosity (μ). Speed of sound in the medium is cs
. We need to find out
lift force experienced by the aerofoil (FL
).
Number of variables (m): 7 Number of fundamental dimensions (n): 3
Number of non-dimensional groups (m-n): 4
Let us choose ρ, V and Lc
as repeating variables.

11. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 11
Buckingham’s Pi Theorem: example 2 (contd)

12. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Application to incompressible flow turbomachines
• Important variables are: flowrate ( ), specific work (W) or head (H),
power (P), speed (N), diameter (D), fluid density (ρ) and viscosity (μ).
12
• Basic dimensions are M, L & T.
• So (7-3)= 4 non-dimensional groups are possible.
We select fluid property (ρ), kinematic variable (N) and geometrical variable
(D) and combine these with other variables.
IIT Madras

13. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Application to incompressible flow turbomachines
• Thus
13
• Similarly
IIT Madras

14. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
• Capacity coefficient/ Flow coefficient
14
• Energy coeff./head coeff./ Pressure coeff.
IIT Madras
Application to incompressible flow turbomachines

15. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
• Power Coefficient
15
N.B. For most of the fluid flow problems in turbomachines, Reynolds number
effect (within a given range of Re) is nominal.
• Reynolds Number
IIT Madras
Application to incompressible flow turbomachines

16. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Affinity Laws
• If the same machine works with the same fluid under different conditions, then D and ρ
are constant.
Note that these are NOT dimensionless numbers
Thus, the performance variables ( , P and W) of a given machine depend on the speed (N) of
a given turbomachine.
16
IIT Madras

17. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
• Thus we understand that for a given blade angle, the shape of the impeller is a function of
N, , and W.
Approach 2: Directly obtaining in a way similar to that of other non-dimensional
parameters.
•So, we can arrive at another non-dimensional number based on these.
Approach 1: By suitable combination of previously-obtained non-dimensional groups.
N in rpm, in m3
/s,
H in m.
N.B. n is in rev/s, in m3
/s, W in m2
/s2
17
IIT Madras
Shape Number

18. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Shape Number & Specific Speed
This non-dimensional group is called shape number (sometimes known as shape
parameter).
Specific speed of a pump: The specific speed (Nq
) of a pump is defined as the speed of a
geometrically similar pump having such dimensions that it delivers a volume flow-rate ( ) of
1 m3
/s while producing a head (H) of 1 m.
Specific speed of a turbine: The specific speed (Ns
) of a turbine is defined as the speed of a
geometrically similar turbine having such dimensions that it produces an output of 1 metric
horse power (HP) (~1 kW) when working under a head of 1 m.
Many times, specific speed is used in place of shape number because in hydroturbomachines
head (H), rather than specific work (W) is used.
Find the conversion factor between 1 mHP and 1 kW!
18
IIT Madras

19. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
Specific Speed
Pump
Turbine
N.B. Specific speed is NOT a dimensionless number!
N.B. N is in rev/min (rpm), in
m3
/s, H in m and PC
in metric HP
19
IIT Madras

20. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
• If the impeller speed is increased further, the diameter has to be decreased more and so the
impeller shape would change as shown below.
20
IIT Madras
Shape Number
(a) (c)
(b) (d) (e)

21. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 21
Summary of today’s discussion
We learnt about the need for non-dimensionalization.
Geometric, kinematic and dynamic similarities are required for any scaling.
Buckingham’s Pi Theorem states that if there are m dimensional variables involving n
fundamental dimensions, then it can be reduced to (m-n) non-dimensional groups.
Non-dimensionalization was carried out for incompressible flow turbomachines
Non-dimensionalization for incompressible flow turbomachines leads to the definition
of shape number which is another means of classification of turbomachines and leads
to change in shapes of impellers/rotating blades.

22. FLUID DYNAMICS AND TURBOMACHINES
Dr. Dhiman Chatterjee
PART-A Module-02 – Dimensional analysis
IIT Madras 22
THANK YOU