Engineering Fluid Mechanics week 12 .pdf

Fluid Mechanics

1. ENGG252 ENGINEERING FLUID
MECHANICS
DIMENSIONAL ANALYSIS AND MODELLING
REF: CENGELAND CIMBALA, 2020 CH: 7)
1

2. OBJECTIVES FOR THIS WEEK
After this week’s classes, you should be able to:
• Understand dimensions, units and dimensional homogeneity
• Understand benefits of dimensional analysis
• Know how to use the method of repeating variables and determine
dimensionless parameters
• Understand the concept of similarity and how to apply it to ex perimental
modelling
2

3. A 1:46.6 scale model
of an Arleigh Burke
class U.S. Navy fleet
destroyer being tested
in the 100-m long
towing tank at the
University of Iowa. The
model is 3.048 m long.
In tests like this, the
Froude number is the
most important
non-dimensional
parameter.
3

4. PRIMARY DIMENSIONS
There are seven primary dimensions(also called fundamental or basic
dimensions): mass, length, time, temperature, electric current, amount of
light, and amount of matter.
All non-primary dimensions can be formed by some combination of the
seven primary dimensions
Dimension Symbol SI unit
MassM kg (kilogram)
LengthLm (metre)
Timet s (second)
TemperatureT K (Kelvin)
Electric currentI A (Ampere)
Amount of lightC cd (candela)
Amount of matterN mol (mole)
Focus on these 3
4

5. DIMENSIONAL HOMOGENEITY
Ex 12-1: Check the dimensional homogeneity of the Bernoulli Equation
constant) (
2
2
H z
g
V
g
P
  

The law of dimensional homogeneity:
Every additive term in an equation must have the same dimension s.
Analysis: Each term is written in terms of primary dimensions.
All three additive terms have the same dimensions
The primary dimension of the Bernoulli constant, H (in the abov e form) is [L]
You can’t add apples and oranges!
5

6. NON-DIMENSIONALISATIONOF EQUATIONS
•Non-dimensional equation:if we divide each term in the equation by a collection
of variables and constants whose product has those same dimensi ons, the
equation is rendered non-dimensional.
•Can Bernoulli's equation be non-dimensionalised?
•Each term in a non-dimensional equation is dimensionless.
• Why is this useful?
It allows us to remove the units from an equation involving phy sical quantities.
Then no matter what size the problem is, comparisons can be made.
2
2
PV
g
zC
PPPP

 
 
2
(constant)
2
V
PgzC



Dimensional Non-dimensional
What is P
∞
?… it is a reference pressure
6

7. NON-DIMENSIONALISATIONOF EQUATIONS
•Normalised equation:if the non-dimensional terms in the equation are of
order unity, the equation is called normalised.
•Non-dimensional parameters:in the process of non-dimensionalising an
equation of motion, non-dimensional parameters often appear – mo st of
which are named after a notable scientist or engineer (e.g. the Reynolds
number and the Froude number).
7

8. Object falling in a vacuum.
Vertical velocity is drawn
positively, so w <0 for a
falling object.
Dimensional variables:Dimensional quantities that change or vary
in the problem. Examples: z (dimension of length) and t (dimension
of time).
Nondimensional(or dimensionless) variables: Quantities that
change or vary in the problem, but have no dimensions. Example:
Angle of rotation, measured in degrees or radians, dimensionless
units.
Dimensional constant:Gravitational constant g, while dimensional,
remains constant.
Parameters:Refer to the combined set of dimensional variables,
non-dimensional variables, and dimensional constants in the
problem.
Pure constants: The constant 1/2 and the exponent 2 in equation.
Other common examples of pure constants are

and e.
Equation of motion:
2
2
dz
g
dt


Dimensional result:
2
0 0
2
1
gt tw zz  
Initial location = z
0
Initial velocity = w
0
Eq: 7-4 from text
8

9. To non-dimensionalise an equation, we need to select scaling parameters, based on the primary
dimensions contained in the original equation.
First, list the Primary dimensions of all parameters:
Only two primary dimensions are used, Land t, so two scaling parameters are needed (from
dimensional constants). Choose w
0
and z
0
(simplest unit options)
Second, use the two scaling parameters to non-dimensionalisezand tand convert them into the
non-dimensional variables z*and t*. This is done by inspection.
0
00
*;*
wt z
zt
zz

Eq: 7-6 from text
      
2 1 1 1 1 1 1
0 0
 
tL L tL t L
g z w t z
z = f(t, w
0
, z
0
, g)
Dimensional
variables
Dimensional
constants
Perform a unit check … both dimensionless
9

10. Rearrange and substitute Eq: 7-6 back into Eq: 7-4
The grouping of dimensional constants above (in blue) is the square of a very well known
dimensionless parameter, the Froude Number
22 222
00
222 2
00 0
2
0
0
(* )**
1
(*) * *
w dzz w dz dz dz
g
dt d z t w z dt dgzt


0
0
w
Fr
gz

2
2
1
*1 * *
2
zt t
Fr
 
2
22
*1
*
dz
dt Fr

Non-dimensionlised equation of motion:
Non-dimensional result:
The two key advantages of non-dimensionalisation of an equation
Eq: 7-8 from text
Eq: 7-10 from text
z
*
= f(t
*
, Fr)
Eq: 7-7 from text
Eq: 7-9 from text
Non-dimensionalised and normalised
10

11. In a general unsteady fluid flow p roblem with a free surface, t he scaling parameters include a
characteristic length L, a characteristic velocity V, a characteristic frequency f, and a reference pressure
difference P
0
P

. Non-dimensionalisation of the di fferential equations of fluid flow produces four
dimensionless parameters: the Reynolds number, Froude number, Strouhal number, and Euler number
11

12. DIMENSIONAL ANALYSIS AND SIMILARITY
• In most experiments, to save time and money, tests are perform ed on a geometrically
scaled model, rather than on the full-scale prototype
• In such cases, care must be taken to properly scale the result s
• Here we introduce a powerful technique called dimensional analysis
The three primary purposes of dimensional analysis are to:
• Generate non-dimensional parameters that help in the
• design of experiments (physical and/or numerical) and
• reporting of those experimental results
• Obtain scaling laws so that prototype performance can be predi cted from model
performance
• (Sometimes) predict trends in the relationship between paramet ers
12

13. The principle of similarity:
Three necessary conditions for complete similarity between a model and a prototype.
(1) Geometric similarity—the model must be the sameshape as the prototype, but may be scaled
by some constant scale factor.
(2) Kinematic similarity—the velocityat any point in the model flo w must be proportional (by a
constant scale factor) to the velocity at the corresponding poi nt in the prototype flow.
(3) Dynamic similarity—When all forcesin the model flow scale by a constant factor to
corresponding forces in the prototype flow ( force-scale equivalence).
DIMENSIONAL ANALYSIS AND SIMILARITY
13

14. In a general flow field, complete similarity between a model and prototype is
achieved only when there is geometric, kinematic, and dynamic similarity.
Kinematic similarity is achieved when, at
all locations, the speedin the model flow
is proportional to that at corresponding
locations in the prototype flow, and points
in the same direction.
Dynamic similarity is achieved when, at all
locations, the forcesin the model flow are
proportional to those at corresponding
locations in the prototype flow, and point
in the same direction.
14

15. DIMENSIONAL ANALYSIS -REPRESENTATION
• Let uppercase Greek letter Pi ( ) denote a non-dimensional parameter
• In general, in dimensional analysis problem, there is one that we call the
dependent , giving it the notation 
1
• The parameter 
1
is in general a function of several other ’s, which we call
independent ’s (e.g.: 
2,

3, …..

k)
• The functional relationship between ’s is:
• Complete similarity is ensured if the model and prototype are g eometrically
similar and all independent

groups must match between model and prototype.
• To achieve complete similarity,
) .......... , , (
3 2 1k
f





p m
pk mk p m p m
,1 ,1
, , ,3 ,3 ,2 ,2
then
.....; ; If
 





 

Eq: 7-12 from text
15

16. CONCEPT OF DIMENSIONAL ANALYSIS
e.g. Design of a new sports car
• Testing of aerodynamics in a wind tunnel
• Drag force is F
D
= f (V, , µ, L)
• Through dimensional analysis, if the flow
is treated as incompressible then we can
reduce the problem to 2 ’s:
) (
2 1

f
D
D
C
LV
F
 
2 2 1

Re
2
  


LV
Eq: 7-13
from text
Geometric similarity between a prototype car
of length L
p
and a model car of length L
m
.
But how do we determine the groups??
V- Velocity; -fluid density; -fluid viscosity; L- characteristic length;
C
D
– Drag coefficient and Re= Reynolds number
16

17. METHOD OF REPEATING VARIABLES
(AND BUCKINGHAM PI THEOREM)
Lets learn how to generate the
non-dimensional parameters
i.e.

’s
There are several methods and
we will use the most popular and
simplest which is the Method of
Repeating Variablesafter
E. Buckingham (1867–1940)
The Method of Repeating Variables
Step 1:List the parameters in the problem and
count their total number n.
Step 2:List the primary dimensions of each of
the n parameters.
Step 3:Set the reduction j as the number of
primary dimensions.
Calculate k, the expected number of ’s
according to Buckingham Pi theorem, k = n –j
Step 4:Choose j repeating parameters.
Step 5: Construct the k number of ’s, and
manipulate as necessary. (Note: I will provide
here a simplified step compared to textbook)
Step 6:Write the final functional relationship and
check your algebra.
A concise summary of the six steps that
comprise the method of repeating variables.
Generally the most
mysterious/confusing
part for students
TABLE 7-3 in text has
Important guidelines
This procedure is best lear nt by examples and practice
17

18. EXAMPLE (PART 1)
Design of a new sports car
• Testing of aerodynamics in a wind tunnel
• Drag force is F
D
= f (V, , µ, L)
• Find the PI groups
18

19. GUIDELINES FOR CHOOSING REPEATING PARAMETERSIN
STEP 4
1) Never pick the dependent variable. Otherwise, it may appear i n all the

’s.
2) Chosen repeating parameters must not by themselves be able to form a
dimensionless group.
3) Chosen repeating parameters must represent all the primary di mensions.
4) Never pick parameters that are already dimensionless.
5) Never pick two parameters with the same dimensions or with dimensions that
differ by only an exponent.
6) Choose dimensional constants over dimensional variables so that only one

contains the dimensional variable.
7) Pick common parameters since they may appear in each of the

’s.
8) Pick simple parameters over complex parameters.
Table 7-3
19

20. GUIDELINES FOR MANIPULATION OF THE ’S IN
STEP 5
You can
i. Impose a constant (dimensionless) exponent on a

or perform a functional
operation on a

.
ii. Multiply a

by a pure (dimensionless) constant.
iii. Form a product (or quotient) of any

with any other

in the problem to
replace one of the

’s.
iv. Use any of guidelines (i) to (iii) in combination.
v. Substitute a dimensional parameter in the

with other parameter(s) of the
same dimensions (e.g. L and D).
Table 7-4
20

21. Table 7-5
21

22. Table 7-5
22

23. SIMILARITY
• Now that you have determined the

groups, the next step is to determine how
to link the model and the prototype.
• This is achieved via similarity and is based on the dimensionl ess PI groups
already determined
Back to Example (Part 2)
23

24. The aerodynamic drag of a new sports car is to be predicted
at a speed of 80 km/h at an air temperature of 25 °C.
Automotive engineers build a one-fifth scale model of the
car to test in a wind tunnel.
It is winter and the wind tunnel is located in an unheated
building; the temperature of the wind tunnel air is only about
5°C.
Determine how fast the engineers should run the wind
tunnel in order to achieve similarity between the model and
the prototype.
For air at atmospheric pressure and
at T = 25°C; = 1.184 kg/m
3
and =1.849 x10
-5
kg/m.s
at T = 5°C; = 1.269 kg/m
3
and =1.754 x10
-5
kg/m.s
EXAMPLE (PART 2)
24

25. 25

26. Now suppose the engineers run the wind tunnel at
354 km/h to achieve similarity between the model
and the prototype. The aerodynamic drag force on
the model car is measured with a drag balance.
Several drag readings are recorded, and the
average drag force on the model is 94.3 N.
Now predict the aerodynamic drag force on the
prototype (at 80 km/h and 25°C).
A drag balance is a device used in a wind
tunnel to measure the aerodynamic drag of a
body. When testing automobile models, a
moving belt is often added to the floor of the
wind tunnel to simulate the moving ground
(from the car’s frame of reference).
EXAMPLE (PART 3)
26

27. 27

28. For air at atmospheric pressure and
at T = 25°C; = 1.184 kg/m
3
and =1.849 x10
-5
kg/m.s
For water at atmospheric pressure and
at T = 20°C; = 998 kg/m
3
and =1.002 x10
-3
kg/m.s
EXAMPLE (PART 4)
Let the engineers use a water tunnel instead of a wind tunnel t o test their one-fifth scale
model. Using the properties of water at room temperature (20 °C is assumed), calculate
the water tunnel speed required to achieve similarity.
28

29. EX 12-7 : LIFT (F
L
) ON AN AIRCRAFT WING (PROBLEM)
Engineers are designing an airplane and
wish to predict the lift produced by their
new wing design.
The chord length L
c
of the wing is 1.12 m,
and its planform area A (area viewed from
the top when the wing is at zero angle of
attack []) is 10.7 m 2
.
The prototype is to fly at V =52.0 m/s
close to the ground where T= 25°C.
They build a one-tenth scale model of the wing to
test in a pressurised wind tunnel. The wind tunnel
can be pressurised to a maximum of 5 atm.
At what speed and pressure should they run the
wind tunnel in order to achieve dynamic similarity?
c - speed of sound
29

30. Step 1:List the parameters in the problem and count their total number n.
F
L
= f (V, L
C
, , , c,

) i.e. n = 7
Step 2: List the primary dimensions of each of the 7 parameters.
Step 3:Take the number of primary dimensions (m, L and t), j=3.
The expected number of k is = n − j = 7 − 3 = 4
Step 4:Choose j=3 repeating parameters V, L
C
and

.
Step 5:A simplified procedure:
 
21 3111

1
LC
FVL c
mLt Lt L mL mL t Lt


 
    
    
Once the repeating variables have been selected, manipulate the repeating
variables (V, L
C
, ) to express them in primary dimensions as:
Mass [m]: use

L
C
3
Length [L]: use L
C
Time [t]: use (L
C
/ V)
Must have a combination
of L, mand t
EX 12-7 : LIFT (F
L
) ON AN AIRCRAFT WING (PROBLEM)
30

31. ),L,V(f F
C L


2 2 2
1 1 3
2 1 1
)() (
C
L
C
C C
L L
LV
F
V
L
L L
F
tLm
F









  
 
C
L
is Lift Coefficient
L
2
L
ified mod,1
C
AV
2
1
F


 
F
L
= f (V, L
c
, , , c,

) ; Find the four dimensionless groups ( s) as follows:
),L,V(f
C



C
1
C 1
C
1 3
C
1 1 1 2
LV
V
L
)L()L(
tLm












 


 
Re
LV
C
ified mod,2



 
Re is Reynolds number
31 1
()
C
C
ccc
Lt VL
L
V
 
  



),L,V(f c
C


Ma
c
V
ified mod,3
  
Ma is Mach number
In the text book, anindicialmethod is used to find the 4s (i.e. in step 5). For this example, that will
involve finding 12 constants (3 constants for each of the 4 s) using various simultaneous equations.
You are likely to make many errors and it takes time. However, in this simpli fied method, allshave
been found without finding the12 constants.


 
00 0 4
tLm
),L,V(f
C



Angle of attack
You should use this simplified method in the Workshop classes a nd any assessments.
31

32. STEP 6:Write the Final Functional Relationship
To achieve dynamic similarity, all three of the independent non -dimensional
parameters must match between the model and the prototype. While it is
trivial to match the angle of attack ( ), it is not so simple to simultaneously
match the Reynolds number and the Mach number.
Assume model at the same temperature.
Now use this velocity for the Ma number
The values are out by a factor of 10 !!



, Re,
2
1
) , , (
2
4 3 2 1
Ma f
AV
F
C ie f
L
L
      
??????
6,???
: ??????
?
L??????
?
??????
?
??????
?
??????
?
??????
?
??????
?
??????
?
L52.0
10
1
L 520??????/??????
??????
6,???
:
??????
???????
M
??????
???????
32

33. A common rule of thumb is that for Mach numbers less than about 0.3,
compressibility effects are practically negligible. Thus, it is not necessary to
exactly match the Mach number; rather, as long as Ma is kept below about 0.3,
approximate dynamic similarity can be achieved by matching the Reynolds
number.
The wind tunnel should be run at approximately 100 m/s, 5 atm, and 25°C.
At constant temperature, density is proportional to pressure, w hile viscosity and
the speed of sound are weak functions of pressure. If the wind tunnel can be
pumped to 10 atm, you can run the wind tunnel at the same speed as the
prototype and get a perfect match of Re and Ma numbers.
33

34. SETUP OF AN EXPERIMENT AND CORRELATION OF
EXPERIMENTAL DATA
Where dimensional analysis can be of real benefit:
• Consider a problem in which there are five original parameters (one of which is
the dependent parameter).
• A complete set of experiments (called a full factorialtest matrix) is conducted.
• This testing would require 5
4
= 625 experiments.
• Assuming that all three primary dimensions are represented in the problem, we
can reduce the number of parameters from five to two ( k =5 - 3 = 2 non-
dimensional groups), and the number of independent parameters from four to
one.
• Thus, for the same resolution we would then need to conduct a total of only
5
1
= 5 experiments instead of 625! (significant time and cost savi ngs!).
34

35. SETUP OF AN EXPERIMENT AND CORRELATION OF
EXPERIMENTAL DATA (CONTD.)
For a two-

problem, we plot the dependent
dimensionless parameter (

1
) as a function of
independent dimensionless parameter (

2
).
The resulting plot can be (a) linear or (b)
nonlinear.
In either case, regression and curve-fitting
techniques are available to determine the
relationship between the

’s.
35

36. EXPERIMENTAL TESTING AND INCOMPLETE SIMILARITY
• There are situations in which complete dynamic similarity is n ot achievable.
• The problem is that it is not always possible to match all the

’s of a model to
the corresponding

’s of the prototype, even if we are careful to achieve
geometric similarity.
• This situation is called incomplete similarity.
• Fortunately, in some cases of incomplete similarity, we are st ill able to
extrapolate model test data to obtain reasonable full-scale pre dictions.
Wind Tunnel Testing
Measurement of aerodynamic
drag on a model truck in a
wind tunnel equipped with a
drag balance and a moving
belt ground plane.
The wind tunnel has a maximum speed of 70 m/s
36

37. INCOMPLETE SIMILARITY (CONTD.)
What do we do? There are several options:
1)use a bigger wind tunnel.Automobile manufacturers typically test with three-
eighths scale model cars and with one-eighth scale model trucks and buses in
very large wind tunnels.
2)we could use a different fluid for the model tests. For example, water can
achieve higher Re numbers, but is also more expensive.
3)we could pressurise the wind tunnel and/or adjust the air temperature to
increase the maximum Reynolds number capability. (limited appli cation)
4)if all else fails, we could run the wind tunnel at several spee ds near the
maximum speed, and then extrapolate our results to the full-scale Reynolds
number.
Fortunately, for many wind tunnel tests the last option is quit e viable.
37

38. INCOMPLETE SIMILARITY (CONTD.)
• The drag coefficient C
D
is a strong function of
Reynold’s number for low Re
• Then levels off after reaching a certain threshold
value
•This results in Reynolds number independence.
•It enables extrapolation to prototype Reynolds
numbers that are outside of the range of the experimental facil ity.
•How much confidence do you have ?
•Comes from lab experience …
38

39. SUMMARY
Several areas have been covered in this lecture:
• Dimensional homogeneity
• Non-dimensionalisation
• Method of repeating variables (6 steps) – Buckingham Pi method
• Simplified step 5 method to avoid solving simultaneous equatio ns
• Dimensional similarity (geometric, kinematic and dynamic)
• Experimental testing and incomplete similarity
Workshop questions this week covering
dimensional analysis and similarity
39