Dimensionless analysis & Similarities

Gandhinagar Institute Of Technology
Subject :- Fluid Mechanics (2130602)
Topic :- Dimensional analysis & similarities PORTION-
2
Name :- Sajan Gohel
Class :- 4D2
En. No :- 160123119010

Dimensionless Numbers• Dimensionless numbers are those numbers which are obtained by dividing the
inertia force by viscous force or Gravity force or pressure force or surface tension
force or elastic force.
• As this is a ratio of one force to the other force, it will be a dimensionless
number
The important dimensionless numbers are,
1) Reynold’s number
2) Froude’s Number
3) Euler's number
4) weber’s Number
5) Mach’s number

• Reynold’s Number (Re)
• It is defined as the ratio of inertia force of a flowing fluid and the viscos
force of the fluid .
• The expression for reynold’s number is obtained as
inertia force =
Viscous force =
Reynold’s number,
2
AVρ
V
A
L
µ• ×
2
i
e
v
F AV V d
R
VF vA
L
ρ
µ
×
= = =
• ×

•Froude’s Number
• The froude’s number is defined as the square root of the ratio of
inertia of a flowing fluid to the gravity force.
Fi from equation =
Fg = force due to gravity
= mass*acceleration due to gravity
=
2
AVρ
A L gρ × × ×
2
i
e
g
F V V
F
F Lg Lg
= = =
i
e
g
F
F
F
=

•Euler’s Number
• It is defined as the square root of the ratio of the inertia force of a
flowing fluid to the pressure force.
Fp = intensity of pressure * area = p*A
Fi =
i
u
p
F
E
F
=
2
AVρ
2
/
i
u
p
F AV V
E
F A p
ρ
ρ ρ
= = =
×

•Weber’s Number
flowing fluid to the surface tension force.
Fi = inertia force
Fs = surface tension force,
i
e
s
F
W
F
=
2
AVρ
Lσ×
2
i
e
s
F AV V
W
F L L
ρ
σ σ ρ
= = =
× ×

•Mach’s Number
flowing fluid to the elastic force.
Fi =
Fe = elastic stress * Area
i
e
F
M
F
=
2
AVρ
2 2
2
/ /
AV V V
M
K L K K
ρ
ρ ρ
= = =
×
2
k L×

Use of dimensionless groups in exprimental
investigation
• Dimensionless analysis can be assistance in experimental investigation by
reducing the number of variables in problem.
• From dimensional analysis, we get n-m number dimensionless group.(n = total
no. of variables)
• This reduction in the num. Of variables greatly reduces the labour of exprimental
investigation.
• Therefore reduction in number of variables is very important.
• Foe example, raynold’s number, can be changed by changing any one
or more quantities .
Re
VDρ
µ
=

Similitude and types of similarities
• “ The relation between model and prototype is known as Simulated.
• The valuable results of obtained at relatively small cost by performing test on
small scale models of prototype.
• The similarity laws help us to understand the results of modal analysis.
• Types of similarity
1.Geometry similarity
2.Kinematic similarity
3.Dynamics Similarity

1. Geometric similarity
• Geometry similarity exist between model and prototype if both of them are
identical in shape but different only in size.
• The ratio of the all the corresponding linear dimension are equal.
• The ratio of dimension of model and corresponding dimension of prototype is
called scale ratio.
m m m m
r
p p p p
l b h d
l
l b h d
= = = =

2. Kinematics similarity
• The kinematic similarity exist between model and prototype, if both
of them have identical motions.
• The ratio of the corresponding velocity at corresponding points are
equal.
( )
( )
( )
( )
( )
( )
1 2 3
1 2 3
m m m
r
p p p
V V V
V
V V V
= = =

3. Dynamic similarity
• The dynamic similarity exist between model and prototyp, if both of
them have identical forces.
• The ratio of the corresponding forces acting at a corresponding points
are equal.
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
g p fi vp p p p p
r
i v g p fm m m m m
F F FF F
F
F F F F F
= = = = =

Model Testing
• Engineers always engaged on the creation of design of Hydraulic
structure or Hydraulic Machines.
• They usually try to find out, in advance, how the structure or machine
would behave when it is actual constructed for this purpose the
engineers have to do experiment
• In fact the experiments cannot be carried out on the full size
structure or machine, which are proposed to be erected.
• Then it is necessary to construct a small scale replica of the structure
of machine and test are performed on it.

Prototype : The actual structure or machine is Call prototype.
•It is full size structure employed in the actual engineering
design & it operate under the actual working condition.
•A Working Prototype represents all or nearly all of the functionality of the
final product.
Model : It is a small scale replica of the actual structure or machine.
•The tests are performed on model to obtain the desired information
•It is not necessary that the models should be smaller than the prototypes,
they may be larger than the prototypes

Advantages of model testing
• The model tests are economical and convenient.
• All the defects of the model are eliminated , efficient and suitable
designed obtained.
• The final results obtained from Model tests useful to modified design of
prototype.
• Model testing can be used to detect and rectify the defects of the existing
structure.

Model laws
• The laws on which the models are designed for dynamic similarity are
called model laws.
• The ratio of the corresponding force is acting at the corresponding
points in the model and prototype should be equal For dynamic
similarity.
• The following are the model law:-
1.Reynold’s model law
2.Froude model law
3.Euler model law
4.Weber model law
5.Mach model law

Reynolds Model Law
• Reynolds model law is the law in which models are based on Reynolds
number.
• Models based on Reynolds number includes:-
• Pipe flow
• Resistance experienced by submarines, airplanes etc.
• Let,
• Vm = velocity of fluid in model
• Pm = density of fluid in model
• Lm = Length dimension of the model
• Um = viscosity of fluid in modeL
[ ] [ ] p p pm m m
e ep p
m p
V LV L
R R or
ρρ
µ µ
= =

Froude Model law
• Froude model law is the law in which the models are based on froude
Number which means to dynamic similarity between the models and
prototype the froude number for both of them should be equal.
• Froude model law is applied in the following fluid flow problems:-
1. Flow of jet from nozzle
2. Where waves are likely to be formed on surface.
Let,
Vm = velocity of fluid in model
Lm = linear dimension
Gm = acceleration due to gravity at a place where model is tested.
( ) ( )mod
pm
e eel prototype
m pm p
VV
F F or
g L g L
= =

Euler Model Law
• Euler’s model law is the law in which The models are designed on
euler's number which means for dynamic similarity between the
model and prototype,the Euler number of model and prototype
should be equle.
• Let,
Vm = velocity of fluide in model
Pm = pressure of fluid in model
Pm = density of Lode in model
Vp,Ppa,ppa = corresponding value in prototype,
/ /
pm
m p pm
VV
p pρ ρ
=

Weber Model Law
• Weber model law is the law in which model are based on weber‘s
number, which is the ratio of the square root of inertia force to
surface tension force
• Let,
M= surface tensile force in model
Lm = length of surface in model
Lp = Corresponding values of fluid in prototype
/ /
pm
m m m p p p
VV
L Lσ ρ σ ρ
=

Mach model law
• Mach model law is the law in which models are designed on Mach
number which is the ratio of the square root of inertia force to elastic
force of a fluid.
• Let,
Km = Elastic stress for model
Pm = density of fluid in model
Vp,Kp and Ppa =Corresponding valued for prototype.
/ /
m m
m m p p
V V
K Kρ ρ
=

Types of Models
• The hydraulic models basically two types as,
1. Undistorted models
2. Distorted models
1.Undistorted model:
• The this model is geometrical is similar to its prototype.
• The scale ratio for corresponding linear dimension of the model and its
prototype are same.
• The behaviour of the prototype can be easily predicted from the result of these
type of model.

Advantages of undistorted model
1. The basic condition of perfect geometrica similarity is satisfied.
2. Predication of model is relatively easy.
3. Results obtained from the model tests can be transferred to directly to the
prototype.
Limitations of undistorted models
1. The small vertical dimension of model can not measured accurately.
2. The cost of model may increases due to long horizontal dimension to obtain
geometric similarity.

Distorted Models
• This model is not geometrical is similar to its prototype the different
scale ratio for linear dimension are adopted.
• Distorted models may have following distortions:
Discussion of hydraulic quantities search is velocity discharge,exc.
 Different materials for the model and prototype.
• The main reason for adopting distored models
 To maintain turbulent flow
 To minimise cost of models

Advantages of distorted models
• Accurate and precise measurement are made possible due to
increase vertical dimension of models.
• Model size can be reduced so its operation is simplified and hence the
cost of model is reduced
• Depth or height distortion is changed wave patterns.
• Slopes bands and cuts are may not properly reproduced in model.
Disadvantage of distorted Models

Dimensionless analysis & Similarities

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