This document discusses dimensional analysis and dimensionless numbers that are important in fluid mechanics. It defines Reynolds number, Froude number, Euler number, Weber number, and Mach number. It explains how dimensional analysis can help reduce the number of variables in experimental investigations. It also discusses similitude and the different types of model testing including undistorted and distorted models. The key uses and advantages of model testing are outlined.
Dimensional analysis Similarity laws Model laws R A Shah
Rayleigh's method- Theory and Examples
Buckingham Pi Theorem- Theory and Examples
Model and Similitude
Forces on Fluid
Dimensionless Numbers
Model laws
Distorted models
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Dimensional analysis Similarity laws Model laws R A Shah
Rayleigh's method- Theory and Examples
Buckingham Pi Theorem- Theory and Examples
Model and Similitude
Forces on Fluid
Dimensionless Numbers
Model laws
Distorted models
Dimension less numbers in applied fluid mechanicstirath prajapati
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Minor losses are a major part in calculating the flow, pressure, or energy reduction in piping systems. Liquid moving through pipes carries momentum and energy due to the forces acting upon it such as pressure and gravity. Just as certain aspects of the system can increase the fluids energy, there are components of the system that act against the fluid and reduce its energy, velocity, or momentum. Friction and minor losses in pipes are major contributing factors.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
Head losses
Major Losses
Minor Losses
Definition • Dimensional Analysis • Types • Darcy Weisbech Equation • Major Losses • Minor Losses • Causes Head Losses
3. • Head loss is loss of energy per unit weight. • Head = Energy of Fluid / Weight • Head losses can be – Kinetic Head – Potential Head – Pressure Head 6/10/2015 4Danial Gondal Head Loss
4. • Kinetic Head – K.H. = kinetic energy / Weight = v² /2g • Potential Head – P.H = Potential Energy / Weight = mgz /mg = z • Pressure Head – P.H = P/ ρ g 6/10/2015 5
5. • (P/ ρ g) + (v² /2g ) + (z) = constant • (FL-2F-1L3LT-2L-1T2) + (L2T-2L1T2)+(L) = constant • (L) + (L) + (L) = constant • As L represent height so it is dimensionally L. 6/10/2015 6 Dimensional Analysis
6. • However the equation (P/ ρ g) + (v² /2g ) + (z) = constant Is valid for Bernoulli's Inviscid flow case. As we are studying viscous flow so (P1/ ρ g) + (v1² /2g ) + (z1) = EGL1(Energy Grade Line At point 1) (P2/ ρ g) + (v2² /2g ) + (z2) = EGL2(Energy Grade Line At point 2) 6/10/2015 7 Head Loss
7. • For Inviscid Flow EGL1 - EGL2= 0 • For Viscous Flow EGL1 - EGL2= Hf 6/10/2015 8 Head Loss
8. MAJOR LOSSES IN PIPES
9. •Friction loss is the loss of energy or “head” that occurs in pipe flow due to viscous effects generated by the surface of the pipe. • Friction Loss is considered as a "major loss" •In mechanical systems such as internal combustion engines, it refers to the power lost overcoming the friction between two moving surfaces. •This energy drop is dependent on the wall shear stress (τ) between the fluid and pipe surface. 6/10/2015 10 Friction Loss
10. •The shear stress of a flow is also dependent on whether the flow is turbulent or laminar. •For turbulent flow, the pressure drop is dependent on the roughness of the surface. •In laminar flow, the roughness effects of the wall are negligible because, in turbulent flow, a thin viscous layer is formed near the pipe surface that causes a loss in energy, while in laminar flow, this viscous layer is non-existent. 6/10/2015 11 Friction Loss
11. Frictional head losses are losses due to shear stress on the pipe walls. The general equation for head loss due to friction is the Darcy-Weisbach equation, which is where f = Darcy-Weisbach friction factor, L = length of pipe, D = pipe diameter, and V = cross sectional average flow velocity.
OPEN CHANNEL FLOW AND HYDRAULIC MACHINERY
Open channel flow: Types of flows – Type of channels – Velocity distribution – Energy and momentum correction factors – Chezy’s, Manning’s; and Bazin formula for uniform flow – Most Economical sections. Critical flow: Specific energy-critical depth – computation of critical depth – critical sub-critical – super critical flows
Non-uniform flows –Dynamic equation for G.V.F., Mild, Critical, Steep, horizontal and adverse slopes-surface profiles-direct step method- Rapidly varied flow, hydraulic jump, energy dissipation
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
Minor losses are a major part in calculating the flow, pressure, or energy reduction in piping systems. Liquid moving through pipes carries momentum and energy due to the forces acting upon it such as pressure and gravity. Just as certain aspects of the system can increase the fluids energy, there are components of the system that act against the fluid and reduce its energy, velocity, or momentum. Friction and minor losses in pipes are major contributing factors.
Fluid Mechanics Chapter 5. Dimensional Analysis and SimilitudeAddisu Dagne Zegeye
Introduction, Dimensional homogeneity, Buckingham pi theorem, Non dimensionalization of basic equations, Similitude, Significance of non-dimensional numbers in fluid flows
Head losses
Major Losses
Minor Losses
Definition • Dimensional Analysis • Types • Darcy Weisbech Equation • Major Losses • Minor Losses • Causes Head Losses
3. • Head loss is loss of energy per unit weight. • Head = Energy of Fluid / Weight • Head losses can be – Kinetic Head – Potential Head – Pressure Head 6/10/2015 4Danial Gondal Head Loss
4. • Kinetic Head – K.H. = kinetic energy / Weight = v² /2g • Potential Head – P.H = Potential Energy / Weight = mgz /mg = z • Pressure Head – P.H = P/ ρ g 6/10/2015 5
5. • (P/ ρ g) + (v² /2g ) + (z) = constant • (FL-2F-1L3LT-2L-1T2) + (L2T-2L1T2)+(L) = constant • (L) + (L) + (L) = constant • As L represent height so it is dimensionally L. 6/10/2015 6 Dimensional Analysis
6. • However the equation (P/ ρ g) + (v² /2g ) + (z) = constant Is valid for Bernoulli's Inviscid flow case. As we are studying viscous flow so (P1/ ρ g) + (v1² /2g ) + (z1) = EGL1(Energy Grade Line At point 1) (P2/ ρ g) + (v2² /2g ) + (z2) = EGL2(Energy Grade Line At point 2) 6/10/2015 7 Head Loss
7. • For Inviscid Flow EGL1 - EGL2= 0 • For Viscous Flow EGL1 - EGL2= Hf 6/10/2015 8 Head Loss
8. MAJOR LOSSES IN PIPES
9. •Friction loss is the loss of energy or “head” that occurs in pipe flow due to viscous effects generated by the surface of the pipe. • Friction Loss is considered as a "major loss" •In mechanical systems such as internal combustion engines, it refers to the power lost overcoming the friction between two moving surfaces. •This energy drop is dependent on the wall shear stress (τ) between the fluid and pipe surface. 6/10/2015 10 Friction Loss
10. •The shear stress of a flow is also dependent on whether the flow is turbulent or laminar. •For turbulent flow, the pressure drop is dependent on the roughness of the surface. •In laminar flow, the roughness effects of the wall are negligible because, in turbulent flow, a thin viscous layer is formed near the pipe surface that causes a loss in energy, while in laminar flow, this viscous layer is non-existent. 6/10/2015 11 Friction Loss
11. Frictional head losses are losses due to shear stress on the pipe walls. The general equation for head loss due to friction is the Darcy-Weisbach equation, which is where f = Darcy-Weisbach friction factor, L = length of pipe, D = pipe diameter, and V = cross sectional average flow velocity.
OPEN CHANNEL FLOW AND HYDRAULIC MACHINERY
Open channel flow: Types of flows – Type of channels – Velocity distribution – Energy and momentum correction factors – Chezy’s, Manning’s; and Bazin formula for uniform flow – Most Economical sections. Critical flow: Specific energy-critical depth – computation of critical depth – critical sub-critical – super critical flows
Non-uniform flows –Dynamic equation for G.V.F., Mild, Critical, Steep, horizontal and adverse slopes-surface profiles-direct step method- Rapidly varied flow, hydraulic jump, energy dissipation
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their fundamental dimensions (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometers, or pounds vs. kilograms vs. grams) and tracking these dimensions as calculations or comparisons are performed.
Properties of Fluids, Fluid Static, Buoyancy and Dimensional AnalysisSatish Taji
The presentation includes a brief view of the basic properties of a fluid, fluid statics, Pascal's law, hydrostatic law, fluid classification, pressure measurement devices (manometers and mechanical gauges), hydrostatic forces on different surfaces, buoyancy and metacentric height, and dimensional analysis.
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The Presentation is divided in two halves: the first half is dimensional effect on engineering systems and the second half deals with the basics of clean room and its classification
Comparision of flow analysis through a different geometry of flowmeters using...eSAT Publishing House
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On the Size-dependent Nonlinear Behavior of a Capacitive Rectangular Micro-pl...Kaveh Rashvand
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Numerical simulations have been undertaken
for the benchmark problem in a Square cavity by using
computational fluid dynamics software. This work aims at
discussing the fundamental numerical and computational
fluid dynamic aspects which can lead to the definition of
the following meshing methods and turbulence models.
The meshes used for the simulation are hexahedral,
hexahedral cell with near wall refinement, tetrahedral
grid, polyhedral, tetrahedral with near wall refinement
and polyhedral mesh with prism layer cells based the near
wall thickness of Y+ less than one. The turbulence models
used for the simulation work are AKN K-Epsilon Low-Re,
Realizable K-Epsilon, Realizable K-Epsilon Two-Layer,
standard K-Epsilon, standard K-Epsilon Low-Re,
Standard K-Epsilon Two-Layer, V2F K-Epsilon,
SST(Menter) K-Omega, and Standard(Wilcox) K-Omega.
From these meshes and turbulence models, we will
conclude the suitable mesh and turbulence for the
recirculation flow by the grid independent test. These
analytical values of results are compared with reference
data which gives an optimization of experimental work.
Unsteady simulation was ran for all the Grid Independent
mesh with the SST k omega model with the time step of
0.01 sec for 40 seconds. The flow nature is studied with
and without the temperature for Reynolds number, 1000
and 10000.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
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By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
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1. Gandhinagar Institute Of Technology
Subject :- Fluid Mechanics (2130602)
Topic :- Dimensional analysis & similarities PORTION-
2
Name :- Sajan Gohel
Class :- 4D2
En. No :- 160123119010
2. 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
3. • 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
×
= = =
• ×
4. •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
=
5. •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
= = =
×
6. •Weber’s Number
• It is defined as the square root of the ratio of the inertia force of a
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
= = =
× ×
7. •Mach’s Number
• It is defined as the square root of the ratio of the inertia force of a
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×
8. 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
=
9. 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
10. 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
= = = =
11. 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
= = =
12. 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
= = = = =
13. 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.
14. 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
15. 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.
16. 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
17. 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
= =
18. 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
= =
19. 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
=
20. 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,
Vm = velocity of fluid in model
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
=
21. 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,
Vm = velocity of fluid in model
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
=
22. 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.
23. 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.
24. 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
25. 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