This document discusses dimensional analysis, which is a mathematical technique used in fluid mechanics to reduce the number of variables in a problem by combining dimensional variables to form non-dimensional parameters. Dimensional analysis allows problems to be expressed in terms of non-dimensional parameters to show the relative significance of each parameter. It has various uses including checking dimensional homogeneity of equations, deriving equations, planning experiments, and analyzing complex flows using scale models. The Buckingham π theorem states that any relationship between physical quantities can be written as a relationship between dimensionless pi groups formed from the variables. Dimensional analysis is applied by setting up a dimensional matrix to determine the minimum number of pi groups needed to describe the relationship.
The derivation of the equation of motion for various fluids is similar to the d derivation of Eular’s equation. However ,the tangential stresses arise during the motion of a real viscous fluid, must be considered
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
The derivation of the equation of motion for various fluids is similar to the d derivation of Eular’s equation. However ,the tangential stresses arise during the motion of a real viscous fluid, must be considered
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
What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
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This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Specific Speed of Turbine | Fluid MechanicsSatish Taji
Watch Video of this presentation on Link: https://youtu.be/I0fHo0z6EgA
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For Video, Visit our YouTube Channel (link is given below).
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Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
#WikiCourses
https://wikicourses.wikispaces.com/Lect04+Multiple+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Multiple+Degree+of+Freedom+%28MDOF%29+Systems
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
Specific Speed of Turbine | Fluid MechanicsSatish Taji
Watch Video of this presentation on Link: https://youtu.be/I0fHo0z6EgA
For notes/articles, Visit my blog (link is given below).
For Video, Visit our YouTube Channel (link is given below).
Any Suggestions/doubts/reactions, please leave in the comment box.
Follow Us on
YouTube: https://www.youtube.com/channel/UCVPftVoKZoIxVH_gh09bMkw/
Blog: https://e-gyaankosh.blogspot.com/
Facebook: https://www.facebook.com/egyaankosh/
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
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.
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
Dimensional analysis is one of the important topic of the fluid mechanics. It is useful for transferring data from one system to other system and also useful in reducing complexity of the equation.
Similar to DIMENSIONAL ANALYSIS (Lecture notes 08) (20)
Earthquake Load Calculation (base shear method)
The 3-story standard office building is located in Los Angeles situated on stiff soil. The
structure of the building is steel special moment frame. All moment-resisting frames are
located at the perimeter of the building. Determine the earthquake force on each story in
North-South direction.
Structural engineering iii- Dr. Iftekhar Anam
Joint Displacements and Forces,Assembly of Stiffness Matrix and Load Vector of a Truss,Stiffness Matrix for 2-Dimensional Frame Members in the Local Axes System,Transformation of Stiffness Matrix from Local to Global Axes,Stiffness Method for 2-D Frame neglecting Axial Deformations,Problems on Stiffness Method for Beams/Frames,Assembly of Stiffness Matrix and Load Vector of a Three-Dimensional Truss,Calculation of Degree of Kinematic Indeterminacy (Doki)
Determine the doki (i.e., size of the stiffness matrix) for the structures shown below,Material Nonlinearity and Plastic Moment,
http://www.uap-bd.edu/ce/anam/
Structural engineering i- Dr. Iftekhar Anam
Structural Stability and Determinacy,Axial Force, Shear Force and Bending Moment Diagram of Frames,Axial Force, Shear Force and Bending Moment Diagram of Multi-Storied Frames,Influence Lines of Beams using Müller-Breslau’s Principle,Influence Lines of Plate Girders and Trusses,Maximum ‘Support Reaction’ due to Wheel Loads,Maximum ‘Shear Force’ due to Wheel Loads,Calculation of Wind Load,Seismic Vibration and Structural Response
http://www.uap-bd.edu/ce/anam/
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Compatible with Backplane mount serial communication.
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• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
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• Compatible with MAFI CCR system.
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• Indigenized local Support/presence in India.
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
1. CHAPTER 8
DIMENSIONAL ANALYSIS
8.1 INTRODUCTION
Dimensional analysis is one of the most important mathematical tools in the study of
fluid mechanics. It is a mathematical technique, which makes use of the study of dimensions
as an aid to the solution of many engineering problems. The main advantage of a dimensional
analysis of a problem is that it reduces the number of variables in the problem by combining
dimensional variables to form non-dimensional parameters.
By far the simplest and most desirable method in the analysis of any fluid problem is
that of direct mathematical solution. But, most problems in fluid mechanics such complex
phenomena that direct mathematical solution is limited to a few special cases. Especially for
turbulent flow, there are so many variables involved in the differential equation of fluid
motion that a direct mathematical solution is simply out of question. In these problems
dimensional analysis can be used in obtaining a functional relationship among the various
variables involved in terms of non-dimensional parameters.
Dimensional analysis has been found useful in both analytical and experimental work
in the study of fluid mechanics. Some of the uses are listed:
1) Checking the dimensional homogeneity of any equation of fluid motion.
2) Deriving fluid mechanics equations expressed in terms of non-dimensional
parameters to show the relative significance of each parameter.
3) Planning tests and presenting experimental results in a systematic manner.
4) Analyzing complex flow phenomena by use of scale models (model similitude).
8.2 DIMENSIONS AND DIMENSIONAL HOMOGENEITY
Scientific reasoning in fluid mechanics is based quantitatively on concepts of such
physical phenomena as length, time, velocity, acceleration, force, mass, momentum, energy,
viscosity, and many other arbitrarily chosen entities, to each of which a unit of measurement
has been assigned. For the purpose of obtaining a numerical solution, we adopt for
computation the quantities in SI or MKS units. In a more general sense, however, it is
desirable to adopt a consistent dimensional system composed of the smallest number of
dimensions in terms of which all the physical entities may be expressed. The fundamental
dimensions of mechanics are length [L], time [T], mass [M], and force [F], related by
Newton’s second law of motion, F = ma.
Prof. Dr. Atıl BULU130
2. Dimensionally, the law may also be written as,
[ ] ⎥
⎦
⎤
⎢
⎣
⎡
= 2
T
ML
F or 1
2
=⎥
⎦
⎤
⎢
⎣
⎡
ML
FT
(8.1)
Which indicates that when three of the dimensions are known, the fourth may be
expressed in the terms of the other three. Hence three independent dimensions are sufficient
for any physical phenomenon encountered in Newtonian mechanics. They are usually chosen
as either [MLT] (mass, length, time) or [FLT] (force, length, time). For example, the specific
mass (ρ) may be expressed either as [M/L3
]or as [FT2
/L4
], and a fluid pressure (p), which is
commonly expressed as force per unit area [F/L2
] may also be expressed as [ML/T2
] using the
(mass, length, time) system. A summary of some of the entities frequently used in fluid
mechanics together with their dimensions in both systems is given in Table 8.1.
TABLE 8.1
ENTITIES COMMONLY USED IN FLUID MECHANICS
AND THEIR DIMENSIONS
Entity MLT System FLT System
Length (L) L L
Area (A) L2
L2
Volume (V) L3
L3
Time (t) T T
Velocity (v) LT-1
LT-1
Acceleration (a) LT-2
LT-2
Force (F) and weight (W) MLT-2
F
Specific weight (γ) ML-2
T-2
FL-3
Mass (m) M FL-1
T-2
Specific mass (ρ) ML-3
FL-4
T2
Pressure (p) and stress (τ) ML-1
T-2
FL-2
Energy (E) and work ML2
T-2
FL
Momentum (mv) MLT-1
FT
Power (P) ML2
T-3
FLT-1
Dynamic viscosity (μ) ML-1
T-1
FL-2
T
Kinematic viscosity (υ) L2
T-1
L2
T-1
With the selection of three independent dimensions –either [MLT] or [FLT]- it is
possible to express all physical entities of fluid mechanics. An equation which expresses the
physical phenomena of fluid motion must be both algebraically correct and dimensionally
homogenous. A dimensionally homogenous equation has the unique characteristic of being
independent of units chosen for measurement.
Equ. (8.1) demonstrates that a dimensionally homogenous equation may be
transformed to a non-dimensional form because of the mutual dependence of fundamental
dimensions. Although it is always possible to reduce dimensionally homogenous equation to a
non-dimensional form, the main difficulty in a complicated flow problem is in setting up the
correct equation of motion. Therefore, a special mathematical method called dimensional
analysis is required to determine the functional relationship among all the variables involved
in any complex phenomenon, in terms of non-dimensional parameters.
Prof. Dr. Atıl BULU131
3. 8.3 DIMENSIONAL ANALYSIS
The fact that a complete physical equation must be dimensionally homogenous and is,
therefore, reducible to a functional equation among non-dimensional parameters forms the
basis for the theory of dimensional analysis.
8.3.1 Statement of Assumptions
The procedure of dimensional analysis makes use of the following assumptions:
1) It is possible to select m independent fundamental units (in mechanics, m=3, i.e.,
length, time, mass or force).
2) There exist n quantities involved in a phenomenon whose dimensional formulae
may be expressed in terms of m fundamental units.
3) The dimensional quantity A0 can be related to the independent dimensional
quantities A1, A2, ......, An-1 by,
( ) 121
1211210 ......,.....,, −
−− == ny
n
yy
n AAKAAAAFA (8.2)
Where K is a non-dimensional constant, and y1, y2,.....,yn-1 are integer
components.
4) Equ. (8.2) is independent of the type of units chosen and is dimensionally
homogenous, i.e., the quantities occurring on both sides of the equation must
have the same dimension.
EXAMPLE 8.1: Consider the problem of a freely falling body near the surface of the
earth. If x, w, g, and t represent the distance measured from the initial height, the weight of
the body, the gravitational acceleration, and time, respectively, find a relation for x as a
function of w, g, and t.
SOLUTION: Using the fundamental units of force F, length L, and time T, we note
that the four physical quantities, A0=x, A1=w, A2=g, and A3=t, involve three fundamental
units; hence, m=3 and n=4 in assumptions (1) and (2). By assumption (3) we assume a
relation of the form:
( ) 321
,, yyy
tgKwtgwFx == (a)
Where K is an arbitrary non-dimensional constant.
Let [⋅] denote “dimensions of a quantity”. Then the relation above can be written
(using assumption (4)) as,
[ ] [ ] [ ] [ ] 321 yyy
tgwx =
or
( ) ( ) ( ) 3221321 22010 yyyyyyy
TLFTLTFTLF +−−
==
Prof. Dr. Atıl BULU132
4. Equating like exponents, we obtain
2
1
1:
0:
yL
yF
=
=
3220: yyT +−= or 22 23 == yy
Therefore, Equ. (a) becomes
210
tgKwx =
or
2
Kgtx =
According to the elementary mechanics we have x=gt2
/2. The constant K in this case
is ½, which cannot be obtained from dimensional analysis.
EXAMPLE 8.2: Consider the problem of computing the drag force on a body moving
through a fluid. Let D, ρ, μ, l, and V be drag force, specific mass of the fluid, dynamic
viscosity of the fluid, body reference length, and body velocity, respectively.
SOLUTION: For this problem m=3, n=5, A0=D, A1=ρ, A2=μ, A3=l, and A4=V. Thus,
according to Equ (8.2), we have
( ) 4321
,,, yyyy
VlKVlFD μρμρ == (a)
or
[ ] [ ] [ ] [ ] [ ]
( ) ( ) ( ) ( )
421432121
4
3
21
4321
224001
1224001
yyyyyyyyy
yyyy
yyyy
TLFTLF
LTLTFLTFLTLF
VlD
−+++−−+
−−−
=
=
= μρ
Equating like exponents, we obtain
421
4321
21
20:
240:
1:
yyyT
yyyyL
yyF
−+=
++−−=
+=
In this case we have three equations and four unknowns. Hence, we can only solve for
three of the unknowns in terms of the fourth unknown (a one-parameter family of solutions
exists). For example, solving for y1, y3 and y4 in terms of y2, one obtains
24
23
21
2
2
1
yy
yy
yy
−=
−=
−=
Prof. Dr. Atıl BULU133
5. The required solution is
2222 221 yyyy
VlKD −−−
= μρ
or
( ) 2
2
2
2
2
l
VVl
KD
y
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
−
ρ
μ
ρ
If the Reynolds number is denoted by Re=ρVl/μ, dynamic pressure by q=ρV2
/2, and
area by A=l2
, we have
( )
qACqA
K
D Dy
== 2
Re
2
where
( ) 2
Re
2
yD
K
C =
Theoretical considerations show that for laminar flow
328.12 =K and
2
1
2 =y
8.3.2 Buckingham-π (Pi) Theorem
It is seen from the preceding examples that m fundamental units and n physical
quantities lead to a system of m linear algebraic equations with n unknowns of the form
mnmnmm
nn
nn
byayaya
byayaya
byayaya
=+++
=+++
=+++
......
..................................................
......
.......
2211
22222121
11212111
(8.3)
or, in matrix form,
bAy = (8.4)
where
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
mnmm
n
n
aaa
aaa
aaa
A
......
......................
.......
.......
21
22212
11211
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
ny
y
y
y
.
2
1
and
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
nb
b
b
b
.
2
1
Prof. Dr. Atıl BULU134
6. A is referred to as the coefficient matrix of order m×n, and y and b are of order n×1
and m×1 respectively.
The matrix A in Equ. (8.4) is rectangular and the largest determinant that can be
formed will have the order n or m, whichever is smaller. If any matrix C has at least one
determinant of order r, which is different from zero, and nonzero determinant of order greater
than r, then the matrix is said to be of rank r, i.e.,
( ) rCR = (8.5)
In order to determine the condition for the solution of the linear system of Equ. (8.3) it
is convenient to define the rank of the augmented matrix B. The matrix B is defined as
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
==
mmnmm
n
n
b
baaa
baaa
baaa
AB
.......
.........................
.......
.......
21
222221
111211
(8.6)
For the solution of the linear system in Equ. (8.3), three possible cases arise:
1) R (A)<R (B): No solution exists,
2) R(A)=R(B)= r = n: A unique solution exits,
3) R (A)=R (B)=r<n: An infinite number of solutions with (n-r) arbitrary unknowns
exist.
Example 8.2 falls in case (3) where
( ) ( ) 43 =<== nBRAR and ( ) ( ) 134 =−=− rn
an arbitrary unknown exits.
The mathematical reasoning above leads to the following Pi theorem due to
Buckingham.
Let n quantities A1, A2,…..,An be involved in a phenomenon, and their dimensional
formulae be described by (m<n) fundamental units. Let the rank of the augmented matrix B
be R (B)= r ≤ m. Then the relation
( 0,.....,, 211 =nAAAF ) (8.7)
is equivalent to the relation
( ) 0,.....,, 212 =−rnF πππ (8.8)
Where π1, π2,…..,πn-r are dimensionless power products of A1, A2,…..,An taken r+1 at a time.
Thus, the Pi theorem allows one to take n quantities and find the minimum number of
non-dimensionless parameter, π1, π2,….., πn-r associated with these n quantities.
Prof. Dr. Atıl BULU135
7. 8.3.2.1 Determination of Minimum Number of π Terms
In order to apply the Buckingham π Theorem to a given physical problem the
following procedure should be used:
Step 1. Given n quantities involving m fundamental units, set up the augmented
matrix B by constructing a table with the quantities on the horizontal axis and the
fundamental units on the vertical axis. Under each quantity list a column of numbers, which
represent the powers of each fundamental units that makes up its dimensions. For example,
ρ p d Q
F 1 1 0 1
L -4 -2 1 3
T 2 0 0 -1
Where ρ, p, d, and Q are the specific mass, pressure, diameter, and discharge,
respectively. The resulting array of numbers represents the augmented matrix B in
Buckingham’s π Theorem, i.e.,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−−=
1002
3124
0011
B
The matrix B is sometimes referred to as dimensional matrix.
Step 2. Having constructed matrix B, find its rank. From step1, since
01
002
024
011
≠=−−
and no larger nonzero determinant exists, then
( ) rBR == 3
Step3. Having determined the number of π dimensionless (n-r) terms, following rules
are used to combine the variables to form π terms.
a) From the independent variables select certain variables to use as repeating
variables, which will appear in more than π term. The repeating variables should
contain all the dimensions used in the problem and be quantities, which are likely
to have substantial effect on the dependent variable.
b) Combine the repeating variables with remaining variables to form the required
number of independent dimensionless π terms.
c) The dependent variable should appear in one group only.
d) A variable that is expected to have a minor influence should appear in one group
only.
Prof. Dr. Atıl BULU136
8. Define π1 as a power product of r of the n quantities raised to arbitrary integer
exponents and any one of the excluded (n-r) quantities, i.e.,
1211
11211
..... += r
y
r
yy
AAAA r
π
Step 4. Define π2, π3,....., πn-r as power products of the same r quantities used in step 3
raised to arbitrary integer exponents but a different excluded quantity for each π term, i.e.,
n
y
r
yy
rn
r
y
r
yy
r
y
r
yy
AAAA
AAAA
AAAA
rrnrnrn
r
r
,2,1,
33231
22221
.....
........................................
.............
.............
21
3213
2212
−−−
=
=
=
−
+
+
π
π
π
Step 5. Carry out dimensional analysis on each π term to evaluate the exponents.
EXAMPLE 8.3: Rework Example 8.1 using the π theorem.
SOLUTION:
Step 1. With F, L, and T as the fundamental units, the dimensional matrix of the
quantities w, g, t and x is,
W g t x
F 1 0 0 0
L 0 1 0 1
T 0 -2 1 0
Where
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
0120
1010
0001
B
Step 2. Since
02
020
110
001
≠=
−
and no larger nonzero determinant exists, then
( ) rBR == 3
Step 3. Arbitrarily select x, w, and g as the r = m= 3 base quantities. The number n-r
of independent dimensional products that can be formed by the four quantities is therefore 1,
i.e.,
tgwx yyy 131211
1 =π
Prof. Dr. Atıl BULU137
9. Step 4. Dimensional analysis gives,
[ ] [ ] [ ] [ ] [ ]tgwx
yyy 131211
1 =π
or
( ) ( ) ( ) ( )TLTFLTLF
yyy 131211 2000 −
=
Which results in
2
1
11 −=y , 012 =y , and
2
1
13 =y
Hence,
x
gt2
1 =π
EXAMPLE 8.4: Rework Example 8.2 using the π theorem.
SOLUTION:
Step 1. With F, L and T as the fundamental units, the dimensional matrix of the
quantities D, ρ, μ, l, and V is.
D ρ μ l V
F 1 1 1 0 0
L 0 -4 -2 1 1
T 0 2 1 0 -1
Where
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−−=
10120
11240
00111
B
Step2. Since
01
101
112
001
≠=
−
−
and no larger nonzero determinant exists, then
( ) rBR == 3
Prof. Dr. Atıl BULU138
10. Step 3. Select l, V, and ρ as the r=3 base quantities. By the π-Theorem (n-r),(5-3)=2 π
terms exist.
μρπ
ρπ
232221
131211
2
1
yyy
yyy
Vl
DVl
=
=
Step 4. Dimensional analysis gives
y11=-2, y12=-2, y13= -1
y21=-1, y22=-1, y23=-1
Hence,
Vl
lV
D
ρ
μ
π
ρ
π
=
=
2
221
8.4 THE USE OF DIMENSIONLESS π-TERMS IN EXPERIMENTAL
INVESTIGATIONS
Dimensional analysis can be of assistance in experimental investigation by reducing
the number of variables in the problem. The result of the analysis is to replace an unknown
relation between n variables by a relationship between a smaller numbers, n-r, of
dimensionless π-terms. Any reduction in the number of variables greatly reduces the labor of
experimental investigation. For instance, a function of one variable can be plotted as a single
curve constructed from a relatively small number of experimental observations, or the results
can be represented as a single table, which might require just one page.
A function of two variables will require a chart consisting of a family of curves, one
for each value of the second variable, or, alternatively the information can be presented in the
form of a book of tables. A function of three variables will require a set of charts or a shelf-
full of books of tables.
As the number of variables increases, the number of observations to be taken grows so
rapidly that the situation soon becomes impossible. Any reduction in the number of variables
is extremely important.
Considering, as an example, the resistance to flow through pipes, the shear stress or
resistance R per unit area at the pipe wall when fluid of specific mass ρ and dynamic viscosity
μ flows in a smooth pipe can be assumed to depend on the velocity of flow V and the pipe
diameter D. Selecting a number of different fluids, we could obtain a set of curves relating
frictional resistance (measured as R/ρV2
) to velocity, as shown in Fig. 8.1.
Prof. Dr. Atıl BULU139
11. 0.015
Water,d=50.80mmWater,d=25.40mmWater,d=62.70mm
Air,d=25.40mm
Air,d=12.70mm
Air,d=6.35mm
0.010
0.005
0
0.01 0.1 1 10 100
v ( ms )-1
R/pv2 All measurements at 15.5°C
Fig. 8.1
Such a set of curves would be of limited value both for use and for obtaining a proper
understanding of the problem. However, it can be shown by dimensional analysis that the
relationship can be reduced to the form
( )Re2
φ
μ
ρ
φ
ρ
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
VD
V
R
or, using the Darcy resistance coefficient 2
2 VRf ρ= ,
( )Reφ
μ
ρ
φ =⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
VD
f
4
-3
10
10 2
2
10
-2
7
2
2 4 7 10
3
10
4
7
-1
5
10
Reynolds number , Re
2 4 7 10
4
42 7 102 4 7
6
Frictioncoefficient,f
Fig. 8.2
Prof. Dr. Atıl BULU140
12. If the experimental points in Fig. 8.1 are used to construct a new graph of Log (f)
against Log (Re) the separate sets of experimental data combine to give a single curve as
shown in Fig. 8.2. For low values of Reynolds number, when flow is laminar, the slope of this
graph is (-1) and f = 16/Re, while for turbulent flow at higher values of Reynolds number,
f = 0.08(Re)-1/4
.
Prof. Dr. Atıl BULU141