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Transcript of Dimensional Analysis 1-Engineering
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CE 2033
1
FLUID MECHANICS II
Dr WCDK FERNANDO
Department of Civil Engineering
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COURSE CONTENT
1. Dimensional and Hydraulic Model Analysis
2. Ideal Flow
3. Boundary Layer Theory
4. Hydraulic Machines
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INTENDED LEARNING OUTCOMES3
Apply the principle of similitude to prepare trueand distorted models in model testing
Sketch streamline plots for common, simple flowfield combinations (eg. Uniform flow and source /
sink, doublet) and estimate the location ofstagnation point/s
Explain concepts of momentum thickness anddisplacement thickness to solve problems in laminar
and turbulent boundary layers Compare the performance characteristics of
centrifugal pumps and calculate efficiency and head
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ASSESSMENT CRITERIA 4
Semester – end Examination – 70%
Assignments – 20%
Practical – 10%
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TYPES OF ASSIGNMENTS5
Class test
Group work
MCQ
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RECOMMENDED READING6
Douglas, J. F., Solving Problems in Fluid
Mechanics, Volume 2, ELBS.
Khurmi, R. S., A textbook of Hydraulics, Fluid
Mechanics and Hydraulic Machines, S. Chand &Company
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DIMENSIONAL & HYDRAULICMODEL ANALYSIS
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1.1 INTRODUCTION8
Dimensional analysis is a mathematical method of obtaining the equations, changing units, determininga convenient arrangement of variable of a physicalrelation.
In an equation expressing a physical relationship between quantities, absolute numerical anddimensional equality must exist.
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MODEL ANALYSIS9
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DIMENSIONS AND UNITS
A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assigna number to that dimension.
Note: All nonprimary dimensions can be formed by somecombination of the seven primary dimensions.
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1.2 APPLICATIONS13
It helps to check whether an equation of any physicalphenomenon is dimensionally homogeneous or not.
It helps to determine the dimensions of a physical
quantity. Dimensional homogeneity helps to convert the units
from one system to another.
The concept of dimensional homogeneity is a step to
dimensional analysis which is fruitfully employed toplan experiments and to present the resultsmeaningfully.
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DIMENSIONAL HOMOGENEITY
The law of dimensional homogeneity, stated as Every additive term in an equation must have the same dimensions.
An equation that is not dimensionally homogeneous is a sure sign of an error.
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DIMENSIONAL ANALYSIS AND SIMILARITY
Primary purposes of dimensional analysis
To generate non-dimensional parameters that
help in the design of experiments (physicaland/or numerical) and in reporting of results.
To obtain scaling laws so that prototype
performance can be predicted from modelperformance.
To predict trends in the relationship betweenparameters.
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1.3 DERIVED DIMENSIONS
Quantity Symbol Dimension
Area
Volume
Velocity
Acceleration
Density Force
Discharge
Shear stress
Dynamic viscosity
Kinematic viscosity
Pressure
Work /Energy
Power
Surface tension
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1.4 BUCKINGHAM’S п THEOREM17
If there are n variables in a problem and these variables contain m primary dimensions (forexample M, L, T) the equation relating all the
variables will have (n-m) dimensionless groups.
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Let A 1, A 2, A 3, ……..A n be the parameters involve suchas pressure, acceleration, velocity etc.
All these parameters are known to be essential to the
solution & hence a fundamental relationship offunctions A 1, A 2, A 3, ……..A n should exist.
If the number of dimensions involve with this set ofparameters is m, there should be (n-m) number ofnon-dimensional quantities.
0A.....A,A,Af n321
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SUMMARY OF STEPS19
Select the relevant variables. Write down the functional relationship.
Find the dimensions involved in the process.
Select the repeating variables. These variablescollectively should contain all the dimensionsinvolved in the process.
Find the number of non-dimensional parametersusing the Buckingham’s pi theorem.
Write Pi parameters in terms of unknown exponents by combining repeating variables with the remaining variables.
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SUMMARY OF STEPS ……20
For each п expression, write the equation of exponent so that the sum of exponents of eachdimension will be zero.
Establish the functional relationship in terms of piparameters.
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Ex 121
The resistance force F of a ship is a function of itslength L, velocity V, acceleration due to gravity g andfluid properties like density ρ and viscosity μ.Formulate a relationship between these variablesusing Buckingham’s Pi theorem.
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Nondimensionalization of Equations
Dimensional homogeneity every term in anequation has the same dimensions.
nondimensional divide each term in theequation by a collection of variables and constants
whose product has those same dimensions. Nondimensional parameters are named after
a notable scientist or engineer (e.g., the Reynoldsnumber and the Froude number).
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1.5 SIGNIFICANCE OF DIMENSIONLESS QUANTITIES
Reynolds number,
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vl
vl
vlR
vll
l
vF
vltll
tllF
F
FR
22
e
2
v
22
2
22
3i
v
ie
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FROUDE NUMBER 24
gl
v
gl
v
gl
vlF
glF
vlF
F
FF
2
3
22
r
3
g
22i
g
ir
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Euler number
Mach number
Weber number
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1.6 THEORY OF HYDRAULIC MODELS
Model-Is generally asmall scalesystem of the
prototype.Prototype- Is the full sizestructure
employed inthe actualengineeringdesign.
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OBJECTIVES OF MODEL STUDIES27
To study the geometrical appearance & relativeproportions of the system
To examine the flow paths over, through or aroundthe objects
To determine the pressure distribution & theresulting forces on the system
To estimate the flow capacity
To ascertain the overall performance of a hydraulicmachine
To check whether the prototype will give theintended performance
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APPLICATIONS OF MODEL STUDIES28
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1.7 SIMILITUDE29
The relationship between model and prototypeperformance is governed by the laws of hydraulicsimilarity .
True models have all significant characteristics of theprototype reproduced to scale i.e. (geometrically similar) and satisfy design restrictions (kinematicand dynamic similitude).
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The concept of dimensional analysis
Three necessary conditions for complete similarity between a model and a prototype.
Geometric Similarity – the model must be thesame shape as the prototype. Each dimension must be
scaled by the same factor. Kinematic Similarity – velocity as any point in the
model must be proportional by a constant scale factor. Dynamic Similarity – all forces in the model flow
scale by a constant factor to corresponding forces in
the prototype flow. Complete Similarity is achieved only if all 3
conditions are met. This is not always possible, e.g.,ship models.