Dimensional Analysis 1-Engineering

7/23/2019 Dimensional Analysis 1-Engineering

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CE 2033

1

FLUID MECHANICS II

Dr WCDK FERNANDO

Department of Civil Engineering



COURSE CONTENT

1. Dimensional and Hydraulic Model Analysis

2. Ideal Flow

3. Boundary Layer Theory

4. Hydraulic Machines

2



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



ASSESSMENT CRITERIA 4

Semester – end Examination – 70%

Assignments – 20%

Practical – 10%



TYPES OF ASSIGNMENTS5

Class test

Group work

MCQ



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



DIMENSIONAL & HYDRAULICMODEL ANALYSIS



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.



MODEL ANALYSIS9



10



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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.



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.



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.



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



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.



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.



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.



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,

23

vl

vl

vlR

vll

l

vF

vltll

tllF

F

FR

22

e

2

v

22

2

22

3i

v

ie



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



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



APPLICATIONS OF MODEL STUDIES28



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).



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.

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Dimensional Analysis 1-Engineering