Unit1_DimensionalAnalysis

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X1 is a function ofX2,X3,...Xn and mathematically, it is expressed as

X1 = f(X2,X3,...Xn). (1)

This equation is a dimensionally homogeneous equation. It contains n variables. If there

are

m

fundamental dimensions then according to Buckingham theorem, equation ??can bewritten in terms of-terms which is equal to nm. Hence equation becomes

F(1,2,3,...nm) = 0. (2)

Each of-terms is dimensionless and independent of the system. Division or multiplication

by a constant does not change the character of the -term. Each of-term contains (m+ 1)

variables, where mis the number of fundamental dimensions and is also called repeating vari-

ables. Let min the above caseX2,X3 andX4 are repeating variables, if fundamental dimensions

(M, L, T) = 3then each-term is written as

1 = Xa12 X

b13 X

c14 X

15, (3)

2 = Xa22 X

b23 X

c24 X

16, (4)

nm = Xanm2 X

bnm3 X

cnm4 X

1n. (5)

Each equation is solved by the principle of dimensional homogeneity and values ofa1,b1, c1

etc., are obtained. These values are substituted in equation ?? and values 1,2,nm are ob-

tained. These values of are substituted in equation ??. The final equation for the phenomenon

is obtained by expressing any one of the -terms as a function of others as

1 = (2,3.....nm),

2 = (1,3.....nm).

Selection of repeating variables

There is no separate rule for selecting repeating variables. But the number of repeating variables

is equal to the fundamental dimensions of the problem. Generally, , and l or , and D are

chosen as repeating variables. It means, one refers to fluid property , one refers to flow property

and the other one refers to geometric property lorD. In addition to this, the following points

should be kept in mind while selecting the repeating variables:

1. The selected variables should not be in dimensionless form.

2. The selected two repeating variables should not have the same dimensions. For example,

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flow velocity and speed of sound have same dimension ofm/s. Then flow velocity and speed of

sound should not be selected as repeating variables.

3. The selected repeating variables should be independent as far as possible.

Steps to be followed in Buckingham method

1. First the variables involved in a given analysis are listed to study about the given phenomenon

thoroughly.

2. Then, these variables are expressed in terms of primary dimensions.

3. Next, the repeating variables are chosen according to the hint given in selection of repeating

variables. Once, the repeating variables should be checked either those are independent or

dependent variables because all should be independent variables.

4. Then the dimensionless parameters are obtained by adding one at a time with repeating

variables.

5. The number of pi-terms involved in dimensional analysis is calculated by using n-m=number

of pi-terms. Where, n = total number of variables involved in given analysis and m = number of

fundamental variables.

6. Finally, each equation in exponential form is solved which means the coefficients of expo-

nents are found by comparing both sides exponents. Then these dimensionless parameters are

recombined and arranged suitably.

LIMITATIONS OF DIMENSIONAL ANALYSIS

1. Dimensional analysis does not give any rule regarding the selection of variables. 2. The

complete information is not provided by dimensional analysis. It only indicates that there are

some relationships between parameters. 3. The values of co-efficient and the nature of function

can be obtained only by experiments or from mathematical analysis. Since the inertia force is

always present in a fluid flow, its ratio with each of the other forces provides a dimensionlessnumber. These have been discussed below.

Similitude

Similitude is defined as the complete similarity between model and prototype. Complete sim-

ilarity is attained, if the following three types is similarity is attained, if the following three of

similarities exist: (1) Geometric similarity, (2) Kinematic similarity and (3) Dynamic similarity.

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Geometric similarity

A model and its prototype are geometrically similar, if the ratios of the corresponding length

dimensions are equal. LetLm,bm,Dm,Am and Lp,bp,Dp,Ap are the length, breath, diameter

and area of a model and a prototype respectively.

For geometric similarity between model and prototype, the relations,LpLm

= bp

bm= Dp

Dm= Lr = length scale ratio.

Ap

Am= Lpbp

Lmbm= L2r = Area scale ratio.

Kinematic similarity

Kinematic similarity is the similarity of motion. It corresponds to the points in the model and

prototype. If the acceleration ratios and acceleration vectors points are same in the same direc-

tion, then two flows are said to be kinematically similar.

Time scale ratio, = Tr = Tp

Tm.

Velocity scale ratio, = Vr =

LpTp

LmTm

=Lr

Tr.

Acceleration scale ratio, = ar =

LpT2p

LmT2m

= Lr

T2r.

Dynamic Similarity

It is the similarity of forces. the flows in the model and prototype are of dynamic similar. In

dynamic similarity, the force polygon of the two flows can be superimposed by change in scale.

let, (Fi)p = inertia force at all points in prototype

(Fv)p = Viscous force at the point in prototype

(Fg)p = gravity force at the points in prototype, and

(Fi)m, (Fv)m, (Fg)mare corresponding values of the force in the model. thus,

(Fi)p

(Fi)m= (Fv)p

(Fv)m= (Fg)p

(Fg)m= Fr = Forceratio (6)

It is very difficult to reproduce flight conditions exactly in wind tunnel experiments, whether the

body is moving through a stationary gas or the gas past a stationary body. Models are therefore

commonly used in wind tunnels of limited dimensions, to predict the behavior of prototypes in

flight.

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1 Dimensionless Parameters

1.1 Reynolds Number

It is defined as the ratio of the inertia force to the viscous force of a flowing fluid denoted by Re

Re = inertiaforceviscousforce

Re = 2

L2

L (7)

1.2 Froude Number

It is defined as the square root of the ratio of the inertia force of a flowing fluid to the gravity

force.

Fr =

inertiaforcegravityforce

Fr =L22

L3g

(8)

1.3 Euler Number

It is defined as the square root of the ratio of inertia force to the pressure force of a flowing fluid.

Eu =

inertiaforcepressureforce

Eu =

L22

pL2 (9)

1.4 Weber Number

It is the ratio of the square root of the inertia force to the surface tension force.

We =

inertiaforcesurfacetensionforce

We =

L22

L (10)

1.5 Mach Number

It is defined as the square root of the inertia force of flowing fluid to the elastic force.

M =

inertiaforceelasticforce

M =

L22

kL2 (11)

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