Dimensional Analysis And Similitude & Model Analysis

Dimensional AnalysisIntroduction:

Dimensional analysis is a mathematical technique making use of study of dimensions.

This mathematical technique is used in research work for design and for conducting model tests.

It deals with the dimensions of physical quantities involved in the phenomenon. All physical quantities are measured by comparison, which is made with respect to an arbitrary fixed value.

Types of Dimensions:

Fundamental Dimensions or Fundamental Quantities Secondary Dimensions or Derived Quantities

Methodology of Dimensional Analysis The Basic principle is Dimensional Homogeneity, which

means the dimensions of each terms in an equation on both sides are equal.

So such an equation, in which dimensions of each term on both sides of equation are same, is known as Dimensionally Homogeneous equation. Such equations are independent of system of units. For example;

Lets consider the equation V=(2gH)1/2

Dimensions of LHS=V=L/T=LT-1

Dimensions of RHS=(2gH)1/2=(L/T2xL)1/2=LT-1

Dimensions of LHS= Dimensions of RHS So the equation V=(2gH)1/2 is dimensionally

homogeneous equation.

Methods of Dimensional Analysis If the number of variables involved in a physical phenomenon are

known, then the relation among the variables can be determined by the following two methods;

Rayleigh’s Method: It is used for determining expression for a variable (dependent)

which depends upon maximum three to four variables (Independent) only.

If the number of independent variables are more than 4 then it is very difficult to obtain expression for dependent variable.

Buckingham’s π(pie)-Theorem: Since Rayleigh’s Method becomes laborious if variables are more than fundamental dimensions (MLT), so the difficulty is overcome by Buckingham’s π-Theorem which states that

“If there are n variables (Independent and Dependent) in a physical phenomenon and if these variables contain m fundamental dimensions then the variables are arranged into (n-m) dimensionless terms which are called π-terms.”

Similitude and Model Analysis Similitude is a concept used in testing of Engineering

Models.

Usually, it is impossible to obtain a pure theoretical solution of hydraulic phenomenon.

Therefore experimental investigations are often performed on small scale models, called model analysis.

A few examples, where models may be used are ships in towing basins, air planes in wind tunnel, hydraulic turbines, centrifugal pumps, spillways of dams, river channels etc and to study such phenomenon as the action of waves and tides on beaches, soil erosion, and transportation of sediment etc.

Model Analysis Model: is a small scale replica of the actual structure. Prototype: the actual structure or machine. Note: It is not necessary that the models should be

smaller that the prototype, they may be larger than prototype (e.g. Water Treatment Plant, Floatation Columns).

Prototype Model

Lp3

Lp1

Lp2Fp1

Fp3

Fp2

Lm

3

Lm1

Lm2Fm

1

Fm

3

Fm

2

Model Analysis Model Analysis is actually an experimental method of

finding solutions of complex flow problems.

The followings are the advantages of the model analysis The performance of the hydraulic structure can be predicted in

advance from its model. Using dimensional analysis, a relationship between the variables

influencing a flow problem is obtained which help in conducting tests.

The merits of alternative design can be predicted with the help of model analysis to adopt most economical, and safe design.

Note: Test performed on models can be utilized for obtaining, in advance, useful information about the performance of the prototype only if a complete similarity exits between the model and the prototype.

Similitude-Type of Similarities Similitude: is defined as similarity between the model

and prototype in every respect, which mean model and prototype have similar properties or model and prototype are completely similar.

Three types of similarities must exist between model and prototype.

Geometric SimilarityKinematic SimilarityDynamic Similarity

Similitude-Type of Similarities Geometric Similarity: is the similarity of shape. It is said to exist

between model and prototype if ratio of all the corresponding linear dimensions in the model and prototype are equal. E.g.

p p pr

m m m

L B DL

L B D

Where: LWhere: Lpp, B, Bpp and D and Dpp are Length, Breadth, and diameter of are Length, Breadth, and diameter of prototype and Lprototype and Lmm, B, Bmm, D, Dmm are Length, Breadth, and diameter are Length, Breadth, and diameter of model.of model.

Lr= Scale ratioLr= Scale ratio

Note:Note: Models are generally prepared with same scale ratios Models are generally prepared with same scale ratios in every direction. Such a model is called true model. in every direction. Such a model is called true model. However, sometimes it is not possible to do so and different However, sometimes it is not possible to do so and different convenient scales are used in different directions. Such a convenient scales are used in different directions. Such a models is call distorted modelmodels is call distorted model

Similitude-Type of Similarities Kinematic Similarity: is the similarity of motion. It is said to exist

between model and prototype if ratio of velocities and acceleration at the corresponding points in the model and prototype are equal. E.g.

1 2 1 2

1 2 1 2

;p p p pr r

m m m m

V V a aV a

V V a a

Where: VWhere: Vp1p1& V& Vp2p2 and a and ap1p1 & a & ap2p2 are velocity and accelerations are velocity and accelerations at point 1 & 2 in prototype and Vat point 1 & 2 in prototype and Vm1m1& V& Vm2m2 and a and am1m1 & a & am2m2 are are velocity and accelerations at point 1 & 2 in model.velocity and accelerations at point 1 & 2 in model.

VVrr and a and arr are the velocity ratio and acceleration ratio are the velocity ratio and acceleration ratio

Note:Note: Since velocity and acceleration are vector quantities, Since velocity and acceleration are vector quantities, hence not only the ratio of magnitude of velocity and hence not only the ratio of magnitude of velocity and acceleration at the corresponding points in model and acceleration at the corresponding points in model and prototype should be same; but the direction of velocity and prototype should be same; but the direction of velocity and acceleration at the corresponding points in model and acceleration at the corresponding points in model and prototype should also be parallel.prototype should also be parallel.

Similitude-Type of Similarities Dynamic Similarity: is the similarity of forces. It is said to exist

between model and prototype if ratio of forces at the corresponding points in the model and prototype are equal. E.g.

gi vp p pr

i v gm m m

FF FF

F F F

Where: (FWhere: (Fii))pp, (F, (Fvv))pp and (F and (Fgg))pp are inertia, viscous and are inertia, viscous and gravitational forces in prototype and (Fgravitational forces in prototype and (Fii))mm, (F, (Fvv))mm and (F and (Fgg))mm are are inertia, viscous and gravitational forces in model.inertia, viscous and gravitational forces in model.

FFrr is the Force ratio is the Force ratio

Note:Note: The direction of forces at the corresponding points in The direction of forces at the corresponding points in model and prototype should also be parallel.model and prototype should also be parallel.

Types of forces encountered in fluid Phenomenon

Inertia Force, Fi: It is equal to product of mass and acceleration in the flowing fluid.

Viscous Force, Fv: It is equal to the product of shear stress due to viscosity and surface area of flow.

Gravity Force, Fg: It is equal to product of mass and acceleration due to gravity.

Pressure Force, Fp: it is equal to product of pressure intensity and cross-sectional area of flowing fluid.

Surface Tension Force, Fs: It is equal to product of surface tension and length of surface of flowing fluid.

Elastic Force, Fe: It is equal to product of elastic stress and area of flowing fluid.

Dimensionless Numbers These are 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 ratio of once force to other, it will be a dimensionless number. These are also called non-dimensional parameters.

The following are most important dimensionless

numbers. Reynold’s Number Froude’s Number Euler’s Number Weber’s Number Mach’s Number

Dimensionless Numbers Reynold’s Number, Re: It is the ratio of inertia force to the viscous force of

flowing fluid.

. .Re

. .

. . .

. . .

Velocity VolumeMass VelocityFi Time Time

Fv Shear Stress Area Shear Stress Area

QV AV V AV V VL VLdu VA A Ady L

2

. .

. .

. .

. .

Velocity VolumeMass VelocityFi Time TimeFe

Fg Mass Gavitational Acceleraion Mass Gavitational Acceleraion

QV AV V V V

Volume g AL g gL gL

Froude’s Number, Re:Froude’s Number, Re: It is the ratio of inertia force to the It is the ratio of inertia force to the gravity force of flowing fluid.gravity force of flowing fluid.

Dimensionless Numbers Eulers’s Number, Re: It is the ratio of inertia force to the pressure force of

flowing fluid.

2

. .

Pr . Pr .

. .

. . / /

u

Velocity VolumeMass VelocityFi Time TimeE

Fp essure Area essure Area

QV AV V V V

P A P A P P

2 2

. .

. .

. .

. . .

Velocity VolumeMass VelocityFi Time TimeWe

Fg Surface Tensionper Length Surface Tensionper Length

QV AV V LV V

L L L

L

Weber’s Number, Re:Weber’s Number, Re: It is the ratio of inertia force to the It is the ratio of inertia force to the surface tension force of flowing fluid.surface tension force of flowing fluid.

Dimensionless Numbers Mach’s Number, Re: It is the ratio of inertia force to the elastic force of

flowing fluid.

2 2

2

. .

. .

. .

. . /

: /

Velocity VolumeMass VelocityFi Time TimeM

Fe Elastic Stress Area Elastic Stress Area

QV AV V LV V V

K A K A KL CK

Where C K

Model Laws or similarity Laws We have already read that for dynamic similarity ratio of corresponding

forces acting on prototype and model should be equal. i.e

g pv s e Ip p p p p p

v s e Ig pm m m mm m

F FF F F F

F F F FF F

Thus dynamic similarity require that

v g p s e I

v g p s e Ip p

Iv g p s e mm

F F F F F F

F F F F F F

FF F F F F

Force of inertial comes in play when sum of all other forces is Force of inertial comes in play when sum of all other forces is not equal to zero which mean not equal to zero which mean

In case all the forces are equally important, the above two In case all the forces are equally important, the above two equations cannot be satisfied for model analysis equations cannot be satisfied for model analysis

Model Laws or similarity Laws However, for practical problems it is seen that one force

is most significant compared to other and is called predominant force or most significant force.

Thus for practical problem only the most significant force is considered for dynamic similarity. Hence, models are designed on the basis of ratio of force, which is dominating in the phenomenon.

Finally the laws on which models are designed for dynamic similarity are called models laws or laws of similarity. The followings are these laws Reynold’s Model Law Froude’s Model Law Euler’s Model Law Weber’s Model Law mach’s Model Law

Reynold’s Model Law It is based on Reynold’s number and states that Reynold’s number

for model must be equal to the Reynolds number for prototype. Reynolds Model Law is used in problems where viscous forces are

dominant. These problems include: Pipe Flow Resistance experienced by submarines, airplanes, fully immersed bodies

etc.

Re Re

1

: , ,

m mP PP m

P m

P P r r

rPm m

m

P P Pr r r

m m m

V LV Lor

V L V L

V L

V Lwhere V L

V L

Reynold’s Model Law The Various Ratios for Reynolds’s Law are obtained as

rr

r

P P P r

m m m r

P Pr

m m

2r

r

sin /

Velocity Ratio: V =L

T L /V LTime Ratio: Tr=

T L /V V

V / VrAcceleration Ratio: a =

V / Tr

Discharge Ratio: Q

Force Ratio: F =

P m

mP P

m P m

P

m

P Pr r

m m

VL VLce and

LV

V L

a T

a T

A VLV

A V

m

2 2 2

2 2 2 3r r rPower Ratio: P =F .V =

r r r r r r r r r r r r

r r r r r r r

a QV LV V LV

LV V LV

Reynold’s Model Law Q. A pipe of diameter 1.5 m is required to transport an oil of specific

gravity 0.90 and viscosity 3x10-2 poise at the rate of 3000litre/sec. Tests were conducted on a 15 cm diameter pipe using water at 20oC. Find the velocity and rate of flow in the model.

p p p p pm m m

m m

2

2

p 2

For pipe flow,

According to Reynolds' Model Law

V D DV D

D

900 1.5 1 103.0

1000 0.15 3 10

3.0Since V

/ 4(1.5)

1.697 /

3.0 5.091 /

5.

m m

m p p p

m

p

p

p

m p

m m m

V

V

V

V

Q

A

m s

V V m s

and Q V A

2

3

091 / 4(0.15)

0.0899 /m s

Solution: Prototype Data:

Diameter, Dp= 1.5m Viscosity of fluid, μp= 3x10-2 poise Discharge, Qp =3000litre/sec Sp. Gr., Sp=0.9 Density of oil=ρp=0.9x1000 =900kg/m3

Model Data: Diameter, Dm=15cm =0.15 m Viscosity of water, μm =1x10-2

poise Density of water, ρm=1000kg/m3

Velocity of flow Vm=? Discharge Qm=?

Froude’s Model Law It is based on Froude’s number and states that Froude’s

number for model must be equal to the Froude’s number for prototype.

Froude’s Model Law is used in problems where gravity forces is only dominant to control flow in addition to inertia force. These problems include: Free surface flows such as flow over spillways, weirs, sluices,

channels etc. Flow of jet from orifice or nozzle Waves on surface of fluid Motion of fluids with different viscosities over one another

e e

/ 1; : ,

m mP PP m

P P m m P m

P P Pr r r r

m mPm

m

V VV VF F or or

g L g L L L

V V LV L where V L

V LLV

L

Froude’s Model Law The Various Ratios for Reynolds’s Law are obtained as

r

P P P r

m m m

P Pr

m m

2 2 5/ 2r

sin

Velocity Ratio: V

T L /V LTime Ratio: Tr=

T L /V

V / VrAcceleration Ratio: a = 1

V / Tr

Discharge Ratio: Q

Force Ratio: Fr=

mP

P m

pPr

m m

r

r

rP

m r

P Pr r r r r

m m

r r

VVce

L L

LVL

V L

LL

La T

a T L

A VLV L L L

A V

m a

2 2 2 2 3

32 2 2 3 2 7 / 2Power Ratio: Pr=Fr.Vr=

r r r r r r r r r r r r r r r

r r r r r r r r r r r r

QV LV V LV L L L

L V V LV L L L

Froude’s Model Law Q. In the model test of a spillway the discharge and velocity of flow

over the model were 2 m3/s and 1.5 m/s respectively. Calculate the velocity and discharge over the prototype which is 36 times the model size.

2.5 2.5p

m

2.5 3

For Discharge

Q36

Q

36 2 15552 / sec

r

p

L

Q m

p

m

For Dynamic Similarity,

Froude Model Law is used

V36 6

V

6 1.5 9 / sec

r

p

L

V m

Solution: Given that

For Model Discharge over model, Qm=2

m3/sec Velocity over model, Vm = 1.5

m/sec Linear Scale ratio, Lr =36

For Prototype Discharge over prototype, Qp =? Velocity over prototype Vp=?

Numerical Problem: Q. The characteristics of the spillway are to be studied by means of a geometrically

similar model constructed to a scale of 1:10. (i) If 28.3 cumecs, is the maximum rate of flow in prototype, what will be the

corresponding flow in model? (i) If 2.4m/sec, 50mm and 3.5 Nm are values of velocity at a point on the spillway, height

of hydraulic jump and energy dissipated per second in model, what will be the corresponding velocity height of hydraulic jump and energy dissipation per second in prototype?

Solution: Given thatFor Model

Discharge over model, Qm=? Velocity over model, Vm = 2.4 m/sec Height of hydraulic jump, Hm =50 mm Energy dissipation per second, Em =3.5 Nm Linear Scale ratio, Lr =10

For Prototype Discharge over model, Qp=28.3 m3/sec Velocity over model, Vp =? Height of hydraulic jump, Hp =? Energy dissipation per second, Ep =?

Froude’s Model Law

p 2.5 2.5

m

2.5 3

p

m

For Discharge:

Q10

Q

28.3 /10 0.0895 / sec

For Velocity:

V10

V

2.4 10 7.589 / sec

r

m

r

p

L

Q m

L

V m

p

m

p 3.5 3.5

m

3.5

For Hydraulic Jump:

H10

H

50 10 500

For Energy Dissipation:

E10

E

3.5 10 11067.9 / sec

r

p

r

p

L

H mm

L

E Nm

Classification of Models Undistorted or True Models: are those which are

geometrically similar to prototype or in other words if the scale ratio for linear dimensions of the model and its prototype is same, the models is called undistorted model. The behavior of prototype can be easily predicted from the results of undistorted or true model.

Undistorted Models: A model is said to be distorted if it is not geometrically similar to its prototype. For distorted models different scale ratios for linear dimension are used.

For example, if for the river, both horizontal and vertical scale ratio are taken to be same, then depth of water in the model of river will be very very small which may not be measured accurately.

The followings are the advantages of distorted models

The vertical dimension of the model can be accurately measured

The cost of the model can be reduced Turbulent flow in the model can be maintained

Though there are some advantage of distorted models, however the results of such models cannot be directly transferred to prototype.

Classification of Models Scale Ratios for Distorted Models

r

r

P

P

Let: L = Scale ratio for horizontal direction

L =Scale ratio for vertical direction

2Scale Ratio for Velocity: Vr=V /

2

Scale Ratio for area of flow: Ar=A /

P PH

m m

PV

m

Pm r V

m

P Pm

m m

L B

L B

h

h

ghV L

gh

B hA

B h

3/ 2

PScale Ratio for discharge: Qr=Q /V

r rH V

P Pm r r r r rH V V H

m m

L L

A VQ L L L L L

A V

Distorted model Q. The discharge through a weir is 1.5 m3/s. Find the discharge through the model of weir if the horizontal dimensions of

the model=1/50 the horizontal dimension of prototype and vertical dimension of model =1/10 the vertical dimension of prototype.

3p

r

r

3/ 2

P

3/ 2

Solution:

Discharge of River= Q =1.5m /s

Scale ratio for horizontal direction= L =50

Scale ratio for vertical direction= L =10

Since Scale Ratio for discharge: Qr=Q /

/ 50 10

V

PH

m

PV

m

m r rH

p m

L

L

h

h

Q L L

Q Q

3

1581.14

1.5 /1581.14 0.000948 /mQ m s

Distorted model Q. A river model is to be constructed to a vertical scale of 1:50 and a

horizontal of 1:200. At the design flood discharge of 450m3/sec, the average width and depth of flow are 60m and 4.2m respectively. Determine the corresponding discharge in model and check the Reynolds’ Number of the model flow.

3

r

r

3/ 2

r P

3/ 2

arg 450 /

60 4.2

Horizontal scale ratio= L =200

Vertical scale ratio= L =50

Since Scale Ratio for discharge: Q =Q /

/ 200 50 7

V

p

p p

PH

m

PV

m

m r rH

p m

Disch e of River Q m s

Width B m and Depth y m

B

B

y

y

Q L L

Q Q

3 3

0710.7

450 /1581.14 6.365 10 /mQ m s

Distorted model

m

VLReynolds Number, Re =

4

/ 60 / 200 0.3

/ 4.2 / 50 0.084

0.3 0.084 0.0252

2 0.3 2 0.084 0.468

0.02520.05385

0.468

Kinematic Viscosity of w

m

m m

m p r H

m p r V

m m m

m m m

m

m

L R

Width B B L m

Depth y y L m

A B y m

P B y m

AR

P

6 2

6

ater = =1 10 / sec

4 4 0.253 0.05385Re 54492.31

1 10

>2000

Flow is in turbulent range

m

m

VR