module-5 dimensional analysis and similitude

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Module-5

Dimensional Analysis and Similitude

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98795 10743

Applied Fluid Mechanics (2160602)

13-11-2019

Prof. Mehul Pujara

9Module-5 Dimensional Analysis and Similitude Darshan Institute of Engineering & Technology, Rajkot

INTRODUCTION:

Dimension Analysis:

It is a mathematical technique whichmake use of the study of dimensionsas an aid to the solution ofengineering problems.

Dimensional analysis helps in determininga systematic arrangement of variables inthe physical relationship and combiningdimensional variables to form nondimensional parameters.

Ex. ρ,v, d, μ to form Reynolds number

It is useful in both analytical andexperimental investigation.


Quantity: numbers or units

Fluid characteristics

Quality: w.r.t primary quantity

Four basic primary or fundamental quantities:

Length – L

Mass – M

Time – T

Temperature – θ

These are basic dimension to describe the secondary quantities.

INTRODUCTION:


Uses:

1. Testing the dimensional homogeneityof any equation of fluid motion.

Ex. v = v0 + at

LT-1 = LT-1 + (LT-2)T

LT-1 = LT-1

2. Deriving equations expressed in termsof non dimensional parameters toshow the relative significance of eachparameter.

Re =ρ v d

μ

INTRODUCTION:


Uses:

3. Planning model tests andpresenting experimental results ina systematic manner in terms ofnon dimensional parameters; thusmaking it possible to analyze thecomplex fluid flow phenomenon.

INTRODUCTION:


PRIMARY or FUNDAMENTAL QUNTITIES:


Those quantities which possess more than on fundamental dimension or quantity.

DERIVED or SECONDARY QUANTITIES:


Determine the dimension of

1. work

2. pressure

3. specific weight

4. kinematic viscosity

5. discharge

6. shear stress

7. angular acceleration

8. angular velocity

Exercise:


Determine the dimension of

1. work ML2T-2

2. pressure ML-1T-2

3. specific weight ML-2T-2

4. kinematic viscosity L2T-1

5. discharge L3T-1

6. shear stress ML-1T-2

7. angular acceleration T-2

8. angular velocity T-1

Exercise:


1. Rayleigh Method

2. Buckingham π- method

METHODS OF DIMENSIOTNAL ANALYSIS:


The model is the small scale replica ofthe actual structure or machine.

The actual structure or machine is calledprototype.

The model analysis is the experimentaltechnique of finding solution of complexproblem.

It has following advantages:

1. The performance of structure ormachine can be easily predicted.

2. Relationship between the variablesinfluencing a flow problem can bedevelop using dimension analysis.

3. To check similarity exists betweenmodel and prototype.

MODEL ANALYSIS:


Similitude : similarity between modeland prototype.

Type of similarity:

1. Geometric similarity

2. Kinematic similarity

3. Dynamic similarity

SIMILITUDE – TYPES OF SIMILARITIES:


1. Geometric similarity:

It is exist if the ratio of all correspondinglinear dimension in the model andprototype are equal.

𝐿𝑝

𝐿𝑚

=𝑊

𝑝

𝑊𝑚

=𝐷𝑝

𝐷𝑚

= Lr = scale ratio

𝐴𝑝

𝐴𝑚

=𝐿𝑝

𝐿𝑚

𝑊𝑝

𝑊𝑚

= Lr Lr = Area’s ratio

𝑉𝑝

𝑉𝑚

=𝐿𝑝

𝐿𝑚

3 =𝑊

𝑝

𝑊𝑚

3 = Volume’s ratio



2. Kinematic Similarity:

It means the similarity of motionbetween model and prototype.

𝑣𝑝1

𝑣𝑚1

=𝑣𝑝2

𝑣𝑚2

= vr = Velocity ratio

𝑎𝑝1

𝑎𝑚1

=𝑎𝑝2

𝑎𝑚2

= ar = Acceleration ratio



3. Dynamic Similarity:

It means the similarity of forces betweenmodel and prototype.

(𝐹𝑖 )

𝑝

𝐹𝑖 𝑚

=(𝐹𝑣 )

𝑝

𝐹𝑣 𝑚

=(𝐹𝑔 )

𝑝

𝐹𝑔 𝑚

= Force ratio



The forces are

1. Inertia force, Fi

2. Viscous force, Fv

3. Gravity force, Fg

4. Pressure force, Fp

5. Surface tension force, Fs

6. Elastic force, Fe

TYPES OF FORCES ACTING ON FLUID:


1. Inertia force, Fi :

It is equal to the product of mass and acceleration of the flowing fluid and acts inthe direction opposite to the direction of acceleration.

2. Viscous force, Fv :

It is equal to the product of shear stress due to viscosity and surface area of theflow.

3. Gravity force, Fg :

It is equal to the product of mass and acceleration due to gravity of the flowingfluid.

4. Pressure force, Fp :

It is equal to the product of pressure intensity and cross-sectional area of theflowing fluid.

5. Surface tension force, Fs :

It is equal to the product of surface tension and length of surface of the flowingfluid.

6. Elastic force, Fe :

It is equal to product of elastic stress and area of the flowing fluid.

TYPES OF FORCES ACTING ON FLUID:


1. Reynold’s number

2. Froude’s number

3. Euler’s number

4. Weber’s number

5. Mach’s number

DIMENSIONLESS NUMBERS:


Ratio of inertia force of a flowing fluid and viscous force of the fluid.

Re = Fi / Fv

Inertia force = Mass * Acceleration of flowing fluid

= Density * Volume * Velocity/time

= ρ * Q * v

= ρ * A * v * v

= ρ * A * v2

Viscous force = Shear stress * Area (τ = μ du/dy)

= μ du/dy * A

= μ v/L * A

1. Reynold Number:


Ratio of inertia force of a flowing fluid and viscous force of the fluid.

Re = Fi / Fv

=ρAv2

μv

LA

=ρvL

μ

Higher the Re, greater the inertia effect. Smaller the Re, greater the viscouseffect.

Examples of such situation:

I. Flow of incompressible fluid in a pipe

II. Motion of submarine

1. Reynold Number:


It is defined as the square root of the ratio of the inertia force andgravitational force.

Fr =Inertia force

gravitational force

=ρAv2

ρALg

=v2

Lg

=v

Lg

Example of such situation:

I. Flow of liquid jet from the orifice

II. Flow over notches, weirs of a dam

2. Froude’s Number:


It is defined as the square root of the ratio of the inertia force and pressureforce.

Fr =Inertia forcePressure force

=ρAv2

p A

=v2

p/ρ

=v

p/ρ


I. Flow through pipe

II. Discharge through orifice and mouthpieces

3. Euler’s Number:


It is defined as the square root of the ratio of the inertia force and surfacetension force.

Fr =Inertia force

Surface tension force

=ρAv2

σ 𝐿

=ρL2v2

σ 𝐿

=v

σ/ρ𝐿


I. Capillarity tube action

II. Flow of blood in veins and arteries

4. Weber’s Number:


It is defined as the square root of the ratio of the inertia force and elasticforce.

Fr =Inertia forceElastic force

=ρAv2

K 𝐿2

=ρL2v2

K 𝐿2

=v

K/ρ

=v

𝐶, K/ρ = velocity of sound in the fluid


I. Aerodynamic testing

II. Water hammer problem

5. Mach Number:


The laws on which the models are designed for dynamic similarity are called model laws or laws of similarity.

Followings are the model laws:

1. Reynold’s model law

2. Froude model law

3. Euler model law

4. Weber model law

5. Mach model law

MODEL LAWS OR SIMILARITY LAWS:


It is law in which models are based on Reynold’s number.

Model based on Reynold’s number includes:1. Pipe flow

2. Resistance experience by sub-marines, airplanes, fully immersed bodies etc.

Fluid flow problems where viscous forces alone are predominant, themodels are designed for dynamic similarity on Reynold’s law.

It states that “The Reynold’s number for the model must be equal to theReynold’s number for the prototype” .

[Re]m = [Re]p

ρmvmLm

μ𝑚

=ρpvpLp

μ𝑝

ρpvpLp

ρmvmLm

*1

μ𝑝/μ

𝑚

= 1

ρrvrLr

μ𝑟

= 1

1. Reynold’s Model Law:

Where,ρr = scale ratio for densityvr = scale ratio for velocityLr = scale ratio for linear dimensionμ 𝑟 = scale ratio for viscosity


Time scale ratio is:

tr =𝐿

𝑟

vr

Acceleration scale ratio is:

ar =𝑣

𝑟

tr

Force scale ratio is:

Fr = mr * ar

= ρrArvr * ar

= ρrL2rvr * ar

Discharge scale ratio is:

Qr = ρrArvr

=ρrL2rvr

1. Reynold’s Model Law:


It is law in which models are based on Froude number.

It is applicable to following fluid flow problems:1. Free surface flow such as flow over weirs, channels etc

2. Flow of jet from an orifice or nozzle

3. Where waves are likely to be formed over one another

4. Where fluids of different densities flow over one another

The law is applicable when the gravity force is only predominant force which controls the flow.

[Fe]m = [Fe]p

v𝑚

Lmgm

= v𝑝

Lpgp

If test performed on the same place then gm = gp equation becomes

v𝑚

Lm

= v𝑝

Lp

Lp

Lm

= v𝑝

v𝑚

v𝑝

v𝑚

= vr = Lr

2. Froude Model Law:


Scale ratio for various physical quantities based on Froude model law are:

1. Scale ratio for time:

Tr = Tp

Tm=

𝐿𝑝

vp

𝐿𝑚

vp

= 𝐿

𝑝

𝐿𝑚

* vm

vp

= Lr * 1

Lr

= Lr

2. Scale ratio for acceleration:

ar = ap

am=

𝑣𝑝

Tp

𝑣𝑚

Tp

= Lr * 1

Lr

= 1

3. Scale ratio for discharge:

Q = A * v = L2 * 𝐿

𝑇= 𝐿3

𝑇

Qr = Qp

Qm=

𝐿3

𝑇 𝑝

𝐿3

𝑇 𝑚

= L3r

* 1

Lr

= L2.5r



4. Scale ratio for force:

F = m * a = 𝜌 ∗ 𝑉 ∗ 𝑎 = 𝜌𝐿3 ∗𝑣

𝑇= 𝜌𝐿2 ∗

𝐿

𝑇* v = 𝜌𝐿2 𝑣2

Fr = 𝐹𝑃

𝐹𝑚

= 𝜌𝑝𝐿𝑝2 𝑣

𝑝2

𝜌𝑚𝐿𝑚2 𝑣

𝑚2

= 𝜌𝑝

𝜌𝑚

∗𝐿𝑝

𝐿𝑚

2 ∗𝑣𝑝

𝑣𝑚

2 { if fluid is same 𝜌𝑝 = 𝜌𝑚

= 𝐿𝑝

𝐿𝑚

2 ∗𝑣𝑝

𝑣𝑚

2

= 𝐿𝑟3

Similarly,

Scale ratio for pressure pr = 𝐿𝑟Torque Tr = 𝐿𝑟

4

Power Pr = 𝐿𝑟3.5



There are two types (i) undistorted model (ii) distorted model

If the models are geometrically similar to its prototype, the models are known as undistorted model.

If the models are having different ratio for the horizontal and vertical dimensions, the models are known as distorted model.

Hydraulic model:


References:

1. Fluid Mechanics and Fluid Power Engineering by D.S. Kumar, S.K.Kataria & Sons

2. Fluid Mechanics and Hydraulic Machines by R.K. Bansal, LaxmiPublications

3. Fluid Mechanics and Hydraulic Machines by R.K. Rajput, S.Chand & Co

4. Fluid Mechanics; Fundamentals and Applications by John. M. CimbalaYunus A. Cengel, McGraw-Hill Publication

module-5 dimensional analysis and similitude

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