1MODEL ANALYSIS

Before constructing or manufacturing hydraulics structures or hydraulicsmachines tests are performed on their models to obtain desired information about theirperformance. Models are small scale replica of actual structure o r machine. Theactual structure is called prototype.

Similitude / SimilarityIt is defined as the similarity between the prototype and its model.

Types of SimilarityThere are three types of similarity.o Geometric similarityo Kinematic similarityo Dynamic similarity

Geometrical SimilarityGeometric similarity is said to exist between the model and prototype if the

ratio of corresponding linear dimensions between model and prototype are equal.

i.e. rm

p

m

p

m

p L............HH

hh

LL

Lr scale ratio / linear ratio

3rm

p2r

m

p LVV

LAA

Kinematic SimilarityKinematic similarity exists between prototype and model if quantities such at

velocity and acceleration at corresponding points on model and prototype are same.

r

m3

p3

m2

p2

m1

p1 V...........VV

VV

VV

Vr Velocity ratio

Dynamic SimilarityDynamic similarity is said to exist between model and prototype if ratio of

forces at corresponding points of model and prototype is constant.

R

m3

p3

m2

p2

m1

p1 F...........FF

FF

FF

FR Force ratio

2 Dimensionless NumbersFollowing dimensionless numbers are used in fluid mechanics.1. Reynolds number2. Froudes number3. Eulers number4. Webers number5. Mach number

1. Reynolds numberIt is defined as the ratio of inertia force of the fluid to viscous force.

v

iRe F

FN

Expression for NReFi = Mass x Acceleration

Fi = x Volume x Acceleration

Fi = x Volume x Timeyin velocitChange

Fi = x Q x VFi = AV2

FV Viscous forceFV = x A

FV = AyV

FV = ALV

NRe =A

LV

AV2

NRe =

VL

In case of pipeline diameter is the linear dimension.

NRe =

VD

32. Froudes Number (Fr)It is defined as the ratio of square root of inertia force to gravity force.

Fr =g

i

FF

Fi = m x a

Fi = x Volume x Acceleration

Fi = AV2

Fg = m x g

Fg = x Volume x g

Fg = x A x L x g

F = gxLxAxAV2

F = LgV2

F =LgV

3. Eulers Number (u)It is defined as the square root of ratio of inertia force to pressure force.

u =p

i

FF

Fi = Mass x Acceleration

Fi = x Volume x TimeVelocity

Fi = x Q x VFi = AV2

Fp = p x A

u = pV

pAAV2

4u =

pv

4. Webers Number (Wb)It is defined as the square root of ratio of inertia force to surface tensile force.

Wb =p

i

FF

Fb = AV2

Fs = x L

Wb =

LVL

AV2

Wb =

L

V

5. Mach Number (M)It is defined as the square root of ratio of inertia force to elastic force.

M =e

iFF

Fi = AV2

Fe = K x AK Bulk modulus of elasticityA Area

M =KAAV2

M =/K

V

M =CV

C Velocity of sound in fluid.

5MODEL LAWS (SIMILARITY LAWS)

1. Reynolds Model LawFor the flows where in addition to inertia force, similarity of flow in model

and predominant force, similarity of flow in model and prototype can be established ifRe is same for both the system.

This is known as Reynolds Model Law.

Re for model = Re for prototype(NRe)m = (NRe)p

pm

VDVD

p

m

ppp

mmm

DVDV

= 1

1DVr

rrr

Applications:i) In flow of incompressible fluids in closed pipes.ii) Motion of submarine completely under water.iii) Motion of air-planes.

2. Froudes Model LawWhen the force of gravity is predominant in addition to inertia force then

similarity can be established by Froudes number. This is known as Froudes modellaw.

(Fr)m = (Fr)p

pmgLV

gLV

rgLV

= 1

Applications:i) Flow over spillways.ii) Channels, rivers (free surface flows).iii) Waves on surface.iv) Flow of different density fluids one above the other.