similitude in hydrosimilitude in hydrodynamicsdynamics

8/13/2019 Similitude in Hydrosimilitude in hydrodynamicsdynamics

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Similitude in Hydrodynamic

Problems

Department of Ocean EngineeringIndian Institute of Technology Madras

Chennai 600 036

Prof.V.G.Idichandy


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Introduction

Hydrodynamic problems need to be separately treated.

Difficult to achieve complete similitude

There are many more problems that require solution using

an empirical approaches based on experimental data. Background for deriving appropriate scaling relationship

for hydrodynamic model tests.

Complete similarity models are models in which the values

of all relevant dimension less parameters in the prototypeare maintained in the model.


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Geometric similarity

Geometrically similar models are undistorted models

Horizontal and vertical scales are the same

The model is a true geometric reproduction.

Definite objectives can be achieved by departing formgeometric similarity.

Such models are called distorted models.


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Kinematic similarity

In hydrodynamic model kinematic similarity is achieved

when the ratio between the components of all vertical

motions for the prototype and the model are the same forall particles at all times.

In a geometrically similar model, the kinematic similarity

gives particles paths that are geometrically similar toprototype.


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Kinematically similar wave motion

What is the scaling criteria necessary for kinematically

similar wave motion for gravity waves whose length is

given by

L is the wave lengthg acceleration due to gravity

T wave period

h water depth

2 2tanh

2

gT hL

L


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2h/L is dimension lessThis ratio for model and prototype should be an invariant

for geometrically similar model.

2h/L is the same, then the hyperbolic tangent will also bethe same.

The scale relationship between L and T is found from theprototype to model ratio of wave length.


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Kinematically similar wave

motion

2

2

2

2

2

tanh(2 / )2

tanh(2 /2

p p

m

m

p p p

m m m

e g T

gTh L

L

L gT h L

L g T

L g T


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g= 1

Therefore the kinematic similarity of wave motions can

be achieved by making

T l


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Dynamic Similarity

Kinematic similarity is achieved without considering any

other properties of the fluid

But this is not so for dynamic similarity.

Dynamic similarity between two geometrically and

kinematically similar system requires that the ratio's of all

vectorial forces in the two systems be the same.

In other words, there must be constant prototype to model

ratio's of all masses and forces acting on the system.


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Dynamic Similarity

For fluid mechanics problems Newton's second law can be written as,

Fi= Fg+ F+ F + Fe+ Fpr

Where Fi inertial force, mass x accelaeration

Fg gravitational force

F viscous forceF surface tension forceFe elastic compression force

pr pressure force


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Dynamic Similarity

Forces being vectors both magnitude and directionmust be represented.

Overall dynamic similarity requires that the ratio of

the inertial forces between prototype and model be

equal to the ratio of the sum of all the active forces

expressed as,

g e pri p p

i g e pr m m

F F F F FF

F F F F F F


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Dynamic Similarity

Perfect similitude requires in addition to the above that all

force ratios between the prototype and model be equal or,

or in terms of scale ratios,

g pri ep p p p p p

i eg prm m m mm m

F FF F F F

F F F FF F

i g e pr F F F F F F


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Dynamic Similarity

No fluid is known that will satisfy all force ratio

requirements, if the model is smaller than the prototype.

So an important task in scale model design is to relate

the important force ratios and to provide justification for

neglecting the others


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Practical aspects

Dynamic similarity is practically unachievable forscale models other than for scale factor 1

For model tests the similarity requirements have

to be violated

In model tests, it should be possible to justifydepartures and where required apply theoretical

corrections

It should also be possible to predict which of theforces are important for a given situation.


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Practical aspects

Almost any major problem can besimplified into the interplay of two majorforces.

Inertia forces are always present in flowproblems. So inertia needs to be balancedby one or more of the other forces.

The first step is to express each of theforces in terms of their physical units.


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Practical aspects

3 22 2

3

2

e

L acceleration L

accleration due to gravity = L g

Velocity VF cos area = V Ldistance L

F unit surface tension length = L

F modules of elastic

i

g

VF mass V

L

F mass

Vis ity L

2

2

ity area = E L

unit pressure area = p LprF

The ratio of the inertial force to any other force

provides the relative influence of the two forces in

hydrodynamic problems. Each force ratio leads

similitude criterion as given below.


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Froude Criterion

It is the parameter that express the relative

influence of inertial and gravitational

forces,expressed as the square root of theratio of forces

2 2

r3Froude Number ( F )

L V V

L g Lg


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Froude Criterion

The Froude criterion for modeling

inertial forces are balanced primarily by gravitational forces,which is true in most cases involving free surface.

Majority of problems in hydrodynamics are scaled accordingto Froude model law and therefore it is the most important

model law .

p m

V V

Lg Lg

p p p

m m m

V g L

V g L

1

1v

g


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Reynolds Criterion

When viscous forces dominate, the important parameter is

the ratio of inertial to viscous forces

Reynolds used this number to distinguish between laminar

and turbulent flows.

Reynolds similitude is achieved when,

2 2

e L V Reynolds Number (R )

L VV L


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Reynolds Criterion

Reynolds law is intended for flows where viscous forces

dominate.

Examples are laminar boundary layer, forces on cylinder for

low values of Re

Lv

1

p m

p p p p

m m m m

v L p

LV

V L

V L


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Weber Criterion

Relative importance of surface tension is given by inertia to surface tension forces

Weber model criteria is used when surface tension forces dominates

Surface tension effects are seldom encountered in ocean engineering problems in

prototype. In very small models surface tension may play some role.

2 2 2int forces vL Weber Number

surface tension forces

ertial V V L

L

2

1p v L


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Cauchy Criterion

An index of the relative importance of inertial forces to

compressive forces

This is of importance where inertial forces are large enough to cause

changes in fluid compressibility. Cauchy number is related to Machnumber (v/c) because the speed of sound c in a fluid is given by

2 2 2

2

inertial forcesCauchy number

elastic forces

L V V

E L E

E 2

2 V

Ea aM C


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Cauchy Criterion

Mach number is used in studies of air flow having high

velocities. Cauchy modeling criterion is,

This criterion has little application in Ocean engineering

problems because the fluid is considered incompressible.

There could application in breaking waves.

2

1v

E


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Euler Criterion

When the pressure forces are dominant Euler

criterion is used

Euler model criterion is

2

2 2 2

Pressure force P L

intertial force L

P

V V

2

v

1

p


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Strouhal Number

Inertial forces in fluid can be caused by two types of accelerations. Convective

accelerations are due to different fluid velocities at different locations in the flow field

and they are represented mathematically as,

Temporal accelerations are changes in flow velocity at a point that occur in time. They

represent the unsteadiness of the flow and can be expressed mathematically as,

/u u x

/u t 3

22

Tenporal inertial force = L

Connective inertial force = L

V

t

V

L


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Strouhal Number

The relative importance of temporal inertia force to connectiveinertial force is given as

which is represented as Strouhal Number .

This dimensional parameter is important in unsteady, oscillatory flows.Where the period of oscillation is given by variable it. Often StrouhalNumber is expressed as (L/V) or (fL/V)

2

22

L

L

V

Lt

V VT

L


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Strouhal Number

For Strouhal Criteria in the model

which states that the velocity scalar ratio is equal to the length scale ratio divided by

time scale ratio. In unsteadyoscillating flows it is important to maintain similarity of

Strouhal Numbers

1

p m

p p p

m m m

L

v t

L L

Vt Vt

L V t

L V t


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Importance of Froude Scaling

Practically most of the ocean engineering problems and fluid flowproblems, where the forces due to surface tension, pressure andelastic compressionare relatively small and can be safely neglected.

This leaves an appropriate hydrodynamic scaling law to an evaluationof whether gravity or viscousforces are dominant.

Therefore either Froude or Reynolds similarity combined with

geometric and Kinematic similarity provides the condition forhydrodynamic similitude in almost all ocean related problems.


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Importance of Froude Scaling

Reynolds similitude is seldom

involved for most models as gravity

forces dominate in free surface flowsand consequently most models are

designed for Froude criteria.

However, the viscous effects must bereducedotherwise dissimilar viscous

forces will constitute scale effect.


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Scaling of A Froude Model

In the study of wave mechanics three non-

dimensional parameters are most

important. They are Froude, Reynolds andStrouhal. Keulegan-Carpenter number also

becomes important in showing the

dependence of inertia and dragcoefficients.


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Except Cm and Cdall terms follow Froude scaling.

The hydrodynamic coefficients Cd& Cmare non-

dimensional and are function of Keulegan-

Carpenter number KC defined as (uT/D).According to Froude's law the velocity and wave

period scale are square root of the scale factor

while linear dimension scale linearly

p m

KC KC

3

2

ewhere as R ep mR


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For current drag, the drag force is proportional to the

square of the velocity. Experiments have shown that flow

characteristics in the boundary layer are most likely to be laminar

at Re < 106 but turbulent for Re > 106. In most model of Froude

scale boundary layer is laminar whereas for prototype it is

turbulent. Thus two scaling laws have to be applied which is ,not

possible. It is convenient to employ Froude scaling in a laboratary

and apply correction for violating Reynolds number.

Once the flow is turbulent the drag coefficient is only

weakly dependent as Re. As a result in many experiments, the

laminar flow is deliberately tripped by some kind of roughness

near the bow of the structure.

similitude in hydrosimilitude in hydrodynamicsdynamics

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