Similarity (fluid dynamic)

8/13/2019 Similarity (fluid dynamic)

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CHAPTER 8

SIMILARITY


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Introduction

Many cases in real life engineering cannot be solved usingequation because it is too difficult to solve or create the equation.

Experiment the only method used to obtaining the reliable data of

information.

In most experiment, to save time and cost, tests are performed on

geometrically scaled model, rather than on the full scale prototype.

Proper scaling law is used to predict the values in the prototype

using result from model testing.

Result from dimensional analysis non dimensional group! is used

as scaling law to predict the values in the prototype.

"here are three necessary conditions for complete similarity

between a model and a prototype which isi. #eometric $imilarity

ii. %inematic $imilarity

iii. &ynamic $imilarity

Geometric Similarity

#eometric similarity exists between model and prototype if the

ratio of all corresponding dimensions in the model and prototypeare equal.

where Lis the scale factor for length.

'or area

(ll corresponding angles are the same.


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Kinematic Similarity

%inematic similarity is the similarity of time as well as geometry.

It exists between model and prototype

i. If the paths of moving particles are geometrically similar.

ii. If the rations of the velocities of particles are similar

$ome useful ratios are)

*elocity

(cceleration

&ischarge

"his has the consequence that streamline patterns are the same.

Dynamic Similarity

&ynamic similarity exists between geometrically and +inematically

similar systems if the ratios of all forces in the model and prototypeare the same.

'orce ratio

"his occurs when the controlling dimensionless group on the right

hand side of the defining equation is the same for model and

prototype.

Scaling La


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hen a hydraulic structure is build it undergoes some analysis in

the design stage.

-ften the structures are too complex for simple mathematicalanalysis and a hydraulic model is build and usually the model is

less than full sie.

Measurements can be ta+en from the model testing and a suitable

scaling law is applied to predict the values in the prototype.

"o illustrate how these scaling laws can be obtained we will use

the relationship for resistance of a body moving through a fluid.

/et say, the resistance, R, is dependent on the following physical

properties)

0 M/12

u 0 /"13

l 0 /

0 M/13"13

n 0 , l, u

"hus, m 0 4, n 0 2 so there are 4 1 2 0 5 groups

'or first group,

/eading to 3as

'or second group,

/eading to 5as


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6otice how 375 is the Reynolds number. e can call this 5a.

$o,

3, 5a! 0 55 luR

,

lu!

8ased on the theory of scaling law,

55lu

R

!m 0 55 lu

R

!p

lu!m 0

lu!p

$o, the Rpcan be predicted by

and

Im!ortant Dimen"ionle"" #um$er" In %luid Mec&anic"

Dimen"ionle""

#um$er

Sym$ol %ormula Im!ortance

Reynolds 6umber eR

du 'luid flow involving viscous and

inertial forces

'roude 6umberRF

dg

u5 'luid flow with free surface

eber 6umber eW

du5 'luid flow with interfacial forces

Mach 6umber aM

c

u #as flow at high velocity

&rag 9oefficientD

C

5

55 du

FD

'low around solid bodies

'riction

9oefficient

f

5

5

u

w

'low though closed conduits


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Pressure

9oefficientpC

5

5u

p

'low though closed conduits.

Pressure drop estimation

E'am!le (

(ssumed that drag forceFacting on sphere flow in air is a function

of air velocity u, density , +inematic viscosity vand the diameter of

sphere d. &etermine the non dimensional groups.

( stationary sphere in water moving at a velocity of 3.:m7s

experiences a drag of ;6. (nother sphere of twice the diameter isplaced in a wind tunnel. 'ind the velocity of the air and the drag

which will give dynamically similar conditions. "he ratio of

+inematic viscosities of air and water is 32, and the density of air

3.5< +g7m2.

$olution =

'rom 8uc+inghams theorem we have m 1 n 0 4 1 2 0 5 non1dimensional groups. and n 0 u, d,

'irst group,

3 ) ua3, db3, c3, &

Matra M,> 0 c3? 3

c30 13

Matra /,

> 0 a3? b31 2c3? 3

1; 0 a3? b3

Matra ",> 0 1a31 5

a30 1 5


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b30 15

$econd group,

3 ) ua5, db5, c5, v

Matra M,> 0 c5

Matra /,

> 0 a5? b51 2c5? 5

15 0 a5? b5

Matra ",

> 0 1a51 3a50 13

b50 13

$o the physical situation is described by this function of

nondimensional numbers,

'or dynamic similarity these non1dimensional numbers are the same

for the both the sphere in water and in the wind tunnel i.e.

'rom 5

5air0 5water


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'rom 3

3air0 3water

E'am!le )

(ssumed that drag forceFacting on tall chimney is a function of

air velocity u, air density , dynamic viscosity and the diameter

of chimney d. &etermine the non dimensional groups.

( cylinder >.3:m in diameter is to be mounted in a stream of water

in order to estimate the force on a tall chimney of 3m diameter

which is sub@ect to wind of 22m7s. 9alculate

a. the speed of the stream necessary to give dynamic

similarity between the model and chimney,

b. the ratio of forces.

#iven

Prototype) 0 3.35+g7m20 3: x 3>1: +g7ms

Model) 0 3>>>+g7m20 < x 3>1;+g7ms

$olution =

'rom 8uc+inghams theorem we have m 1 n 0 4 1 2 0 5 non1

dimensional groups and n 0 u, d, .


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'irst group,

3) u3a, d3b, 3c,

Matra M,

> 0 c3? 3

c30 13

Matra /,

> 0 a3? b31 2c31 3

15 0 a3? b3

Matra ",

> 0 1a31 3

a30 13

b30 13

$econd group,

5) u5a, d5b, 5c, '

Matra M,

> 0 c5? 3

c50 13

Matra /,

> 0 a5? b51 2c51 3

12 0 a5? b5

Matra ",

> 0 1a51 5

a50 1 5

b50 13


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$o the physical situation is described by this function ofnondimensional numbers,

'or dynamic similarity these non1dimensional numbers are the

same for the both water and air in the pipe.

'rom 33air0 3water

"o find the ratio of forces for the different fluids use 5

Similarity (fluid dynamic)

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