flow around cylinder with circulation

7/28/2019 Flow Around Cylinder With Circulation

1/27

FLOW AROUND CYLINDER

WITH CIRCULATION


2/27

FLOW PAST A CYLINDER WITH CIRCULATION

(DOUBLET, VORTEX & UNIFORM FLOW)

2

cos

ln2

sin

2

ln

22

sin

2

0

2

0

0

0

r

RrU

rrRrU

U

kR

rr

kyU

vortexdoubletuniform

STREAM FUNCTION

VELOCITY POTENTIAL


3/27



0

2

sin2

0

2

sin2

0

21sin

cos1

0

0

2

2

0

2

2

0

r

Uq

q

r

Uq

q

rrRU

rq

r

RrU

rq

r

r

r

VELOCITY COMPONENTS

AT SURFACE OF CYLINDER

r = R

AT STAGNATION POINT


4/27



2/3&;2/&0

1sin0

4

;0

0sin

0

4sin

0

0

RU

RUCONDITION FOR STAGNATION

FREE VORTEX FLOW IS ABSENT

LOCATION OF STAGNATION POINTS

STRENGTH OF FREE VORTEX IS SMALL

LOCATION OF STAGNATION POINTS


5/27



NO ROTATION OF CYLINDER


6/27



SMALL ROTATIONAL SPEED OF CYLINDER


7/27



1sin

4

2/3,2/1sin

4

0

0

RU

RU CORRESPONDS TO SINGLESTAGNATION POINT,

STAGNATION POINT IS ATTHE BOTTOM OF DIVIDING

STREAMLINE

STRENGTH OF FREE VORTEXIS LARGE. IS IMAGINARY.

STAGNATION POINT IS NOT ON

THE CIRCULAR STREAMLINE BUT IT

IS LOCATED BELOW IT


8/27



CRITICAL ROTATIONAL SPEED OF CYLINDER


9/27



02

sin2

0

0

rUq

qr

REPRESENTS A PURE

CIRCULATORY MOTION

2

0

2

00

22

00

2sin2

2

1

2

1

2

1

RUUpp

qpUp

ACTING NORMAL

TO THE SURFACE


10/27



RU

B

U

pp

BBUpp

pp

RUB

BBUBUpp

pp

bt

b

b

t

t

0

2

0

22

00

0

22

0

22

00

48

2

1

432

1

2

2

432

121

2

1

2

PRESSURE AT TOP

PRESSURE AT BOTTOM

PRESSURE DIFFERENCE


11/27

LIFT ON CYLINDER KUTTA

JOUKOWSKI THEOREM

FL

FD

pRsind

pRcosd

F=pRd

d

2

0

sin

cos

sin

dpRF

pRddF

pRddF

L

D

L

TOTAL LIFT


12/27

LIFT ON CYLINDER KUTTA

JOUKOWSKI THEOREM

oddnd

dRU

dRU

dUdUdRpRF

d

RU

UdUdRpRF

RUUpp

n

L

L

0sin

sin4

sin2

sin42

1sin

2

1sin

sin

2

sin2

2

1sin

2

1sin

2sin21

2

1

2

0

2

0

2

0

2

0

2

0

2

0

22

0

2

2

2

0

32

0

2

00

2

0

2

0

2

0

2

0

2

0

2

00

2

0

2

00


13/27

LIFT ON CYLINDER KUTTA JOUKOWSKI

THEOREM

MAGNUS EFFECT

0

2

0

0

2

0

0

2

0

2

0

2

0

4

2sin

2

1

2

2cos1

sin2

2

1

U

U

dU

dRU

URFL

LIFT FORCE/LENGTH. ACTS PERPENDICULAR TO FLOW


14/27

MAGNUS EFFECT

The Magnus effect is the phenomenon whereby aspinning object flying in a fluid creates a whirlpool offluid around itself, and experiences a force perpendicularto the line of motion. The overall behaviour is similar tothat around an aerofoil (lift force) with a circulationwhich is generated by the mechanical rotation, ratherthan by aerofoil action.

A spinning object creates a kind ofwhirlpool of rotatingair about itself. On one side of the object, the motion ofthe whirlpool will be in the same direction as thewindstream that the object is exposed to. On this sidethe velocity will be increased. On the other side, themotion of the whirlpool is in the opposite direction of thewindstream and the velocity will be decreased. Thepressure in the air is reduced from atmospheric pressureby an amount proportional to the square of the velocity,so the pressure will be lower on one side than the othercausing an unbalanced force at right angles to the wind.
http://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Whirlpoolhttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Whirlpoolhttp://en.wikipedia.org/wiki/Aerofoilhttp://en.wikipedia.org/wiki/Lift_forcehttp://en.wikipedia.org/wiki/Circulation_%28fluid_dynamics%29http://en.wikipedia.org/wiki/Aerofoilhttp://en.wikipedia.org/wiki/Lift_forcehttp://en.wikipedia.org/wiki/Circulation_%28fluid_dynamics%29http://www.wordiq.com/definition/Whirlpoolhttp://www.wordiq.com/definition/Whirlpoolhttp://en.wikipedia.org/wiki/Circulation_%28fluid_dynamics%29http://en.wikipedia.org/wiki/Lift_forcehttp://en.wikipedia.org/wiki/Lift_forcehttp://en.wikipedia.org/wiki/Lift_forcehttp://en.wikipedia.org/wiki/Aerofoilhttp://en.wikipedia.org/wiki/Whirlpoolhttp://en.wikipedia.org/wiki/Fluid


15/27

MAGNUS EFFECT


16/27

COMPLEX FORM

zi

zUzw ln

2

Uniform flow

Doublet flow

Circulation flow

We have already seen that combination

of Uniform flow & Doublet results in acircle of radius a=(/U) or=a2U with

stream function =0 as the dividingline


17/27

COMPLEX FORM

In order to get the same circle of

radius a with =0, we shouldmodify the complex potential due to

circulation.

0

ln2

ln2

;ln2

ar

a

r

iariw

reza

ziw

i Modified function


18/27

COMPLEX FORM

a

zi

z

azUw ln

2

2

COMBINED COMPLEX FUNCTION

THIS COMBINED FUNCTION REPRESENTS A

FLOW DUE TO A UNIFORM FLOW IN WHICH A

ROTATING CYLINDER HAS BEEN PLACED.FOR THIS FUNCTION =0 FOR r=a


19/27

COMPLEX FORM

01

21

01

21

00;0

12

1

ln2

sinsin

2

2

2

2

2

2

1

2

zUiz

a

zi

zaU

ivuvu

zi

zaUivu

dzdw

ka

r

r

arU

FOR STAGNATION POINTS


20/27

COMPLEX FORM

aU

aUaU

aUaUi

a

z

a

z

aU

i

a

z

zU

iaz

c

c

4;1

14

044

2

442

01

2

02

222

2

222

2

2

2

22

CASE I


21/27

CASE I


22/27

CASE II


23/27

CASE III


24/27

COMPLEX FORM

22

0

2

0

2

22

2

1

2

1

2

1

2

1

2sin2

arar

arar

ar

arar

qUpp

Upqp

aUq

vuq

PRESSUREDISTRIBUTION


25/27

COMPLEX FORM

2

2

0

22

0

2sin2

2

1

2

1

2

1

2

1

aUUpp

qUpp

ar

arar

THEREFORE PRESSURE VARIES WITH ANGLE MADE WITH THE +VE X AXIS


26/27

COMPLEX FORM

d

p

a

dF= p*ad*1

dFx= -p*ad*cos

dFy= -p*ad*sin

0cos

2

sin2

2

1

2

1

cos

2

0

2

2

0

2

0

2

0

da

a

UUpF

paddFF

x

xx


27/27

COMPLEX FORM

00

sin2

sin22

1

2

12

0

2

2

0

yx

y

y

FF

UF

daa

UUpF

LIFT CAUSED DUE TO CIRCULATIONMAGNUS EFFECT

NO CIRCULATION , NO FORCE

flow around cylinder with circulation

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