Potential Flow Theory : Incompressible Flow

8/12/2019 Potential Flow Theory : Incompressible Flow

1/42

Potential Flow Theory : Incompressible Flow

P M V Subbarao

Professor

Mechanical Engineering Department

IIT Delhi

A mathematical Tool to invent flow Machines.. ..


2/42

Basic Elements for Construction Flow Devices

Any fluid device can be constructed using following Basicelements.

The uniform flow: A source of initial momentum.

Complex function for Uniform Flow : W = Uz

The source and the sink : A source of fluid mass.

Complex function for source : W = (m/2p)ln(z)

The vortex : A source of energy and momentum.

Complex function for Uniform Flow : W = (ig/2p)ln(z)


3/42

THE DIPOLE

Also called as hydrodynamic dipole.

It is created using the superposition of a source and a sink ofequal intensity placed symmetrically with respect to the origin.

Complex potential of a source positioned at (-a,0):

The complex potential of a dipole, if the source and the sink arepositioned in (-a,0) and (a,0) respectively is :

)ln(

2

azm

W

p Complex potential of a sink positioned at (a,0):

)ln(2

azm

W p


4/42

Streamlines are circles, the center of which lie on they-axis and

they converge obviously at the source and at the sink.

Equipotential lines are circles, the center of which lie on thex-axis.


5/42

THE DOUBLET

A particular case of dipole is the so-

called doublet, in which the quantity a

tends to zero so that the source and

sink both move towards the origin.

The complex potential of a doublet

is obtained making the limit of the dipole potential for vanishing a

with the constraint that the intensity of the source and the sinkmust correspondingly tend to infinity as aapproaches zero, the

quantity

zW

p2

ma2


6/42

Uniform Flow Past A Doublet

The superposition of a doublet and a uniform flow gives the complex

potential

zUzW

p

2

zUzWp

p2

22

) )iyxiyxUW

p

p2

22

) ) ) ) )iyxiyxiyxiyxUW

pp

22

2

) ) ) )22

2

2

2

yx

iyxiyxUW

p

p

) ) )22

2232223

2

222

yx

iyxxyiiyyixyixxyxU

W

p

p


7/42

) ) )22

2232223

2

222

yx

iyxxyiiyyixyixxyxUW

p

p

) ) )22

3223

2

2

yx

iyxiyyixxyxUW

p

p

) ) )22

3223

2

22

yx

yyyxUixxyxUW

p

pp

) ) )223223

222yx

yyyxUixxyxUW p

pp

)

)

)

)22

32

22

23

2

2

2

2

yx

yyyxUi

yx

xxyxUW

p

p

p

p


8/42

)

)

)

)

p

p

p

pi

yx

yyyxUi

yx

xxyxUW

22

32

22

23

2

2

2

2

)

)

)

)2232

22

23

2

2&

2

2

yx

yyyxU

yx

xxyxU

p

p

p

p

)222

yx

yUy

p

Find out a stream line corresponding to a value of steam function is zero

)22

2

0

yx

yUy

p


9/42

yyxUy p 2220 )2220

yx

yUy

p

p 2220 yxU

Uyx p

222

222

2 R

Uyx

p

There exist a circular stream line of radium R, on which value of

stream function is zero.

Any stream function of zero value is an impermeable solid wall.

Plot shapes of iso-streamlines.


10/42

Note that one of the streamlines is closed and surrounds the origin at a

constant distance equal to

U

R

p

2


11/42

Recalling the fact that, by definition, a streamline cannot be

crossed by the fluid, this complex potential represents the

irrotational flow around a cylinder of radiusRapproached by a

uniform flow with velocity U.

Moving away from the body, the effect of the doublet decreases so

that far from the cylinder we find, as expected, the undisturbed

uniform flow.

In the two intersections of thex-axis with the cylinder, the velocity

will be found to be zero.

These two points are thus called stagnation points.

zUzW

p

2


12/42


13/42

Velocity components from w


14/42

To obtain the velocity field, calculate dw/dz.z

UzWp

2

22 zU

dz

dW

p

21

2 zU

dzdW

p

)

22222

22

4

22 yxyx

ixyyxUdzdW

p

) )

2222222222

22

422

42 yxyxxyi

yxyxyxU

dzdW

p

p

ivudz

dW


15/42

)

22222

22

42 yxyx

yxUu

p

)

22222

4 yxyx

xyv

p

222 vuV

) )

2

22222

2

22222

222

442

yxyx

xy

yxyx

yxUV

p

p

Equation of zero stream line:

UR

p

2 222 yxR with


16/42

Cartesian and polar coordinate system

sincosryrx

sincos

VvVu


17/42

)

4

4

2

222 2cos21

r

R

r

RUV

On the surface of the cylinder, r = R, so


18/42

V2Distribution of flow over a circular cylinder

The velocity of the fluid is zero at = 0oand = 180o. Maximum

velocity occur on the sides of the cylinder at = 90oand = -90o.


19/42

Pressure distribution on the surface of the cylinder can

be found by using Benoullis equation.

Thus, if the flow is steady, and the pressure at a greatdistance isp,


20/42

Cpdistribution of flow over a circular cylinder


21/42


22/42


23/42

Development of an Ultimate Fluid machine


24/42

Anatomy of an airfoil

An airfoil is defined by first drawing amean camber line.

The straight line that joins the leading andtrailing ends of the mean camber line iscalled the chord line.

The length of the chord line is called

chord, and given the symbol c. To the mean camber line, a thicknessdistribution is added in a direction normalto the camber line to produce the finalairfoil shape.

Equal amounts of thickness are added

above the camber line, and below thecamber line.

An airfoil with no camber (i.e. a flatstraight line for camber) is a symmetricairfoil.

The angle that a freestream makes withthe chord line is called the angle of attack.


25/42

Conformal Transformations

P M V Subbarao

Professor

Mechanical Engineering DepartmentIIT Delhi

A Creative Scientific Thinking .. ..


26/42

INTRODUCTION

A large amount of airfoil theory has been developed by

distorting flow around a cylinder to flow around an airfoil.

The essential feature of the distortion is that the potential

flow being distorted ends up also as potential flow.

The most common Conformal transformation is the

Jowkowski transformation which is given by

To see how this transformation changes flow pattern in

the z (or x - y) plane, substitute z = x + iy into the

expression above to get


27/42


28/42

This means that

For a circle of radius r in Z plane x and y are related as:


29/42

Consider a cylinder in z plane

In zplane


30/42


31/42

C=0.8


32/42

C=0.9


33/42

C=1.0


34/42


35/42


36/42

Translation Transformations

If the circle is centered in (0, 0) and the circle maps into

the segment between and lying on thexaxis;

If the circle is centered in (xc,0), the circle maps in an

airfoil that is symmetric with respect to thex' axis;

If the circle is centered in (0,yc

), the circle maps into a

curved segment;

If the circle is centered in and (xc, yc), the circle maps in

an asymmetric airfoil.


37/42

Flow Over An Airfoil


38/42


39/42


40/42


41/42


42/42

Documents

Potential Flow Theory : Incompressible Flow