02 Examples of Potential Flows

SOURCE FLOW

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02 Examples of Potential Flows

u

SOURCE FLOW

x

yr

xu

yv

Cartesian coordinates

Potential Function

02

u

rQur

rrQur

2

01

ru

Polar coordinates

rur

r

u 1

xu

yv

Cartesian coordinates

rrQ

2

01

r

rQur 2

sincos ryrx

ry

yrx

xr

ruvu sincos

rrQ

2

01

r

cfrQ

ln

2

crf

rQ ln2

rQ

rur

2

1

0

r

u

2Q

cf

2Q

crfQ

2

For a SINK flow, Q will be negative

Stream Function

Kundu’s book p. 69

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02 Examples of Potential FlowsIRROTATIONAL

VORTEX

02

rur

u

The circulation along any circle around the origin is a constant

edrd ˆ

constdrudu

2

0

2

0 22ddr

r

As a potential flow, it is irrotational & incompressible

011

ur

urrr

u r

011

rur

urrr

u

02

rur

u

IRROTATIONALVORTEX

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02

rur

u

IRROTATIONAL VORTEX Potential Function

0

r

ur cf

rru

21

c

2

02

rur

u

VORTEX Stream Function

rru

2

01

rur

cfr

ln

2

crf

DOUBLET (Sources and Sinks)

x

y

source sink Both have strength Q. The flow field can be obtained by combining the potential function for the sink φ2 and the source φ1. Laplace’s function is linear – linear superposition is valid.

21

2222 ln2

ln2

yxQyxQ

0, 0,

221 ln

4yxQ

222221 lnln

4

Oyx

xyxQ

Taylor expansion

221 ln

4yxQ

2

2222

12ln

4

O

yxxyxQ

2

2222

22ln

4

Oyx

xyxQSimilarly

2

22214

4

Oyx

xQ

Q

rrr

yxx

coscos

222

22 yxx

In the same way ψ2 and ψ 1 can be added because Laplace’s function is linear – linear superposition is valid.

xy

xyQ 11 tantan

2

222

11 tantan

O

yxy

xy

xy

222

2 yxyQ

22 yxy

Q

22 yxy