0 2 Examples of Potential Flows SOURCE FLOW All rights reserved by don moorcroft
02 Examples of Potential Flows
SOURCE FLOW
All rights reserved by don moorcroft
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
All rights reserved by don moorcroft
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