Flow at high Re
upgDtuD
2
Inviscid Flow
pguu
tu
Euler’s equation
zuw
yuv
xuu
tu
xpuu
tu
1
zvw
yvv
xvu
tv
ypvu
tv
1
zww
ywv
xwu
tw
zpgwu
tw
1 0
pguu
tu
Euler’s equation
uuuuu
2
21
Can use the vector identity:
zww
ywv
xwu
zvw
yvv
xvu
zuw
yuv
xuu
uu
zww
zvv
zuu
yww
yvv
yuu
xww
xvv
xuu
zw
zv
zu
yw
yv
yu
xw
xv
xu
u222
222
222
2
21
21
uuuuu
2
21
yu
xv
xw
zu
zv
yw
wvuzyx
kji
u
ˆˆˆ
yu
xv
xw
zu
zv
yw
wvukji
uuu
ˆˆˆ
zvv
ywv
xwu
zuu
yuu
xvu
zvw
yww
xww
zuw
yuv
xvv
uuuu 2
21
221 u
zww
zvv
zuu
yww
yvv
yuu
xww
xvv
xuu
u
zvv
ywv
xwu
zuu
yuu
xvu
zvw
yww
xww
zuw
yuv
xvv
pguu
tu
uuuuu
2
21
Can use the vector identity:
pguutu
2
21
alternate form ofEuler’s equation zgpuu
tu
1
21 2
Look at component along a streamline zgpuu
tu
1
21 2
1
2
d
dzgpdudud
tu
utouto
||
2
2
02
2
zgpu
stu ss
irrotational flow
02
2
zgpu
stu ss
For steady and irrotational flow :
02
2
zgpus
s
Bzgpus 2
2
Bernoulli’s equation
B is constant everywhere
For unsteady and irrotational flow, we get another form of Bernoulli’s equation
For irrotational flows, the velocity vector can be written as:
u
The gradient of a scalar potential – the velocity potential
02
2
zgpu
stu ss
Inserting the velocity potential in:
02
2
zgput
s
02
2
tFzgput
s
Bernoulli’s equation for unsteady and irrotational flow
Bzgpus 2
2
Bernoulli’s equation for steady and irrotational flow
http://commons.wikimedia.org/wiki/File:Graphic1.png
Bzgpus 2
2
http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/FLUID-MECHANICS/lecture-15/15-1a_mesure_flow.htm
Bzgpus 2
2
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