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