6 7 irrotational flow

Irrotational Flow(Potential Flow)

• Bernoulli Equation

• Velocity Potential

• Two-Dimensional, Irrotational, Incompressible Flow

• Elementary Plane Flows

• Superposition of Elementary Plane Flows

Irrotational Flow

Fluid elements moving in the flow field do not undergo any rotation

0 ,0 V

0

y

u

xx

w

z

u

zy

w

0111

rzrr V

rr

rV

rr

V

z

V

z

VV

r

Rectangular coordinate

Cylindrical coordinate

Bernoulli Equation

If the flow is irrotational,

VVkgp

ˆ1

VVVVVV

2

1= 0

2

2

1

2

1ˆ1VVVkgp

rd

dzkdyjdxird ˆˆˆ

zk

yj

xi

ˆˆˆ

2

2

1Vdgdz

dp

0

2

2

gz

Vpd

constant2

2

gzVp valid between any two points

in an irrotational flow

0 V

Example: Flow field with tangential motion

- Forced Vortex rfVVr and 0

Forced Vortex (rigid body rotation)

rrfV

rzz

V

rr

rV

rV

11

2

1

2

1

rrr

r

r 2

2

11

2

1 2

a

b rarb

arbr

arbrba rdrdr

r

ppp 2

02

222

ba rr

Example: Flow field with tangential motion

- Free Vortex rfVVr and 0

Free Vortex (irrotational vortex)

r

r

r

CrfV a

2

rzz

V

rr

rV

rV

11

2

1

2

1

01

2

1

r

C

r

a

b rarb

222 abba VVpp

22

2 11

2 ab rr

C

then If 2 arC 02

2222

babba rrrpp

Velocity Potential

If the flow is irrotational, a potential function, , can be formulated to represent the velocity field.

From the vector identity, 0

The velocity field of an irrotational flow can be defined by a potential function so that

V

zw

yxu

zV

rV

rV zr

1

Stream Function and Velocity Potential

For a two-dimensional, incompressible, irrotational flow, the velocity field can be expressed in terms of both and .

xy

u

yx

u

According to the irrotationality condition,

0

y

u

x0

2

2

2

2

yx

According to the continuity equation,

0

yx

u0

2

2

2

2

yx

Laplace’s equation

Solution of Laplace’s equation represents a possible 2-D, incompressible, irrotational flow field.

Slope of Velocity Potential Line

Along a line of constant , d=0 and

0

dyy

dxx

d

The slope of a line of constant is given by

u

y

x

dx

dy

uuy

x

dx

dy

1

u

u

dx

dy

dx

dy

The slope of a line of constant is given by

Lines of constant and constant are orthogonal.

Example: Determine the Velocity Potential Line of a Flow The flow streamline function is .2axy

yyxxy

u

xz2 022

axy

ayx

ayx

axy

u 2 2

The flow is irrotational.

In term of velocity potential, the velocity components are

ayy

axx

u 2 2

222 2 2 caxxgax

dx

dguxgayay

y

122 2 2 cayyfay

dy

dfyfaxax

xu

caxay 22

Elementary Plane Flows

Uniform Flow

3c2c1c

01c2c3c

3k 2k 1k 01k 2k 3k

U

x

y

x

y

Uu

0

Uy

Uy

u

0

x

Ux

Ux

u

0

y

=0 around any closed curve

Inclined uniform flow

3k2k

1k 01k 2k 3k

x

y cosVu

sinV

xVyV sincos

cosVy

u

sinVx

xVyV cossin

cosVx

u

sinVy

V

x

y

0

3c

2c1c

1c2c

3c


Source Flow

r

qVr

2

0V

2

q

rq

ln2

x

y

3c2c

1c

04c

5c6c

7c

1k2k

x

y

Origin is singular point

q: volume flow rate per unit depth


Sink Flow

r

qVr

2

0V

2

q

rq

ln2

x

y

3c2c

1c

04c

5c6c

7c

1k2k

x

y


q: volume flow rate per unit depth


Irrotational Vortex

x

y 3c2c

1c

4c

3k

2k

1k

04k

5k

6k

7k

x

y

r

KV

2

0rV

2

K

rK

ln2


K: strength of the vortex

=K around any closed curve enclosing origin

=0 around any closed curve not enclosing origin

Doublet

3c

2c

1c

0

1c

2c

3c

2k

1k 1k

2k

x

y

x

y

sin2r

V

cos2r

Vr r

cos

r

sin


: strength of the doublet


Superposition of Elementary Plane Flows

and satisfy Laplace’s equation.

• Laplace’s equation is linear and homogeneous.

• Solutions of Laplace’s equation may be added together (superposed) to develop more complex flow patterns.

• If 1 and 2 satisfy Laplace’s equation, so does 3 = 1 + 2 .

• Any streamline contour can be imagined to represent a solid surface (there is no flow across a streamline).

Source + Uniform Flow

sin22flow uniformsource Urq

Uyq

cosln2

ln2flow uniformsource Urr

qUxr

q

x

y

U

Stagnation point

Flow past a half body

Solid surface formedby two streamlines

Source + Sink Flow

2121sinksource 222

qqq

1

221sinksource ln

2ln

2ln

2 r

rqr

qr

q

Source and sink with equalstrength, origin separated 2a apart

Source + Sink+Uniform Flow

sin222 2121flow uniformsource Urq

Uyqq

cosln2

ln2

ln2 1

221flow uniformsource Ur

r

rqUxr

qr

q

x

y

U

Stagnation point

(Flow past a Rankine body)Solid surface formedby two streamlines

Stagnation point

Doublet+Uniform Flow

2flow uniformdoublet 1sinsinsinsin

r

UUrUr

rUy

r

2flow uniformdoublet 1coscoscoscos

r

UUrUr

rUx

r

U

Stagnation point

Flow past a cylinderSolid surface formedby two streamlines

Stagnation point

aU

a

Doublet+Vortex+Uniform Flow

rK

r

aUrUrr

K

rln

21sinsinln

2

sin2

2

flow uniformvortexdoublet

21coscos

2

cos2

2


K

r

aUrUr

K

r

a

Ua

Flow past a cylinderwith circulation

UaUK 44

Sink+Vortex

rKq

ln22vortexsink

2

ln2vortexsink

Kr

q

Vortex Pair1

221vortex2vortex1 ln

2ln

2ln

2 r

rKr

Kr

K

1221vortex2vortex1 222

KKK

Example: Flow past a cylinder

2flow uniformdoublet 1sinr

UUr

2flow uniformdoublet 1cosr

UUr

U

Stagnation point (a,)

Stagnation point (a,0)

a Ua

2

2

2

2

1cos1cosr

aU

r

aUr

rrVr

2

2

2

2

1sin1cos11

r

aU

r

aUr

rrV

Example: Flow past a cylinder- pressure distribution on the surface

U

a

Ua

2

2

1cosr

aUVr

2

2

1sinr

aUV

p

gzV

pgzU

p 22

22

22222 sin4UVVV arrar

= 0

2222222 sin41

2

1sin4

2

1

2

1UUUVUpp

2

2

sin41

21

U

pp

Pressure distribution on the surface of the cylinder

2

21

U

pp

2

2

sin41

21

U

pp

Pressure drag force acting on the cylinder

U

p

ApdFd

a

2

22

12 U

VUpp

2

0

22pressure cossin41

2

1dabUpF

Drag

2

0

322

0

22

0 sin3

4

2

1sin

2

1sin abUabUabp

0

2

0pressure coscos dpabpdAFADrag

Lift force acting on the cylinder

U

p

ApdFd

a

2

0pressure sinsin dpabpdAFALift

2

0

22pressure sinsin41

2

1dabUpF

Drag

2

0

32

2

0

22

0 cos43

cos4

2

1cos

2

1cos abUabUabp

0

Example: Flow past a cylinder with circulation

rK

r

aUr ln

21sin

2

2


21cos

2

2


K

r

aUr

a

Ua

UaUK 44

2

2

1cosr

aU

rVr

r

K

r

aU

rV

21sin

12

2

Stagnation points

a

KUV ar

2sin2

4

sin and at 0 1-

Ua

KarV

Flow past a cylinder with circulation- Surface pressure distribution

a

Ua

UaUK 44

a

KUV ar

2sin2

2

222

2sin2

a

KUVVV arrar

01cos2

2

a

aUV arr

2

22

12 U

VUpp

22

2

2

2sin

2sin4

Ua

K

Ua

K

U

V

22

2

2sin

2sin41

2 Ua

K

Ua

KUpp

Pressure distribution on the surface of the cylinder

2

21

U

pp

aUK

aUK 2

aUK 3

22

2

2sin

2sin41

2 Ua

K

Ua

KUpp

Pressure drag force acting on the cylinder

p

2

22

12 U

VUpp

2

0

222

pressure cos2

sin2

sin412

1dab

aU

K

aU

KUpF

Drag

ApdFd

2

0pressure coscos dpabpdAFADrag

a

2

0

22 cossin412

1dabUp

2

0

22 cos

2sin

2

2

1dab

Ua

K

Ua

KU

0

0sin22

sin22

0

22

abUa

K

Ua

Kab

Lift force acting on the cylinder

p

2

22

12 U

VUpp

2

0

222

pressure sin2

sin2

sin412

1dab

aU

K

aU

KUpF

Lift

ApdFd

2

0pressure sinsin dpabpdAFALift

a

2

0

22 sinsin412

1dabUp

2

0

22 sin

2sin

2

2

1dab

Ua

K

Ua

KU

0

bU

KUb

U

KUab

Ua

K

Ua

Kab 2

22

2

2cos

24

sin

2

2 222

0

22

Circulation around on the cylinder

p

2

0

2

0

2

0ˆˆˆˆ ˆˆˆ eerdVeerdVerdeVeV rrrrr

sdV

a

2

0 2sin2 rd

Ur

KU

0

r

KUV

2

sin2

2

2

1cosr

aUVr

KK

Ur

2

0

20 2

cos2

U

KU

b

FLift 2

2

1 2

UUK

Home work: 6.51, 6.66, 6.76, 6.93, 6.96, 6.98

Technology

6 7 irrotational flow