FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW

Bernoulli’s Equation

Inviscid flow, no body forces, momentum equation given by eq. (2.104a) becomes

x

p

z

uw

y

uv

x

uu

t

ux

p

Dt

Du

.............(3.1)

For steady flow

dxx

pdx

z

uwdx

y

uvdx

x

uu

dx

x

p

z

uw

y

uv

x

uu

1

by multiply

1

,

.............(3.2)

Consider the flow along a streamline in 3-D space. The equation of streamline is given by eq. (2.108a to c)

0

0

udyvdx

wdxudz

.......(3.3)

.............(2.108b)

.............(2.108c)

dxx

pdz

z

udy

y

udx

x

uu

dxx

pdz

z

uudy

y

uudx

x

uu

1

1

(3.3), eq. into ngsubstituti

.............(3.4)

.............(3.5)du

dxx

pud

dxx

pudu

1

2

1

1

2 .............(3.6)

dzz

pwd

dyy

pvd

zy

1

2

1

1

2

1

component and for

2

2 .............(3.7)

.............(3.8)

Bernoulli’s Equation

dzz

pdy

y

pdx

x

pwvud

1

2

1

(3.8) through (3.6) eq. adding

222

.............(3.9)

dpdzz

pdy

y

pdx

x

p

wvu

2222 V

however,

.............(3.10)

.............(3.11)

VV

V2

1

(3.9), into (3.11) and (3.10) eq. ngsubstituti

2

ddp

dpd

.............(3.12)

Euler’s equation

2

22

2

11

2

1

2

212

V2

1V

2

1

2

V

2

V

VV

constant,

2

1

2

1

pp

pp

ddpp

p

V

V

.............(3.13)

Bernoulli’s equation

streamline along V2

1 2 constp

flow t the throughou V2

1 2 constp

Bernoulli’s equation for irrotional flow

.............(3.14)

.............(3.15)

The venturi and Low-Speed Wind Tunnel

xAA

1

1

1

A

V

2

2

2

A

V

1 2

Quasi-one-dimensional flow in a duct

S

V S

dV

0dSV

flowsteady for

0dSVt

eq., continuity theof form Integral

...........(2.39)

.............(3.16)

1 2

0dSVdSVdSV

duct, theapply to

A A wall

.............(3.17)

wall

0dSV

0dSV

wall, thealong

.............(3.18)

1

111VdSV

1station at

A

A .............(3.19)

2

222VdSV

2station at

A

A .............(3.20)

222111

222111

VV

00VV

(3.17) into (3.20) to(3.18) ngSubstituti

AA

AA

.............(3.21)

2211VV

flow, ibleincompress

AA .............(3.22)

The venturi and Low-Speed Wind Tunnel

1

1

1

A

p

V

2

2

2

A

p

V

x

p

Throat

Pressure is aminimum atthe throat

12

V

V2

V

VV

V2

V

2

21

2121

21

2

2

112

21

12

12

2212

21

AA

pp

A

App

A

A

pp

.............(3.23)

.............(3.24)

.............(3.25)

.............(3.26)

The venturi and Low-Speed Wind Tunnel

1

1

1

A

V

22

2

A

V

,3

3

3

A

V

model

Settlingchamber

nozzle Test section diffuser

fan

2121

22

233

222

211

23

23

12

12

V2

V

V2

1V

2

1V

2

1

VV

VV

pp

ppp

A

A

A

A

.............(3.27)

.............(3.28)

.............(3.29)

.............(3.30)

2

12

2122

22

2

1

221

22

1

2V

V2

V

AA

pp

A

App

.............(3.31)

.............(3.32)

Pitot TubeTotal pressure Static pressure

oB

B

pp

V

0

1V

B

A

101

02

11

22

2V

pressure totalpressure dynamic press static

0V2

1

V2

1V

2

1

pp

pp

ppBBAA

.............(3.32)

.............(3.34)

Pressure coefficient

2V2

1

q

where

q

ppC p

.............(3.36)

2

2

2

22

22

22

V

V1

V21

VV21

(3.36), into ngsubstituti

VV2

1

V2

1V

2

1

flow, ibleincompress

p

p

C

q

ppC

pp

pp

.............(3.37)

.............(3.38)

0V

0V0

constant flow, ibleincompress

0Vt

equation, continuity

0V

flow, ibleincompress

.............(3.39)

.............(2.43)

Condition on Velocity For Incompressible Flow

.............(3.39)

Governing Equation For Irrational, Incompressible Flow: Laplace’s Equation

0

0

,irrational and ibleincompress

V

, potential velocity a flow, irrational

0V

flow, ibleincompress

2

.............(3.39)

.............(2.145)

.............(3.40)

Laplace’s Equation

0

s,coordinatecartesian

2

2

2

2

2

22

zyx

zyx

,,

.............(3.41)

011

s,coordinate lcylindrica

2

2

2

2

22

zrrr

rr

zr

,,

.............(3.42)

011

s,coordinate spherical

22

2

sinsinsin

sin

,,

rr

rr

r

.............(3.43)

Governing Equation For Irrational, Incompressible Flow: Laplace’s Equation

0yx

(3.44), into b) and (2.141a ngsubstituti

0V

function, stream flow, ibleincompress D,2

22

xyyxxy

y

v

x

ux

v

yu

0

0yx

(2.122), into b) and (2.141a ngsubstituti

0

irrational flow, ibleincompress D,2

2

2

2

2

yx

yx

y

u

x

v

.............(3.44)

.............(2.141a)

.............(2.141b)

.............(3.45)

.............(2.122)

.............(3.46)

Conclusions:1.Any irrotational, incompressible flow has a velocity potential and stream function

(for 2-D) that both satisfy Laplace’s Equation.2. Conversely, any solution of Laplace’s equation represents the velocity potential or

stream function (2-D) for an irrotational, incompressible flow.

V V

V

V

xfyb

Boundary Conditions

0

V

infinity,at

xyv

yxu

Boundary conditions on velocity at infinity

.............(3.47a)

.............(3.47b)

surface

b

u

v

dx

dy

orS

n

0

or

0

0nnV

wall,at the

.............(3.48a)

.............(3.48b)

.............(3.48c)

.............(3.48e)

Boundary conditions on velocity at the wall

x

y

Uniform Flow:1st Elementary Flow

h

l

C curve

V

cont

cont

rx

y

0

V

vy

and

ux

.............(3.49a)

.............(3.49b)

x

x

then

xgconst

yfx

V

const.V

(3.49b), eq from

V (3.49a), eq from

)

.............(3.52)

........(3.50).......(3.51)

.............(3.53)

Velocity potential for a uniform flow

y

vx

and

uy

V

then

0

V

.............(3.54a)

.............(3.54b)

.............(3.55)

Stream function for an incompressibleuniform flow.

sin

cos

sincos

r

and

r

ryrx

V

V

and s,coordinatepolar in

.............(3.56)

.............(3.57)

Uniform Flow:1st Elementary Flow

0

000dSV

C curve closeconsider

dSV

flow, uniform ain n circulatio

hlVhlVC

C

0

0,V,irrational is flow uniform

dSV

0

00VdSVdSV

V

S

CC

const.

.............(2.127)

.............(3.58)

.............(2.128)

Source/Sink Flow: 2nd Elementary Flow

0

Vr

cVr

r

rr

rlm

v

drllrdm

V2

rate flow volume

VV

flow mass total

2

0

2

0

r

rl

v

r

r

2V

V2

length unit per rate flow volume

...............3.59a)

...............3.59b)

. .....3.60)

.....3.60)

.....3.62)

.....3.61)

Source/Sink Flow: 2nd Elementary Flow

0V1

2V

r

potentialvelocity 2

r

r

c

r

rfcont

fr

respect to gintegratin2

r respect to gIntegratin

ln

rln

2

0V

2V

1

function stream the

r

rr r

rf

2

respect to w/ gintegratin

2

0dSV S

.....3.63)

.....3.64)

.....3.65)

.....3.66)

.....3.67)

.....3.68)

.....3.69)

.....3.70)

fconst

r

respect to w/ gintegratin

.....3.71)

.....3.72)

Combination of a uniform flow w/ a source and sink

const.2

streamline2

function stream

sin

sin

rV

rV .....3.74)

.....3.75)

sin

cos

Vr

V

rV

rVr 2

1

fieldvelocity

.....3.76)

.....3.77)

,

sin

cos

Vr,θ

r

Vr

V

2

at located exists,point stagnation

one that find we, and for solving

0

02

points stagnation

.....3.78)

.....3.79)

3.17 fig.in ABC

curve asshown is streamline This const.2

by described ispoint stagnation the

throughgoes that streamline theHence,

const2

const22

(3.75) eq. intopont stagnation ngsubstituti

sinV

V

21

21

2

22

function stream

sin

sin

rV

rV

const2

bygiven is streamline theofequation The

such that located are points stagnation

21

2

sinrV

V

bbOBOA

.....3.80)

.....3.81)

.....3.82)

022

streamline stagnation

Bpoint at 0

Apoint at

streamline stagnation

21

21

21

sinrV .....3.83)

Doublet flow

strengthdoublet

2const

0

limit

doublet. as definedpattern flow

special aobtain weconstant,remain while0 as22

isfunction stream the

flow in the point any At distance aby separated

-strength ofsink a and strength of source a

21

:

.

d

l

l

ll

Pl

.....3.84)

.....3.85)

l

P

d

a

br

3.85 into 3.86 eq. ngSubstituti

Hence

cossin

cos

sin

lr

l

b

ad

b

ad

lrb

la

.....3.86)r

lrl

l

lr

l

l

l

sin

cossin

cossin

2

or

2const

0

limit

or

2const

0

limit

.....3.87)

Doublet flow

sin

sin

cos

cr

cr

r

2

or

const2

flowdoublet for streamline the2

potential velocity the

.....3.88)

.....3.89)

Nonlifting flow over a circular cilinder

221

or 2

function stream the

rVrV

rrV

sin

sinsin

.....3.91)

2

2

2

1

as written becan 3.91 eq ,V2let

r

RrV

R

sin .....3.92)

0 1

0 1

points stagnation oflocation the

1

1 2

1

1 11

fieldvelocity

2

2

2

2

2

2

2

2

3

2

2

2

2

2

sin

cos

sin

sinsin

cos

cos

Vr

R

Vr

R

Vr

RV

Vr

R

r

RrV

rV

Vr

RV

r

RrV

rrV

r

r

.....3.93)

.....3.94)

.....3.95)

.....3.96)

sin

sin

VV

V

Rr

VR

r

RrV

r

2

0

with 3.94 and 3.93 eq.

by given iscylinder theof theof

surface on theon distributiVelocity

2

01

is points stagnation he through tpasses that streamline ofequation

2

2 .....3.97)

.....3.98)

.....3.99)

.....3.100)

2

2

41

3.38 and 3.100equation combining

1

tcoefficien pressure

sin

p

p

C

V

VC .....3.38)

.....3.101)

dyCC

cc

dxCCc

c

TE

LElpupa

c

uplpn

1

1

become 1.16 and 1.15equation

0

,,

,, .....3.102)

.....3.103)

Vortex flow

2

2

2dSV radius

of streamlinecircular given a aroundn circulatio

C

andr

V

or

rVr

r

constV

C,

.....3.105)

.....3.106)

.....3.104)

dS

πCor

dSπC

dSdS

r

dSπC

S

SS

S

2 V

V2

3.109 and 3.108 eq combining

V V

0Let

VdSV2

flow of plane thelar toperpendicuboth

direction, same in the are dS and VBoth

dSVC2

2.137 and 3.106 eq. combining

origin. at theexcept alirrotation is flowvortex

.....3.107)

.....3.108)

.....3.109)

.....3.110)

V

become 1103eq hence

00

.

,,dSr

π

Γφ

rV

r

Vr r

2

b and 3.111a eq. gintegratin2

1

0

flow for vortex potential velocity the

.....3.111a)

.....3.111b)

.....3.112)

rπ

Γ

rV

r

Vr r

ln2

b and 3.113a eq. gintegratin2

01

flow for vortexfunction stream the

.....3.113a)

.....3.113b)

.....3.114)

Lifting Flow Over A Cylinder

2

2

1 1

radius

cylinder aover flow liftingnon

r

RrV

R

sin .....3.92)

R

r

π

Γ

R

rπ

Γ

ln

ln

ln

2

2constant and,

constant 2

strength w/ for vortexfunction stream the

2

2

.....3.115)

.....3.116)

.....3.117)

R

r

r

RrV lnsin

21

2

2

21

.....3.118)

RVθ

rV

r

RV

rVr

RV

rV

r

RV

rVr

RV

r

r

4

obtain we, for solving and

3.122 eq. intoresult thisngsubstituti R,r 3.121, eq. from

02

1

01

points stagnation

21

1

fieldvelocity

2

2

2

2

2

2

2

2

arcsin

sin

cos

sin

cos .....3.119)

.....3.120)

.....3.121)

.....3.122)

.....3.123)

22

22

2

241

2211

tcoefficien pressure2

2

cylinder, theof surface on thevelocity

RVRVC

orRVV

VC

RVVV

Rr

p

p

sinsin

sin

sin

TE

LE

lp

TE

LE

upd

TE

LE

lpupad

dyCc

dyCc

c

or

dyCCc

cc

1

1

1

,,

,,

0

0

2

1

2

1

2 and

3.127 into 3.128 eq. ngsubstituti

n

coordpolar

to3.127 eq. converting

dC

dCc

Rc

dRdyR siy

lp

upd

cos

cos

cos,

,

,

.....3.125)

.....3.126)

.....3.127) .....3.129)

.....3.128)

0

obtainy immedietel we

0 and

0

0

thatnoting and 3.130 into 3.126 eq. ngsubstituti

2

1

2

1

2

1

2

0

2

0

2

2

0

2

0

2

0

d

pd

ppd

plpup

c

d

d

d

dCc

dCdCc

CCC

cossin

cossin

cos

cos

coscos

, ,,

.....3.130)

.....3.131a)

.....3.131b)

.....3.131c)

.....3.132)

c

up

c

lpnl dxCc

dxCc

cc

00

1

1

,,

2

0

02

2

1

2

1

2

1

3.133 into 3.134 eq. ngsubstituti

coordpolar toconverting

dCc

dCdCc

dRdxR x

pl

uplpl

sin

sinsin

sin,cos

,,

.....3.133)

.....3.134)

.....3.135)

.....3.136)

RVcl

obtainy immedietel we

0 and

0

0

thatnoting and 3.136

into 3.126 eq. ngsubstituti

2

0

2

2

0

3

2

0

d

d

d

sin

sin

sin .....3.137a)

.....3.137b)

.....3.137b)

.....3.138)

VL

RVRVLRS

ScVScqL

L

ll

'

',

'

'

22

1 12

2

1

span perunit lift

2

2 .....3.139)

.....3.140)

The lift per unit span is directly proportional to circulation

Kutta-Joukowski theorem

The Kutta-Joukowski Theorem and The Generation of Lift