Chapter8 Latest Combined

8/13/2019 Chapter8 Latest Combined

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058:0160 Chapter 8Professor Fred Stern Fall 2009 1

Chapter 8: Inviscid Incompressible Flo: a !sef"l Fantas#

8.1 Introduction

For hi$h %e e&ternal flo abo"t streamlined bodies visco"s effects are confined tobo"ndar# la#er and a'e re$ion( For re$ions here the )(* is thin i(e( favorable press"re$radient re$ions+ ,isco"s-Inviscid interaction is ea' and traditional )(* theor# can be"sed( For re$ions here )(* is thic' and-or the flo is separated i(e( adverse press"re$radient re$ions more advanced bo"ndar# la#er theor# m"st be "sed incl"din$visco"s-Inviscid interactions(

For internal flos at hi$h %e visco"s effects are ala#s important e&cept near the

entrance( %ecall that vorticit# is $enerated in re$ions ith lar$e shear( .herefore+ o"tsidethe )(* and a'e and if there is no "pstream vorticit# then /0 is a $ood appro&imation(ote that for compressible flo this is not the case in re$ions of lar$e entrop# $radient(lso+ e are ne$lectin$ noninertial effects and other mechanisms of vorticit# $eneration(

Potential flow theory

13 4etermine from sol"tion to *aplace e"ation02 =

)(C:

at BS : ( 0 0V nn

= =

at S : V =

Note:F: S"rface F"nction

1


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10 ( 0 (

DF F FV F V n

Dt t F t

= + = =

for stead# flo ( 0V n=

23 4etermine V from V = and p&3 from )erno"lli e"ation

.herefore+ primaril# for e&ternal flo application e no consider inviscid flo theor# 0= 3 and incompressible flo const= 3

Euler equation:

( 0

( 3

V

DVp g

Dt

VV V p z

t

=

= +

+ = +

2

(2

: 2

VV V V

Where V vorticity fluid angular velocity

=

= = =

21 32

0 0 :

Vp V z V

t

If ie V then V

+ + + =

= = =

16 3

2p z B t

t

+ + + =

Bernoullis Equation for unsteady incompressible flow notf(x)

Contin"it# e"ation shos that 74 for is the *aplace e"ation hich is 2ndorderlinear P4 ie s"perposition principle is valid( *inear combination of sol"tion is also asol"tion3

2

1 2

2

2 2 2 2 1

1 2 1 2 2

2

0

00 3 0 0

0

V

= = == +

= = + = + =

=.echni"es for solvin$ *aplace e"ation:

13 s"perposition of elementar# sol"tion simple $eometries323 s"rface sin$"larit# method inte$ral e"ation33 F4 or F3 electrical or mechanical analo$s53 Conformal mappin$ for 24 flo363 nal#tical for simple $eometries separation of variable etc3

2


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058:0160 Chapter 8Professor Fred Stern Fall 2009

8.! Elementary plane"flow solutions:

%ecall that for 24 e can define a stream f"nction s"ch that:

x

y

v

u

=

=

03636 2 ==

== yxyxz

yxuv

i(e( 02 =lso recall that andare ortho$onal(

yx

xy

v

u

==

==

udyvdxdydxd

vdyudxdydxd

yx

yx

+=+=

+=+=

i(e(const

const

dx

dyv

u

dx

dy

=

=

==

1

8.! Elementary plane flow solutions

#niform stream

yx

xy

v

constUu

===

==== 0

i(e(yU

xU

=

=

ote: 022 == is satisfied(;V U i = =

Sa# a "niform stream is at an an$le tothe &


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( 0

1 16 3 6 3 0r

V

rv vr r r

=

+ =

i(e(:

rv

rrvr

=

==

===

r

1velocit#.an$ential

1velocit#%adial

S"ch that 0V = b# definition(

.herefore+

rv

rrvr

=

==

=

==

r

1

0

1

r

m

i(e(x

y""

yx"r"

1

22

tan

lnln

==

+==

$oublets:

.he do"blet is defined as:

==so"rce

2sin'

1so"rce

2sin'

1 3 "$ "

2tan

1tan1

2tan

1tan

tan21

tan%%

%%&

"'&%'%

+

==

1 2

2 2

sin sintan tancos cos

sin sin

2 sincos costan6 3sin sin

1cos cos

'r rr a r a

r rarr a r a

r r" r a

r a r a

= = +

+ = =

+ +

For small val"e of a

5


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1

2 2 2 2 20

2 sin 2lim tan 3a

ar a"y y"

r a r x y

= = = + cos

2 3a" Doulet Strengthr

= =

)# rearran$in$:222

22 3

23

2

=+++

= yxyx

y$

It means that streamlines are circles ith radi"s

2=( and center at 0+


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058:0160 Chapter 8Professor Fred Stern Fall 2009 A

S"ppose that val"e of theand for the so"rce are reversal(0

1

rv

)v

r r r

=

= = = P"rel# recirc"latin$ stead# motion+ i(e( ( )v f r= ( inte$ration res"lts in:

ln *+constant

)%

$ ) r

==

24 vorte& is irrotational ever#here e&cept at the ori$in here V and V are infinit#(

,irculation

Circ"lation is defined b#:

c* closed contour

+ V d s

=

=

For irrotational flo

Br b# "sin$ Sto'es theorem: if no sin$"larit# ofthe flo in 3

( -c A A

+ V d s V d A ndA= = = =

.herefore+ for potential flo 0= in $eneral(oever+ this is not tr"e for the point vorte& d"e to the sin$"lar point at vorte& corehere Vand V are infinit#(

If sin$"larit# e&ists: Free vorte&r

)=

{ {

2 2

0 0; ; 3 2

2 and

V d s

)v e rd e rd ) )

r

= = = =

ote: for point vorte&+ flo still irrotational ever#here e&cept at ori$in itself hereV+ i(e(+ for a path not incl"din$ 0+03 0 =

A


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lso+ e can "se Sto'es theorem to sho the e&istence of :

D

*

AB* AB *

V d s V d s = = SinceD

( 0AB*B A

V d s=.herefore in $eneral for irrotational motion:

(V d x =

=here: se "nit tan$ent vector alon$ c"rve &Since se is not Eero e have shon:V = i(e( velocit# vector is $radient of a scalar f"nction if the

motion is irrotational( 0V d s = 3.he point vorte& sin$"larit# is important in aerod#namics+ since+

e&tra so"rce and sin' can be "sed to represent airfoils and in$s and e shall disc"ssshortl#( .o see this+ consider as an e&ample:

an infinite ro of vortices:

==

=

32

cos2

cosh2

1ln

2

1ln

1 a

x

a

y)r)

i

i

=here ir is radi"s from ori$in of ithvorte&(

"all# speed and e"al stren$th Fi$ 8(11 of .e&t boo'3

For y a? the flo approach "niform flo ith

a

)

yu

=

=

: belo & a&is


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orte) sheet:

From afar i(e( y a? 3 loo's that thin sheet occ"rs hich is a velocit# discontin"it#(

4efine ==a

)

2stren$th of vorte& sheet

dxa

)dxuudxudxud

ulul

23 ===

i(e(dx

d= Circ"lation per "nit span

ote: .here is no flo normal to the sheet so that vorte& sheet can be "sed to sim"late abod# s"rface( .his is be$innin$ of airfoil theor# here e let 36x= to represent bod#$eometr#(

,orte& theorem of elmholtE: important role in the st"d# of the flo abo"t in$s3

13 .he circ"lation aro"nd a $iven vorte& line is constant alon$ its len$th23 vorte& line cannot end in the fl"id( It m"st form a closed path+ end at a

bo"ndar# or $o to infinit#(3 o fl"id particle can have rotation+ if it did not ori$inall# rotate

,ircular cylinder /without rotation0:

In the previo"s e derived the folloin$e"ation for the do"blet:

2 2

sinDoulet

y

x y r

== = +

=hen this do"blet is s"perposed over a

"niform flo parallel to the &< a&is+ e $et:

2

sin 1sin 1 sinU r U r

r U r

= =

=here: = do"blet stren$th hich is determined from the 'inematic bod# bo"ndar#condition that the bod# s"rface m"st be a stream s"rface( %ecall that for inviscid flo it is

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no lon$er possible to satisf# the no slip condition as ares"lt of the ne$lect of visco"sterms in P4s(

.he inviscid flo bo"ndar# condition is:F: S"rface F"nction

10 ( 0 0DF F FV F V nDt t F t

= + = = = for stead# flo3

.herefore at r%+ V(n0 i(e( (rrv ==0 (

; ;r rV v e v e = + +

2 2

; ;;

r

r

r

F Fe e

F rn eF F F

+ = = =

+

2

11 cosrV n v U

r U r

= = =

0

2U ( =

If e replace the constantU

b# a ne constant %2+ the above e"ation becomes:

2

21 sin

(U r

r

=

.his radial velocit# is Eero on all points on the circle r%( .hat is+ there can be novelocit# normal to the circle r%( .h"s this circle itself is a streamline(

=e can also comp"te the tan$ential component of velocit# for flo over the circ"lar

c#linder( From e"ation+ 2

21 sin

(v U

r r

= = +

Bn the s"rface of the c#linder r%+ e $et the folloin$ e&pression for the tan$ential andradial components of velocit#:

2 sinv U =

0=rv

o abo"t press"re p loo' at the )erno"lliDs e"ation:

( )2 2 21 12 2r

ppv v U

+ + = +

fter some rearran$ement e $et the folloin$ non


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( )2 2

22

+ 11

2

r

p

v vp p* r

UU

+= =

For the circ"lar c#linder bein$ st"died here+ at the s"rface+ the onl# velocit# componentthat is non


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bodies( circ"lar c#linder is not a streamlined bod# li'e as airfoil3 b"t rather a bl"ffbod#( .h"s+ as e shall determine no the potential flo sol"tion is not realistic for thebac' portion of the circ"lar c#linder flos 3906 > (For $eneral:

F(I*-I!.S)I.D(A/F!(0DDD

*** +=

D(A/F!(0D* : dra$ d"e to visco"s modification of press"re destr"ction

F(I*-I!.S)I.D* : 4ra$ d"e to visco"s shear stress

:

:DS) DF!

DF! DS)

Strea"lined Body * *

Bluff Body * *

> >

,ircular cylinder with circulation:.he stream f"nction associated ith the flo over a circ"lar c#linder+ ith a point vorte&of stren$th placed at the c#linder center is:

sin

sin ln2U r rr

= From ,(n0 at r%: 2U ( =

.herefore+2 sin

sin ln2

U (U r r

r

=

.he radial and tan$ential velocit# is $iven b#:2

2

11 cos

r

(v U

r r

= =

2

21 sin

2(v U

r r r

= = + + Bn the s"rface of the c#linder r%3:

2

2

11 cos 0r

(v U

r (

= = =

2 sin2

v Ur (

= = +

e&t+ consider the flo pattern as a f"nction of ( .o start lets calc"late the sta$nationpoints on the c#linder i(e(:

2 sin 02

v U(

= + =

sin - 2 2

)

U ( U (

= = =

ote:2

))

U (

= =

12


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So+ the location of sta$nation point is f"nction of (

2

)

U ( U (

= = s

sta$nation point3

0 0sin = 3 0+1801 5(0sin = 3 0+1502 1sin = 3 90?2 1sin > 3 Is not on the circle b"t here 0rv v= =

For flo patterns li'e above e&cept a3+ e sho"ld e&pect to have lift force in K#direction(

%ummary of stream and potential function of elementary !"$ flows:In Cartesian coordinates:

yx

xy

v

u

==

==

In polar coordinates:

rv

rrvr

=

=

=

=

r

1

1

Flo !niform Flo xU yUSo"rce m?03Sin' m@03

ln" r "

1


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4o"bletr

cos

r

sin

,orte& ) 1 ln) r90 Corner flo 362-1 22 yxA Solid


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Meep in mind that this is the potential flo sol"tion and ma# not ell represent the realflo especiall# in re$ion of adverse p&(

he *utta 2 3ou&ows&i lift theorem:

Since e 'no the tan$ential component of velocit# at an# point on the c#linder and theradial component of velocit# is Eero3+ e can find the press"re field over the s"rface ofthe c#linder from )erno"lliHs e"ation:

22 2

2 2 2

r vv p Up

+ + = +

.herefore:

2 22 2 2 2

2 2

2

1 12 sin 2 sin sin

2 2 2 8

sin sin

Up p U U p U U

( ( (

A B *

= + = + +

= + +

here

22

2 2

1

2 8A p U

(

= +

U

B(

=

22* U =

Calc"lation of *ift: *et "s first consider lift( *ift per "nit span+ * i(e( per "nit distancenormal to the plane of the paper3 is $iven b#:

=Lo2er upper

pdxpdxL

Bn the s"rface of the c#linder+ & %cos( .h"s+ d&


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.his is an important res"lt( It sa#s that cloc'ise vortices ne$ative n"merical val"es of3 ill prod"ce positive lift that is proportional to and the free stream speed ithdirection 90 de$rees from the stream direction rotatin$ opposite to the circ"lation( M"ttaand No"'os'i $eneraliEed this res"lt to liftin$ flo over airfoils( "ation = uL is 'non as the M"tta


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058:0160 Chapter 8Professor Fred Stern Fall 2009 1A

,elocit# ratio:a

U

.heoretical and e&perimental lift and dra$ of a rotatin$ c#linder

&periments have been performed that sim"late the previo"s flo b# rotatin$ a circ"larc#linder in a "niform stream( In this case (v = hich is d"e to no slip bo"ndar#condition(

< *ift is "ite hi$h b"t not as lar$e as theor# d"e to visco"s effect ie floseparation3

< ote dra$ force is also fair# hi$h

Fletttern 1923 "sed rotatin$ c#linder to prod"ce forard motion(

1A


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18


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19


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8.3 Method of Images

.he method of ima$e is "sed to model Oslip all effects b# constr"ctin$ appropriateima$e sin$"larit# distrib"tions(

Plane Boundaries:

2


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%pherical and ,ur(ilinear Boundaries:

.he res"lts for plane bo"ndaries are obtained from consideration of s#mmetr#( Forspherical and circ"lar bo"ndaries+ ima$e s#stems can be determined from the Sphere Q

Circle .heorems+ respectivel#( For e&ample:

Flo field Ima$e S#stemSo"rce of stren$th R at c o"tside sphere ofradi"s a+ c?a So"rces of stren$th c

"a atc

a 2 and line

sin' of stren$th a" e&tendin$ from center

of sphere toc

a 2

4ipole of stren$th at l o"tside sphere ofradi"s a+ l3a dipole of stren$th la

9

at la2

So"rce of stren$th m at b o"tside circle ofradi"s a+ b?a e"al so"rce at

a 2 and sin' of same

stren$th at the center of the circle

21


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7ultiple Boundaries:

.he method can be e&tended for m"ltiple bo"ndaries b# "sin$ s"ccessive ima$es(

13 For e&ample+ the sol"tion for a so"rce e"all# spaced beteen to parallel planes

( )[ ] ( )[ ][ ]

( ) ( ) ( ) ( ) ( ) ( )[ ]

++++++++++++=

+++= =

azazazazazaz"

anzanz"z2"

2lnln6lnln2lnln

2lnln3+2+1+0

22


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23 s a second e&ample of the method of s"ccessive ima$es for m"ltiple bo"ndariesconsider to spheres and ) movin$ alon$ a line thro"$h their centers atvelocities !1and !2+ respectivel#:

Consider the 'inematic )C for :( ) ( ) 2222+ azyytxtxF ++=

1 1;; ;0 or cos( ( (

DFV e U 4 e U

Dt = = =

here 2 cos( = +

2 cos

2(Ua

( = =

Similarl# for ) 2 cos D( U =

.his s"$$ests the potential in the form

1 1 2 2U U = +

here 1and 2both satisf# the *aplace e"ation and the bo"ndar# condition:

1 1

D

cos + 0D( a ( ( (

= =

= = 3

2 2

D

0+ cos DD( a ( ( (

= =

= = 3

2


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1 potential hen sphere moves ith "nit velocit# toards )+ ith ) at rest2 potential hen sphere ) moves ith "nit velocit# toards + ith at rest

If ) ere absent(9

01 2 2

cos cos2

a

( (

= = +

2

9

0

a=

b"t this does not satisf# the second condition in 3( .o satisf# this+ e introd"ce theima$e of 0 in )+ hich is a do"blet 1 directed alon$ ) at 1+ the inverse point of ith respect to )( .his ima$e re"ires an ima$e 2 at 2+ the inverse of 1ith respectto + and so on( .h"s e have an infinite series of ima$es 1+ 2+ T of stren$ths 1 +

2 + 9 etc( here the odd s"ffi&es refer to points ithin ) and the even to points ithin(

*et Qn nf AA AB c= =

c

cf

2

1 = +1

2

2f

af = +

2

2

9fc

cf

= +T

=

9

9

01c

+

=

9

1

9

12f

a+

( )

=

9

2

9

29fc

+T

here 1 ima$e dipole stren$th+ 0 dipole stren$th 99

sistance

radi"s

0 1 1 2 21 2 2 2

1 2

cos cos cos

( ( (

= K ith a similar development proced"re for 2(

ltho"$h e&act+ this sol"tion is of "nield# form( *etHs investi$ate the possibilit# of anappro&imate sol"tion hich is valid for lar$e c i(e( lar$e separation distance3

22 2 2 2

2

1 12 22 2

2 2

2D 2 cos 1 cos

1 1 11 2 cos 1 2 cos

D

c c( ( c cr (

( (

c c ( (

( ( ( ( c c c

= + = +

= + = +

2


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Considerin$ the former representation first definin$c

(= and cosu =

[ ] 21

221

1

D

1 += u

((

)# the binomial theorem valid for 1

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