Lefs J�( ve., -tk �d.xv �;tGt.\-t c.- (or a.ef'o st:aJ\ c)
�ak\on . Coh s\ dex- �.e..- ba.t �.J. rtl.\) i t-a.tiol'\ .Por�.
·
\AJe- htLre..
o�r st-o..'"'t-;V\.� �OtV\t is Su.lex·'s t:���on �
Dv :::: 0 c t\ 0 CA.CCe-[�; OV\)
bt
=fo-\7f
_.,
0
"P��6
�
,-
\] p ':
�x
(� }
o(j
E.£. ) Q_t
�� )
-r
�
fa :=.��X ) o'a', d:z J
d t \Jp
-50� l �f�
•
I
I X
I
'2-
2
fh-t = -�oh
�f -=- ?o -
fOh
Lek 15 "'ff\ 'j tk boWA-tLxy 0>n J..�� o"' f -tk �u..-+tv.
vJ"-.ue- ="" \
0 �
f ::= fr.\::::: f a.....Jol�t
f""'�Stu-e.-
0... -=-
f�-=- Po- gdhl
\� fo""' Po.- + f 6"'-t W"'ert- fo '=:: r�SlU"e- ,J- � bo�·bo�-r.
? :=: ?o- +�6h, - )�k
o-r ih t-e'f'V't\.S: e+ tlL Jq� � ::::. 'n, -"'
3
A ss \Jv'M � t-\ $ ·• o�
\. sta�'\ e.t s.We... ef cv..... id� 64-d
@ p -=- � RT o� f = ft.
f
dp-=- - RT 6d:z.
® �=-LJ.::z.
f Jri. "'ttoVI +
'01s}
l
Buic Oow equ3tio:-.s Usdulconce{IU for the
(conlninlng the fund!amlinc t-
f-
H Energy ('IQI.:y 1\.z. �z
-:: +
Cross P.roc:kJ: " "'
' "'
I.
� ( "
A.,.. A'l A·:. A2 l6y) \·
=
= (Ay B� - ·l-
B)Y f::. r (t I9 ,'f)
\1?: � � +
,
;,x ll � + 'll �
�'(3 0 ';?) � == (�
� )
� 0·
v -o
'
-=- � e-.,. + � e.e
r v--aa
-+ �
�;t.
�
i
_?�ev:�.e...u. o(· a.. ve.obco.- ,�&J.
'
• ' 0\Clk \/· ....
v ::. I :i.� 2.."\
c� "+ ()� �: + ";):Z.11/'P
�
rsine
---
a'f
�
1--
---
·P,-sic.J. "'e� o� euw-1 0 'rs t1...e Q)...a.._e
..- velodty ...
� o.f � \u.i.J e,\.t.w\�ct
o. . We. 1.u\\l s\...ow KJ::
iS ..L
-=: \ V F'o"' "'c\Ms
c.u..- . '(e.uOY\ �� e.«\ i s .Js"
�
-ro+:v
z �
coltM.oted
Co..lled � v-o'tor o� V cu...J.
V X� �a � f2) )>;j 'd.z.
Vx Vy V-z.
-"
v -=
A A
I"
v're.-.+ Vee9 -1- Vrp,€'f'
"' A
e e'f
"
rea
•
er r�t�e "Of
Vr rV6 Y"si" e "'f
-
.... -?
...
E \ e-1-,.e-.,t of cve.o... =- d� -= tr-. a A1 1 t;\ev..��t ot \e«-fft�.:::-llsidll-.. \..c>.\.f "- Sf\,U'f.. S bot.Md.-J.
ka0 1).. �se\\:- Rt. o" d: ·
f.II ,;� !
. ..
ci.-c..b. o.....d � t,J..f!_ s.f�e.�S a. $ u.r F�� ("'�eil� c ve.c.to\'" �lu.r- oV�
�"'-.c.Q.,Se.J S'l1.i"�"-·ce.) h ho�;o.� ·i\'\1: �
a. Vo
.,
j A-ds= -fff. (V -"$...)df
s ·ry>
We...e 'M;�d �-i..t._ sow..\,o\ ��· � vol�e.. siii��loV. �tv'e w.ie\4; J, .,r fl'it"((..S ·SI>ILI!_ l...it.. 1 • , ..
"
ns 1>
�
l 'U(Xi
\jJ'e: S '1 . fi ·:S e.e: :1!1 it>\.
t "
-> I D � (_('A1r>)'
. " . " .L v V· "=
�."r t>.t- .
___:...'
·
i� �a.:f:Je..ol. ,Vo� """{J.
t
p,7
Gv-o..�� -t�..�o(tA>v ( A"'-"-\.o o-f t� (' ll.u..t
� 0i·v). r1Mf
"'('
D{��>-= � c-v·v)dr:r= �- �) ��
·Dt-
�"'f
'!/. · .,..,..
' .v ::. \ � ,(?Y')
..
�""'(' t>t
• -
], ·t>, Is. � t\i.e. I'T\�r-i·d 'J.er� v-q_ \?ive +l--e.
ov-
Dt .� l(.b�ti� f vo\��
� -::. UMtt \lee-to'N\.J.. � ot �-\;
v�h s�o����. 'T\..e'('e.fore...
dmc.v �
- - ,, \C.$
_
dt
M cs -: � �>{y �) d A ·
c.s
-�s��c-1[> = ��cv.�)dA
cv cs
1"� c:ll�e...�.J ..Po.-m of � c.ortl't � ex:�;VI
Sinu. � eon*.�o\ vo\l..VW\_e. ls .r\xe.J. i� stll(:JL �
t� \oot.tM.� Q.fe. �(xed I
fu i:lW\e. olertvctkt ve.. .:£.t
C"-"". be. W�\�� \\1\S\de. �
.. til\� 3
§ � dMf � g Cii-�J dA + =o
cv �t cs
Usi no � G-a.u..s� �ve..-� � Wtc � I
re.wV.t� � su..-+o..� t'V\.� e� � k vol...t.....sL
11- "'-
·,1'\� ot- V• (�v)
f�i J� § 'V· (� V) d\ Vo�� ove-r OJ
�[� v· C�ih] de-y:>=
� o
cv
s·rt\ce fu CA./ is of a.'t'-'btk\-o- s��e..- �J.. �\ ;t:e...
4
�+V·(fV) =0 ,
A, ?Ck0-'� tL:.tf��o...t �/}JU �� --(PD£) �rpli�ble. f.o· lwl
?0\"-t tY\ � �{ow�c:JLcl��-- I I
..
·
f-o'r sh.� .s t.J:e.. �low � "-o..Je.
'd� o V\o tlVV\e,. va.rtctAloV\_t o.f +�e. d..�it-a.--
-tt.e_ oKe. r �Y"o feA-'' � C9 .f- � �d.
-=.
0t
0 v-
�lt&Acl fYt>f�e.-!. � s�l\\ vcuo w( � �
·
�e.
$� C.:.ord.\n ..A� '1. e �;::: � (X 1� 12.) I
v-=- v ex ,6, :z) d-e..
·
§ g{y· �)dA -::: o
c.s
v ·LiV) = 0 v-=-- (u.J liT, M)
'J • (fV):. � (Pll)
�X }
+ �
'?>
(ol\r) � w)-=:. 0
� \..S + CJ ;;t. (f
rov-- the, � of' in com press.tble.- ..P/ot.u (+his
i� c.kks s� � 1.\Av)-\LA..� .P.lo10s)
we �we.. ._
g = eons.t Cl�u.w.p��ilol�
0�- 0
-
�t
' I 4
�1$ Su.ond La:;. of- Mo-HoV\ .A,r- o.. f-1� P�4-
A\so kY\OwV\ o...s t�e. YV\OW\e.�t� �loY\
t_�e, t"l.1.-W\ i:h e.o retn
vnov\ hJi !_ol�e of-F(
o v-
.J
� oW\ e.\'\. .
(l)
'
'
•
; I
.
�
col\e..J-i o+ �lui J. f \:w- �{ �>+ tno\fii!M-�
) -\:k �t
Ieo�k\ s�-t"t � (..o\'\.�{ � �)
I
d(tnv)=
4
I
1
I l
dt -1
I N�t: 1:-otca.�
..L ll
of e�d.W\�V''Ej
Cont\ �e.��\ov. 1 we.. J +o rewri c..cu,.._ �'to c..u
-t\...e.. i t��
\'\ cs Q.. �t.,_� 0" ts � �.tb\e....- �d. visc.o� f21.ow
sre.UJL
.. .PfoW U.(X,(11%.)
� =
I. �te.� sh\:-e. ?t. = 0 U.=
4�
(fV V) - Vp f't
� -T
'\!• = + +f"vi$
\\1\\ s is a..\ so K'f\oWn a.. s tke... ue.-r-D(f
e...�io""
euncl &e... f\�t \...our of l'�ex·vnoc!on�t�S .
fo-r o... -\:\-.e.\'1"(\ o �,.,...u,c..Q. s 1'te-wv o{ P; xeJ.
�0..$�
I
We. \( V\ OW t�a.t- � Q.,W.. 0 U.o\.t.l- l) f he.� � 'J
a..olde.J \;o -\i.e_ S(jst� �J. � wcr� �w d.�
OV\." 1::\...e_ 51>t-� \.a- -l'l-.e $ � """"- � cJ..,. CIM.� I
.\:k i>�h'("�oJL � E e.(. Yke. $'f>t�
$;1- + h
o.f � � e_J; t-\owl '6 i "-\;o -l:t-e CN p Lu .s �e.J
n.
��oiul. b� \...� Col'l.k.c.hovt cu.. d Vi.S co u..5 cL:� i ?.Ji t1
*' J .f� d. cf
�
C)./
-1- Qvis
T"'e. wo.-� {llt..u:J. CoN\..sts ot! fu Wor-k.
ot\ �
ol� lo{t � �� o..ck-t"-Q o"'.\ \:\,_e. {1-(t.UJ J
?.M...cl �oW"Ie.J:t.uM �· .
S � e o:cJ._ CaAtD OJ('� ••
'. �v W h_l cl_ �
4... 0
r
-=-
No bo"! t,-rte..-S � 0.
1-.--..
_
S. -=.
F I t.u:O. kI '1\ t. ""a:t.:c s : \
I P.- · ·-
T�e stM.� .,+ tl�d rnoti oY\ wi�ott,t lies c.r\ Lit\} tL.�
.f.oY'c.tS
� \/e\o U.,ta �t'elJ. ·,s. kltowY\ Q.u.� �\v� \a� •.
a.-"6- �
4 � ':
\J :: "'-X\ - LL = CA;>(
1\f' = - 0.. '}t
E!i-�--�--1 d
dx o..� X u.. -
� hh�v-�)we aet:
- -
�=-
0
.
Jl,v. 0 �X -1- C
:o
xd:: e
-
� )C.�-- c b., c.o�o.�t-
X
c-==
(b) f'�:�.-1:� \ i he : Tl.... �d.je �.t.'"� J.esc '"i \..� l..zr p.2:i a..
tll..U.J. pQ..rb.'c\e. o..� ft tnovt. t\..v-ou..� �e �low h'e\d..
(c) ��o(f- - p�>i>�.T. >\:\'"ea.I t-l'""e.ll. �>' \l"" e.s. .
.
\'h�e.. two �� Q.,f�fo� \\1\ � .QiMJa- of -9�d P/ow,
ctve
-l\...e La..�'fcw.."r �f•oc:t.M..J.. � Ec..ler Clffroo...d,. .
r
LQ..�r�rQM... a..rroa.� T�� W\Ot{o� ot �lt.v.:.d i..s : a....
..l.e.scril.eJ \,(} �olloLU\11\(J ra..ttfc./ . of fi>.. ...
Q.V\Q. d.�t"ertDKti cA� a...
MeV\�� o...\o'V\6' �e>..t� \\�e
o... A tlYV\t. t -t"'-e. ��rtic.[(.. 'rt
.
loc.oXe4 ()..t poi�t ?. Its �o>it;.on ved:ov--; aMJ.. Ve\oc.lty
V Q.v-e. Y":::(X)�>;:z.) �.a V V ( Y" ,t).
� 4 -4 � ....:::1
=
The pa: 0 V\ e.t\.-\: 0 t -;: l s,
�
a�
a.x-::: 'du. + u. 'Ou.. + rv ou. � w
-ot 'C)x �;(..
In tTo�ce � s\.ort'v-t� '1\.0-\;o..,-\:-ioY\
1?. -2- +AT� � 2 V· \1
-a�
+ Ll 'd + w -= +
Dt- 'dt ox 'az -ot
This o pev-�'tor is k�oW\'l Cl..S t�e 0'\�ev-\oJ. J.ev-'l volt ve J
or
tt_e & 1.1.b J4 (\i.J ole
.. .. I \1 O.�i,.¥e. o Y de i Illa).; ivt. �\low\ 11.a
�e �\ow,
r---- -----�-- �
Pv-o'b\� 4-.l4 ( Mun[on eA-. a.( �yo\ Ed.') ��
'
>
A 2- a ve\o�\:1 tielJ. l.s a'"� \:,()':
C ,Y
� �
\A..:: C x "V = w��v-e
\ C ::::
�
c.cn�L.
U A
� a.� J
A.
L Dete�mlt'\e. �e. �\u4J. o..cce\e ra.:�-\ o� = �x \ �
D4 = �v
__!
�
4
rov
� 4
0..-== + u. �v + IS"
Dt -ot -ax
tq�
a.x �+ u rou. + �ou.
-a0
=
ot ax
Cl.6 -:: �+ �+ IV '0 t\T"
�� '()X
U.
at
T"' ve\o�t� Com poV\4!nt� u.. OM-a AT �
�
\�e"' 1 So we.
c� cAt.t"e..,.m\�� t�ei� det'\VA.t\ \It� :
'()u.: 2CX � = o o.v:: 0 �-: 2.C (1-
�x
?>� -ox
-a(;
S lA.bsti.t""·t\""t' 1:L.c.. ci�\ v�\v�:
a.x = u � -::. (ex') (l.C x) -== 2 c�X a
�X
a.y �-:=.. (_cd�) ( 2 c'O)
=- 1\) :::: 2 c'2 a 3
'd(1-
2. A-\: w�a.t po\�t.s tlV\ R& .0/oc.u t'd'J t.s 0: = 0 ?
�\ V\ \- ( �, �) ::::. (_ 0 I 0 J .
�-- ------'--'-- --·-··-·
�.
CoY\.Stde'f � 'lwo-c\.\m.e..¥\.S:to�c:J �{ow. Choose ot\e. c.oord.lnct"te'
'
�\o�0 t�t .stf'e.A-m \\Y\e. GlY\d. �� o-l�v- c,oov-�V\afe... perfet\dtc.u.lq..r
to the. s.tY'e.t:tW\ \\Y\e. .
� �
A ""
Detihe un\t
'h.
ve.c.tor \)
�....,..._�
,A
t "-l\ �e.\'\t-
A
S Md.. M �ct
rerpend� �\�r to the.
Str-eo. � \\ S -ce s ��c. t \ v' \ 0 0 � e tA �e. t t\u �
x:
t\ e c:l e, . ;
J
o.. v Cl � o
s1st� i� t� 4T v l.s �\ Vtlf.., \,'a- s\m�le. e�f"essao� :. �
-? A
v = v s
0.... _.;,
�oY" �v; 2-d �lo'V 4
0.. -= �s
1\
S -+ a."
A
It\. = 'D't_
Dt
V -:: l V 1 V (�I tfL)
:= -:::. '(n tLQ V\.\ tu. cle 0 � �
V
.
S ( s If\.)
.A
d.\ 'Cec;t'
......,
A r
S -:= 1 = 'on 't V
�
Cl -
1)�
- -
1)(V�) Dv A
s: + V\)S -
15t
ot Dt
_ _
_
Pt
- - -
Js-
n-
v
0
0
B
• FIGURE 4.9
A
n· Relationship between
the unit vector along
the streamline, s, and
A· the radius of curvature
o of the streamline, CJt.
"
A f\ os. rv A A'o P:,1 Si �\ \a.r tt-\�� \e> .
li-= tsst
\.__.,;
T 'r..tY"C .C«e.. � I fS I = (fi 1
�
ls t
-
t
-
Tktl\ ( �� J I
S's ([
�"'a ..t�e o...cc..eleY'"'Q..tto� \ s 0 I veV\ bO':
a..
� -=.V�s-J--n
0 A \12.. ""
o> (R_
O..s -=. V
oV is i:ke. CoV\Vec..t I ve ac,elenJ;\ ��
�.s
Tke ronen-1: �'l. i.s. ti-e centri.�e.t.J �c..e!en:tf"�on
I
c..o� o..h =.
A.clva..-.t�a� of c-l,oosi � -1:-l..e s:t"..-eawdine s�.ste..- o� j
I
T�e. ...
Coof"dl Y\o....-\ es \ s :
I. l�f �cc.e\�tiCn Q,\0\f\d �e.. d.i,v-evtlOV\ 0� �e. $-tQct\t\ lih� �
i � �� to Ke. '-t-v-t a. \In \\'t\e .
2. T� o..cc-z\ero..\ion ?�V"�end.\c.aJ.o.v -to *"'-e sT-v-e'""'lt'�e.> ·,s
�e. toRt. C(M.tri�ta..l efh�c.J· > \fer0 si�llo..r- .\"b � ettt!cf
Y"� �\ J. ��y-\-\c,lt. t,-r�d. t" cl.GM.� 'lt.S J..:�cl1oV\ Q,lo�
,
ct � o �
c..u.rv-ea f� . 'T�\s �.s crt pri_,"' f('ov't d.e.s
o.., better fh0.si cJ \
u.nderstQ.�d.\V\.� .
a...
PrcL,l� 4 4- (
a.. � \ i � iv�.,
·
$-tl rt�c..e.. o-f. V\ der is
V\orrna..\ ponent.c at
CoM � le.Y"a..ticn. of f/�d
�c.c:..e a..
-pa.rt,c\e .Qiow't�o u.lo�d �� su.�fc...t.e. of tle. �l\vtd.er.
n A
S:oh.tT\ oY\ In
s
: kl� strea.rn\\nt
>ca.steW\ of C...Oord.\.nC\..-te.s;:
i�e. a.c..c.efera.. tio\1\ is. �\ven b0 ·
.
OV'. tke S K� (Jl\nd�, tle.. 'C''l� of C,u.rVa.:b"'re..
l.u·fa.c.e E't
l S � q,;( to � 'ro...cll.uA of � CO\ t\11 d.
·
(]C e� -=- a._
An eletw\e��o+ \en�t� d> o...\o�(f ot- c.l.rc..le.(.s. G\M.. Q..r(.. Q...
%\vtM., b� ds=:. a..de
?JV 'OV J_ �-= l. 2. (2Vosit19)::: 2Vo Gos8
=iS -
=
-oCQ..e) Gl.. -oe o..... 'Oe
_
o.-
� '2. v 'Z. 1\.
2 Vo �n 2B
I\..
C\ � 6 + - h .
0....
CD ·
A .stre.a.M H'f\e. Is o... C&AKVe. wn.: c"-. Ls pct.ra..lle-\ tc t�e.. ��OW �
c:U('"ec..tton .poi\'\t" ·. a...
oiX' d�- d.� 'Z.
(A 1"\T w
For
--- ....._.__
2-D .t(o\4J: X d
dx
---d�
u..
_
f\T
_____:;, �-=-
dx
�
LL
Q e I ..± (.,._ be:l:wet.v� tLe. hea..mlir.e.
s QJ.A..d._ -1:i...e. c.o llti...W:I:-(1 e1..
V·V =- -()u.. + at\)=..
a�
o
�x
.
Ihtf'o�c.e.. 0- s:� .func.t-lOl'\. 4' = 4'CX,'(t) tcr s� 1
-0low.s or 1./J = lj.J (X1'(1 J t) � UX\t� �w.s '"-� -tt....d::,
LA- = � - 0 4'
o�
1\) -::::.
·
'OX
ea..S(t +o Ve.t\� tl� t�\.s d.e.t\"-it\o'¥\ e\ '\)
Jt't.s
scJisf�e.� � C..0\1\tl��� �c:ttioV\. S.Ln CJL ·. >
-v·1-= �+roN
� a"4' :::::::
0� axo(l -oxoe1 -
o
ox
_
-r�e �W d.�'�ev-t.n\;-\oJL �\ tp \s 6\"eM..- h(J- �
@
d 'f -:::. � dx + � d.� ::::- � ...v-d x lL d.�
o(t o
+
-ox
It dtp-=:O J
i.e. YJ�t:Pns.t tl€-V\ -1\f"Q)(+Lldd=O
OY .:!..4 � tk �o\1\ of. 0... � h-eCUM-ll� .
_.1 - u...
)
_
.
o...lo�
�x
w� (p� c..tu..ck.. tl..a.k- � ( � c..,� s.:t-� a.. �t'('EJCl·�·dihe
P �' i � V't\et\. n \ � �+ � st"Q� +u..v...c.:tioV\ .
Con�ide.r- -bl.e. >iWYtpll-k'ecl. w J
\N\...e.y-e � .-0tv-e.o..""\\ne; �ctY'a....\\-e.\
a.t"e..
b.t�0�t- l\r\e_s. .J so t� V'e\o �tcr C.O\Y\ pone.vtt LL =..u.c0)
� (() ) .d,If d ljl d.(t
lA. -=>
0� lL = u. --'!> iL :::
d.� 2.
= LL
c�h.
Tnte.or� �cro..ss d-'f.,_a.. line_s
.... lfz.-4{==-fd�==- Su.dd-l dt
l\J'l. \�e. Vo lllm e:f:ri +loW \tCte.. �
s')j� c.La-- {low r�
c:..
q_
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-::: =
r � u.nic dept-h..
()I
X
l�ere. 4'2-- l\Jl =:}
i.e. tle dtt.fe"o\ a\v� b(j :
·
\Vl a..v- c..oord.thaXe..-' l�
V·-: = _!_ �
e> r-
(rrv-y-)-+ J_ �Ne =- o
r- r- ae
It i.r e.as(t to ckevk.. �� tk a.,lr,ove. d4'n�ttoV1 of tke.
&.t-rea. �c.tt'o n 4'
vn & cdis. �te.> Ke. C:..OY\1: i n.u.L-t0 e.ttlA..atioY\
l de tiCA..\\ a .
t'\
Ext:U\'\p\e: I. � hoLU t�� tke.. .flow d.esc:d t,eA. bet- tk
� llow i "-(r vel o C:.t0"
o � e l� .s. w �o....t r = 'f0 c..ov..rt l..s ct... .st'\e.a..m h'\'\e
·�..� a'VeAA b� :
-=
Th� vel()cl.t(t .P.ietcl
2 2
t\T� =:: V0 (\- � \ c.ose �d Ne =..-\fa(\+ Y""o )sin
f2) f2
he� Vo \ s C..O\'\,t-� '('ete-ce-v\c..e. >peed·
.
w o.
� o (u..ttort
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..
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Q.. 2- D &. low � �o\ CLr r
�
2
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'f -ae r'2. r
'2..
�-= Vo �
(r-+ 'ine �tV::: 'IoCr--'§.. s-ine ..t--f,(S)+
t)
_ _ 2..
o'Cr, e)-= (
"
Vo r- �) s·me
)�et-ch\"� ..so�e st-'fe�U"e.s. ·. .
,_
�. For -r ::. ('0 1...J> ( ...-0 e )
> Va ( ""o =. - +.) s i11 e = o
r -:=_ '("o ....:;> 4J c e ) ::: Q) \ 5 0...
Y'0 1 S tre. �c.u ko.-., fu o r-t 0 t't"\
� Cb\ter o{- �e- COHndeK"") �
O
wh.\� i s Cl... i v-e.n
�c\� e:
V-: \V\ = JrJ"; Ni
+ = (v� c..os. 'l.€7 + V0"'- slvi'"e J -:::.Vo
..
f().r � ,A-oil'\ \-k c.a-t\'1\d.G crl.ba
eon ()..
�low oJo olf Y'ot "t
va. \ V VXV :=. c..u.r
.
=
k
r.-
J
""
l
\Jxv - - $_ =i c�- �J-ac�- �')
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'd �
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=-
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u.
�IVeh
t
w +k(�-�)
a(1 ax
If tk�vvelout0 c..owq>o ne n±,s. (lv-e .cu.� fka_t-
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point In .t\.e �low .Pie/d. �IL.Ud. pct.r tA' cl� J t�a.t
; Q. o
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pot t does not V'o-\:a.te. 0-f'Ol.C.N\cl � ro\nt ct.S rl tn�\ ckl'" Vo�tt.)( ptodM.ced- when d.rcU_,\'\\v\j-
J
Q..
0.. b�th tu..b. The. �luid. W�t��"' -t� c.o-re o� � vort�>'
j s. t"�to..t\.o�Q..\ > w he�a,s � C.lol.O ou.t�i d.� � c..o� L�
\frotct.'t\ot\c:J. , r"is c.o....-.. loe. verl{l'� �(1 �\ o_u 'r\CT �
D f(oQ.\:-i�� obaect l\1 Re. How: r"' tk� (..ore)
t� ok�ect rotcd-e.s ri�1d. bo� . Ot.t.t.side.
a..s. �
t-lc. Cov-e. tl€. olo�ec.t w\\\ ke�p t't-> ori.e.�td.t'on
J
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CPm ponem;> .sa-fl.cf� � toU ow i n.(t rel�\ on s
-aw
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oX '0'0
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_
ox � ;z.
bet-IV\t. o_.. �$\o� y; suc:.k t�4.t- V-== Vy; == dro..dt/'
i.e. u-= 'dt.P 1\T=. �� w=. 'dlf
�(1
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LeF 5 J....ec:...k_. � c:t.bov� \''fv-otQ.t-to� o..lt td- re.lo.:t"t'o� s:
'C) w 'd'2.t_p
'(?)0 = o-o o;L Th.\.s c..ko,'c.e. o+ tf Sct+,'s F,'es. t-k�
lrro t t�ol'lo...l: tb Gol'\ck t; ide nh'c.q(0>1
a?...p
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::
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l al:-ex-- . If we. kt\ow f e-verQ- w� J
we. � ob� �
ve\o� �t&
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1 -� �
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s khoW� tt� +l.e_ �orfhoJ.;t-'0- e.ond..i. tl oV\ .
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ctu "" -:::.o
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U \'\\ forW\ .C.l ow d
Co r\. d.e.lf"
. �� p o +-e ... f:-� lA -== A -==- u be. Cllv--c:l 1\) =- o
-= C.ovu.t � 'f = u"' a + �(X
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J Q.. Q_
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r =
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2.:trr
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-:::
o-c zlt''r z 1(
F{'O� � �� �e.W\.� ru Jio�
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\J
q> , w� o'b t�\.� \..P :::: Q.,e
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r
z rrtf
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r (} 9- 21\'" r
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)
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pr-ln � r\e. �
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et �r� e o.lr-fo\ l � .
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p'('o b\� t.>
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po\ nt .
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w o rk cl t-- -
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l n-\:e.,.y...t\ � .., ;_s �ee.t-. � f-wo poi hts.
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Source or sink
' 2Jtr
u
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, _
(see Fig.
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Fig.
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a.�
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un't Pc>rM. �[o�
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r
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., _: sh·f j
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of � t coo-c&.o... s�� o.l � \oec:JioV\ of. +l� Joll.b\
�
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ror � �u..b\ek X
t..p -::. sin e-_ ! r
2.11:. '{"
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f""�t�U..�\e- o� �
of � toW\ b 1 wt-i ot ltM..l f:or
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2..tc. Y'
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�
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21(r
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r o €1 'f
v& -=- - � _-U� s\ "& - � 2- s\ � e
y- - 2tt r
st�"a±�QV\ �o\ Y\,t>
+
1 At � V-c o..v.A \.J 9
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:::
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o :::- o
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dO vv- -== u lx) c.os e- - -=- o
2-ocr�
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'l 2rr ''2..
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Fro rn (2)
�... 0...,. (I) (U b6 - ..!._.,) CD $ 9 :=-0
� 2:1( 1"
ihe. -1:-�oro �� ... $ � to b� solv� StMu.\.to.� '2r
-Per � Q.M.� e . S i Y\ u.. j 70 �a.. r- '7 0 1 � �
(U � + r J iV\ E, . (f) �n at VO-Mi s� �a w� ha.v
' 21{f4
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Vy- = U D'> co s e = o ...& &
V& ::: - U �>- s·, .._e (1 Sf.,J + = -2 U!l, t i t\ e
lo obt� � �.s wre- Jl st-n btvtlon ()� � t\(,�
of- � �\\1\J.t'J �J.� t
o... t:le_ avJ:eK" of &e.
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lb'b � f'r\1-.ttp\e_ ot- V..� «fosl \;i
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o ,
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l � -= U� c.os. & ( l - - J
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r ae r"'
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= -
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Vr -=-
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. rY\ J �"" ·.
- 2. Ut>() s\ n & - L = o
2-rr R.
S\ n e -=. - \'
lfn- \.J �R sl
r 7o .s ·, t\ e L.. 0 e \� t-�\ r& � � �culr� .
S \ "� s\ "' e is bou.AA � ��w � _\ � t 1 r �\
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r R�9t.. t e
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t> � ����U. f4Y � leM.�
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oV\ f\
a.ss�\-..� o � � � � -P1f)w is \V\V'Lr ci J , As � ��e��n
we. ob�� Uow �Jtl w\� � * - �Is a.o O
Q..
ft�WW J- � '3-�"-\4\.e.�' • LIJ..s i-1 � lVII u �o...\v-J,el't f
W ._ \.,....e,.
., $e-. lo e. f.c.,.e. t-\.,. � C.,,.. ;,._lt!Q r : 0
I
j , e. Co
�e. H fHV\j' � 1 fu )(-�)(is � o
t\ o W\ t.u � o... �l�o_
�.f >1 Mrhett"� o..v--et ��v-e. L 0 =. 'Z..ex-b li ft; .
Th� pn �S U.f'e.. cL: ,\:ri b l.l.\;-to\1\ � Wtl:> o 1.-t�
�r -t\..e l'\ oV\ \ l Ftt�o- � \.Vo (>Q .S" i n � L = 0
c_...,... r IAJ i tt-- tw.. 1 i '-t-i ...6 � :
�2- p �
l .)
P U � - 1 > ( 2.. U
2.. P s in e + _c_ \�� L /o
2-
2 tt R. J
:
-= t>o
T� � wrr�ri \, u,�loV\ to � \ \ � w\ l\ be. okt-� from k
olt{C«"� �dwu..M- � two p�.>� �eQJs, .
g_ - '(1 -=: - ..\. � [ !t U "" � i 11 8 + [_ \.,_ -1- if [lfU� s i n'e)
2 tt � )
:
2.
-::: -i�( lf uJ si "1.e -1- If u r d "e- + [� ,_"\ -1- .t (t,u: si \9)
� llt.� )
..
2 tt R . '
2
f� -B' -l p ( (f. uc. r \
� )
= s. i l\ e + r'-
ct "'tl""t
2. )
2. TC R '1.-
2� 2�
L -J ( F-.. -p1') P- s.\ ct e = + �f S�U.J's i r.'I.e � Lsi �Js
-::: 1'\ e
o Ylt� )
2v o 1t ·
Jr s- 1. " e de
2
1 \- S
�.. '
L\ eJ-{1 -=. o
� \� � 1(
0
0
2.�
L -= $- �u r s i l'l"e Je
+ ... f u l). f' § u li'..
fi
) cUs h--i b u..b� 0"'
vo.-ti.:;ty..._ o f s�tl.. t
of
o � tle. CU.'ru a.. b .
Fov- cw .k(f� J.s J � r.t�� �f � v ov-te)(. olist-n bu.-f·i
o.ton t d s i > 't ds: . for inf i ni tesi m �
... d� 1 t"th �
toe.- � � c1.txeJ. o.- c.ort '-eM- tv-tt.i:e:J "D"" t-�)(. , w k� r ,�
vo..-\:-\ �� �� � l� .
t- s
J.
_I v -::. - td s. d. 'f-:::. e
0\ - 2. "lt
21tx-
T"e.. [oc.J_s-\:;r£0\� 't
s\,.eek is �.J. h � l o ud ®
�t> j.._ �'b:a.Q. ��� � o..U0 �5 -l\...e.. Vort--&jw vo�\;Qx e-t�� 'td. s .
r = u. , ds v"t dV\ - lAt. cls 4- v, dn -:::. �\ -u.z)ds -1- (v,-v. )d
u.
( ll2.) c\ s t,) dY\ .
-
r :::. \ d. s = \ - -(.. cv l - v
-rc ob� tk� �r Au.. -::. u l - 1.4.. d.CtbSS tL-.e.
sW J let c). "' -? 0
r -=. �Js � ct.c. l - u,__) d.s
@
1\.e.. ""�
�y-ro � ·. s� rt\X'cve� � e..� s of
thi ck�t.-S!. QM...Ei C.C\,M.Jo e(' . FoY" o.. �\ n a.i.V'�o\\ , \ 0\.1\.o� �
eU�J- • � �i e.k.. ll\ e?S COW\r\etetd: � t...ko. t'" b a...= c �
tker effe.vt rl � �f- Jt-� cu-d ��r .
Bul\ d. \:1.- e. fuw p•o .l.... \\ ot :zero
\:\....; �" �� haJJ t "(J 1:\.e. l>"'-"fe. f)f � e._a.o.....loe.r l i ll e. .
t"-e_. t'("() \?\e.Mv now is t-o �e.r-rrdl'l� a... cl.t.' s. h··i l.:. u.ttot'\ of
Vo'ftl �� a.to�o tt....__ �V"' ( i � e.. su.L Ka.t f:t..e.... �e.r
\ i �e.. kl Ctij S fu b ot.c.M-- el � ��cL:hcV\ {!)f rLo f{DW
tlf't>'-'-t � c�u Une. , \ � oth� WoY"d.S 1 tV e. r-e.. �re.....
Kwt � c..� � l L �� k 0- st-� u\'\�
� b ct. y :X -:. "%.. ()( )
;;}_ tcs) vvlcs)
-�___,...- 0 c.
Uoo
� o rte)( s keA-
() \\ tl(.. �b� � c.-hof'd. \ i ne,
1"\-.t. b.,� Col\ � OW\ t\..._. S" w-� o-{... {:k. tU.r{.o\1
jS 1'\0 �\)W � f.eM-J.,: o.&x- \-c tk � li
0\., + w' (s) =. 0 C>.X wtklj ?"'hk.
� s.
Li\'\ e 1.. i} \:....,._ �-\; to @
�t. c:M.-r.fo\\ �t o.... ro\n t- A
�S 7o � � 40
d. X.
r-------1--'-'���-
�a-. e\ "'"'\�\
X.
c. b-..t-
V.o si" c� + e)
oc OM.luk- .
W (X.) dW tJ. �
-z.
=- -
tJ 2� lX-:f)
c ®
w tx) =- s tC!) 4_3,
.
�"{f ()(- �)
_
0
-n,,� l,o�� Cc"'-c\l.t� O"'- D \r'\ �«- .S O..� fctUL. � �
0
�r�i\
u �J V\ VJ (' )()
whi � t$ """"' iV\ tur-J ��.M:o'-'\ �r f(�) .
c:...
J_ f tc g) .&� -= u� c� - £)
2 lt 0 ;%:u ()( - ! ) olx
1hi .s �?-cth:oY\ � � K� Vo'f"H ct� d;.st-r\ \o�'ot-\ tc_5)
t-o � � (;� cJt-o..ti .. W\�5. ;
r tC�) d '
c.
_l_ \/noble
2� � x- �
�
w�i � �� �v-
is �e. e.x.�t c:V.� � B.u��)
a.,
)o\v\�0 �{ �cJ:io\\ w\\,\. ?rotlu.o.. � .� t>t ��le.. of cJ:tt
\h ot'olunoV\ J
h-ow,.... ,.........
ci.eJ.ud k�
· .
C.GM. 'o(; s\{_lo stl. �novt \"" � cU.("fo\ \ �on:
t l&}==:. 'J.. V�,o cJ.. ( I + eos El l
Sl \\B
lhe. v oJ.w.. o � t at -\:l...t.. 1:;rol.\\ "Q e.!.. 7t
•
S u..b s\-\ \:\.t.h OIJ\ �f. 8 -=.-oc. \V\ � �oh.(.tl o� tof"' t-o Q_
0
6�� o� � km (J+ (.(>se)js'1 n e
r (1t) -:::::. � v0() t>l. �IV\ - st � e == 0
e _,.1L CD S 9
R� &� t LTE) -= LL , - u. 2 --= 0 o...t R� TE
s o 'tl1t) -= o s oJ:ts �\.� \:'h.e. \ +�r m tU:ior) � -:::. � (\ - c.o s e) �
X ':::: I ( l - (.()S So) to rt-W t-\ te. � lY\� l't-t tf!N'.W\.J of e QM..a &.
"lt-
_l__ (' t(e') .,t..._ l:) d.� Voo ( J "�"' 't ( 1t.') = o
Fa.- �L so"'"" tr-k o.l...CO\ \ "'� � 0::. 0 tke. toUcwi"r
·. �
so\�� \NCl.G de,��
+ U)se
(9)
..t--
's� -
') V�o( \---
c:;;f\ e
,z
_
h'4 � Cc.uA.Jo� ol-S e - c..o s eo
"l(
� c;\'(\Y\e �t'n e d.f7 -=- - tt. Co s t\ eo .
0 - c..o�e Cc �s� in . � o.1r�'tt l\1\,tL_�v-ni �4 "\ \.Ve dek �
oO
Ac - '2. AY\ c..o s n eo -::: oJ.... - d.:z..
��\ dx
t>O
M -:::. ct.-Ao -t- 2 AV\ U:>s � eo
0.)
?\'oblfM.-- ..t- �� -\:\... ��w..J; of- Fo......: er f. � �V\
�(G-) ,
1t:. 1\
e,o ::0 -jj �0�) J & 'b l'\ -=- � � p Ce) Co�V\& dtr
0
0
Ct��o·, r j.lft- �� L-. 1 � cceffi � of l i (t CL c
c_ �
r ""
o
r \L5,) J� ::. � S f(e) s i n e de-
0
S �b&t� tu..{e. thel. .e.'>'p�l oil\ f-er 't(8)
t(�) =. � V�>o (1\o \+,(.J)�e
S lr\
+ � t>.,._� i nn0
t\.0:.\ ) I
t'r\ � r ��
,c. "1C
r "'" cvb l Ao ro+c.ose)ae-
.L ;.. , ( SiMeSin.e ae
J
+ ..
.tor-
��t
o
't\ "1t. t'l ': r
�
�(l + u..c....
T
CQ 21t- ( +-:k � � (eo >eo -l) d&o)
=. J.
o c�
$ I ope. ot fu. l\ Ct- turve :
so �,. ��n'�
t\ C� -
-
2. 1t.. Sl ope. -::. L'[t. toY both
�--¥---- t:J..
d ol..
o a.x-e.. C1 ;t 1t" � Cl.M..d. C.J. -=. :t_ it (ol.. - ol.. oRJ
�
VJ �� � .o.M-f ot £ttto..ek k -zero l t'ft- o ( Ao I + .c.o se
Sln 6
+
i An
n =\
c; In ne)
C m1Ls. -=- 1 [_ A�+ A 1 - �)
·
c Q -= -rr ( 2. Ao + A 1)
Cm,� � - I� -1- � (A1-Az)1
-h,-r o. >- � '��� ekrt c.. fu.rfoi I A l A.,_ o = =
�o�� cJ, � � t��- a.kor-d
Leading edge
Mean camber line
Chord line
�--- Chord c-----�"'1 � Trailing
edge
Figure 4.3 Airfoil nomenclature.
Airplane Airfoil
Beechcraft Sundowner NACA 63A4 1 5
Beechcraft Bonanza NACA 230 1 6.5 (at root)
NACA 230 1 2 (at tip)
Cessna 1 50 NACA 241 2
Fairchild A- 1 0 NACA 67 1 6 (at root)
NACA 67 1 3 (at tip)
Gates Learjet 24D NACA 64A109
General Dynamics F- 1 6 NACA 64A204
Lockheed C-5 Galaxy NACA 00 1 2 (modified)
NACA 24 1 2 airfoil
c, 2.0
1 .6
1 .2 (!)!1
Lift coefficient
0.8
0.4
c m, c/4
0 0
-0.4 -0. 1
-0.8 -0.2
- 1 .2 -0.3
® Re = 3 . 1 X 1 06
I!] Re = 8 . 9 X 1 06 -0.4
-8 0 8 16 24
a, degrees
Figure 4.5 Experimental data for lift coefficient and moment
coefficient about the quarter-chord point for an NACA
24 1 2 airfoil. (Source: Data obtained from Abbott and
von Doenhoff, Reference 1 1 .) Also shown is a
comparison with theory described in Section 4 . 8 .
cd
r-
®
0.024
0.020 �
(!]
e
0.0 1 6 �
(!]
e
0. 0 1 2 1- (!) [!]
®
� a
®
®
0.008 �
� f�
[!] r:il
il coefficient
'-.... Drag
G G
�
0.004 �
cm, ac
0 0
B ll rl f!l l •••• ij} a m � -
-0.05
� Moment
coefficient - -0. 1
® Re = 3 . 1 X 1 06 - -0. 1 5
[!] Re = 8 . 9 X 1 06
I I I I I I I
r
-12 -8 -4 0 4 8 12 16
a, degrees
Figure 4.6 Experimental data for profile drag coefficient and
moment coefficient about the aerodynamic center for
the NACA 24 1 2 a irfoil. (Source: Abbott and von
Doenhoff, Reference 1 1 .)
2.4
2.0
NACA 001 2 airfoil
1 .6
1 .2
0.8
0.4 C m, c/4
-0.4 -0. 1
-0.8 -0.2
- 1 .2 ® Re = 3.0 X 1 06 -0. 3
8 Re = 9.0 X 1 06
-0.4
- 1 .6
-24 - 1 6 -8 0 8 I6 24 32
a, degrees
Figure 4.20 Comparison between theory and
experiment for the lift and moment
coefficients for an NACA 00 1 2 airfoil.
(Source: Abbott and von Doenhoff,
Reference 1 1 .)
.: 1.2
..;
c
·u
u
!.::
0.8
8
...
�
u
.!:!
c
-
u
u
rn
J o-q
I
5 \
u I
I
·-
IE
§ \
J
Section angle of attack a = 0°
o a = 4°
A a = 8°
o a = 1 2°
-3.0
-2.0
X
c
1 .0
f-··.. �
0.6
0 B 16 24
Angle of attack (deg)
(c)
Figure 6.11 Relative importance of separation effects: (a) analysis of geometry
alone; (b) analysis with boundary layer modeled; (c) analysis with boundary layer
and separated wake modeled. (From Ref. 6.4.)
�
Theory-geometry alone
.
: Theory-separated
-8
wake modelled
-6
coefficient, Cp
Pressure
-4
-2
0
1 .2
2
Chord fraction, xlc
0 0.4 0.8 1 .6
Angle of attack • 1 2.5°, flap angle - 40°
NASA GAW-1 airfoil, 30% Fowler flap,
Figure 6.20 Comparison of experimen
tal and calculated pressure distributions
on a two-element airfoil with separation
Flap wake from both surfaces. (From Ref. 6.4.)
1- - - - - - - c1 = 0.35
- - - - - - -l
c, = 0.6
a = 2°
��3 D ---
-
-
-
(a) (b)
c, = 1.6
(d)
(c)
(I!')
figure 4.34 Example of leoding-edge stall. Streamline patterns for on NACA 44 1 2 airfoil at different angles of
Reference 50.) Re = 2 . 1 x 1 05 and V00 8 m/s in air. The corresponding experimentally
attack. (The streamlines are drawn fo scale from experimental data given by Hikaru Ito in
measured lift coefficients are indicat.d by orr� at the right of each streamline picture, where the
=
length of each arrow indicates the relaliw magnitude of the hft. The lift coefficient is olso shown in
port If).
c1• 1.2
c1= 0.6
r--
�
==--
-
D
-
(a) (b)
Ct = 1 .35
D
(c) (d)
Figure 4.35 Example of trailing-edge stall. Streamline patterns for an NACA 442 1 airfoil at different angles
Reference 50.) Re 2 . 1 x 1 05 and V00 = 8 m/s i n air.
of attack. (The streamlines are drawn to scale from the experimental results of Hikaro Ito in
=
-
-
- -
-
:::::-----=:==
c1 = 0.25
Q �
-
-
�-- -
-
D
-
(a) (b)
c1= 0.75 c1= 0.7
a= 9° a = 15°
0 D
(c) (d)
Figure 4.37 Example of thin airfoil stall. Streamline patterns for a flat plate at angle of attack. (The
streamlines ore drown to scale from the experimental data of Hikoru Ito in Reference 50.)
1..5
t:;
1.0
...:
=
IS
·;::;
«.l
�
(,)
�
,g
u
«.l
en Thin airfoil (flat plate)
10 20
a, degrees
-0.�
lift-coefficient curves for three airfoils with different
aerodynamic behavior: trailing-edge stall (NACA .4.42 1
airfoil), leading-edge stall (NACA 44 1 2 airfoil), thin
airfoil stall (flat plate).
0.04
4 8 12 16
-0.2
F1pn 6.1$ Experimental life curve, drag polar, and pitching curve for a high-lift,
single-element airfoil, Rec = 3 X 106• (From Ref. 6.7.)
Re
0 9.0 x 1 0 6
0 6.0
0 3.0
2.0 6 Standard roughness 6 x 1 06
1 .2
Airfoil thickness i n percent of chord
Figure 4.�8 Effect of airfoil thickness on maximum l ift
coefficient for the NACA 63-2XX series of
airfoils. (Source: Abbott and von Doenhoff,
Reference 1 1 .)
/- ,
/
0 = 10
/
I
I
I
I - 1 0-
I I (b)
-....
I
/---f I
I
II I
:
I
I
1I
11c, -
_j_ 1
-
�
I
�
-
I �"
I �
-
::--.;;;:;
(c)
(a)
...ure 4.39 Effect of flap deflection on streamline shapes.(The streamlines are drawn to scale from
coefficient. (bt Streamline pattern with no flop deflection. lcJ Streamline pattern with o
the experimental data of Hikaru fto in Reference 50.} (aJ Effect of flop deflection on lift
1 5° ftop deAection.
LE slat �
LE droop �
Kruger flap ;jr in \1\ to m �.. s H
�� >Ie.. Fl o w ( Low �p�)
Dow � �o..t� : ��"" �o\1\ f)( w\�� \-t �
Se.-c.ou...,- � �
\} o v- t'l c.�
I
�oW t\Wax-d Ve-lo�1 w � fo�� w\ l\ �L
A-
\�d.u,.CLct 0� � t�� �ot.U o Ver � Wl�6
� t:le. effe &l vt- �\� of o..tt- � to h€.
·
t�\ S w\L\
\e.�S t'�AM. � ����\ c.- OM._Ole. o t �.:ltA..JL �� k�� . o·�
e.f+� J � l i �t L i s p e"�ol.:� � � \\n e.
t> f. �L \ o� re ( �l ve.. w tV\ d V [ �Dt � V 0o l ; kQ.. th
{:;k � - J alr-4i \) liM..$ VIe�� 0... �'--{'ot.-td
..l)i �� t-\..e �,.fc.,{.t ()\1\ W�l � l S. k\toWll\ 4.:$
•
c:> t- v�
\ie. i" oLu.uJ oL>r...a . f).. oLr...d
. k.e to tift 1 su C·�"e .
_ .: f
•
L
Free-trailing vortex
1
p
'
�X
Replace finite
-- �
win& with
b
bound vortex
-2 L-----1-....:::L.-
Free-trailing vortex
Finite win& Horseshoe vortex
Fl1vre S. I O Replacement of the finite wini with a bound vortex.
· : .... .
.
.... ... . ·--· ·
_-;.� ----- -- -· - - -.;- - �·"":""-.-- - .
-- -
. .
.
-·-..--�-
;
'. � It""
1'
.....
.- -�_.;._
·, _______ Tnllinl _..; .
_.. - - - - ---�-
'""' - \IOrt -
. · J: !
.
.
' .
�
�- -
-"'·
. '�:'�
·� ...... •• 1 1 Dow.twath d'illributKHl e6ong the y axia
for 0 lingle honeahoe '*'-"·
,:.
:.�
. ' . �..��..-:";'
"
�
.• _ .1
i·"
Co� !.\ d.t-Y � \) o"' te.x �� \ a.W\�
o' r.+:r&�()
4
d.Q :x. '('
--4
r
V -=
- b4
f "ifn. LY \1.
lt tL.L � ' t�� i.s
-t- 't-1"'ev.:.okt- l i
0- �� e.)C. r:��a- �
- bO +-o +110 tLe. l\\ �ud \iUo C..:..ht ..
n-lt..c.� {, '
+ t>o
v -
r
- z� h
is
�
y r e� � � c!A �ax- h
�e. p r � d. e. �· � � c:L b}
K�. I i t\. e. QM..J. � \ �t f
1t � l '""� � t-�J.$ �
0 to � �
\ - tA
v -::. 4.-n h
r\ �J..tl t...\ �t\'V\6- - L',�� l��oY'�
s i "o\e. to o�&. vo'�"t�>'
'd
� 0- 1.
�- �
0
f. "-� t'v\ C.i"'� te ��r Q� hoY"S� S hoe... Vo'(t-(
s� F,'a . ). l?:,
1/t.
we�.) = -�rr S �r/djj) �
- b/2- '1 o- 'if
f. "'
�
)( 1} :;. .... 1 �
6. '"
)(
1 «- )( ....., f. \ ,... ...., b
" ("
�
� '\ .....
I �_;. ....- r ,.... ....- A- -- -- � di\ -- --
� I 12 -- -- --.
-- -- -- +-
,......
'
I
00
F
1
I
I 00
..... I I
I dr3
-- -- -- � -- -- --.
� -- -- -- � -- --
,.... I D
/
( I
00
I
00
..... 5. 1 2 · Superposition of a finite number of horseshoe vortices along the lifting
line.
.
"u.
r.o r .,.
......- '1 ' -f tin\ \:e.. �\� � pY"o �tL tkls \if-t cL: stri bl.Ltion �
elf
di -= - -b,_
4 r;
_ �y. l-:.)_�
Q __ 11
__
_ ;_
'Pit.
wc1.) � -� ) (ctr/.J�) d.'a;
-� 1o - �o
s: v.b.Cbl t:..H "'-6' cl r j-,_ -\:l..e. l'f\ te.o � .......d. '\_..
ttsi ll � � CoS e
(}
=
W� c bt-� 0...
� �
�t cloW f\\V�lA
w Ce.) � - f.. over tt..... wi� �f...._
u
'2.b
T� \n. Juf�� rw._f- r> .f- �� o{ (
oZ \ ':::: -� = r: t.o� s t-� o\Je;r � s rcw-
v� 2bVIIc)
AR ,. b2 /S
High AR (low
c J induced drag)
1- b
�
Low AR (high
induced drag)
FlguN 5. 1 4 Schematic of high- and
low-ospe�t-ratio wings.
z
/
""
� -""
"' /
/
y
rklt' d:;- a.a -::.. 2. )
T �� f lti- per �� S f4A-- IS ;,lv� t,�
C�) c(�) C.i
'2-
"2 ) V�
I
J
L =-
(� ) L.: ('(l )
)
C =
b wJ- 1- .. C_Q :::- c.cns.t-
� � Cl_
OM.cL L' (OJ -::. g YO :::. C po + CL e d �r eUitti � ""' '<3 .
-n. e. Af<--
ol Ct... ::: Dl -=: _ _
a.. o
J. cA \
-
ll..o
+
1t AtL
6
0.2 0.4 0.6 0.8 1 .0
Taper ratio, c1/c,
....... 5. 1 8 Induced drag foetor c5 as a function of taper
ratio. (Source: McCormick, B. W.,
Mechanics, John Wiley & Sons, New York,
C a
Aerodynamics, Aeronautics, and Flight
• .
1 979.)
I
/."7��/
120
�
Pt.: =!::::
,....
"'
h�v ·..;o 0 / y
�......
v
/00
�
��
· v �-"
/vv �/......
80 �
....
60 /;,� v / /
40 �vvv
llj v
20 '/
r-- Cw
���...
0
/0 20
��
-20
Figure 5. 1 9 Prondtl's classic rectangular wing data
for seven different aspect ratios from 1
to 7; variation of lift coefficient versus
drag coefficient. For historical interest,
we reproduce here Prondtl's actual
Co lift coefficient and Cw drag
graphs. Note that, in his nomenclature,
= =
coefficient. Also, the numbers on both
the ordinate and abscissa ore 1 00
coefficients. (Source: Prondtl, L.,
limes the actual values of the
"Applications of Modern
Hydrodynamics to Aeronautics, "
NACA Report No. 1 1 6, 1 92 1 .)