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CHAPTER 1
1
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CHAPTER 2
2
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CHAPTER 3
3
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Bernoullis Equation
An other application of momentum eq is the Bernoullis Eq given as:
Consider x component of Momentum eq
or inviscid flo! !ith no "od# force it reduces to
$
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Bernoullis Equation
%
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&ovem"er 21' 2(11 )
Bernoullis Equation
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*
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+
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,
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1(
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-he equation
or rotational flo!
or irrigational flo!
11
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EXEMPLE 3.1
e"ruar# 1$' 2((% AE 2+(: Chapter 1: .ntroductor#-houghts
12
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&ovem"er 21' 2(11 13
INCOMPREIBLE !LO" IN A #$CT
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&ovem"er 21' 2(11 1$
INCOMPREIBLE !LO" IN A #$CT
For constant density the above eq reduces to
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&ovem"er 21' 2(11 1%
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
%enturi
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&ovem"er 21' 2(11 1)
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
%enturi
/ Consider flow entering a CD Nozzle as shown below/ This type of nozzles is called a VENTU!
/ "tation where area is #ini#u# is called throat/ !f ratio of inlet to throat area is $nown and pressure difference
%&'%( is e)peri#entally #easured then any velocity can be
#easured
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APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
0sing Bernoulli eq
1*
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&ovem"er 21' 2(11 1+
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
&Lo' (ee) 'in) Tunnel*
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&ovem"er 21' 2(11 1,
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&ovem"er 21' 2(11 2(
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0sing manometer the pressure difference is measured as:
21
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&ovem"er 21' 2(11 22
E*+,%-E ./&
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&ovem"er 21' 2(11 23
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&ovem"er 21' 2(11 2$
E)a#ple ./0
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2%
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&ovem"er 21' 2(11 2)
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
Pitot Tu+e
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&ovem"er 21' 2(11 2*
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
Pitot Tu+e
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APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
Pitot Tu+e
2+
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&ovem"er 21' 2(11 2,
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
Pitot Tu+e
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&ovem"er 21' 2(11 3(
APPLICATION O! INCOMPREIBLE !LO" IN A #$CT
Pitot Tu+e
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31
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32
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C(
33
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C( e,a-(le
3$
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C( E,a-(le
3%
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3)
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3*
1hich is -aplace equation2 Thus !2 !C flow is described by -aplace equation
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3+
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3,
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.f .C flo! is . also then
$(
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La(lae Equation
$1
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$2
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E,a-(le o/ a))ition o/ solutions o/ LP#
$3
-%D 3 linear partial differential equation
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Boun)ar0 Con)itions
$$
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$%
At infinity
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$)
At wall ( on surface )
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$*
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$+
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$NI!ROM !LO"
$,
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%(
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%1
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%2
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%3
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%$
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%%
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%)
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%*
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%+
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%,
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)(
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)1
$ i/ !l &1 t El t !l *
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1231% AE 2+(: Chapter 3 4art A5 )2
$ni/or- !lo' &1stEle-entar0 !lo'*
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)3
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)$
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)%
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))
oure !lo' &2n) Ele-entar0 !lo'*
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1231% AE 2+(: Chapter 3 4art A5 )*
oure !lo' &2n)Ele-entar0 !lo'*
By definition, a source flowis one inwhich the streamlines are radial lines
emanating from the origin. Also,(3.59a)
where cis a constant. i.e., thevelocity varies inversely as thedistance from the origin
In a source flow the velocity is direc-ted away from the origin. In a sinkflow, the velocity is directed towardthe origin. In fact, sin flow is anegative source flow
A source is really a line along thez-
a!is. "he constant ccan #e o#tainedfrom considerations of mass flowacross the surface of a cylinder(r,,z) and of de$th l.
If dsis an element of arc which su#-
tends an angle don the circle ofradius r, the corres$onding elementalsurface area on the cylinder of de$th lis%
"herefore, the mass flow across dSis
"hus the total mass flow across thesurface of the cylinder is
0,/ == VrcVr
)( rdlldsdS ==
lrdVSdVmd r==
2 2
0 0( / ) 2 .rm V lrd c r lrd c l
= = = &
oure !lo' &2n) Ele-entar0 !lo'*
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oure !lo' &2n)Ele-entar0 !lo'*
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1231% AE 2+(: Chapter 3 4art A5 ),
oure !lo' &ontinue)*
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1231% AE 2+(: Chapter 3 4art A5 *(
oure !lo' &ontinue)*
&olume flow rate '
ow the volume flow rate $er unit
de$th is defined as the sourcestrength . "hen, sing
or (3.*+)
om$aring (3.59a) with (3.*+) gives
e shall now determine , , and for this flow
lcmv 2/ ==
2 .r
vrV
l =
&
rVr
2
=
2/=c
/rV c r=
oure !lo' &ontinue)*
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oure !lo' &ontinue)*
&olume flow rate '
ow the volume flow rate $er unit
de$th is defined as the sourcestrength . "hen, sing
or (3.*+)
om$aring (3.59a) with (3.*+) gives
e shall now determine , , and for this flow
(a) Velocity potential : e have,
$on integrating, these give%
(3.*)
"hus, e/ui$otentials are the curves
i.e. r = const
i.e., they are circles with center at theorigin (see fig 3.+0)
lcmv 2/ ==
2 .r
vrV
l =
&
rVr
2
=
2/=c
01
,2 ====
V
rrV
rr
rln2
=
constr=
= ln2
/rV c r=
oure !lo' &ontinue)*
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1231% AE 2+(: Chapter 3 4art A5 *2
oure !lo' &ontinue)*
(a) Stream Function : e have,
$on integrating, these give%
(3.+)
"he e/uations of the streamlines are
i.e., = const
i.e., they are radial lines originating at
the origin (see fig 3.+0)
ote% streamlines and e/ui$otentialsare mutually $er$endicular, as they
should #e
(c) Circulation :
1ince the flow is irrotational, the
circulation is 2ero. i.e., ' 0
0,2
1
==
==
VrrVr r
2
=
const==
2
#ou+let !lo' &3r) Ele-entar0 !lo'*
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1231% AE 2+(: Chapter 3 4art A5 *3
#ou+let !lo' &3r)Ele-entar0 !lo'*
onsider a source-sin $air (of
strength ) distance la$art as
shown in 4ig 3.+a) 1tream function atPdue to the
source is
and due to the sin is
"hus the total stream function at
Pis
where
ow let
this is shown in fig 3.+#
hen you tae this limit, the source-
sin $air is called a doublet. ithout
this limit, it is not a dou#let, it is 6ust asource-sin $air
"he $roduct lis called the strengthof the doubletdenoted #y a$$a
112
=
222
=
=
=+= 2)(2 2121
12 =
constlwhilel =0
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1231% AE 2+(: Chapter 3 4art A5 *$
l cos4
b 5 r ' l cos4
a 5 l sin4
"in d4 5 d4 5 a6b
#ou+let !lo' &ont)*
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#ou+let !lo' &ont)*
"he stream function for the dou#let is
(3.75)
4rom fig 3.+#,
"herefore,
i.e., (3.7)
In a similar fashion, the velocitypotential for a dou#let is given #y
(3.77)
Note% If the sin is $laced to the left ofthe source, then the signs in
e/uations (3.7) and (3.77) will #e
reversed
= == dconstll 2lim0
sin , cosa l b r l = =
cos
sinsin
lr
l
b
add
==
==
cos
sin
2lim
0 lr
l
constl
r
sin
2=
r
cos
2=
2
cos
2rV
r r
= =
2
1 sin
2V
r r
= =
#ou+let !lo' &ont)*
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1231% AE 2+(: Chapter 3 4art A5 *)
#ou+let !lo' &ont)*
8/n of the streamlines is ' const
i.e.,
i.e.,
this is an e/n of a circle with diameter
don the vertical a!is and with center
at d+ directly a#ove the origin. 1ee
4ig 3.+5
By convention, we designate the
direction of the dou#let #y an arrow
drawn from the sin to the source
8/n of the e/ui$otentials is ' const
i.e.,
i.e.,
this is an e/n of a circle with diameter
don thex-a!is and with center at d+
to the right of the origin. :irection of dou#let is donated #y arrow
drawn from sin to source . :irection and
sign will change #y changing $lacement
c
r
=
sin
2
sinsin
2d
cr ==
cconstr
==
cos
2
coscos
2d
cr ==
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1231% AE 2+(: Chapter 3 4art A5 **
%orte, !lo' &t Ele-entar0 !lo'*
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1231% AE 2+(: Chapter 3 4art A5 *+
%orte, !lo' & Ele-entar0 !lo'*
4rom the definition of Vorte Flow,
we have
i.e., streamlines are concentric circles
centered at the origin
(a) Circulation! % By definition,
"herefore,
1u#stituting for V , gives
Also,
(3.;0*)
and
(3.;05)
rCVVr == ,0
= C sdV erdsdeVV , ==
drVerdeVsdV == )()(
==2
0
2 CdC
2
=C
rV
=
2
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%orte, !lo' &ont)*
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%orte, !lo' &ont)*
(a) Velocity "otential x,y) % By
definition, we have
Integrating these e/uations gives,
(3.;;+)
8/uation of the e/ui$otentials is%
' constant, i.e.,
i.e., = const "hus, e/ui$otentials are radial lines
originating from the origin
(a) Stream Function x,y) % By
definition, we have
Integrating these e/uations gives,
(3.;;)
8/uation of the streamlines is%
' constant, i.e.,
i.e., r = const "hus, streamlines are circles with
center at the origin
rV
rV
r r
2
1,0 ==
==
2
=
const=
2
rV
rV
r r
2,01 ==
==
rln2
=
constr=
ln2
Ta+le o/ Ele-entar0 !lo's
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1231% AE 2+(: Chapter 3 4art A5 +1
Ta+le o/ Ele-entar0 !lo's
1ince elementary flows form the #asis of more com$licated flows, we
summari2e their results in "a#le 3.; #elow
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AE 24Essentials o/ Aero)0na-is
Ca(ter 3 &Part B*
!lo' Co-+inations
$ni/or- !lo' 'it a oure an) in5
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$ni/or- !lo' 'it a oure an) in5
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+$
$ni/or- !lo' 'it a oure an) in5
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$ni/or- !lo' 'it a oure an) in5
"he velocity field is o#tained from
(3.*)
(3.)
ote that the velocity com$onents
of the resultant flow are the
sum of the corres$onding velocity
com$onents of the individual flows
#uestion% Is there (or are there) any
stagnation $oint(s) in the resultant
flow >
$nswer% 4ind out #y setting the
velocity com$onents to 2ero.
i.e., (3.7) (3.9)
4rom (3.9), ' . "hen from (3.7)
"hus, a stagnation $oint e!ists atB%
rV
rV
r
2cos
1
+=
=
sin=
= Vr
V
),( VVr02cos =
+ rV
0sin = V
cos( ) 0,2 2V rr V
+ = =
( , ) ( 2 , )r V =
$ni/or- !lo' 'it a oure an) in5
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$ni/or- !lo' 'it a oure an) in5
"hus the stagnation $oint is at a
distance
directly u$stream of the source
%bserve% if the source strength increases, the stagnation $oint moves
farther away from the source. If thefree stream velocity of the
uniform flow increases, the stagnation
$oint moves closer to the source. "his
is clearly intuitively o#vious
ow let=s find the e/n of the stream-line which contains the stagnation
$oint. 1u#stitute the coordinates of
the stagnation $oint in ' const
"his streamline is shown as the curve
ABCin fig 3.++
=V
DB2
V
constV
V =
+
=
2sin
2
const==2
$ni/or- !lo' 'it a oure an) in5
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$ni/or- !lo' 'it a oure an) in5
&mportant conclusion%
e now that no flow occurs across a
streamline. "hus, any streamline can#e re$laced with a solid surface of the
same sha$e.
In $articular, since the streamline
ABCcontains the stagnation $oint, it
is a dividing streamline. i.e., itse$arates the fluid coming from the
free stream and the fluid emanating
from the source
All the fluid outsideABCis from the
free stream and all the fluid insideABCis from the source
"herefore, as far as the free stream is
concerned, the entire region inside
ABCcould be replaced with a
solid body of the same shape andthe free stream flow would not feel
the difference
$ni/or- !lo' 'it a oure an) in5
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1231% AE 2+(: Chapter 3 4art B5 ++
$ o o t a ou e a )
"he streamline e!tends
downstream to infinity, forming a
semi'infinite body "herefore, if we want to construct the
flow over a solid semi-infinite #ody
descri#ed #y the curveABC, then all
we need to do is tae a uniform
stream with velocity and add to ita source of strength at $ointD.
he resulting flow will then
represent the flow over the
prescribed solid semi'infinite body
of shapeABC ow consider a source () and a sin
(? ) $laced res$ectively at a distan-ce bto the left and right of the origin.
!!!
2=
V
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+,
$ni/or- !lo' 'it a oure an) in5
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1231% AE 2+(: Chapter 3 4art B5 ,(
ow su$erim$ose a uniform stream
with velocity as shown
"hen the stream function from thecom#ined flows is
e can now determine the velocity
flow field using the e/uations
#ut you will have to e!$ress
in terms of first. "his can #e done
with the geometry of fig 3.+3 "hen, #y setting velocity ' 2ero, we
can determine the stagnation $oints.
"here are two of them,AandB
"he location of the stagnation $oints
is found to #e
(3.7;)
"he e/n of the streamlines is
V
)(2
sin 21
+= rV
rV
rVr
=
=
,
1
),( 21
+==VbbOBOA
2
$ni/or- !lo' 'it a oure an) in5
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1231% AE 2+(: Chapter 3 4art B5 ,1
"he e/n of the streamlines is
"he e/n of the s$ecific streamline
which $asses through the stagnation
$oints is o#tained #y setting a$$ro-
$riate stagnation values in the a#ove
e/n. ote that at $ointA%
and at $ointB%
ith these values, the constant in the
streamline e/n is 2ero. "hus the e/n
of the stagnation streamline is
"his is the e/n of an oval and is alsothe dividing streamline
"he oval is called a ankine %val
constrV =
+= )(2
sin 21
)( 21 ===
)0( 21 ===
0)(2
sin 21 =
+
rV
$ni/or- !lo' 'it a #ou+let&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
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1231% AE 2+(: Chapter 3 4art B5 ,2
&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
"he flow over a circular cylinder can
#e $roduced with a combination of
uniform flow and a doublet. "his isa classic $ro#lem in aerodynamics
onsider the dou#let $lus uniform
flow com#ination shown in 4ig 3.+*.
the direction of the dou#let is
u$stream, facing into the uniform flow. "he stream function for the com#ined
flow is
or
(3.9;)
If we let then,
(3.9+)
"he velocity field is o#tained #y
differentiating this e/n as follows
rrV
sin
2sin =
=
22
1sinrV
rV
VR 22
= 2
2
1)sin(r
RrV
=
= 22
1)cos(11
r
RrV
rrVr
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,3
$ni/or- !lo' 'it a #ou+let&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
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1231% AE 2+(: Chapter 3 4art B5 ,$
&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
"hus,
(3.93)
Also,
i.e.,
or (3.9)
1tagnation $oints are located #y
setting (3.93) and (3.9) to 2ero%
(3.95)
(3.9*)
"he solution yields two stagnation
$oints ?A(R,0) andB(R,) in 4ig
3.+*
"he e/n of the streamline throughBcan #e o#tained #y su#stituting the
coordinates ofBinto the e/n '
const. "he result is% const ' 0
cos12
2
= V
r
RVr
+ )sin(1
2)sin(
2
2
3
2
Vr
R
r
RrV
rV = =V
sin1 2
2
+= V
r
RV
0cos12
2
=
V
rR
0sin12
2
=
+ V
r
R
$ni/or- !lo' 'it a #ou+let&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
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1231% AE 2+(: Chapter 3 4art B5 ,%
&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
1imilarly, the e/n of the streamline
throughAcan #e o#tained #y
su#stituting the coordinates ofAintothe e/n ' const. "he result is
again% const ' 0
"hus, the streamline ' 0 $asses
through #oth stagnation $oints and it
is the dividing streamline. "he e/n ofthis streamline is o#tained from (3.9+)
as
(3.9)
#uestion% hat is the sha$e of this
streamline >
$nswer% @oo at e/n (3.9). It is satisfied
for r' for all values of thus solution of
this e/n is% r = R' const
"his e/n re$resents a circle in $olar
coordinates with radiusRand center
at the origin
"herefore, e/n (3.9), i.e., the divid-ing streamline, re$resents a circle of
radiusRas shown in 4ig 3.+* a#ove
01)sin(2
2
=
=
r
RrV
$ni/or- !lo' 'it a #ou+let&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
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1231% AE 2+(: Chapter 3 4art B5 ,)
&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
Cther solutions of the streamline e/n
(3.9) are ' and ' 0.
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&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
1ince the flow is $erfectly symmetric
a#out hori2ontal and vertical a!is,
thus the $ressure distri#ution over the u$$er nad lower surfaces of the
cylinder is com$letely #alanced. "hus,
there is no net normal force (lift )
com$onent in the vertical direction
1imilarly $ressure distri#ution is #alanced
on front and rear halves thus no net a!ial(drag) 4orce is generated.
In other words, there is no net lift and
no net drag. "hat is why we call this
case the Dnonlifting flow over a
circlar c!linderE. "his is nown as
Dd*$lembert*s paradoE
Because in real life we are aware that
there always is drag associated with
an o#6ect immersed in a moving fluid
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&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
$nswer% Because the flow is viscous
only within the #oundary layer which
is very thin. Cutside the #oundarylayer the flow is inviscid. 1o all of this
analysis will a$$ly to the region of
flow outside the #oundary layer. "his
is also im$ortant in aerodynamics
@et us calculate the velocity distri#u-tion and the $ressure distri#ution on
the surface of the cylinder
"he velocity distri#ution on the
surface of the cylinder is o#tained
from the velocity e/ns (3.93) and(3.9) with r = R, resulting in
(3.99)
(3.;00)
"herefore,
(3.;00a)
the minus sign is included to mae
the e/n consistent with the sign
convention for (i.e., $ositive
counterclocwise, ne!t slide)
0=rV
sin2 = VV
2 sinV V V = =
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&ovem"er 2(11 AE 2+(: Chapter 3 ,,
2 sinV V V = =
Velocity in direction of
decreasing angle thus it is
negative on upper surface
Velocity in direction of
increasing angle thus it is
positive on lower surface
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"he velocity is ma!imum at i.e., at the to$ and the #ottom of the cylinder
&ovem"er 2(11 AE 2+(: Chapter 3 1((
2 =
$ni/or- !lo' 'it a #ou+let&Nonli/tin6 !lo' O7er a Cirular C0lin)er*
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1231% AE 2+(: Chapter 3 4art B5 1(1
& 6 0 *
"he $ressure coefficient C" is
(3.3*) A$$lying Bernoulli=s e/uation to an
ar#itrary $oint and a $oint in the free
stream, we have
from which we have
(3.3)
1u#stituting (3.3) into (3.3*) gives
(3.37)
But, from (3.;00a) we have
1u#stituting this into (3.37) gives
(3.;0;)
#
""C"
2
212
21
+=+ V"V"
2 212
( )" " V V =
2
1
=
VVC"
sin2=VV
2sin41="C
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&ovem"er 2(11 AE 2+(: Chapter 3 1(2
E,a-(le 3.8 9 (a6e 22
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1231% AE 2+(: Chapter 3 4art B5 1(3
+ample ,-. / page 001:
onsider the nonlifting flow over a
circular cylinder. alculate the loca-tions on the surface of the cylinder
where the surface $ressure e/uals
the freestream $ressure
Solution:
At , we have from (3.;0;)%
i.e.,
"hese results are shown in 4ig 3.30
At stagnation, C"' ;, and
"he minimum $ressure occurs at the
to$ and #ottom i.e., at
At the to$ and #ottom, C"' -3, and
the corres$onding $ressure is
= ""212 sin,0sin41 === "C
0000 330,210,150,30=
+= #""
090= #" 3
Li/tin6 !lo' O7er a C0lin)er
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1231% AE 2+(: Chapter 3 4art B5 1($
In the uniform-flow-$lus-dou#let com-
#ination we $roduced the flow around
a circular cylinder with 2ero lift (anddrag).
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1(%
Li/tin6 !lo' O7er a C0lin)er
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"he resulting stream function for the
flow of 4ig 3.3+ is F i.e.,
GHHHHH (3.;;7)
"he resulting $attern given #y (3.;;7)is setched at the right of 4ig 3.3+
If r = Rin (3.;;7), then ' 0 for all
values of . "hus, the dividing
streamline is the circle of radiusR
ote that the streamlines are nolonger symmetrical a#out the
hori2ontal a!is. "hus, there will #e a
lift force #ut still no drag force
"he velocity field can #e o#tained
from the stream function of (3.;;7).
"he result is%
(3.;;9)
(3.;+0)
21 +=
R
r
r
RrV ln
21)sin(
2
2
+
=
rV
r
RV
2sin1
2
2
+=
cos12
2
= V
r
RVr
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1231% AE 2+(: Chapter 3 4art B5 1(*
"he stagnation $oints are o#tained #y
setting the velocity com$onents to
2ero, thus
(3.;+;)
GHHHHHH (3.;++) 8/n (3.;+;) gives, r = R. "hen e/n
(3.;++) gives
(3.;+3)
1ince is $ositive, e/n (3.;+3) tellsus that must lie in the th/uadrant
"hus, there are two stagnation $oints
on the #ottom half of the cylinder as
shown #y $oints ; and + in 4ig 3.33a.
"hese results are valid only when
#ecause otherwise, e/n (3.;+3) is
meaningless
02
sin12
2
=
+=
rV
r
RV
0cos12
2
=
= V
r
RVr
=
RV
4sin 1
RV
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1231% AE 2+(: Chapter 3 4art B5 1(+
If
there is only one stagnation $oint on
the surface of the cylinder, namely,$oint (R, -+) la#eled as $oint 3 in
4ig 3.33#
4or the case of
we return to e/n (3.;+;). It was satis-
fied #y r = RF however, it is alsosatisfied #y
1u#stituting into (3.;++)
and solving for r, gives
(3.;+)
"hus, there are two solutions and
therefore two stagnation $oints
Cne is inside and the other is outside
the cylinder , and #oth on the vertical
a!is, as shown #y $oints and 5 in
4ig 3.33c. @et=s e!$lain this a littlemore
RV= 4
RV> 4
2 =2 =
2
2
44R
VVr
=
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1231% AE 2+(: Chapter 3 4art B5 1(,
"he resulting stream function for the "hus, for steady flow we would write
Li/tin6 !lo' O7er a C0lin)er &Potos*
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1231% AE 2+(: Chapter 3 4art B5 11(
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
E,a-(les 3.14: 3.11 9 (a6e 22
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1231% AE 2+(: Chapter 3 4art B5 111
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
;utta
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1231% AE 2+(: Chapter 3 4art B5 112
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
#ra6 7ersus Re0nol)s Nu-+er
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1231% AE 2+(: Chapter 3 4art B5 113
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
#ra6 7ersus Re0nol)s Nu-+er
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1231% AE 2+(: Chapter 3 4art B5 11$
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
#ra6 7ersus Re0nol)s Nu-+er &Poto*
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1231% AE 2+(: Chapter 3 4art B5 11%
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
#ra6 7ersus Re0nol)s Nu-+er &Poto*
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1231% AE 2+(: Chapter 3 4art B5 11)
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
#ra6 7ersus Re0nol)s Nu-+er &Poto*
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1231% AE 2+(: Chapter 3 4art B5 11*
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
Pressure #istri+ution O7er a C0lin)er
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1231% AE 2+(: Chapter 3 4art B5 11+
"he following e/uations re$resent a
flow field
"hus, for steady flow we would write
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11,
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12(
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121
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