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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 1
Chapters 3 & 4: Integral Relations for a Control ol!"e
and #ifferential Relations for Fl!id Flo$
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 2
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 3
Re%nolds ransport heore" 'R(
)eed relationship *et$een ( )sysBdtd
and +hanges in
==CVCV
ddmcvB ,
1 - ti"e rate of +hange of . in C - =CV
ddt
d
dt
cvdB
2 - net o!tfl!/ of . fro" C a+ross CS -
R
CS
V n dA
dAnVddt
d
dt
dBR
CSCV
SYS +=
eneral for" R for "oing defor"ing +ontrol ol!"e
Spe+ial Cases:
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 4
1( )ondefor"ing C "oing at +onstant elo+it%
( ) dAnVdtdt
dBR
CSCV
SYS +=
2( Fi/ed C
( ) dAnVdtdt
dB
CSCV
SYS +=
reens heore":CV CS
b d b n dA =
( ) ( )
+
= dVtdt
dB
CV
SYS
Sin+e C fi/ed and ar*itrar% 0li"d gies differential e,
3( Stead% Flo$: 0=
t
4( nifor" flo$ a+ross dis+rete CS 'stead% or !nstead%(
=CSCS
dAnVdAnV (- inlet, + outlet)
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 5
Contin!it% !ation:
. - - "ass of s%ste"
7 - 1
0=dt
dM*% definition s%ste" - fi/ed a"o!nt of "ass
Integral For":
dAnVddt
d
dt
dM
CS
R
CV +== 0
dAnVddt
d
CS
R
CV
=
Rate of decrease of mass in CV net rate of mass outflo! across CS
)ote si"plifi+ations for nondefor"ing C fi/ed Cstead% flo$ and !nifor" flo$ a+ross dis+rete CS
In+o"pressi*le Fl!id: 9 - +onstant
dAnVddt
d
CS
R
CV
=
+onseration of ol!"e;
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 6
Spe+ial +ase C for" +ontin!it% e!ation:
Fi/ed C
0CV CS
d V n dAt
+ = and !nifor" flo$ oer dis+rete inlet
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 =
0
1=
+
+
+
unit$erc%an&eofrate
'
!
y
v
(
u
V
unit$er
c%an&eofrate
#t
#
Called the +ontin!it% e!ation sin+e the i"pli+ation is that
9 and Vare +ontin!o!s f!n+tions of /,
In+o"pressi*le Fl!id: 9 - +onstant
0
0
=
+
+
=
'
!
y
v
(
u
V
P3,15 >ater ass!"ed in+o"pressi*le flo$s steadil%thro!gh the ro!nd pipe in Fig, P3,15, he entran+e
elo+it% is +onstant 0u )= and the e/it elo+it%
appro/i"ates t!r*!lent flo$ ( )1 =
"a/ 1u u r R= , #eter"ine
the ratio "a/) u for this flo$,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 8
Stead% flo$ nondefor"ing or "oing C one inlet
!nifor" flo$ and one o!tlet non!nifor" flo$( )
1 =2
0 "a/0
0 1 2R
) R u r R rdr = + 2 2
0 "a/
4?0
60) R u R
= +
0
"a/
4?
60
)
u=
( ) ( )15 =
"a/ "a/0
2 2
1 12 2 1 1
1 12 1
= =
R
u rdr u r R r R
R R
=
+ +
2
"a/
= =2 0
15 8u R
= 2
"a/4?260
u R=
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?
P3,12 he pipe flo$ in Fig, P3,12 fills a +%lindri+al tan@
as sho$n, At ti"e t-0 the $ater depth in the tan@ is
30+", sti"ate the ti"e re!ired to fill the re"ainder of
the tan@,
nstead% flo$ defor"ing C one inlet one o!tlet
!nifor" flo$
1 20CV
dd " "
dt = +
2 2
1 20 4 4CV
d d d
d V Vdt
= +( ) ( )
2
4
#t % t
=
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 10
( )2 2
2 104 4
# d% dV V
dt
= +
( )
2
2 1 0,0153d% d V Vdt # = =
0,=46
0,0153 0,0153
d%dt s= = =
Stead% flo$ one inlet and t$o e/its $ith !nifor" flo$
1 2 30 " " "= + +
( )2
3 1 2 1 24
d" " " V V
= =
( )
2
2
1 2
4
4
#d%
dtd"
V V
= =
( )
2
1 2
#d%d
V V
=
3
A
*" V n dA
s=
ol!"e of fl!id per !nit ti"e thro!gh A
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 11
P4,1= A reasona*le appro/i"ation for the t$o
di"ensional in+o"pressi*le la"inar *o!ndar% la%er on
the flat s!rfa+e in Fig,P4,1= is
2
2
2y yu )
= for
y
$here 1 2C(= C const='a( Ass!"ing a noslip +ondition at the $all find an
e/pression for the elo+it% +o"ponent ( )v ( y for y ,'*( he find the "a/i"!" al!e of v at the station 1( m= for the parti+!lar +ase of flo$ $hen 3) m s= and
1,1cm= ,
0u v( y
+ =
( )2 2 32 2v u
) y yy ( (
= = +
( )2 2 30
2y
(v ) y y dy =
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 12
'a(2 3
2 32
2 3(
y yv )
=
1 2
C(= 1 22 2
(
C(
(
= =
'*( Sin+e 0yv = at y =( )"a/
2 1 1
2 2 3
)v v y
(
= = =
3 1,10,0055
6 6
)m s
(
= = =
o"ent!" !ation:
. - V- "o"ent!" 7 - V
Integral For":
{
' (
31 2
R
CV CS
d MV d V d V V n dA +
dt dt
= + = 1 4 2 4 3 1 44 2 4 43
+ - e+tor s!" of all for+es a+ting on C
- F.B Fs
F.- .od% for+es $hi+h a+t on entire C of fl!id d!e
to e/ternal for+e field s!+h as grait% or
ele+trostati+ or "agneti+ for+es, For+e per !nit
ol!"e,
Fs- S!rfa+e for+es $hi+h a+t on entire CS d!e to
nor"al 'press!re and is+o!s stress( and
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 13
tangential 'is+o!s stresses( stresses, For+e per
!nit area,
>hen CS +!ts thro!gh solids Fs"a% also in+l!de FR-
rea+tion for+es e,g, rea+tion for+e re!ired to hold nole
or *end $hen CS +!ts thro!gh *olts holding nolehere ( ) V
V Vt t t
= +
andDD D
V V VV ui V v,V !- V = = + + is a tensor' ( ' ( ' ( ' ( ' (V V VV uV vV !V
( y '
= = + +
VVVV += ('
- 0 +ontin!it%
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 14
( )CV
VV V V V d +
t t
+ + + =
Sin+eV #V
V Vt #t
+ =
= +d#t
V#
CV
= f#t
V# $er elemental fluid volume
sbffa +=
bf - *od% for+e per !nit ol!"e
sf - s!rfa+e for+e per !nit ol!"e
.od% for+es are d!e to e/ternal fields s!+h as grait% or
"agneti+ fields, Eere $e onl% +onsider a graitational
field that isd(dyd'&+d+
&ravbody = =
and D& &-= for
i,e, Dbody
f &-=
S!rfa+e For+es are d!e to the stresses that a+t on the sides
of the +ontrol s!rfa+es
i,i,i, $ +=
++
+=
'''y'(
y'yyy(
('(y((
$
$
$
is+o!s stre)or"al press!re
g
S%""etri+ i, ,i =
2ndorder tensor
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 15
As sho$n *efore for p alone it is not the stresses
the"seles that +a!se a net for+e *!t their gradients,
S%""etr% +ondition fro" re!ire"ent that for ele"ental
fl!id ol!"e stresses the"seles +a!se no rotation,
fff
$s+=
Re+all $f$ = *ased on 1storder S, f is "ore +o"ple/
sin+e i, is a 2ndorder tensor *!t si"ilarl% as for p the
for+e is d!e to stress gradients and are deried *ased on
1storder S,
GGG
GGG
GGG
-,i
-,i
-,i
'''y'('
y'yyy(y
('(y(((
++=
++=
++=
' ( ' ( ' (s ( y '+ d(dyd'( y '
= + +
Res!ltant stress
on ea+h fa+e
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 16
' ( ' ( ' (( y 'sf
( y '
= + +
and si"ilarl% for
G' ( ' ( ' (
G' ( ' ( ' (
G' ( ' ( ' (
(( y( '(s
(y yy 'y
(' y' ''
f i( y '
,( y '
-( y '
= + +
+ + +
+ + +
i, i,s,
f(
= =
/
%
dyd'd((
((
((
+
y(
y( dy d(d'y
+
y(d(d'
((dyd'
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 1=
' ( ' ( ' (
' ( ' ( ' (
' ( ' ( ' (
s( (( (y ('
sy y( yy y'
s' '( 'y ''
f( y '
f ( y '
f( y '
= + +
= + +
= + +
P!tting together the a*oe res!lts
Di,
#Va &-
#t
= = +
)e/t $e need to relate the stresses Hito the fl!id "otion
i,e, the elo+it% field, o this end $e e/a"ine the
relatie "otion *et$een t$o neigh*oring fl!id parti+les,
J .: VdrVdVV +=+ 1storder a%lor Series
,i,
'y(
'y(
'y(
d(e
d'
dy
d(
!!!
vvv
uuu
VdrdV =
==
*od% for+e d!e
to grait%"otion
s!rfa+e for+e - p B is+o!s ter"s
'd!e to stress gradients(
.
A '!$( - V
relatie "otion defor"ation rate
tensor - i,e
dr
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 18
1 1
2 2
, ,i i ii, i, i,
, , i , i
u uu u ue
( ( ( ( (
symmetric $art anit symmetric $arti, ,i i, ,i
= = + + = +
= =
1 442 4 43 1 442 4 43
1 10 ' ( ' (
2 2
1 1' ( 0 ' (
2 2
1 1' ( ' ( 0
2 2
y ( ' (
i, ( y ' y
( ' y '
u v u !
v u v ! ri&id body rotation
of fluid element
! u ! v
= =
6 47 48
142 43
1 42 43
!%ere rotation about ais rotation about y ais. rotation about ' ais
)ote that the +o"ponents of iare related to the orti+it%e+tor define *%:
DD D' ( ' ( ' (
2 22
y ' ' ( ( yV ! v i u ! , v u -
= = + + 14 2 4314 2 43 14 2 43
- 2 ang!lar elo+it% of fl!id ele"ent
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 1?
1 1' ( ' (
2 2
1 1' ( ' (
2 2
1 1' ( ' (
2 2
i,
( y ( ' (
( y y ' y
( ' y ' '
rate of strain tensor
u u v u !
v u v v !
! u ! v !
=
+ +
= + + + +
( y 'u v ! V + + = - elon&ation (or volumetric dilatation)
of fluid element
1 #
#t
=
('2
1(y
vu + - distortion $rt '/%( plane
('2
1('
!u + - distortion $rt '/( plane
('2
1y'
!v + - distortion $rt '%( plane
h!s general "otion +onsists of:
1( p!re translation des+ri*ed *% V
2( rigid*od% rotation des+ri*ed *% K /
3( ol!"etri+ dilatation des+ri*ed *% V
4( distortion in shape des+ri*ed *% i i
It is no$ ne+essar% to "a@e +ertain post!lates +on+erning
the relationship *et$een the fl!id stress tensor 'Hi( andrateofdefor"ation tensor 'ei(, hese post!lates are
*ased on ph%si+al reasoning and e/peri"ental
o*serations and hae *een erified e/peri"entall% een
for e/tre"e +onditions, For a )e$tonian fl!id:
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 20
1( >hen the fl!id is at rest the stress is h%drostati+ and
the press!re is the ther"od%na"i+ press!re
2( Hiis linearl% related to eiand onl% depends on ei,
3( Sin+e there is no shearing a+tion in rigid *od%
rotation it +a!ses no shear stress,
4( here is no preferred dire+tion in the fl!id so thatthe fl!id properties are point f!n+tions '+ondition of
isotrop%(,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 21
sing state"ents 13
i,i,mni,i, -$ +=
imn- 4thorder tensor $ith 81 +o"ponents s!+h that ea+h
stress is linearl% related to all nine +o"ponents of Li,
Eo$eer state"ent '4( re!ires that the fl!id has no
dire+tional preferen+e i,e, Hiis independent of rotation of
+oordinate s%ste" $hi+h "eans imnis an isotropi+ tensor- een order tensor "ade !p of prod!+ts of 0i,
i,mn i, mn im ,n in ,m- = + +
scalars=('
Mastl% the s%""etr% +ondition Hi - Hire!ires:
@i"n- @i"n N - O - is+osit%
{2i, i, i, mm i,$
V
= + +
and O +an *e f!rther related if one +onsiders "ean
nor"al stress s, ther"od%na"i+ p,
3 '2 3 (ii
$ V = + +
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 22
1 2
3 3ii$ V
$ mean
normal stress
= + +
=123
2
3$ $ V
= +
In+o"pressi*le flo$: $$= and a*sol!te press!re is
indeter"inant sin+e there is no e!ation of state for p,
!ations of "otion deter"ine $ ,
Co"pressi*le flo$: $$ and - *!l@ is+osit% "!st *e
deter"ined ho$eer it is a er% diffi+!lt "eas!re"ent
re!iring large1 1# #
V#t #t
= =
e,g, $ithin sho+@
$aes,
Sto@es E%pothesis also s!pported @ineti+ theor%
"onotoni+ gas,
$$=
= 3
2
2 23
i, i, i,$ V = + + eneraliation dy
du= for 3# flo$,
+
=i
,
,
i
i,
(
u
(
u ,i relates s%ear stress to strain rate
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 23
2 12 2
3 3
i iii
i i
u u$ V $ V
( (
normal viscous stress
= + = + + 1 4 442 4 4 43
>here the nor"al is+o!s stress is the differen+e *et$een
the e/tension rate in the /i dire+tion and aerage
e/pansion at a point, Qnl% differen+es fro" the aerage -
+
+
'
!
y
v
(
u
3
1 generate nor"al is+o!s stresses, For
in+o"pressi*le fl!ids aerage - 0 i,e, 0V = ,
)on)e$tonian fl!ids:
i,i, for s"all strain rates
$hi+h $or@s $ell for
air $ater et+, )e$tonian fl!ids
{ {
ni, i, i,
tnon linear %istory effect
+
)on)e$tonian
is+oeslasti+ "aterials
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 24
)on)e$tonian fl!ids in+l!de:
'1( Pol%"ers "ole+!les $ith large "ole+!lar
$eights and for" long +hains +oiled together
in spong% *all shapes that defor" !nder shear,
'2( "!lsions and sl!rries +ontaining s!spended
parti+les s!+h as *lood and $ater
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 25
22 22
,i i
i, i
, , , i , ,
uu uu V
( ( ( ( ( (
= + = = =
For in+o"pressi*le flo$ 0V =
2D
D D
#V&- $ V
#t$ !%ere $ $ '
$ie'ometric $ressure
= +
= +1 42 43
For O - 0
D#V & - $#t
= !ler !ation
)S e!ations for 9 O +onstant
2D#V $ V#t
= +
2D#V
V V $ V #t
+ = +
1on-linear 2ndorder 3#4, as is t%e case for 5, 6 not constant
Co"*ine $ith V for 4 e!ations for 4 !n@no$ns V pand +an *e al*eit diffi+!lt soled s!*e+t to initial and
*o!ndar% +onditions for V p at t - t0and on all
*o!ndaries i,e, $ell posed; I.P,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 26
Appli+ation of C o"ent!" !ation:
RCV CS
d
+ V d V V n dAdt = +
SB +++ += 'in+l!des rea+tion for+es(
)ote:
1, e+tor e!ation
2, n - o!t$ard !nit nor"al: RV n 0 inlet 0 o!tlet
3, 1# o"ent!" fl!/
( ) ( )i ii iout inCS
V V n dA m V m V =
& &
>here iV i are ass!"ed !nifor" oer dis+rete inlets
and o!tlets
i i ni im V A=&
{ ( ) ( )i ii i inout
CV
d+ V d m V m V
dtnet force inlet momentumoutlet momentum
time rate of c%an&eonCV flu(flu(of momentum in CV
= + & &1 42 431 4 2 4 31 4 2 4 3
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 2=
4, o"ent!" fl!/ +orrelation fa+tors
2 2
avu V n dA u dA AV
a(ial flo! !it%non uniformvelocity $rofile
= =
14 2 43
>here
2
1
avCS
udA
A V
=
1av
CS
"V u dA
AA= =
Ma"inar pipe flo$:
12
2
0 021 1r r
u ) )R R =
0,53avV )= 34=
!r*!lent pipe flo$:
m
R
r)u = 10 1 1? 5
m
( )0
2
1 '2 (avV )
m m=
+ + =1=m Vav 7829
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 28
( ) ( )
(22('21'2
21 22
mm
mm
++++= m:;< = :792
5, Constant p +a!ses no for+e herefore
se pgage- pat"pa*sol!te
0$CS CV
+ $n dA $ d= = = for p - +onstant
6, At et e/it to at"osphere pgage- 0,
=, Choose C +aref!ll% $ith CS nor"al to flo$ and
indi+ating +oordinate s%ste" and +on C si"ilar
as free *od% diagra" !sed in d%na"i+s,
8, an% appli+ations !s!all% $ith +ontin!it% and
energ% e!ations, Caref!l pra+ti+e is needed for"aster%,
a, Stead% and !nstead% deeloping and f!ll%
deeloped pipe flo$
*, "pt%ing or filling tan@s
+, For+es on transitions
d, For+es on fi/ed and "oing anes
e, E%dra!li+ !"p
f, .o!ndar% Ma%er and *l!ff *od% drag
g, Ro+@et or et prop!lsion
h, )ole
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 2?
i, Propeller
, >aterha""er
First relate !"a/to 0!sing +ontin!it% e!ation
( ) += R m
drrRruR)
0"a/
2
0 210
( )0 "a/20
11 2
Rm
avr) u r dr V
RR
= =
T
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 30
"a/
2
'1 ('2 (av
V um m
=+ + " - 1
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 31
22
0
2
21 02,(' R)R$$+
turb =
Re+onsider the pro*le" for f!ll% deeloped flo$:
Contin!it%:
0in out
in out
m m
m m m
+ == =
& &
& & & or U - +onstant
o"ent!":
2
1 2
1 1 2 2
2 1
' (
' ( ' (
' (
0
(+ $ $ R + uV A
u u A u u A
" u u
= =
= +=
=
( ) 21 2 2 0!$
$ $ R R (
=14 2 43
=
d(
d$R!
2 or for s"aller C r R
=
d(
d$r
2
(valid for laminar or turbulent flo!, but assume laminar)
===
d(
d$r
dr
du
dy
du
2 % - Rr '!all coord7)
=
d(
d$r
dr
du
2
Co"plete anal%sis
!sing CF#V
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 32
cd(
d$ru +
=
4
2
0(' ==Rru
= d(d$R
c 4
2
=
d
d$rRru
4('
22
=
d(
d$Ru
4
2
"a/
=
2
2
"a/ 1('
R
ruru
==
d(
d$Rdrrru"
R
82('
4
0
2
"a/
28ave
" R d$ uV
A d(
= = =
2
8 4
2 2
ave ave!
V VR d$ R
d( R R
= = =
2
8 32 64 64
Re
!
ave ave ave
fV RV V #
= = = =
Re aveV #
=
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 33
/a+t sol!tion of )S for la"inar f!ll% deeloped pipe
flo$
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 34
a@ing "oing C at speed s- WR X en+losing et and
*!+@et:
Contin!it%: 0in out m m + =& &
Rin out
CS
m m m V n dA= = = & & &
Inlet DR ,V V R n i= =
Q!tlet G' (R ,V V R n i= =
o"ent!":
> buc-et out in+ + mu mu= = & &
2
' ( ' (
2 ' (
2 ' (
buc-et , ,
,
, ,
+ m V R V R
m V R
A V R
=
=
=
&
' (, ,m A V R
=
22 ' (buc-et , ,
3 R+ A R V R= =
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 35
0=d
d3 for
3
,VR = 3"a/
8
2= , ,
3 A V=
If infinite n!"*er of *!+@ets: , ,m A V=&
3
"a/
2 ' (
2 ' (
10
2 2
buc-et , , ,
, , ,
,
, ,
+ A V V R
3 A V R V R
Vd3for R 3 A V
d
=
=
= = =
C +ontin!it% e!ation for stead% in+o"pressi*le flo$
one inlet and o!tlet A - +onstant
in out
V ndA V ndA m " = = = &
in out " "=
( ) ( )ave avein out V A V A=
For A - +onstant ( ) ( )ave avein out V V=
( ) ( )in out
+ V V n dA V V n dA = + Pipe:
( ) ( )(in out
+ u V n dA u V n dA = + ( ) ( )2 2ave ave
in out AV AV = +
( )ave out in"V = +hange in shape !
all et "ass flo$
res!lt in $or@,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 36
ane:
( )out in out in+ m V V V V= = &
( ) ( )2( out in in+ m u u m u= = & &
+hange in dire+tion !
Appli+ation of differential "o"ent!" e!ation:
1, )S alid *oth la"inar and t!r*!lent flo$ ho$eer
"an% order of "agnit!de differen+e in te"poral and
spatial resol!tion i,e, t!r*!lent flo$ re!ires er%s"all ti"e and spatial s+ales
2, Ma"inar flo$ Re+rit-)
1000
Re Re+rit insta*ilit%
3, !r*!lent flo$ Retransition 10 or 20 Re+rit
Rando" "otion s!peri"posed on "ean +oherent
str!+t!res,
Cas+ade: energ% fro" large s+ale dissipates at
s"allest s+ales d!e to is+osit%,Yol"ogoro h%pothesis for s"allest s+ales
4, )o e/a+t sol!tions for t!r*!lent flo$: RA)S #S
MS #)S 'all CF#(
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 3=
5, 80 e/a+t sol!tions for si"ple la"inar flo$s are
"ostl% linear 0V V =
a, Co!ette flo$ - shear drien
*, Stead% d!+t flo$ - Poise!ille flo$
+, nstead% d!+t flo$
d, nstead% "oing $alls
e, As%"ptoti+ s!+tion
f, >inddrien flo$sg, Si"ilarit% sol!tions, et+,
6, Also "an% e/a+t sol!tions for lo$ Re Sto@es and
high Re .M appro/i"ations
=, Can also !se CF# for non si"ple la"inar flo$s
8, AF# or CF# re!ires $ell posed I.P therefore
e/a+t sol!tions are !sef!l for set!p of I.P ph%si+s
and erifi+ation CF# sin+e "odeling errors %ield
S- 0 and onl% errors are n!"eri+al errors S)
i,e, ass!"e anal%ti+al sol!tion - tr!th +alled
anal%ti+al *en+h"ar@
M'(-M'(-MA'A(-0 si"ilarl% for IC&.C
ZS- S [ - ZS)B ZS
S2- S)
2B S2
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 38
nerg% !ation:
. - - energ%
7 - e - d
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 3?
vd? dA V = & - viscous force velocity
v
CS? V dA=
&
( )' (
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 40
#efine @ineti+ energ% +orre+tion fa+tor
32
21' (2 2
ave
aveA A
VVdA V V n dA mA V
= =
Ma"inar flo$:
=
2
0 1
R
r)u
Vave97A = ; D2
!r*!lent flo$:m
R
r)u
= 1
0
( ) ( )3 3
1 2
4'1 3 ('2 3 (
m m
m m
+ +=
+ +
m:;< D:79A8 as !it% =, DE: for
turbulent flo!
2 2
D D' < ( ' < (2 2
s ave aveout in
? V V"u $ &' u $ &'
m m = + + + + + +
& &
& &
Met in - 1 o!t - 2 V- Vae and diide *% g
2 21 1 2 21 1 2 2
2 2$ t *
$ $V ' % V ' % %
& & & &
+ + + = + + + +
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 41
$s tt $
?? ?% %
&m &m &m= =
&& &
& & &
2 1
1' (
*
"% u u
& m&=
&
&
%* t%ermal ener&y (ot%er terms re$resent mec%anical ener&y
1 1 2 2m AV A V = =&
Ass!"ing no heat transfer "e+hani+al energ% +onerted
to ther"al energ% thro!gh is+osit% and +an not *e
re+oered therefore it is referred to as head loss 0
$hi+h +an *e sho$n fro" 2ndla$ of ther"od%na"i+s,
1# energ% e!ation +an *e +onsidered as "odified.erno!lli e!ation for hpht and hM,
Appli+ation of 1# nerg% e!ation f!ll% deeloped pipe
flo$ $itho!t hpor ht,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 42
1 21 2
2
2
' ( ' (
2
1
8
2
*
! !
ave
ave*
$ $ * d% ' ' $ ' * d(
& & d(
*f& R
V
V*% f
# &
= + = + =
= =
=
For la"inar flo$
2
8 32!
ave ave
fV RV
= =
2
32 ave*
*V%
#
= Vae e/a+t sol!tionV
For t!r*!lent flo$ Re+rit ] 2000 Retrans] 3000
f-f 'Re @
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 43
%* %f+ F%m $here
2
2m
V% @
&
@ loss coefficient
=
=
h"- so +alled; "inor losses e,g, entran+e
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 44
.erno!lli negle+t g p2-pa2 2
1 1 2 2
1 1
2 2$ V $ V + = + %*9, 'constant
( )2 21 2 2 11
2$ $ V V= + 2 21
,=? ??8101 000 '5,22 2,0? (2
$ = +
1 110020$ 3a=
)ote: 2 2 2
2 2 3 3 4 42 2 2
$ V $ V $ V
+ = + = +
2 3 4 2 3 4a$ $ $ $ V V V= = = = =
2 2 3 3 4 40CS
V A A V A V A V = = +432
AAA +=
3 3 3 4 4 40 ' (yCS
+ VV A V V A V V A = = = + 2 2
3 3 4 4V A V A = 43 AA =
'*( For the ro!nd et gien in the pro*le" state"ent
2 2
2 2
2
,=? ?8? ,02 4254
(+ + mV V 1
A
= = = & 142 43
12 3
2 41,4 < 10,3
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 45
Contin!it% e!ation *et$een points 1 and 22
21 1 2 2 1 2
1
#V A V A V V
#
= =
2
1
241,4
5V
=
1 6,63
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 46
'a(2
21 2
2
V' '
&= + 01101 212 ==== ''%*
2 1 22 ' (V & ' ' = 11_81,?_2= sm
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 4=
'+(
2 2
2 21 2 2
2 2
V V*' ' f
& # &= + + D2:
( )2
21 2 1 here ( )i,#V
V V & V #t
=
D' ( ii,
,
ii,
,
u#u- J
#t (
u$ V
(
= +
1 2 3
D' < ( ' ( 0ii,,
u# #$u $ - J dissi$ation function
#t #t (
+ = + + =
123
++=#t#$J-
#t#% ('
S!""ar% # for +o"pressi*le non+onstant propert%
fl!id flo$
Cont, ' ( 0Vt
+ =
o", i,#V
& $#t
= +
D
i, i, i,V
& &-
= +
=
' < ($ # #
$#t #t
= +
"o"ent!" e!ation
+ontin!it% e!ation
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 50
nerg% ++= (' J-#t#$
#t
#%
Pri"ar% aria*les: p V
A!/iliar% relations: 9 - 9 'p( O - O 'p(
h - h 'p( @ - @ 'p(
Restri+tie Ass!"ptions:
1( Contin!!"2( )e$tonian fl!ids
3( her"od%na"i+ e!ili*ri!"
4( f - 9g D-
5( heat +ond!+tion follo$s Fo!rier`s la$
6( no internal heat so!r+es
For in+o"pressi*le +onstant propert% fl!id flo$
Dvdu c dJ = cv, 6, , 5 E constant
+= J-#t
#Jc
v
2
For stati+ fl!id or Vs"all
J-t
Jc
$
2=
%eat conduction eIuation (also valid for solids)
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 51
S!""ar% # for in+o"pressi*le +onstant propert% fl!id
flo$ '+] +p(
0V =
2D#V &- $ V#t
= + ellipti+;
+= J-#t
#Jc
$
2 $here,
i
i,
(
u
=
Contin!it% and "o"ent!" !n+o!pled fro" energ%
therefore sole separatel% and !se sol!tion post fa+to to
get ,
For +o"pressi*le flo$ 9 soled fro" +ontin!it% e!ation
fro" energ% e!ation and p - '9( fro" e!ation of
state 'eg ideal gas la$(, For in+o"pressi*le flo$ 9 -
+onstant and !n+o!pled fro" +ontin!it% and "o"ent!"
e!ations the latter of $hi+h +ontains $ s!+h that
referen+e p is ar*itrar% and spe+ified post fa+to 'i,e, for
in+o"pressi*le flo$ there is no +onne+tion *et$een p and
9(, he +onne+tion is *et$een$
and0V
= i,e, asol!tion for p re!ires 0V = ,
)S 21 D
#V$ V
#t
= + D$ $ '= +
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 52
('1S 'See deriation details on p,81(
2 21 ,i
, i
uu#V $#t ( (
= +
For 0V = :
i
,
,
i
(
u
(
u$
= 2
Poisson e!ation deter"ines press!re !p to additie
+onstant,
Appro/i"ate odels:
1( Sto@es Flo$
For lo$ Re 1 ] 0)*
V V
=
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 53
0V =21V
$ Vt
= +
0(' 2 = $1S
2( .o!ndar% Ma%er !ations
For high Re 1 and atta+hed *o!ndar% la%ers or
f!ll% deeloped free shear flo$s '$a@es ets "i/ing
la%ers( v )
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 54
3( Inis+id Flo$
( ) 0
b
' ( '
Vt#V
& $ 4uler 4Iuation nonlinear %y$erboli#t
#% #$- J $ V J un-no!ns and % - f $
#t #t
+ =
=
= + =
4( Inis+id In+o"pressi*le Irrotational
2
0
0 0 b b
V V
V linear elli$tic
= =
= =
!ler !ation .erno!lli !ation:
2
2$ V &' const
+ + =
an% elegant sol!tions: Mapla+e e!ation !sing
s!perposition ele"entar% sol!tions separation of
aria*les +o"ple/ aria*les for 2# and .o!ndar%
le"ent "ethods,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 55
Co!ette Shear Flo$s: 1# shear flo$ *et$een s!rfa+es of
li@e geo"etr% 'parallel plates or rotating +%linders(,
Stead% Flo$ .et$een Parallel Plates: Combined Couette
and 3oiseuille lo!7
0
0
0
( y '
(
V
u v !
u
=+ + =
=2D
#V$ V
#t = + 0=+++
'y( !uvuuu
t
u
D0 ( yy$ u= ++= J-
#t
#Jc
$
2 0( y '
JuJ vJ !J
t
+ + + =
2
0yyy
u-J +=
(note inertia terms vanis% in%erently and 5 is absent
from eIuations)
2 2 2
2 2
2
2 2 2
' ( ' ( ' (
' (
( y '
( y y ' ' (
( y '
y
u v !
v u ! v u !
u v !
u
= + +
+ + + + + +
+ + +
=
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 56
1on-dimensionali'e eIuations, but dro$ N
)uu
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 5=
Sol!tion depends on2
D(
%B $
)= :
. 0 D($ is opposite to
. 0,5 *a+@flo$ o++!rs near lo$er $all
c.c 1 flo$ approa+hes para*oli+ profilePress!re gradient effe+t
22 3 4Pr Pr Pr 1 '1 ( '1 ( ' ( '1 (
2 8 6 12
c c c4 4 B 4 BJ y y y y y= + + +
G55555555555555555555555H
1ote usually 3r4cis Iuite small
Substance 3r4 c dissi$ation
Air 0,001 er% s"all
>ater 0,02d
Pr
Brin-man
4Brc
=
=
Cr!de oil 20 large
P!re
+ond!+tion
rises d!e to
is+o!s dissipation
#o"inant ter"
for .
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 58
Shear Stress
1( D 0($ = i,e, p!re Co!ette Flo$
2 21 1
2 2
y
f
uC
= =_
_ 1< 2y
u =
%)% Re
1==
2
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 5?
_ _21 '1 (2
u B y= __ _y
u By= y%
B)u
y 2= u)=
>here"a/2D
(
u%B $
) )= =
#i"ensional for" ( )2 2
"a/
1D 1
2
% yu $
%
u
=
14 2 43 "a/34
%udyu"%
%
==
"a/
3
2
2u
%
"u ==
%u
%
u
%
B)
lo!er%
B)
u$$er%
B)u
!
%yy!
32
"a/
===
+=
===
6Re6
2
1 02 ==== %f!
f C3or%u
)
C
/+ept for n!"eri+al +onstants sa"e as for +ir+!lar pipe,
2
,
,
u lam
u turb
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 60
Rate of heat transfer at the $alls:
%
)JJ
%
-J-I
%yy!
4
('
2
2
01 ==
B - !pper - lo$er
Eeat transfer +oeffi+ient:
( )1 0!I
J J=
212
Br-
%1u ==
For .r 2 *oth !pper & lo$er $alls "!st *e +ooled to
"aintain 1and 0
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 61
Conseration of Ang!lar o"ent!": "o"ent for" of
"o"ent!" e!ation 'not ne$ +onseration la$V(
0
sys
B O r V dm= = = an&ular momentum of system aboutinertial coordinate system 97
r V=
( ) ( )
0
CV CS
d O d
r V d r V V n dAdt dt = + == 0M vector sum all eternal moments a$$lied CV
due to bot% B and S7
For !nifor" flo$ a+ross dis+rete inlet
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 62
a@e inertial fra"e 0 as fi/ed to earth s!+h that CS
"oing at Vs- R i
2 0D DV V i R i= 2 Dr R,=
1 0
DV V -= 1 D0r ,=
0$i$e
"V
A=
2 12 10 inD ' ( ' (P outM J - r V m r V m= = & &
out inm m "= =& &
0D D' (' (oJ - R V R - " =
0 0
2
V J
R "R
= interestin&ly, even for J99, /maV9;R
Retarding tor!e d!e to*earing fri+tion
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 63
#ifferential !ation of Conseration of Ang!lar
o"ent!":
Appl% C for" for fi/ed C:
'&- ang!lar a++elerationQ - "o"ent of inertia
2 2 2 2'
d( d( dy dyQ a dy b dy c d( d d( = + &
( )' y yQ ddy = &
Sin+e3 3 2 2
12 12Q d(dy dyd( d(dy d( dy
= + = +
2 2
12 ' (y y(d( dy
+ =
&
0 0li"
d( dy y((y = si"ilarl% '((' = 'yy' =
i,e ,ii, = stress tensor is symmetric (stressest%emselves cause no rotation)
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 64
.o!ndar% Conditions for is+o!sFlo$ Pro*le"
he # to *e dis+!ssed ne/t +onstit!te an I.P for
a s%ste" of 2ndorder nonlinear P# $hi+h re!ire
.C for their sol!tion depending on ph%si+al pro*le"
and appropriate appro/i"ations,
%pes of .o!ndaries:
1, Solid S!rfa+e2, Interfa+e
3, Inlet
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 65
1, Solid S!rfa+e
a, Mi!id
- "ean free path fl!id "otion therefore
"aros+opi+ ie$ is no slip; +ondition i,e, no
relatie "otion or te"perat!re differen+e *et$een
li!id and solid,
liIuid solid V V= solidliIuid JJ
=
/+eption is for +onta+t line for $hi+h anal%sis is
si"ilar to that for gas,
*, as
S"ooth $all
Ro!gh $all
!
!
dy
dulu =
Spe+!lar refle+tionConseration of
tangential "o"ent!"
!$-0-fl!id elo+it% at$all
#iff!se refle+tion,
Ma+@ of refle+tedtangential "o"ent!"
*alan+ed *% !$
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 66
!
!
dy
du =
al
32
= lo! density limit
!
!
au
2
3=
a)Ma = 21
2
!f
C
)
=
f! CMa)u =5,< =
Eigh Re: Cf E 9799A
Say Ma E 29
Mo$ Re: Cf E 7HRe-:;2 Re;
2
1
Re
4,
(
! Ma
)
u=
Signifi+ant slip possi*le at lo$ Re high a:
E%personi+ M Pro*le";
Si"ilar for :
Eigh Re: J&as J!
Mo$ Re( )
,8=&as !
f
r !
J JMaC
J J
=
air- driing
01,0hi+h leads to:
Eel"holt heore" 3: )o fl!id parti+le +an hae
rotation if it did not originall% rotate, Qr e!ialentl% in
the a*sen+e of rotational for+es a fl!id that is initiall%
irrotational re"ains irrotational, In general $e +an
+on+l!de that orti+es are presered as ti"e passes, Qnl%
thro!gh the a+tion of is+osit% +an the% de+a% ordisappear,
Yelins Cir+!lation heore" and Eel"holt
heore" 3 are er% i"portant in the st!d% of inis+id
flo$, he i"portant +on+l!sion is rea+hed that a fl!id that
he +ir+!lation of a
"aterial loop neer
+hanges
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 =8
is initiall% irrotational re"ains irrotational $hi+h is the
!stifi+ation for idealflo$ theor%,
In a real is+o!s fl!id orti+it% is generated *%
is+o!s for+es, is+o!s for+es are large near solid
s!rfa+es as a res!lt of the noslip +ondition, Qn the
s!rfa+e there is a dire+t relationship *et$een the is+o!s
shear stress and the orti+it%,
Consider a 1# flo$ near a $all:
12 12
22 22
32 32
2 0
0
u v u
y ( y
v
y
! v
y '
= = + =
= = =
= = + =
>hi+h sho$s that
dy
u(
= 0== 'y
Eo$eer fro" the definition orti+it% $e also see that
he is+o!s stresses are gien *
i, ,n $here i, i, =
11 1 12 2 13 3
21 1 22 2 23 3
31 1 32 2 33 3
(
y
'
n n n
n n n
n n n
+ + =
+ + =
+ + =)Q: the onl% +o"ponent of
is , A+t!all% this is a
general res!lt in that it +an *e
sho$n that s!rfa+eis
perpendi+!lar to the li"iting
strea"line,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 =?
( '
u
y
= =
i,e, the $all orti+it% is dire+tl% proportional to the $all
shear stress, his anal%sis +an *e easil% e/tended for
general 3d flo$,
i, , i, ,n n = at a fi/ed solid $all
r!e sin+e at a $all $ith +oordinate /21 3
0( (
= =
and
fro" +ontin!it%2
0v
(
=
Qn+e orti+it% is generated its s!*se!ent *ehaior is
goerned *% the orti+it% e!ation,
)S ( ) 2here0
V
V
=
= V =
If 0V V V = + =
hen 2 0 = J%e 7d7e7 foris t%e *a$lace 4I7
And V A = Since ( ) 0= A
2 ' (
V A
A A
= =
= +
he irrotational part of
the elo+it% field +an *e
e/pressed as the gradient
of a s+alar
Again *% e+tor identit%
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 84
i,e = A2
he sol!tion of this e!ation is
= dR
A
4
1
h!s 31
4
RV d
R
=
>hi+h is @no$n as the .iotSaart la$,
he .iotSaart la$ +an *e !sed to +o"p!te the elo+it%
field ind!+ed *% a @no$n orti+it% field, It has "an%
!sef!l appli+ations in+l!ding in ideal flo$ theor% 'e,g,
$hen applied to line orti+es and orte/ sheets it for"s
the *asis of +o"p!ting the elo+it% field in orte/latti+eand orte/sheet liftings!rfa+e "ethods(,
he i"portant +on+l!sion fro" the Eel"holt
de+o"position is that an% in+o"pressi*le flo$ +an *e
tho!ght of as the e+tor s!" of rotational and irrotational
+o"ponents, h!s a sol!tion for irrotational part V
represents at least part of an e/a+t sol!tion, nder +ertain+onditions high Re flo$ a*o!t slender *odies $ith
atta+hed thin *o!ndar% la%er and $a@e Vis s"all oer
"!+h of the flo$ field s!+h that V is a good
appro/i"ation to v , his is pro*a*l% the strongest
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 85
!stifi+ation for idealflo$ theor%, 'incom$ressible,
inviscid, and irrotational flo!),
)oninertial Referen+e Fra"e
h!s far $e hae ass!"ed !se of an inertial referen+e
fra"e 'i,e, fi/ed $ith respe+t to the distant stars in
deriing the C and differential for" of the "o"ent!"
e!ation(, Eo$eer in "an% +ases noninertial referen+efra"es are !sef!l 'e,g, rotational "a+hiner% ehi+le
d%na"i+s geoph%si+al appli+ations et+(,
i rel
dVa a
dt= +
i rel
dV+ ma m adt
= = +
rel
dV+ ma m
dt = 1 2 3
rRSi +=
i,e )e$ton`s la$
applies to non
inertial fra"e $ith
addition of @no$n
inertial for+e ter"s
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 86
i
d RV V r
dt= + +
2
22 ' (i
rel
dV d R d a r V r
dt dt dt
dVa
dt
= + + + +
= +
2
2
dt
Rd acceleration (,y,')
rdt
d - an&ular acceleration (,y,')
2 V - Coriolis acceleration
(' r - centri$etal acceleration (-W2*, !%ere * normal
distance from r to ais of rotation W)7
Sin+e R and W ass!"ed @no$n altho!gh "ore
+o"pli+ated $e are si"pl% adding @no$n
inho"ogeneities to the "o"ent!" e!ation,
C for" of o"ent!" e!ation for noninertial
+oordinates:
rel R
CV CV CS
d+ a d V d v V n dAdt
= +
#ifferential for" of o"ent!" e!ation for noninertial
+oordinates:
rdterm from fact t%at
(,y,') rotatin& at W(t)7
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 8=
2D 23
rel ,i i,
,
#Va & $ V
#t
+ = +
( ) vrrRa rel +++=
2
All ter"s in rela seldo" a+t in !nison 'e,g, geoph%si+al
flo$s(:
R ] 0 earth not a++elerating relatie to distant stars
] 0 for earth
( )r ] 0 g nearl% +onstant $ith latit!de
v2 "ost i"portantV
1
0 '2 (idVa R Vdt
= + 00
tVVV tV *
= =
2
0 00
0
V * V
R RossbyV *
= = =
if M is large i,e, +o"para*le to the
order of "agnit!de of the earth
radi!s R01 then Coriolis ter" is
larger than the inertia ter"s and is
i"portant,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 88
/a"ple of )oninertial Coordinates:
eoph%si+al fl!ids d%na"i+s
At"osphere and o+eans are nat!rall% st!died !sing non
inertial +oordinate s%ste" rotating $ith the earth, $o
pri"ar% for+es are Coriolis for+e and *!o%an+% for+e d!e
to densit% stratifi+ation 9 - 9'(, .oth are st!died !sing
.o!ssines appro/i"ations '9 - +onstant e/+ept
( ) DJ &- ter" and O @ Cp- +onstant( and thin la%er on
rotating s!rfa+e ass!"ption *O)? ] ,
#ifferen+es *et$een at"ospheri+al o+eans: lateral
*o!ndaries '+ontinents( in o+eans +!rrents in o+ean 'g!lf
and Y!roshio strea"( along $estern *o!ndaries +lo!ds
and latent heat release in at"osphere d!e to "oist!re
+ondensation o+ean- 0,1]1 or 2 "
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 8?
Coriolis for+e - 2 V
-!vu
-,i
'y(
GGG
- ( )
+ +ossinsin+os2
GGG
u-u,v!i
-GGG
+os2 -u,fuifv + sin2=f
f 0 northern he"isphere
f 0 so!thern he"isphere
f - at poles
0 sin+e $
- planetar%orti+it%
- 2 _ erti+al
+o"ponent WPerson
spins at
W
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?0
f - 0 at e!ator
!ations of otion
0V =
!
&
'
$
#t
#!
vy
$fu
#t
#v
u(
$fv
#t
#u
2
00
2
0
2
0
1
1
1
+
=
+=+
+
=
('100
JJ= p 9 - pert!r*ation fro" h%drostati+
+ondition
eostrophi+ Flo$: !asistead% larges+ale "otions in
at"osphere or o+ean far fro" *o!ndaries
(
$fv
=
0
1
y
$fu
=0
1
2
] 0#V )
#t *
] 0 ' (f V f) M - horiontal s+ales
Ross*% n!"*er - f*)
At"osphere: ] 10 "
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?1
herefore negle+t#V
#tand sin+e there are no *o!ndaries
negle+t2V ,
"o"ent!" $
&'
=
*aro+lini+ 'i,e, p - p'((
and +an *e !sed to eli"inate p in a*oe e!ations
$here*% '!( - f''(( $hi+h is +alled ther"al $ind *!t
not +onsidered here,
If $e negle+t 9-9'( effe+ts '!( - f'p( and +an *e
deter"ined fro" "eas!red p'/%(, )ot alid near thee!ator 'B 3o( $here f is s"all,
( )0
1D D D D D D$ $ $ $u i v , $ i , i ,f y ( ( y
+ = + +
- 0
i,e Vis perpendi+!lar to $ horiontal elo+it% is
along 'and not a+ross( lines of +onstant horiontal
press!re $hi+h is reason iso*ars and strea" lines
+oin+ide on a $eather "apV
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?2
@"an Ma%er on Free S!rfa+e: effe+ts of fri+tion near
*o!ndaries
is+o!s la%ers:
S!dden a++eleration flat plate: t yyu u=
3,64 t =
Qs+illating flat plate: yyt uu =
6,5
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?3
='
u at - 0
0='
v at - 0
0(' =vu at -
!ltipl% e!ation *% 1=i and add to !e!ation:
2
2
d V i f V
dt
= V u i v
com$le( velocity
= +
='1 ( < '1 (
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?4
F, )ansen '1?02( o*sered drifting ar+ti+ i+e drifted 20
400to the right of the $ind $hi+h he attri*!ted to
Coriolis a++eleration, Eis st!dent @"an '1?05( deried
the sol!tion,
Re+all f 0 in so!thern he"isphere so the drift is to the
left of \,
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058:0160 Chapters 3 & 4
Professor Fred Stern Fall 2006 ?5
Si"ilar sol!tion for i"p!lsie $ind:
0(0'00 ====== 'u'u'uuu'''t
tu
20=
la"inar sol!tion0
0' 6 < 20 (
!indu V m s J C = = = 97H m;s after one min7, 27 m;s
after one %our
t!r*!lent sol!tion(more realistic)u9972 m;s after : %r ( Y v!ind)
For @"an la%er si"ilar +onditions m - 400)
Ma"inar sol!tion !0- 2,= "
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