stream lines
DESCRIPTION
Stream LinesTRANSCRIPT
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2103351 Fluid Mechanics I
4
4.1
( , , , )V V x y z t =
(4.1)
( , , , ) ( , , , ) ( , , , )x y z t x y z t xiV y z tu wj k= + +
(4.2)
V 2 2 2 1/ 2( )V u w= + +
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4.1.1 Eulerian Lagrangian
2
Lagrangian
Eulerian
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4.1 Eulerian (velocity sensor) (hot-wire)0
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x
x
y
y
c
o
4.1 Eulerian Lagrangian
()
0( ,x x=
0,y y=
0 )z z=
0 0 0( , , , )V V x y z t =
1 ( , , , )V V x y z t =
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Lagrangian
c c c
c c
V V= ( t) Lagrangian
(steady flow)
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0t
=
( , , )x y z =
0
V
t
=
( , , )V V x y z=
(unsteady flow)
4.1.2
4.1 ( ) 3 x,y,z
( , , , )V V x y z t =
(4.1)
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r
x
4.2
x
r
4.3
no-slip
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(velocity profile) 4.2 4.3
(uniform flow) 4.4 4.5 4.4 rx 4.5
x
r
4.4
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x
r
4.5
4.1.3 Pathline, Streakline
Streamline
streamline streakline pathline
streakline pathline
pathline
streakline
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streamline
streamline
streamlinestreamline
streamline xydy/dx tan
dy
dx u
=
u,xy
streamline
dx dy
u
= (4.3)
streamline
dx dy dz
u w= = (4.4)
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pathline, streakline streamline
4.2
(surrounding)
(control volume)
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4.2.1 0
system
dM
dt
=
(4.5)
system
system
M d
= (4.6)
4.2.2 ( )
system
d mVF
dt
=
(4.7)
P mV=
Msystem system
P mV Vdm V d
= = = (4.8)
4.2.3 ( )d r mV
Tdt
=
(4.9)
H r mV=
( ) ( )M system system
r mV r Vdm r V d
= =
(4.10)
( )
s shaft
M system
T r F r gdm T = + +
(4.11)
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4.2.4
Q W dE = (4.12)
(path function) (point
function)
system
Q W dE
dt dt dt
= (4.13)
( ) ( )
system
M system system
E edm e d
= = (4.14)
2
2
Ve u gz= + + (4.15)
4.2.5 Q
dST
1
system
dS Q
dt T dt
(4.16)
( ) ( )system
M system system
S sdm s d
= = (4.17)
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4.3 4.3.1
t
t + t
4.6
0lim
s ss t t t
t
B BdB
dt t
+
=
(4.18)
4.6 sB
cvB
( ) ( )s II III t t cv I III t t t tB B B B B B+ ++ = + = +
s cvt tB B=
4.18
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0
( )lim
cv I III t t cvs t
t
B B B BdB
dt t
+
+ =
0 0 0lim lim lim
cv cv III I s t t t t t t t
t t t
B B B BdB
dt t t t
+ + +
= +
b
0lim
cv cvt t t cv
t cv
B B Bb d
t t t+
= =
dA
V
III
t t+
4.7 III
dIII cos d III
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dIII d
( )III III
III III t tt t t t cs cs
B dB b d ++ += =
cosIII
cs t t
b d +
=
IIIcs III
0 0
coslim lim
III
III t t
t tcs
B b d
t t
+
=
0lim cos
III
tcs
b dt
=
0lim
tV
t
=
d d =
0lim cos
III
III t t
tcs
B b V dt
+ =
I
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dA
V
I
t t+
4.8 I
0 0 0lim lim lim
I I
I I
t t tcs cst t t t
B dB b d
t t t
+ +
= =
0
limcos
Icst
b dt
=
cosI
cs
b V d =
B
cos cosI III
s
cv CS CS
dBb d b V d b V d
dt t
= + +
I
cs
IIIcs
cs
B
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s
cv cs
dBb d b V d
dt t
= +
(4.19)
4.19 Bs
B(net flux) B
(Reynolds transport theorem)
4.4
0system
dM
dt= (4.5)
Bs M
bB1
0cv cssystem
dMd V d
dt t
= + =
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0cv cs
d V dt
+ =
(4.20)
4.20
4.5
Bs mV b
BsV
S B
cv cs
F F F V d V V d
t
= + = +
(4.21)
SF
BF
( )
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( )
xyz
x Sx Bxcv cs
F F F u d u V d t
= + = +
y Sy Bycv cs
F F F d V d t
= + = +
(4.22)
z Sz Bzcv cs
F F F w d w V d t
= + = +
u, , wxyz
4.6
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XYZ ()
xyz
XYZ cv xyzV V V= +
(4.23)
XYZ cv xyza a a= +
(4.24)
XYZ
V
XYZ
a
XYZ.
cvV
cva
XYZ
xyz
V
xyz
a
xyz
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(4.7) 4.23 4.24
system
dF mV
dt=
(4.7)
( ) ( ) ( )XYZ cv xyz
d d dF mV mV mV
dt dt dt = = +
( ) ( )XYZ cv xyz
d dF mV ma mV dt dt
= = + (4.25)
cv
ma
cv cv cv
system
ma a dm a d
= =
4.25
( )cv xyzsystem
dF a d mV
dt
=
0t=
on system cv on CV cv
system CV
F a d F a d
=
4.25
( )cv xyzcv
dF a d mV
dt
=
(4.26)
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4.6.1
4.26
( )xyzd
F mVdt
=
(4.27)
4.19 Bs
xyzmV
b xyzV
4.27
S B xyz xyz xyz
cv cs
F F F V d V V d t
= + = +
(4.28)
4.21
xyzV
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4.20 4.20
0xyzcv cs
d V dt
+ =
(4.29)
4.6.2
4.26 4.19
S B cv xyz xyz xyzcv cv cs
F F F a d V d V V d t
= + = +
(4.30)
3
Sx Bx cvx xyz xyz xyzcv cv cs
F F a d u d u V d t
+ = +
Sy By cvy xyz xyz xyzcv cv cs
F F a d d V d t
+ = +
(4.31)
Sz Bz cvz xyz xyz xyz
cv cv cs
F F a d w d w V d
t
+ = +
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,xyz
u ,xyz
xyz
w x,y,z xyz
4.6.3
XYZ cv xyzV V r V = + +
(4.32)
2XYZ cv xyz xyza a r r V a = + + + +
(4.33)
x
x
y
z
y
z
r
4.9 XYZ
xyz
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4.7
( )
system
d mV
F dt
=
(4.7)
XYZXYZ
dVF m ma
dt= =
4.33 2
XYZ cv xyz xyzF ma m a r r V ma = = + + + +
2cv xyz
m a r r V + + +
2 2cv xyz cv xyzm a r r V a r r V dm + + + = + + +
2cv xyz
system
a r r V d
= + + +
xyzxyz dmVmadt
=
0t= 4.7
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( )XYZ
dF mV
dt=
2 ( )cv xyz xyzcv
da r r V d mV dt = + + + +
2cv xyz
cv
F a r r V + + +
( )xyz
dd mV
dt =
(4.34)
4.19
2S B cv xyz
cv
F F F a r r V = + + + +
d
xyz xyz xyzcv csV d V V d t
= +
(4.35)
4.7
4.9 4.11
( )s
M system
r F r g +
( )shaft
ddm T r mV
dt+ =
Bsr mV bBs
r V
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s shaft cv cv cs
r F r g d T r V d r V V d t
+ + = +
(4.36)
4.8
4.13, 4.14 4.15
system
Q W dE
dt dt dt
= (4.13)
systemsystemE e d
= (4.14)2
2
Ve u gz= + + (4.15)
E be 4.13
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corrected addcv cs
Q We d e V d
dt dt t
= +
(4.37)
corrected shaft norm shearal other WW W W W
dt dt dt dt dt
= + ++
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shaftW
dt
normalWdt
corrected normalcs cs
W ppV d V ddt
= =
shearWdt
shear
cs
W
Vddt
=
V
shaftW
dt
V
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V
V
otherWdt
shaft shear other
cs
Q W p W W V d
dt dt dt dt
2
( )2cv cs
Ve d u gz V d
t
= + + +
2
( )2
shaft shear other
cv cs
Q W W W p V e d u gz V d
dt dt dt dt t
= + + + +
(4.37)
2
2
Ve u gz= + +
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4.9
4.2.5 4.16 4.17 Bs S b s
1 1
cv cs cs
Qs d s V d d
t T A dt
+
(4.35)