Chapter 2 Reynolds Transport Theorem (RTT)
description
Transcript of Chapter 2 Reynolds Transport Theorem (RTT)
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Chapter 2 Reynolds Transport Theorem (RTT)
2.1 The Reynolds Transport Theorem
2.2 Continuity Equation
2.3 The Linear Momentum Equation
2.4 Conservation of Energy
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2.1 The Reynolds Transport Theorem (1)
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2.1 The Reynolds Transport Theorem (2)
ρ ( ) (inflow if negative) (11-41)net out in
CS
B B B b V n dA
42)-(11 ρ dV bBCV
CV
43)-(11 )(ρρ CSCV
dAnVbdV bdt
d
dt
dB :General sys
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2.1 The Reynolds Transport Theorem (3)
Special Case 1: Steady Flow
Special Case 2: One-Dimensional Flow
44)-(11 )(ρ CS
dAnVbdt
dB :flowSteady sys
:flow ldimensiona-One
45)-(11 ρ-ρρin
exiteach for out
exiteach for CV
iiiieeeesys AVbAVbdV b
dt
d
dt
dB
46)-(11 -ρinoutCV
iieesys bmbmdV b
dt
d
dt
dB
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2.2 Continuity Equation (1) An Application: The Continuity Equation
47)-(11 )(ρρ0 CS
dAnVdVdt
d :equation Continuity
CV
48)-(11
in
iout
e mm :flowSteady
49)-(11 or AA 221 AVAVVV
:constant stream,Single
22111
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2.3 The Linear Momentum Equation (1)
..50)-(11 )V(m
dt
d
dt
Vdm amF
51)-(11 ρdVVdt
dF
sys
52)-(11 )(ρρ)(
CSCV
sys dAnVVdVVdt
d
dt
Vmd
53)-(11 (ρρ )dAnVVdVVdt
dF:General
CSCV
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2.3 The Linear Momentum Equation (2)
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2.3 The Linear Momentum Equation (3)
Special Cases
54)-(11 )(ρ dAnVVF :flowSteady CS
55)-(11 V -Vρ ie
in
i
out
e
CV
mmdVVdt
dF
:flow ldimensiona-One
56)-(11 V -V ie
in
i
out
e mmF
:flow ldimensiona-oneSteady,
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2.3 The Linear Momentum Equation (4)
57)-(11 )V-V(mF :exit)-oneinlet,-(one
flow ldimensiona-oneSteady,
12
58)-(11 )V-V(mF :coordinate x Along x1,x2,x
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2.4 Conservation of Energy
2
) V dA
V
2
system
cv cs
shaft normal shear other
dEQ W e dV e
dt t
e u gz
W W W W W
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Chapter 3 Flow Kinematics
3.1Conservation of Mass
3.2 Stream Function for Two-Dimensional
Incompressible Flow
3.3 Fluid Kinematics
3.4 Momentum Equation
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3.1 Conservation of mass• Rectangular coordinate system
x
y
z
dx
dy
dzo u
v
w xA udydz
yA dxdz
zA wdxdy
surface control thethrough
outflux mass of rateNet 0
surface control theinside
change mass of Rate
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x
y
z
dx
dy
dzo u
v
w xA udydz
x(left)A udydz
dydzdx
x
uu
dx
x
22
dxdydzx
u
xuudydz
2
1
x(right)A udydz
dydzdx
x
uu
dx
x
22
dxdydzx
u
xuudydz
2
1
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y(bottom)A dxdz
dxdzdy
y
dy
y
22
dxdydzyy
dxdz
2
1
y(top)A dxdz
dxdzdy
y
dy
y
22
dxdydzyy
dxdz
2
1
x
y
z
dx
dy
dzo u
v
w
yA dxdz
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z(back)A wdxdy
dxdydz
z
ww
dz
z
22
dxdydzz
w
zwwdxdz
2
1
z(front)A wdxdy
dxdydz
z
ww
dz
z
22
dxdydzz
w
zwwdxdy
2
1
dx
dy
dzo u
v
w
x
y
z
zA wdxdy
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Net Rate of Mass Flux
x(left)A udydz
x(right)A udydz
y(bottom)A dxdz
y(top)A dxdz
z(back)A wdxdy
z(front)A wdxdy
dxdydzx
u
xuudydz
2
1
dxdydzx
u
xuudydz
2
1
dxdydzyy
dxdz
2
1
dxdydzyy
dxdz
2
1
dxdydzz
w
zwwdxdz
2
1
dxdydzz
w
zwwdxdy
2
1
CS AdV
dxdydzz
w
zw
yyx
u
xu
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Net Rate of Mass Flux
CS AdV
dxdydzz
w
zw
yyx
u
xu
dxdydzz
w
yx
u
Rate of mass change inside the control
volume
dxdydzt
Vdt
V
dxdydzt
0
dxdydzz
w
yx
u
t
0
z
w
yx
u
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Continuity Equation
t
0
z
w
yx
u
zk
yj
xi
ˆˆˆ
Vz
w
yx
u
0
Vt
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3.2 Stream Function for Two-Dimensional
Incompressible Flow• A single mathematical function (x,y,t) to
represent the two velocity components, u(x,y,t) and (x,y,t).
• A continuous function (x,y,t) is defined such that
xyu
and
The continuity equation is satisfied exactly
0
xyyxyx
u
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Equation of Streamline
• Lines drawn in the flow field at a given instant that are tangent to the flow direction at every point in the flow field.
dyjdxijuirdV ˆˆˆˆ0
dxudyk ˆ
0 dxudy Along a streamline
0
ddyy
dxx
dxx
dyy
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Volume flow rate between streamlines
u
v V
21, yxB
11, yxA
22 , yxC1
23
x
y
Flow across AB
21
21
yy
yy dy
yudyQ
Along AB, x = constant, and dyy
d
1221
21
yy ddy
yQ
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Volume flow rate between streamlines
u
v V
21, yxB
11, yxA
22 , yxC1
23
x
y
Flow across BC,
21
21
xx
xx dx
xdxQ
Along BC, y = constant, and dxx
d
1221
12
xx ddx
xQ
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Stream Function for Flow in a Corner
Consider a two-dimensional flow field
0
w
Ay
Axu
xyu
and
yAxu
xfAxyxfdy
y
dx
dfAy
x
0
dx
df
cAxy
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Motion of a Fluid Element
Translation
x
y
z
Rotation
Angular deformationLinear deformation
3.3 Flow Kinematics
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Fluid Translation
x
y
z
Fluid particle pathAt t At t+dt
r
rdr
tzyxVVtp ,,,
dtt
Vdz
z
Vdy
y
Vdx
x
VVd pppp
t
V
dt
dz
z
V
dt
dy
y
V
dt
dx
x
V
dt
Vda pppp
p
t
V
z
Vw
y
V
x
Vu
dt
Vda p
p
t
VVV
Dt
VDa p
p
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Scalar component of fluid acceleration
t
u
z
uw
y
u
x
uu
Dt
Duaxp
tzw
yxu
Dt
Dayp
t
w
z
ww
y
w
x
wu
Dt
Dwazp
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Fluid acceleration in cylindrical coordinates
t
V
z
VV
r
VV
r
V
r
VV
Dt
DVa rr
zrr
rr
rp
2
t
V
z
VV
r
VVV
r
V
r
VV
Dt
DVa z
rrp
t
V
z
VV
V
r
V
r
VV
Dt
DVa zz
zzz
rz
zp
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Fluid Rotation
x
y
aa'
b
b'
o
x
y
t
x
t ttoa
00
limlim
txx
ttxx
xx
xt
xtxx
toa
0
lim
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t
y
t ttob
00
limlim
tyy
ututy
y
uu
aa'
b
b'
o
x
y
xx
u
yy
uu
y
u
t
ytyyu
tob
0
lim
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aa'
b
b'
o
x
y
xx
u
yy
uu
xoa
y
uob
y
u
xoboaz
2
1
2
1
Similarily, considering the rotation of pairs of perpendicular line segments in yz and xz planes, one can obtain
zy
wx
2
1
x
w
z
uy 2
1
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Fluid particle angular velocity
zyx kji ˆˆˆ
y
u
xk
x
w
z
uj
zy
wi
ˆˆˆ2
1
wuzyx
kji
V
ˆˆˆ VV
curl
2
1
2
1
V
2 Vorticity: A measure of fluid element rotation
rzrzr
V
rr
rV
rk
r
V
z
Ve
z
VV
reV
11ˆˆ1
ˆ
Vorticity in cylindrical coordinates
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Fluid Circulation, c sdV
x
y x
x
u
yy
uu
c
y
xo
b
a
yxyy
uuyx
xxu
yxy
u
x
yxz 2
A zA zC dAVdAsdV
2
Circulation around a close contour
=Total vorticity enclosed
Around the close contour oacb,
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Fluid Angular Deformation
x
y
aa'
b
b'
o
x
y
xx
u
yy
uu
dt
d
dt
d
dt
d
dy
du
x
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Fluid Linear Deformation
x
y
yy
uu
a a'
bb'
o
x
y
xx
u
t
y
dt
dy
t
yy
0
limdilation of Rate
tyy
tyy
tyy
2
1
ydt
d yy
tyy
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txx
utx
x
uutx
x
uu
2
1
t
xx
txx
00 limstrain of Rate
x
uxx
0
z
wzz
0
tx
u
x
tz
w
z
tzz
wtz
z
wwtz
z
ww
2
1
t
zz
tzz
00 limstrain of Rate
yy
uu
a a'
bb'
o
x
y
xx
u
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t
VVV
t
0
limrate dilation Volume
zyxzyxVV
Ozyyxzx
tz
wt
yt
x
u
z
w
yx
ut
VVV
t
limrate dilation Volume0
V
zyx
zyx
zyyxzx
V
VV
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Rate of shearing strain(Angular deformation)
y
u
xyxxy
zy
wzyyz
x
w
z
uxzzx
x
uVxx
2
3
2
yVyy
2
3
2
z
wVzz
2
3
2
Rate of Strain
Rate of normal strain
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3.4 Momentum Equation
z
Vw
y
V
x
VuVd
Dt
VDdmFdFdFd sB
t
V
z
Vw
y
V
x
Vu
Vd
Fd
t
VVV
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x
y
z
xxxy
xz
zy
zxzz
direction plane jiij
yy
yzyx
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Forces acting on a fluid particle
x
y
z
x-direction
2
dz
zzx
zx
2
dy
yyx
yx
2
dz
zzx
zx
2
dx
xxx
xx
2
dx
xxx
xx
2
dy
yyx
yx
SxdF dydzdx
xdydz
dx
xxx
xxxx
xx
22
+ dxdzdy
ydxdz
dy
yyx
yxyx
yx
22
dxdydz
zdxdy
dz
zzx
zxzx
zx
22
+
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Forces acting on a fluid particle
x-direction SxdF dydzdx
xdydz
dx
xxx
xxxx
xx
22
+ dxdzdy
ydxdz
dy
yyx
yxyx
yx
22
dxdydz
zdxdy
dz
zzx
zxzx
zx
22
+
SxdF dxdydzzyxzxyxxx
SxBxx dFdFdF dxdydzzyx
g zxyxxxx
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Components of Forces acting on a fluid element
x-direction
Vd
dF
Vd
dF
Vd
dF SxBxx
zyx
g zxyxxxx
Vd
dF
Vd
dF
Vd
dF SyByy
zyx
g zyyyxyy
Vd
dF
Vd
dF
Vd
dF SzBzz
zyx
g zzyzxzz
y-direction
z-direction
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Differential Momentum Equation
zyxg zxyxxx
x
zyxg zyyyxy
y
zyxg zzyzxz
z
z
uw
y
u
x
uu
t
u
z
wyx
ut
z
ww
y
w
x
wu
t
w
l element/Vo fluid on the acting Forces
naccleratio Fluid
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Momentum Equation:Vector form
Dt
VDg
zxAyxAxxAAAAxSx kji
zk
yj
xi
Vd
dF
ˆˆˆˆˆˆ
A
A
A
zzyzxz
zyyyxy
zxyxxx
k
j
i
ˆ00
0ˆ0
00ˆ
zyxzxyxxx
is treated as a momentum flux
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Stress and Strain Relation for a Newtonian Fluid
y
u
xxyyxxy
zy
wyzzyyz
x
w
z
uzxxzzx
x
uVpp xxxx
2
3
2
yVpp yyyy
2
3
2
z
wVpp zzzz
2
3
2
Newtonian fluid viscous stress rate of shearing strain
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Surface Forces
zyxVd
dF zxyxxxSx
y
u
xyxyx
x
w
z
uzxzx
x
uVpp xxxx 2
3
2
zxyxxxSx
zyp
xVd
dF
x
w
z
u
zy
u
xyV
x
up
x
3
22
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Momentum Equation:Navier-Stokes Equations
x
w
z
u
zy
u
xyV
x
u
xx
pg
Dt
Dux
3
22
y
w
zzV
yyxy
u
xy
pg
Dt
Dy
3
22
Vz
w
zy
w
zyz
u
x
w
xz
pg
Dt
Dwz
3
22
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Navier-Stokes Equations
For flow with =constant and =constant
0 V
2
2
2
2
2
2
z
u
y
u
x
u
x
pg
Dt
Dux
2
2
2
2
2
2
zyxy
pg
Dt
Dy
2
2
2
2
2
2
z
w
y
w
x
w
z
pg
Dt
Dwz
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3.5 Conservation of Energy
iij
j
Dh Dp udiv k T n
Dt Dt x
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Summary of Basic Equations
t
D
Dtg + p
Dh
Dt
Dp
Dtk T
u
x
ij
iji
j
div V
V'
div '
0