continuity equation
DESCRIPTION
Continuity Equation. Continuity Equation. Net outflow in x direction. Continuity Equation. net out flow in y direction,. Continuity Equation. Net out flow in z direction. Net mass flow out of the element. Continuity Equation. Time rate of mass decrease in the element. - PowerPoint PPT PresentationTRANSCRIPT
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Continuity Equation
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Continuity Equation
dxdydz x
)u( dz dy u - dy dz dx x
)u(u
Net outflow in x direction
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Continuity Equation
net out flow in y direction,
dxdydz y
)v( dz dx v - dx dz dy y
)v(v
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Continuity Equation
Net out flow in z direction dxdydz
zwdydxw dx dydz
zww )( - )(
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Net mass flow out of the element
dxdydz z
)w( y
)v( x
)u(
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Time rate of mass decrease in the element
dxdydzt
-
Net mass flow out of the element =
Time rate of mass decrease in the control volume
dxdydzt
dxdydz zw
yv
xu )( )( )(
Continuity Equation
![Page 8: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/8.jpg)
sec3m
kgm 0 z
)w( y
)v( x
)u( t
0 . V
t
The above equation is a partial differential equation form of the continuity equation. Since the element is fixed in space, this form of equation is called conservation form.
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0 )( )( )(
0
sec 0 )( )( )( 3
zw
yv
xu
t
mkgm
zw
yv
xu
t
If the density is constant
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0 )( )( )(
0 )( )( )(
zw
yv
xu
zw
yv
xu
This is the continuity equation for incompressible fluid
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Momentum equation is derived from the fundamental physical principle of Newton second law
Fx = m a = Fg + Fp + Fv
Fg is the gravity force Fp is the pressure force Fv is the viscous force Since force is a vectar, all these forces will have three components.
First we will go one component by next component than we will assemble all the components to get full Navier – Stokes Equation.
MOMENTUM EQUATION
[NAVIER STOKES EQUATION]
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Fx – Inertial Force
Inertial Force = Mass X Acceleration derivative. Inertial Force in x direction = m X
represents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.
DtDu
DtDu
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u ).V( tu
DtDu
zu w
yu v
xu .u
tu
vma
zuw
yuv
xuu
tu
DtDua
vm
DtDu
Inertial force per unit volume in x direction =
Is called Material derivative or
Substantial derivative or
Acceleration derivative
‘u’ is variable
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Inertial force / volume in y direction
DtDv
zv w
yv v
xv u
tv
Inertial force / volume in z direction Dt
Dw
zw w
yw v
xw u
tw
DtDuInertial force / volume in x direction
zuw
yuv
xuu
tu
![Page 16: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/16.jpg)
Body forces act directly on the volumetric mass of the fluid element. The examples for the body forces are
Eg: gravitationalElectricMagnetic forces.
Body force =
Body force in y direction
Body force in z direction
xx g
dxdydzdxdydz
vmg
g
yg
zg
Body force per unit volume
![Page 17: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/17.jpg)
Pressure on left hand face of the element
Pressure on right hand face of the element
Net pressure force in X direction is
Net pressure force per unit volume in X direction
dydzP
dydzdxxpP
dxdydzxpdydzdx
xpPP
xp
dxdydzdxdydz
xp
Pressure forces per unit volume
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Net pressure force per unit volume in X direction
Net pressure force per unit volume in Y direction
Net pressure force per unit volume in Z direction
Net pressure force in all direction
Net pressure force in 3 direction
xp
yp
zp
zp
yp
xp
zp
yp
xp
P
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Viscous forces
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Resolving in the X direction Net viscous forces
dxdy dz z
dxdz dy y
dydz dx
dx
zxzx
zx
yxyx
yxxxxx
xx
![Page 21: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/21.jpg)
dxdydz z
y
x
F zxyxxxv
a z
y
x
zxyxxx
b z
y
x
zyyyxy
c
zyxzzyzxz
Net viscous force per unit volume in X direction
Net viscous force per unit volume in Y direction
Net viscous force per unit volume in Z direction
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UNDERSTANDING VISCOUS STRESSES
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LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE
strainlinear of rate average local x 2 x xxxx
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Linear strain in X direction
xuexx
yveyy
zwezz
zzyyxxe e e
zw
yv
xu
V divor V . Volumetric strain
![Page 33: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/33.jpg)
Three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality. 1. dynamic viscosity.
2. relate stresses to volumetric deformation.
V divxu2xx
V divyv2yy
V divzw2zz
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[ Effect of viscosity ‘ ’ is small in practice.
For gases a good working approximation can be obtained taking
Liquids are incompressible. div V = 0]
3/2
In this the second component is negligible
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SHEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE
n.deformatioangular rate average x 2 x yxxy
xv
yu
yxxy
xw
zu
zxxz
yw
zv
zyyz
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z
y
x
F xzxyxxvx
z
y
x
F xzyyyxvy
z
y
x
F zzzyzxvz
![Page 37: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/37.jpg)
zu
xw
zyu
xv
xFvx
y .V
xu 2
.V 2
zv
yw
zyw
yyu
xv
xFvy
.V. 2
yw
zzv
yw
yxw
zu
xFvz
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Having derived equations for inertial force per unit volume, pressure force per unit volume body force per unit volume, and viscous force per unit volume now it is time to assemble together the subcomponents.
vgfx F F F F
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Assembly of all the components
z
y
x
g xp
DtDu zxyxxx
x
z
y
x
g yp
DtDv yzyyyx
y
zyx
gzp
DtDw zzzyzx
z
X direction:-
Y direction:-
Z direction:-
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xw
zu
z
yu
xv
y
.V xu 2
x g
xp
zu w
yu v
xu u
tu x
X direction:-
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zv
yw
z .V
yv 2
y
yu
xv
x g
yp
zv w
yv v
xv u
tv y
Y direction:-
![Page 42: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/42.jpg)
.V zw 2
z
zv
yw
y
xw
zu
x g
yp
zw w
yw v
xw u
tw z
Z direction:-
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z
y
x g
x
tDu xzxyxx
x
+
. uV u V. Vu .
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CONVERTING NON CONSERVATION FORM ONN-S EQUATION TO CONSERVATION FORM
Navier-stokes equation in the X direction is given by
zxz
yxy
xxx xg
x
tDu
uV. . . VuuV
VuuVu . . V.
Divergence of the product of scalar times a vector.
![Page 45: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/45.jpg)
t
u tu
tu
t
u tu
tu
Taking RHS of N-S Equation we have
u.V
tu u.V
tu
zu w
yu v
xu u
tu
DtDu
![Page 46: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/46.jpg)
V . u uV . t
u tu
V . t
u uV . tu
0 u uV . tu
DtDu
CONTINUITY zw
yv
xu
t
since
Is equal to zero
![Page 47: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/47.jpg)
zxz
yxy
xxx xg
xp uV .
tu
zyz
yyy
xyx yg
yp uV .
tv
zzz
yzy
xzx zg
zp uV .
tw
![Page 48: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/48.jpg)
CONSERVATION FORM:-
zxz
yxy
xxx xg
xp
zuw
yuw
x
2u tu
zyz
yyy
xyx yg
xp
zvw
y
2v xuv
tv
zzz
yzy
xzx zg
zp
z
2w yvw
xuw
tw
![Page 49: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/49.jpg)
xg xw
zu
z
yu
xv
y
xu 2 .V
x
xP
zuw
yuv
x
2u tu
SIMPLICATION OF NAVIER STOKES EQUATION
xg xzw2
2zu2
2y
u xyv2
xu 2
zw
yv
xu 3
2 x
xP
zuw
yuw
x
2u tu
If is constant
![Page 50: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/50.jpg)
xg xzw2
2zu2
2yu
xyv2
xu 2
zw
yv
xu 3
2 x
xP
zuw
yuw
x
2u tu
xg xzw2
2zu2
2yu2
xyv2
2xu2
2 zx
w2 3
2 yxv2
32
2xu2
32
xP
zuw
yuw
x
2u tu
![Page 51: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/51.jpg)
xzw2
31
xyv2
31
2zu2
2yu2
xu2
311
xP
zuw
yuv
x
2u tu
zw
x 3
1 yv
x 3
1 xu
x 3
1
2yu2
2xu2
xP
zuw
yuv
x
2u tu
zw
yv
xu 3
1 2zu2
2yu2
2xu2
xP
zuw
yuv
x
2u tu
![Page 52: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/52.jpg)
V. 31
2zu2
2yu2
2xu2
xP
zuw
yuv
x
2u tu
2zu2
2yu2
2xu2
xP
zuw
yuv
x
2u tu
For Incompressible flow
0 V .
![Page 53: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/53.jpg)
Energy EquationEnergy is not a vector
So we will be having only one energy equation which includes the energy in all the direction.
The rate of Energy = Force X velocity
Energy equation can be got by multiplying the momentum equation with the corresponding component of velocity
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dQ = dE + dW dE = dQ - dW = dQ + dW [Work done is negative] because work is done on the system.
Work done is given by dot product of viscous force and velocity vector.
for Xdirection
V.vF
dxdydz
zzxu
yyx.u
xxxu
xup
![Page 55: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/55.jpg)
for Y direction
V.vF
dxdydz yzu yyyv
xyxv
yvp
for Z direction
dxdydz xxw
yzyw
xzxw
zwp
V.vF
![Page 56: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/56.jpg)
Body force is given by dxdydz V.g
wzg vyg uxg
![Page 57: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/57.jpg)
![Page 58: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/58.jpg)
Total work done
dxdydz V.f
dxdydz
zzzw
yyzw
xxzw
zzyv
yyyv
xxyv
zzxu
yyxu
xxxu
z
wp
yvp
xup
C
Net Heat flux into element = Volumetric Heating + Heat transfer across surface.
Volumetric heating dxdydz .q
![Page 59: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/59.jpg)
Heat transfer in X direction = dydz dx
x
.xq
xq x q
dxdydz x
.q
dxdydz z
.zq
y
.yq
x
.xq
Heating of fluid element
![Page 60: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/60.jpg)
dQ = B = dxdydz z
.zq
y
.yq
x
.xq
.q
dQ = B dxdydz zTk
z
yTk
y
xTk
x q
z
wp yvp
xup
zTk
z
yTk
y
xTk
x q
2
2V e DtD
![Page 61: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/61.jpg)
f.V zzzw
yyzw
xzw
zzyv
yyyv
xxyv
zzxu
yyxu
xxxu
z
wp yvp
xup
zTk
z
yTk
y
xTk
x q
2
2V e DtD
![Page 62: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/62.jpg)
Energy EquationNonconservation form
z
wp yvp
xup
zTk
z
yTk
y
xTk
x q
2
2V e DtD
f.V
zzzw
yyzw
xxzw
zzyv
yyyv
xxyv
zzxu
yyxu
xxxu
![Page 63: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/63.jpg)
Non conservation:-
z
wp yvp
xup
zTk
z
yTk
y
xTk
x
q V 2
2V e . 2
2V e DtD
f.V
zzzw
yyzw
xxzw
zzyv
yyyv
xxyv
zzxu
yyxu
xxxu
![Page 64: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/64.jpg)
Conservation:-
.V p 2zT2
k 2yT2
k
2xT2
k q z
Tpc w
y
Tpcu
x
Tpcu
x
Tpc
.V p 2zT2
k 2yT2
k
2xT2
k q z
wT y
vT x
uT pc x
Tpc
![Page 65: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/65.jpg)
xfzxz
yxy
xxx
xp
tDuD
fyz
yzyyy
xyx
yp
tDvD
fzz
zzy
zyx
zxzp
tDwD
Momentum Equation Non conservation form
X direction
Y direction
Z direction
![Page 66: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/66.jpg)
Momentum Equation
Conservation form
X direction
Y direction
Z direction
xfzxz
yxy
xxx
xpVu
tDuD
)(.
fyz
yzyyy
xyx
ypVv
tDvD
).(
fzz
zzy
zyx
zxzp
VwtDwD
)(.
![Page 67: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/67.jpg)
Vfz
zzwy
zywx
zxwz
xzz
yzvy
yyvx
yxvz
xzuy
xyuxxxu
zwp
yvp
xup
zTk
zyTk
yxTk
xqVe
tDD
.)()()(
)()()()()()(
)()()(2
2)()()()(
Energy Equation
Non conservation form
![Page 68: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/68.jpg)
Vfz
zzwy
zywx
zxwzxz
zyzv
yyyv
xyxv
zxzu
yxyu
xxxu
zwp
yvp
xup
zT
kzy
Tk
yxT
kx
qVeVet
V
.)(
)()()()()(
)()()()()()(2
2.
2
2)()()(])([])([
Energy equation
Conservation form
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FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD
energytotalofFluxVVe
energyInternalofFluxVemomentumofcomponentzofFluxVwmomentumofcomponentyofFluxVvmomentumofcomponentxofFluxVu
fluxMassV
)(2
2
![Page 70: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/70.jpg)
Solution vectar
)(2
2Ve
wvu
U
![Page 71: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/71.jpg)
Variation in x direction
xzwxyvxxuxTkupuVe
xzuwxyuv
xxpuu
F
)(2
2
2
![Page 72: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/72.jpg)
Variation in y direction
zywyyvxyuyTkvpvVe
zyvwyypv
yxvuv
G
)(2
2
2
![Page 73: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/73.jpg)
Variation in z direction
zzwyzvxxzuzTkwpwVe
xzzpwxyzwv
xzwuw
H
)(2
22
![Page 74: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/74.jpg)
Source vectar
qzfwyfvxfuzfyfxf
J
)(
0
![Page 75: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/75.jpg)
Time marching
JzH
yG
xF
tU
Types of time marching
1. Implicite time marching
2. Explicite time marching
![Page 76: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/76.jpg)
Explicit FDM
![Page 77: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/77.jpg)
Implicit FDM
![Page 78: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/78.jpg)
Crank-Nicolson FDM
![Page 79: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/79.jpg)
Space marching
JzH
yG
xF
![Page 80: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/80.jpg)
![Page 81: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/81.jpg)
![Page 82: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/82.jpg)
![Page 83: Continuity Equation](https://reader031.vdocument.in/reader031/viewer/2022012303/56815be3550346895dc9d32a/html5/thumbnails/83.jpg)