Download Inertial force per unit volume in x direction

Continuity Equation
Continuity Equation
Net outflow in x direction

(u )
 (u )

dx  dy dz - u  dy dz 
dxdydz
u  x
x


Continuity Equation
net out flow in y direction,


(v)
(v)
dy dx dz - v dx dz 
dxdydz
v 
y
y


Continuity Equation
Net out flow in z direction

 ( w)
 ( w)



dz  dx dy - w dx dy 
dxdydz
 w  z
z


Net mass flow out of the element
 (u )
 
 x
(v)

y

(w ) 

z

dxdydz
Continuity Equation
Time rate of mass decrease in the element

 dxdydz
t
Net mass flow out of the element =
Time rate of mass decrease in the control volume
 ( u)

 x
( v)

y

( w) 

dxdydz
dxdydz  

z
t


(u )

t
x
(v)

y

t

(w )
kgm
 0
z
m3 sec
 . V   0
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.

 ( u )

t
x
 ( v)

y

 ( w)
kgm
 0
z
m 3 sec
If the density is constant

0
t

 (u )
x
 (v )

y

 ( w)
z

0

 (u )
x
 (u )
x
 (v )

y
 (v )

y


 ( w)
z

 0
 ( w)
0
z
This is the continuity equation for incompressible fluid
MOMENTUM EQUATION
[NAVIER STOKES EQUATION]
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.
Fx – Inertial Force
Inertial Force = Mass X Acceleration derivative.
Inertial Force in x direction = m X
Du
Dt
Du
Dt
represents instantaneous time rate of change of velocity
of the fluid element as it moves through point through
space.
Du
Dt
Is called Material derivative or
Substantial derivative or
Acceleration derivative
‘u’ is variable
Du
u

Dt
t
u
u
u
u
 (V.) u  t  u. x  v y  w z
Inertial force per unit volume in x direction
=
ma
v
m
Du
u
u
u 
 u

a  
  
 u
 v
 w

v
Dt

t

x

y

z


Inertial force / volume in x direction

Inertial force / volume in y direction

Inertial force / volume in z direction
Du

Dt
u
u
u 
 u
 t  u x  v y  w z 
Dv

Dt
v
v
v 
 v
 t  u x  v y  w z 
Dw

Dt
w
w
w 
 w
 
 u
 v
 w
x
y
z 
 t
Body force per unit volume
Body forces act directly on the volumetric mass of the fluid element.
The examples for the body forces are
Eg:
gravitational
Electric
Magnetic forces.
Body force =
g x dxdydz
mg

 g x
v
dxdydz
Body force in y direction
 g y
Body force in z direction
 g z
Pressure forces per unit volume
Pressure on left hand face of the element
Pressure on right hand face of the element
 P dydz

p

 P 
dx  dydz
x


Net pressure force in X direction is

p
p

 P  P
dx  dydz  
dxdydz
x
x


Net pressure force per unit volume in X direction
p dxdydz
p
 
 
x dxdydz
x
Net pressure force per unit volume in X direction
p
 
x
Net pressure force per unit volume in Y direction
p
 
y
Net pressure force per unit volume in Z direction
p
 
z
Net pressure force in all direction
 p
  
 x
p

y
p 


z 
Net pressure force in 3 direction
p
 
x
  P
p

y
p

z
Viscous forces
Resolving in the X
direction
Net viscous forces
 xx dx


  xx 
  xx  dydz 
dx




 zx 
   zx 
dz    zx  dxdy
z





 yx 
dy    yx  dxdz
  yx 


y



et viscous force per unit volume in X direction
 yx
  xx
 zx
Fv  


y
z
 x
 yx  zx
  xx
 


y
z
 x

 dxdydz


 a

Net viscous force per unit volume in Y direction
 yy zy
  xy



y
z
 x

 b

Net viscous force per unit volume in Z direction
 yz  zz
  xz
 


y
z
 x

 c

UNDERSTANDING VISCOUS
STRESSES
LINEAR STRESSES = ELASTIC
CONSTANT X STRAIN RATE
 xx   xx   x 2 x local average
rate of linear strain
Linear strain in X direction
e xx
u

x
Volumetric strain
e yy
v

y
 . V
e zz
w

z
or div V
 exx  e yy  e zz
u
v
w



x
y
z
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.
u
 xx  2
  div V
x
v
 yy  2
  div V
y
w
 zz  2
z
  div V
In this the second component is negligible
   2/3 
[ Effect of viscosity ‘ ’ is small in practice.
For gases a good working approximation can be obtained taking
Liquids are incompressible. div V = 0]
SHEAR STRESSES = ELASTIC
CONSTANT X STRAIN RATE
 xy   yx   x 2 x average rate angular deformatio n.
 u
v
 xy   yx   

x
 y



 u w 
 xz   zx   


x 
 z
 v w 

 yz   zy   

 z y 
 xy  xz
 xx
Fvx 


x
y
z
Fvy 
 yx
x

 yy
y
 xz

z
 zy
 zx
 zz
Fvz 


x
y
z

  u

Fvx 
2
 .V 

x  x
 y



 v u 
   w u 
 




 

y  z   x
z 
 x
 
Fvy 

x 
 
 v u 
 
w
 


2
 .V 


y 
y 
y
 x
 z
 
Fvz 

x 

 u w 
 
 
x 
y
 z






 w v 

 
z 
 y

 w v    w


  
2
 ..V

z  z 
y
 y

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.
Fx  Ff  Fg  Fv
Assembly of all the components
X direction:-
 yx
  xx
 zx 
Du
p

 
 g x  



Dt
x

x

y

z


Y direction:-
 yy
 yz 
  yx
Dv
p

 
 g y  



Dt
y

x

y

z


Z direction:-
 zy
  zx
Dw
p
 zz 

 
 g z  



Dt
z

x

y

z


X direction: u
u
u
u 
p
  u

   u
 v
 w  
 g x 
2




.V


x
y
z 
x
x  x

 t
   v u     u w 
 





 

y   x
y  z   z
x 
Y direction: v
 
 t


y
v
v
v 
p
   v
u 

 u
 v
 w  
 g y 

 
x
y
z 
y
x   x
y 

 v
   w
v 

 2
  .V  

 
z 
 y
 z   y
Z direction: w
 
 t


y
w
w
w 
p
   u
w 
 u
 v
 w
 g z 



 

x
y
z 
y
x   z
x 
  w
v 
  w

 

  .V 
 2
 
z  z  z

  y
+
 xx  xy
 xz
Du



 g x 


t
x
x
y
z
. uV  
u . V
  V. u
CONVERTING NON CONSERVATION FORM ON
N-S EQUATION TO CONSERVATION FORM
Navier-stokes equation in the X direction is given by
 xx  xy
 xz
Du


 
 g x 


t
x
x
y
z
Divergence of the product of scalar times a vector.
. uV   u.V  V.u
 V. u  . uV   u . V
 u 
u

 
 u
t
t
t
u
 u 


t
t

 u
t
Taking RHS of N-S Equation we have
 u

Du
u
u
u 
u
 u

   u
v
 w      V.u   
Dt
x
y
z 
t
 t

 t
 V.u
 u 


 u
 .  uV   u . V
t
t


  u 
 
 
 .  uV   u 
 . V 
 t
 t



Du
 u 
 
 
 .  uV   u 0
Dt
 t

since

u 
v 
w 
CONTINUITY 



t
x
y
z
Is equal to zero
 xx  xy  xz
u 
p
 .  uV   
 g x 


t
x
x
y
z
 yx  yy  yz
v
p
 .  uV     g y 


t
y
x
y
z
zx zy  zz
w 
p
 .  uV   
 g z 


t
z
x
y
z
CONSERVATION FORM:-
 
 xx
u   u 2
uw  uw 
p



 
 g x 
t
x
y
z
x
x

 xy
y

 xz
z
 
 yx
v 
uv 
 v 2
vw 
p



 
 g y 
t
x
y
z
x
x

 yy

y
 yz
z
 

w  uw  vw   w 2
p



 
 g z  zx
t
x
y
z
z
x

 zy
y

 zz
z
SIMPLICATION OF NAVIER STOKES EQUATION
 
u 
 u 2
uv 
uw 
P



 
t
x
y
z
x
 
u 
  v u 
  u w 
 
 



  g x
 .V  2  

x 
x  y  x y  z  z
x 
If
 is constant
 
 u 
 u 2
 uw 
 uw 
P



 

t
x
y
z
x

x




2
 3

 u
v
w 
u 
2v
u






2





2
y
z 
x 
yx

y
 x

 2u
 2w
 

 g x
zx
z 2
 
 u 
 u 2
 uw 
 uw 
P



 

t
x
y
z
x

x




2
 3

 u
v
w 
u 
2v
u






2





y
z 
x 
yx
y 2
 x

 2u
 2w
 

 g x
2

z

x
z
 
 u 
 u 2
 uw 
 uw 
P



 
t
x
y
z
x
 2
 2u
 2v
 2w
 2u
 2v
2
2





 2

3 x 2
3 xy
3 xz
2
yx
x
 2u
 2u
 2w
 

 
 g x
zx
y 2
z 2
 
  2u
 u 
 u2
 uv 
 uw 
P
1



 
1

3  x
t
x
y
z
x

 



 2u
 2u 1
 2v
2w

 
1 
3 yx
3 zx
y 2
z 2
 
u 
 u2
uv 
uw 
P
 2u
 2u



 


t
x
y
z
x
x 2
y 2
 u 1
 v
 w
1 
 
 1 
3 x x
3 x y
3 x z
 
u 
 u2
uv 
uw 
P
 2u
 2u



 


t
x
y
z
x
x 2
y 2
 2u 1



3
z 2
 u v w 
 x  y  z 


 
u 
 u2
uv 
uw 
P



 
t
x
y
z
x
 
  2u
 2u
 2u  1



  3  . V 
2
2
2
y
z 


 x



For Incompressible flow . V  0


 
u 
 u2
uv 
uw 
P



 
t
x
y
z
x
  2u
 2u
 2u 




2
2
2
y
z 
 x
Energy Equation
Energy 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
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
 
Fv . V


 u. yx
 up 
u xx 
u zx 
 



 dxdydz
x
y
z 
 x
for Y direction
 
Fv . V




 v yx
 v yy
 vp 
 


  u yz

y

x

y



 dxdydz


for Z direction
 

 w zy 
 wp 
w zx 


 w xx  dxdydz
Fv . V  

z

x

y


Body force is given by
 g.V dxdydz
 g x u  g y v  g z w
Total work done



C 





 up vp wp  
u xx   u yx
u zx 




 x  y  z   x

y
z





 dxdydz
 v xy
 v yy
 v zy
w xz   w yz
w zz  






x
y
z
x
y
z 

 
 



 f .V dxdydz
Net Heat flux into element = Volumetric Heating + Heat transfer across surface.
Volumetric heating
.
  q dxdydz
Heat transfer in X direction
Heating of fluid element


.




 qx
 q x   q x 
dx  dydz

x






.
 .
.
 q

q

q
y
x 
z
  

y
z
 x




 dxdydz



.
q

dxdydz
=
x
dQ = B =
dQ = B
D

Dt


 .
 q 



.
 .
.
 q
 q y  q z 
x
 dxdydz



y
z 
 x




 
  T    T 
  T 


  q 
k

k


 dxdydz

k



x  x  y  y 
z  z 

2



 e  V    q    k T     k T   
 y 


2

x

x

y
z






up 
vp 
wp 


x
y
z
 T 

k
 z 
D

Dt

2



 e  V    q    k T     k T   
 y 


2

x

x

y
z






 T 

k
 z 
up 
vp 
wp 


x
y
z
 u xx

x

 v xy

x



 u yx
y






 v yy

y
 w yz
  w xz  
y
 u zx
z


 v zy

z


 w zz
z

 f.V
Energy Equation
Nonconservation form
D

Dt


2

V
  T 
  T 
  T 

e 


 q 
k


k



k





2 
x  x  y  y 
z  z 

up 
vp 
 wp 


x
y
z


 u xx   u yx
 u zx 



x
y
z







 v xy
 v yy
 v zy



x
y
z

 w yz
 w xz 
 w zz 



 f.V
x
y
z
Non conservation:-

D  
V 2 
  . 
 e 


Dt  
2 




2 

V
 V   q 
e 

2  


  T    T 
  T  up 
vp 
wp 
 
 k
 
k




k

x  x  y  y 
z  z 
x
y
z


 u xx   u yx
 u zx 



x
y
z







 v xy
 v yy
 v zy



x
y
z

 w yz
 w xz 
 w zz 



 f.V
x
y
z
Conservation:-

c p T
x
 
 
 

 cpT
 cpT 
  cpT
 2T
  u
u
 w
  q  k
x
y
z 
x 2

 2T
 2T
 k
k
   p .V 
2
2
y
z

c p T
x

 uT 
 vT 
 wT 
 2T
 c p 


  q  k 2
y
z 
x
 x
 2T
 2T
 k
k
   p .V 
2
2
y
z
Momentum Equation
Non conservation form
X direction
Du
 p   xx   y x   z x





  fx
Dt
x
x
y
z
Y direction
Dv
 p   x y   yy   z y





  fy
Dt
y
x
y
z
Z direction

Dw
 p x z  y z  z z




  fz
Dt
z
x
y
z
Momentum Equation
Conservation form
X direction

Du
 p   xx   y x   z x
  .( u V )  



  fx
Dt
x
x
y
z
Y direction

Dv
 p   x y   yy   z y
  .(  v V )  



  fy
Dt
y
x
y
z
Z direction

Dw
 p x z  y z  z z
  . (  wV )  



  fz
Dt
z
x
y
z
Energy Equation
Non conservation form

2
D
V
e
Dt
2
(
)   q  x ( k Tx )  y ( k  Ty )  z ( k Tz )   (upx )  (vpy )   (wpz )

 (u xx)  (u yx) ( u  z x)  (v x y)  (v y y) ( v z y)   z x






x
y
z
x
y
z
z
 ( w x z)  ( w y z) ( w z z)



  f .V
x
y
z
Energy equation
Conservation form
2

V
 e
t
2
[ (
2
V
 T
 T
 T

 .  e 
V  q 
k

k

k
2
x x y y z z
 (up)  (vp)  (wp)  (u xx)  (uyx) ( u  z x)






x
y
z
x
y
z
 (v x y)  (vy y) ( v  z y)   z x  (wx z)  ( w y z)






x
y
z
z
x
y
(w z z)

  f .V
z
)] [ (
) ]
( )
( )
( )
FORMS OF THE GOVERNING EQUATIONS
PARTICULARLY SUITED FOR CFD
V
 uV
 vV
 wV
 eV
Mass flux
Flux of x component of momentum
Flux of y component of momentum
Flux of z component of momentum
Flux of Internal energy
2
V
 e
V
2
(
)
Flux of total energy
Solution vectar

 

 u

U  v
 w

V2

  e  2
(
)











Variation in x direction



 u



  u 2  p   xx


F    v u   xy

  wu  

xz


2
T
V



e

u

p
u

k

u


v


w

xx
xy

xz 
2

x


(
)
Variation in y direction

 v

  u v   yx

G    v 2  p   yy
  wv  
yz

T
V2


e

v

p
v

k
 u  yx  v  y y  w y z

2
y

(
)











Variation in z direction

 w

  u w  z x

H    v w   zxy
  w2  p  
zxz

V2
T


e

w

p
w

k
 u  xx
z  v  z y  wz z

2
z

(
)











Source vectar
0

  fx

J    fy
 f
 z
  u f  v f  w f   q
x

y
z
(
)









Time marching
U
 F G  H



 J
t
x y z
Types of time marching
1. Implicite time marching
2. Explicite time marching
Explicit FDM
Implicit FDM
Crank-Nicolson FDM
Space marching
F
G  H


 J
x
y z