Computational fluid mechanics

1. 1
Computational Fluid
Dynamics and Heat transfer
(CFDHT)
3. Conduction Heat Transfer
Dr. M. R. Nandgaonkar
Department of Mechanical Engineering
College of Engineering, Pune

2. 2
Conduction Heat Transfer

3. Discretization Techniques :FDM,
FVM & FEM
 What is Discretization?
 Analytical Solution : Continuous
 Numerical Solution : Discrete
 Types of Discretization Technique
 FDM (Finite Difference Method): Most popular during
the early days of CFD
 FVM (Finite Volume Method): Gives greater flexibility in
handling complex geometry
 FEM (Finite Element Method): More commonly used in
Solid Mechanics rather than Fluid Mechanics
3

4. Discretization Techniques :FDM,
FVM & FEM
 Finite Difference Method
 Discrete grid points
 Finite Difference Quotient
 Finite Difference Equations and Solution
Methodology
 One-Dimensional Steady State Conduction
 One-Dimensional Unsteady State Conduction
4

5. 5

6. FDM: Discrete Grid Points
6
P
(i,j) (i+1,j)
(i-1,j)
(i,j+1)
(i-1,j+1)
(i,j-1)
(i-1,j-1)
(i,j)
P
i
j

7. FDM: Finite Difference Quotient
3
2
4
2
1
2
2
( ) sin 2
0.2, 0.02 ( ) 0.9511
( ) ( )
0.9
0.9824 %
( ) ( )
2
823 3
0.9
.176%
899 %
(
(0.775
(
E
Error
0.01 Error
rr
)
)
)
or
f x x
x x f x x
f x f x x
f x x f x x
f f
x
f
f
x


      
  
      


 
 
7
I Guess Add to
Capture Slope
1 2
3
4
x
Taylor Series Expansion
Example :
f(x)
Add to account
for Curvature

8. FDM: Finite Difference Quotient
8
Taylor Series Expansion
2 2 3 3
1, , 2 3
, , ,
2 2 3 3
1, , 2 3
, , ,
2 3
1, ,
2 3
, ,
2 3
2 3
2
i j i j
i j i j i j
i j i j
i j i j i j
i j i j
i j i j
x x
x
x x x
x x
x
x x x
x
x x x x
  
 
  
 
 
  



   
    
 
     
   
 
  
     
   
    
 
     
   
 
  
     
   
 
   
 
  
    
 
   
      
 
2
,
1, ,
,
3
O
i j
i j i j
i j
x
x
x x
 
 
 
 

 

 

 

 

 
  
 
 
 
   
Finite Difference Quotient T.E (Truncation Error)
I Order Accurate
I Order Forward Difference

9. FDM: Finite Difference Quotient
9
 
2 3 2
, 1,
2 3
, , ,
, 1,
,
3 2 5 4
1, 1,
3 5
, , ,
2 3
2
2 3 5
O
i j i j
i j i j i j
i j i j
i j
i j i j
i j i j i j
x x
x x x x
x
x x
x x
x x x x
 
  
 

 
  


 
 
    
 
    
 
   
 
     
 
   
   
     
 

 

 
  
 
 
 
   

    
 
    
 
  

     
 
   
       

 
1, 1, 2
, 2
O
i j i j
i j
x
x x
 
  


 


 

 
  
 
 
 
   
II Order Accurate
 I Order Backward Difference
 II Order Central Difference
I Order Accurate

10. FDM: Finite Difference Quotient
 Higher Order Accurate Difference Quotients
 Disadvantage : By requiring more neighboring
grid points, results in more computational time
for each grid point computation
 Advantage : Smaller number of total grid
points in a flow solution to obtain comparable
overall accuracy
 II Order accuracy has been accepted in the
vast majority of CFD applications
10

11. FDM: Finite Difference Quotient
11
 
2 4 2 6 4
1, , 1,
2 2 4 6
, , ,
2
1, , 1, 2
2 2
,
2
2, 1, , 1, 1,
2
,
2
2
4 6
2
16 30 16
1
i j i j i j
i j i j i j
i j i j i j
i j
i j i j i j i j i j
i j
x x
x x x x
x
x x
x
  
  
  

    

 
 
   
 
 
     
 
    
   
 
 
     
   
 
 
     
 
 
   

  
 
 
 
 
 
    
 


 

 
O
 
 
4
2
2
x
x
 
 
 
 

 
 
O
 II Order Central Second Difference
 IV Order Second Difference
II Order Accurate
IV Order
Accurate

12. FDM: Finite Difference Equation
and Solution Methodology
12
1, , 1, 1, 1,
,
2
2 1
3
4
5
2
6 2
2
2
1
2
0
2
2 1 0 0 0
0
1 2 1 0 0
0
At 0, ;
0
0 1 2 1 0
0
0 0 1 2 1
0 0 0 1 2
,
i j i j i j i j i j
b b
i j
b
b
T
x
x T T x L T T
T T T T T
T
x
T T
T
T
T
T T
   
  
  

    
 
   
 
     
 
   
  
 
   
 
     
 
   
 





 
 
 
  
 1-D Steady State Conduction
 G.E:
 B,C’s:
 F.D.Eq. :
 Tridiagonal Matrix:
 Solver
 Jacobi
 Gauss-Seidel
Tb1 Tb2
i=1 2 3 4 5 6 7
L

13. FDM: Finite Difference Equation
and Solution Methodology
13
 
1
1, , 1, 2
2
2
2
0
2
T.E=O( t, x )
As t 0 & x 0 T.E 0 F.D.Eq. is Consistent
At t 0;
n n n
n n
i j i j i j
T T T
T T
T T
t x
T T
t x



 
 


 

 
 
 
   



 
1-D Unsteady State Conduction
G.E:
I.C & B.C’s:
F.D.Eq. :
Reliability of num. results (within T.E.) Vs analytical results
If the difference equation is consistent
If the numerical algorithm is stable
If the B.C’s are handled properly
Tb1 Tb2
i=1 2 3 4 5 6 7
Tb1 Tb2
Tb1 Tb2
n+
1
n
n-
1

14. 14

15. 15

16. 16

17. 17

18. 18

19. 19

20. 20

21. 21

22. 22

23. 23

24. 24

25. 25

26. 26

27. 27

28. 28

29. 29

30. 30

31. 31

32. 32

33. FVM: Finite Volumes
33
i
j
E
S
Ncell j
Ncell i
P
1
1
δxe
δys
δyn
Fictitious Cells:
i=1 & Ncell i; j=1,Ncell j
j=1 & Ncell j; i=1,Ncell i
Real Cells:
i=2,Ncell i-1; j=2,Ncell j-1
W
n
Δx
Δy
N
δxw
e
s
w

34. FVM: Finite Volume Equations
and Solution Methodology
34
 2-D Steady State Conduction
 G.E:
 Finite Volume Discretization:
2 2
2
2 2
2 2
, ,
, ,
0
0 0
ˆ
. 0
0
f f
f f
V V
f f
S S S
f e w f n s
f f
S S
f e w f n s
f f
P W N P P S
E P
e w n
e w n s
T T
T
x y
TdV TdV
T T
T ndS dS dS
x y
T T
dS dS
x y
T T T T T T
T T
S S S
x x x x
   

  
 
 
 
 
   
 
    
 
    
 
 
 
 
 
 
 
 
   
  

     
 
 
  
 
 
0
s
S
 

35. FVM: Finite Volume Equations and
Solution Methodology
35
 2-D Unsteady State Conduction
 G.E:
 Finite Volume Discretization:
 Explicit Method
 Implicit Method
2 2
2 2
1
mean P
1
1
T
t
T T T
t t t
P
n n
P P
P P P
V
n n n n n n
n n n n
P W N P P S
P P E P
P e w n s
e w n s
n n
P P
P
T T
x y
T T
dV V V V
t
T T T T T T
T T T T
V S S S S
t x x x x
T T
V
t


   




 
  
 
 
  
 
   
   
     
   
   
   
 
  
 
        
 
  

 


1 1 1 1 1 1
1 1 n n n n n n
n n
P W N P P S
E P
e w n s
e w n s
T T T T T T
T T
S S S S
x x x x

   
     
 
 
  

      
 
 

36. 36
FVM: 1-D Steady State Heat Conduction
G.E:

37. 37
FVM: 1-D Steady State Heat Conduction

38. 38
FVM: 1-D Steady State Heat Conduction

39. 39
FVM: 1-D Steady State Heat Conduction

40. 40
FVM: 1-D Steady State Heat Conduction

41. 41
FVM: 1-D Steady State Heat Conduction

42. 42
FVM :1-D Unsteady State Conduction

43. 43
FVM :1-D Unsteady State Conduction

44. 44
Discrete Representation of Domain:
Grid Generation (Finite CV)

45. 45
Discrete Representation of Domain:
Grid Generation (Finite CV)

46. 46
Grid Generation

47. 47
Grid Generation

48. 48
Derivation of PDE or LAE

49. 49
Derivation of DIFFERENTIAL Equation:
Steady State Heat Conduction

50. 50
Derivation of Algebraic Equation:
Steady State Heat Conduction

51. 51
Derivation of Algebraic Equation:
Steady State Heat Conduction

52. 52
Derivation of Algebraic Equation:
Steady State Heat Conduction

53. 53
A Plate subjected to
Constant Temperature on the Boundary

54. 54
FVM: 2-D Steady State Heat Conduction

55. 55
.
FVM: 2-D Steady State Heat Conduction

56. 56
FVM: 2-D Steady State Heat Conduction

57. 57
Solution Algorithm:
2-D Steady State Conduction

58. 58
Solution Algorithm:
2-D Steady State Conduction

59. 59
Results: 2-D Steady State Conduction

60. 60
Results: 2-D Steady State Conduction

61. 61
Results: 2-D Steady State Conduction

62. 62
FVM : 2-D Steady State Conduction

63. 63
FVM : 2-D Steady State Conduction

64. 64
A Plate subjected to different Types of
Thermal Boundary Conditions

65. 65
Discrete Representation of
Boundary Conditions

66. 66
Results: 2-D Steady State Conduction

67. 67
Results: 2-D Steady State Conduction

68. 68
Results: 2-D Steady State Conduction

69. 69
Results: 2-D Steady State Conduction

70. 70
FVM : 2-D Unsteady State Conduction

71. 71
Derivation of Algebraic Equation:
Unsteady State Heat Conduction

72. 72
Derivation of Algebraic Equation:
Unsteady State Heat Conduction

73. 73
FVM : 2-D Unsteady State Conduction

74. 74
Unsteady State Heat Conduction:
Computational Stencil

75. 75
2-D Unsteady State Conduction:
Explicit Vs Implicit Approach

76. 76
Grid points for
Heat Fluxes and Temperature

77. 77
A Pseudo Code:
2-D Unsteady State Conduction

78. 78
Solution Algorithm:
2-D Unsteady State Conduction

79. 79
Solution Algorithm:
2-D Unsteady State Conduction

80. 80
FVM :1-D Unsteady State Conduction
with Heat Generation

81. 81
FVM :1-D Unsteady State Conduction
with Heat Generation

82. 82
FVM :1-D Unsteady State Conduction
with Heat Generation

83. 83
FVM :1-D Unsteady State Conduction
with Heat Generation

84. 84
FVM : 2-D Unsteady State Conduction

85. 85
FVM : 2-D Unsteady State Conduction

86. 86
FVM : 2-D Unsteady State Conduction

87. 87
Special Topics: Computational Multi-Solid
Heat Transfer (CMSHT)

88. 88
Special Topics: CMSHT –
Grid Generation

89. 89
Special Topics: CMSHT –
Interface Treatment

90. 90
Special Topics: Result-CMSHT

91. 91
Special Topics: CMSHT

92. 92
Special Topics: CMSHT

93. 93
Special Topics: Nonlinearity in
Conduction Heat Transfer

94. 94
Special Topics: Nonlinearity in
Conduction Heat Transfer

95. 95
Special Topics: Nonlinearity in
Conduction Heat Transfer

96. 96
Derivation of DIFFERENTIAL Equation:
Unsteady State Heat Conduction

97. 97
Derivation of ALGEBRAIC Equation:
Unsteady State Heat Conduction

98. 98
Derivation of Algebraic Equation:
Steady State Heat Conduction

99. 99
Derivation of Algebraic Equation:
Steady State Heat Conduction

100. 100
Discrete Representation of Domain:
Grid Generation (Finite CV)

101. 101
Grid points for
Heat Fluxes and Temperature

102. 102
A Pseudo Code:
2-D Unsteady State Conduction