P. R. Naren
School of Chemical & Biotechnology
SASTRA Deemed to be University
Thanjavur 613401
E-mail:
[email protected]
at
One Week
National Workshop on Advances in Computational Mechanics
School of Mechanical Engineering
SASTRA Deemed to be University
Thanjavur 613401
Outline
•
Conservation equations and control volume
–
Eulerian and Lagrangian framework
–
Integral form of conservation equation
•
FVM approach
–
Steady state diffusion equation in 1D
–
Convective term
Governing Equations
•
Conservation of mass
•
Conservation of momentum
•
Conservation of energy
•
Concept of CV
*
m
lim
Framework
•
Eulerian
–
Fixed reference
•
Infinitesimally small
control volume
–
Differential form
•
Lagrangian
–
Moving reference
•
Finite control volume
–
Integral form
–
Gross behaviour
Advection
i
V
N
T
P
i
V
N
T
P
i
u
N
T
P
i i
u
u
N
N
T
T
P
x
P
i
m
u A
P m u
Q mC T
N
Pi
1
u
C T
x
P
m
P
u
Q
C T
N
Generic Transport Equation
•
Transport equation for a quantity
f
Accumulation + Net outflow = Net Diffusion + Net source
div( V )
div
grad
S
t
f f
f
f
Equation
Specific quantity
f
per unit mass)
S
fMass balance
1
0
0
Momentum
balance
u
m
g
Energy balance
C
pT
k
-D
H
RUA
D
T
Species balance i
x
iD
r
iP
Mass Balance
•
Mass
d i v (
)
0
t
U
u
v
w
0
t
x
y
z
Momentum Balance
•
Navier Stokes
M
D
div
div
S
Dt
U
p
Mx
u
p
div( u )
div
grad u S
t
x
-
m
U
Navier
Numerical Techniques
•
Finite Difference
•
Finite Element
•
Finite Volume
Illustration: 1D heat Conduction
Steady state 1D heat conduction in a solid rod with ends kept at fixed temp
Model Formulation
•
Axisymmetric
–
q
–
L >> D
–
Radial variation ignored
–
No heat loss thro’ surrounding
•
Governing equation
•
BC: z = 0 T = T
A; z = L T = T
Bwith constant thermo-physical properties
T
AT
BL
T
AT
Bd
d T
0
d z
d z
axisymmetric
Solution to Heat Conduction in Rod
•
Analytical Solution
–
Continuous
•
Numerical solution
–
Finite Difference
–
Solution at discrete points
–
Affected by
D
z
T
AT
BT
AT
BT
AT
BFinite Diff
Grids
Soln
B
A
A
T
T
T
T
z
L
-
i 1
i
i 1
1
A
T
2 T
T
0
i
2 to N 1
T
T
-
Issues with Finite Difference
•
Solution discrete vs. governing equation continuous
•
Treatment of local source terms
•
Discontinuities in solution
Integral Form of Conservation Equation
S
x
dx
d
dx
d
u
dx
d
f
f
f
f
f
f
f
div
u
div
grad
S
t
f
u
div
grad
f
S
fdiv
xd
u d
d
d
d
S d
d x
d x
d x
fD D D
f
f
Gau
b
xd
u S. d x
d
d
S. d x
S
S.d x
d x
d x
d x
ff
f
For steady state flow
For 1D flow
Integral Form of Momentum Equation
FVM Approach for Heat Conduction
•
Computational domain for
heat conduction in rod
–
Temperature at node is
average of CV around the
node
–
Boundary nodes
–
Internal nodes
T
A
T
B
T
AT
B
S
xdx
d
dx
d
u
dx
d
f
f
f
dV
S
dV
dx
d
dx
d
dV
u
dx
d
V x V
V
D f D
D
f
f
P
E
W
w
e
s
n
S
N
Finite Volume Formulation
f
f
f
f
S
grad
div
u
div
t
f
u
div
grad
f
S
fSome Mathematics !!
•
Taylor Series
P
E
W
w
e
s
n
S
N
Finite Volume Formulation . . .
dV
S
dV
dx
d
dx
d
dV
u
dx
d
V x V
V
D f D
D
f
f
e
w
V
u
u
dV
u
dx
d
f
f
-
f
D
u
u
2
u
u
2
u
u
W E
P W
P E
f
-f
f
f
-f
f
P
E
Finite Volume Formulation . . .
Finite Volume Formulation
P
E
W
w
e
s
n
S
N
f
f
f
f
a
a
s
a
P
P
W
W
E
E
f
f
f
f
f
f
a
a
a
a
s
a
P
P
W
W
E
E
N
N
S
S
dV
S
dV
dx
d
dx
d
dV
u
dx
d
V
x
V
V
D
f
D
D
f
e
w
V
p
p
dV
x
p
-
-
D
2
p
p
2
p
p
2
p
p
W
E
W
P
P
E
-
-
P
E
W
w
e
s
n
S
N
Difficulty in pressure term discretization
Checker board
Solution?
Suhas V Patankar
Summary
•
Finite volume
–
Solution represent average over the region NOT a point
solution
•
Finite volume approach applied to Integral form of
Conservation equation
•
Discretization of diffusion and advective terms
–
How to get values of advected quantities at face
f
e
f
w
?
Other alternatives
Resources
•
Chung T. J. (2002)
Computational Fluid Dynamics
. Cambridge University Press
•
Date A. W. (2005).
Introduction to Computational Fluid Dynamics
. Cambridge University Press
•
Fox, R. O. (2003)
Computational Models for Turbulent Reacting Flows
. Cambridge University
Press
•
Hoffmann K. A. and Chiang S. T. (2000).
Computational Fluid Dynamics
Vol1, 2 and 3.
Engineering Education System, Kansas, USA.
•
John F. W., Anderson, J.D. (1996)
Computational Fluid Dynamics: An Introduction
Springer
•
Patankar, S. (1980)
Numerical Heat Transfer and Fluid Flow
. Taylor and Francis
•
Ranade, V.V. (2002).
Computational Flow Modeling for Chemical Reactor Engineering
,
Academic Press, New York.
Web Resources
•
http://www.cfd-online.com
•
http://en.wikipedia.org/wiki/Computational_fluid_dynamics
•
http://www.cfdreview.com/
•
https://confluence.cornell.edu/display/SIMULATION/FLUENT
+Learning+Modules
•
http://weblab.open.ac.uk/firstflight/forces/#
•
NPTEL
Gratitude
•
Dr. Vivek V. Ranade – My Mentor Guide and Teacher
–
iFMg - Research group at NCL, Pune
•
Audience
THANK YOU
A person who never made a
mistake never tried anything new
- Albert Einstein
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