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Progress Through Quality Education
P. R. NarenSchool of Chemical & Biotechnology
SASTRA University
Thanjavur 613401
E-mail: [email protected]
at
Faculty Development Program on Computational Fluid DynamicsSchool of Mechanical Engineering
SASTRA University
Thanjavur 613401
03 June 2015
Finite Volume Method
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
• Issues with collocated grid
3-Jun-15 Finite volume method 2
Governing Equations
• Conservation of mass
• Conservation of momentum
• Conservation of energy
• Concept of CV
*
mlim
δ∀ → ∀
δρ =δ∀
Transport Equations
3-Jun-15 Finite volume method 3
Framework
• Eulerian – Fixed reference
• Infinitesimally small control volume – Differential form– No discontinuity
• Lagrangian– Moving reference
• Finite control volume– Integral form– Gross behaviour
Samimy et al., 2003
3-Jun-15 Finite volume method 4
Advection
i
V
N
T
P
ρ
i
V
N
T
P
ρ
i
u
N
T
P
ρ
i i
u u
N N
T T
P
+ δρ + δρ
+ δ+ δ
xδP
i
m u A
P mu
Q mC T
N
= ρ=
=
ɺ
ɺ ɺ
ɺ ɺ P
i
1
u
C T
x
P
ii i
m
Pu
QC T
NC x
= ρδ∀
= ρδ∀
= ρδ∀
= = ρδ∀
ɺ
ɺ
ɺ
Unit Mass Unit VolumeBalance
3-Jun-15 Finite volume method 5
Generic Transport Equation
• Transport equation for a quantity φ
Accumulation + Net outflow = Net Diffusion + Net source
( ) ( )div( V ) div grad St φ
∂ ρφ+ ρ φ = Γ φ +
∂
��
Equation Specific quantity φ ( per unit mass)
Γ Sφ
Mass balance 1 0 0
Momentumbalance
u µ ρg
Energy balance CpT k −∆HR UA∆T
Species balance i xi D ri
P∇
3-Jun-15 Finite volume method 6
Mass Balance
• Mass
d i v ( ) 0t
∂ ρ + ρ =∂
U
( ) ( ) ( )u v w0
t x y z
∂ ρ ∂ ρ ∂ ρ∂ ρ + + + =∂ ∂ ∂ ∂
3-Jun-15 Finite volume method 7
Momentum Balance
• Navier Stokes
M
Ddiv div S
Dtρ = + +U
p τ
( ) ( ) Mx
u pdiv( u ) div grad u S
t x
∂ ρ ∂+ ρ = − + µ +∂ ∂
U
Navier
Stokes
3-Jun-15 Finite volume method 8
Integral Form
( ) xSdx
d
dx
du
dx
dφ+
φΓ=ρφ
( ) ( ) ( ) φ+φΓ=ρφ+∂ρφ∂
Sgraddivudivt
( ) ( ) φ+φΓ=ρφ Sgraddivudiv
( ) x
d d du d d S d
d x d x d x φ∆ ∀ ∆ ∀ ∆ ∀
φ ρ φ ∀ = Γ ∀ + ∀
∫ ∫ ∫
Gauβ
( ) x
x x x
d d du S. d x S. d x S S.d x
d x d x d x φ∆ ∆ ∆
φ ρ φ = Γ +
∫ ∫ ∫
3-Jun-15 Finite volume method 9
Geometry
3-Jun-15 Finite volume method 10
3-Jun-15 Finite volume method 11
Numerical Techniques
• Finite Difference
• Finite Element
• Finite Volume
Taylor
3-Jun-15 Finite volume method 12
Finite Volume
( ) ( ) ( )
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂+
∂ρ∂
y
u
yx
u
xx
p
y
vu
x
uu
t
u
( ) ( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uudt
t
u
VVVVVt∫∫∫∫∫∫
∆∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂+
∂ρ∂
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
3-Jun-15 Finite volume method 13
Some Mathematics !!
• Taylor Series
( ) ( ) ( ) ( ) ...xf!2
hxhfxfhxf ''
2' +++=+
( ) ( ) ( ) ( ) ...xf!2
hxhfxfhxf ''
2' −+−=−
( ) ( ) ( ) ( ) ...xf!2
h2xf2hxfhxf ''
2
++=−++( ) ( ) ( ) ( )...xf!3
h2xhf2hxfhxf "'
3' +=−−+
( ) ( ) ( ) ( )...xf!3
h
h
1
h2
hxfhxfxf "'
3' −−−+= ( ) ( ) ( ) ( )
...h
hxfxf2hxfxf
2'' −−+−+=
3-Jun-15 Finite volume method 14
( ) xSdx
d
dx
du
dx
dφ+
φΓ=ρφ
( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ
Finite Volume Formulation
( ) ( ) ( ) φ+φΓ=ρφ+∂ρφ∂
Sgraddivudivt
( ) ( ) φ+φΓ=ρφ Sgraddivudiv
3-Jun-15 15
P EW ew
s
n
S
N
Finite Volume Formulation . . .
( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ
( ) ( ) ( )we
V
uudVudx
d ρφ−ρφ≈ρφ∫∆
( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ
( ) ( )[ ] ( ) ( )[ ]
( ) ( )[ ]2
uu
2
uu
2
uu
WE
PWPE
ρφ−ρφ=
ρφ+ρφ−ρφ+ρφ=
3-Jun-15 16
P EW ew
s
n
S
N
Finite Volume Formulation . . .
( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ
weV dx
d
dx
ddV
dx
d
dx
d
φΓ−
φΓ≈
φΓ∫∆
( ) ( )[ ] ( ) ( )[ ]PW
WP
EP
PE
xx ∆φΓ−φΓ−
∆φΓ−φΓ=
3-Jun-15 17
Finite Volume Formulation
P EW ew
s
n
S
N
φ+φ+φ=φ saaa EEWWPP
φ+φ+φ+φ+φ=φ saaaaa SSNNEEWWPP
( ) dVSdVdx
d
dx
ddVu
dx
d
V
x
VV∫∫∫
∆φ
∆∆
+
φΓ=ρφ
3-Jun-15 Finite volume method 18
( ) ( )we
V
ppdVx
p −≈∂∂−∫
∆
[ ] [ ]
[ ]2
pp
2
pp
2
pp
WE
WPPE
−=
+−+=P EW ew
s
n
S
N
Difficulty in pressure term discretization
Checker board Solution?
3-Jun-15 Finite volume method 19
Suhas V Patankar
Professor Emeritus Univ. of Minnesota
Summary
• Finite volume approach applied to Integral form of Conservation equation
• Discretization of diffusion and advective terms
3-Jun-15 Finite volume method 20
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.
• Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to computational Fluid Dynamics
- The Finite Volume Method. Longman Scientific and Technical
3-Jun-15 Finite volume method 21
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
– Balchandra Puranik and Atul Sharma
– Srinivaas Jayanthi
3-Jun-15 Finite volume method 22
Gratitude
• Dr. Vivek V. Ranade – My Mentor Guide and Teacher
– iFMg - Research group at NCL, Pune
• Audience
– For patient hearing and for their thirst in knowledge
3-Jun-15 Finite volume method 23
THANK YOU
A person who never made a mistake never tried anything new
- Albert Einstein - 1879 -1955
3-Jun-15 Finite volume method 24
Concept of staggered grid
3-Jun-15 Finite volume method 25
( ) ( ) ( )
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂+
∂ρ∂
y
u
yx
u
xx
p
y
vu
x
uu
t
u
( ) ( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uudt
t
u
VVVVVt∫∫∫∫∫∫
∆∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂+
∂ρ∂
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
3-Jun-15 Finite volume method 26
U Control volume
3-Jun-15 Finite volume method 27
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
( ) ( ) ( )we
V
uuuudvx
uu ρ−ρ≈∂ρ∂
∫∆
( ) ( )
( ) ( )[ ] ( ) ( )[ ]
( ) ( )[ ]2
uuuu
2
uuuu
2
uuuu
uuuu
J,1iJ,1i
J,1iJ,iJ,iJ,1i
J,1IJ,I
−+
−+
−
ρ−ρ=
ρ−ρ−
ρ−ρ=
ρ−ρ=
X Momentum Equation
3-Jun-15 Finite volume method 28
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
( ) ( ) ( )sn
V
vuvudVy
vu ρ−ρ≈∂ρ∂
∫∆
( ) ( )
( ) ( )[ ] ( ) ( )[ ]
( ) ( )[ ]2
vuvu
2
vuvu
2
vuvu
vuvu
1J,i1J,i
J,i1J,i1J,iJ,i
j,i1j,i
−+
−+
+
ρ−ρ=
ρ−ρ−
ρ−ρ=
ρ−ρ=
X Momentum Equation . . .
3-Jun-15 Finite volume method 29
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
( ) ( ) J,1IJ,I
V
ppdVx
p−
∆
−≈∂∂−∫
X Momentum Equation . . .
3-Jun-15 Finite volume method 30
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
weV x
u
x
udV
x
u
x
∂∂µ−
∂∂µ≈
∂∂µ
∂∂
∫∆
( )[ ] ( )[ ]
( )[ ]δ
+−µ=
δ−µ
−δ−µ
=
∂∂µ−
∂∂µ=
−+
−+
−
2
uu2u
2
uu
2
uu
x
u
x
u
J,1iJ,iJ,1i
J,1iJ,iJ,iJ,1i
J,1IJ,I
X Momentum Equation . . .
3-Jun-15 Finite volume method 31
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
snV y
u
y
udV
y
u
y
∂∂µ−
∂∂µ≈
∂∂µ
∂∂
∫∆
( )[ ] ( )[ ]
( )[ ]δ
+−µ=
δ−µ
−δ−µ
=
∂∂µ−
∂∂µ=
−+
−+
+
2
uu2u
2
uu
2
uu
y
u
y
u
1J,iJ,i1J,i
1J,iJ,iJ,i1J,i
j,i1j,i
X Momentum Equation . . .
3-Jun-15 Finite volume method 32
( )J,1IJ,IJ,iJ,i ppauua −−+= ∑
( ) ( )dV
y
u
ydV
x
u
xdV
x
pdV
y
vudv
x
uu
VVVVV∫∫∫∫∫
∆∆∆∆∆
∂∂µ
∂∂+
∂∂µ
∂∂+
∂∂−=
∂ρ∂+
∂ρ∂
Final form
3-Jun-15 Finite volume method 33
