HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 1
5.2 Finite-Volume Method
Closely related to Subdomain MethodBut without explicit introduction of trial or
interpolation functionApproximate the flux terms directly (rather than
the function itself)Use the integral form of PDEs (instead of
weighted residuals)
“Numerical Heat Transfer and Fluid Flows,” S.V. Patankar, McGraw-Hill, 1980.
Navier-Stokes Equations 2D Compressible N-S equations
General Form
i
j
j
iij
yyyxxyt
yxyxxxt
yxt
x
u
x
u
2
1
0pvvuvv
0uvpuuu
0vu
;
)()()(
)()()(
)()(
yyxy
xyxx
pvvG ,uvF ,vq:momentumy
uvG ,puuF ,uq:momentumx
vG ,uF ,q :continuity
0y
G
x
F
t
q
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 2
Green’s Theorem
3D: Volume integral <==> Surface integral
2D: Surface integral <==> Line integral
) ,( ,:
) ,( ,:
) ,( , :
)(et
yyxy
xyxx
pvvuvHvqmomentumy
uvpuuHuqmomentumx
vuHqcontinuity
y
G
x
FHF, GHl
CV CS
dSnHdH
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 3
Triangular elements
5-sided or 6-sided control volumes
5.2.1 First Derivatives
vG ,uF ,cq:ibleincompress
vG ,uF ,q:lecompressib0
y
G
x
F
t
q
NW
W
SW
P
N
NE
E
SES
AB
C
D e
n
s
w
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 4
First Derivatives
Evaluate surface integrals for arbitrary control surfaces
CS
CV
dSnH
dH
First Derivatives
AB
C
Dn
n
n
n
A
B
n
dxdy
n
dx
dy
dy
dx
n
n
dx
dy
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 5
First Derivatives
dxdy
ds
dx
ds
dy
dS dx dy( , )
nds dy dx( , )
dS n n dS 0
H ndS F G dy dx Fdy Gdx( , ) ( , )
Finite-Volume Method
0xGyFxGyF
xGyFxGyFqAdt
d
0xGyFxGyF
xGyFxGyFqAdt
d
xGyFqAdt
d0dSnHqd
dt
d
DAwDAwCDnCDn
BCeBCeABsABsPP
DACD
BCABPP
DA
ABPP
CV CS
)()(
)()()(
)()(
)()()(
)()(
0yyyy
0xxxx
yyyxxx
yyyxxx
yyyxxx
yyyxxx
DACDBCAB
DACDBCAB
DADADADA
CDCDCDCD
BCBCBCBC
ABABABAB
;
;
;
;
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 6
Linear Interpolation
)(
)(
)(
)(
)(
)(
)(
)(
PWwDA
PNnCD
PEeBC
PSsAB
PWwDA
PNnCD
PEeBC
PSsAB
GG2
1GG
GG2
1GG
GG2
1GG
GG2
1GG
FF2
1FF
FF2
1FF
FF2
1FF
FF2
1FF
0xGyF2
1xGyF
2
1
xGyF2
1xGyF
2
1
dt
dqA
DAWDAWCDNCDN
BCEBCEABSABSP
P
)()(
)()(
Uniform Cartesian Grids
0y2
GG
x2
FF
dt
dq
0GG2
xFF
2
yyx
dt
dq
SNWEP
SNWEP
)()()(
yy0x
0yxx
yy0x
0yxx
DADA
CDCD
BCBC
ABAB
;
;
;
;
ueuw
vn
vs
0y
GG
x
FF
dt
dq snweP
or
A B
CD
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 7
General Curvilinear Coordinates
For curvilinear grids, FVM provides a discretization in Cartesian coordinates without explicit (boundary-fitted) coordinate transformation
Grid curvature terms are ignoredAlternatively, one may apply FVM in transformed plane
5.2.2. Second DerivativesConvective Transport Equations
x y x y xx yyu v u v S( ) ( ) ( )
: ; / Re,
: ; / Re,
: ; / Re,
: ; ,
: ;
u x t
v y t
t
x momentum u 1 S p u
y momentum v 1 S p v
vorticity 1 S
stream function u v 0 S
temperature T 1
/
: ; /
Pe
concentraction C 1 Pe
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 8
Convective Transport Equations
x y x y xx yyu v u v S( ) ( ) ( )
x
y
F u F Glet S 0
G v x y
F GS dxdy 0
x y( )
P P s AB s AB e BC e BC
n CD n CD w DA w DA
A S F y G x F y G x
F y G x F y G x 0
( ) ( ) ( )
( ) ( )
Line integral for element centered at P: ABCDA
Second Derivativess x s x s s
s y s y s s
F u u
G v v
( ) ( ) ( )
( ) ( ) ( )
NW
W
SW
P
N
NE
E
SES
A
B
C
D e
n
s
w
AB
Line integral for element centered at s
AewBA
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 9
Second Derivatives -- First derivatives centered at s -- came from second derivatives centered at PUse Green’s theorem again for elements centered
at s, e, n, and w (not centered at P )
x s y s( ) , ( )
x y x
CV CS
x y y
CV CS
H F G 0 H F G
Hd H ndS Fdy Gdx dy
H F G 0 H F G
Hd H ndS Fdy Gdx dx
( , ) ( , )
( )
( , ) ( , )
( )
x s x S A B B B e P ew A wAs s s
y s y S A B B B e P ew A wAs s s
1 1 1 dxdy dy y y y y
A A A
1 1 1 dxdy dx x x x x
A A A
( ) ( )
( ) ( )
NW
W
SW
P
N
NE
E
SES
A
B
C
D e
n
s
w
AB
Line integral for element centered at s
AewBA
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 10
Bilinear InterpolationIgnore grid expansion or contraction
Bilinear interpolation
wASPeB
wASPeB
weABBA
weABBA
yyy
xxx
yyy
xxx
A S P W SW B S P E SE
1 1
4 4( ); ( )
2 2x s AB y s AB AB AB S P
s
AB SP AB SP E W SE SW
AB S P AB E W SE SW
1y x x y
A
1 y y x x
41
Q P4
( ) ( ) ( )( )
( )( )
( ) ( )
Flux Evaluation
s AB s AB AB S P AB E W SE SW
P S AB P S AB
AB S AB S AB S
F y G x Q P4
1 1 u u y v v x
2 21
Q u y v x 2
( ) ( )
( ) ( ) ( ) ( )
( )
AB P AB P AB P
AB E W SE SW
1 Q u y v x
2
P4
( )
( )
Diffusive and convective fluxes
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 11
Convective Transport Equation
NW
W
SW
P
N
NE
E
SES
e
n
s
w
A’B’
Line integral for elements centered at s, e, n, wfour overlapped surface (line) integrals
Convective Transport Equations
P E E W W N N S S NE NE NW NW SE SE SW SW P P
S AB DA BC S AB S AB SE AB BC
E BC AB CD E BC E BC NE BC CD
N
C C C C C C C C C S
1 1C Q P P u y v x ; C P P
D 4 2 4 D
1 1C Q P P u y v x C P P
D 4 2 4D
1C
D
( )
( ) ( ) ( )
( ) ( ) ; ( )
CD BC DA N CD N CD NW CD DA
W DA CD AB W DA W DA SW DA AB
PP we sn sn we
1Q P P u y v x C P P
4 2 4D
1 1C Q P P u y v x C P P
D 4 2 4D
A 1C x y x y
D D
( ) ( ) ; ( )
( ) ( ) ; ( )
( )
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 12
Convective Transport Equations
wWPDAWPDADAw2DA
2DADA
nNPCDNPCDCDn2CD
2CDCD
eEPBCEPBCBCe2BC
2BCBC
sSPABSPABABs2AB
2ABAB
AyyxxP AyxQ
AyyxxP AyxQ
AyyxxP AyxQ
AyyxxP AyxQ
/)(;/)(
/)(;/)(
/)(;/)(
/)(;/)(
)(
)
()(
DACDBCAB
DAPDAPCDPCDPBCPBCP
ABPABPDACDBCAB
QQQQ
xvyuxvyuxvyu
xvyu2
1QQQQD
Uniform Cartesian Grid
P E E w w N N S S NE NE NW NW SE SE SW SW P P
xE EE cell2 2 2
xW WW cell2 2 2
N
C C C C C C C C C S
1 x y u y 1 1 u 1C 1 0 5R
D x 2 D x 2 x D x
u y u1 x y 1 1 1C 1 0 5R
D x 2 D x 2 x D x
1 x yC
D
( )
.
.
yN Ncell2 2 2
yS SS cell2 2 2
NE NW SE SW P
2 2 2 2
v x 1 1 v 1 1 0 5R
y 2 D y 2 y D y
v x v1 x y 1 1 1C 1 0 5R
D y 2 D y 2 y D y
x y 1C C C C 0 C
D D
1 1 D 1 1D 2 x y D 2
x y x y x y
.
.
;
;
x ycell x cell y
u x v yR P R P,
Cell Reynolds (Peclet) number
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 13
Central Difference
x y xx yy
E P W N P S E E W W N N S SP2 2
u v S
2 2 u u v vS 0
x y 2 x 2 y
( ) ( ) ( )
( )
Identical to the finite-volume method
WEP P E W2 2 2 2
SNN S P2 2
uu1 1 1 12 D
x y x 2 x x 2 x
vv1 1 1 S
y 2 y y 2 y( )
ycell
xcell R
2
1yv
2
1
2
yvR
2
1xu
2
1
2
xu
Re ;Re
Unphysical solutions may occur if cell Reynolds (Peclet) numbers > 2
Exponential Scheme Ref: “Numerical Heat Transfer and Fluid Flow,” by S.V. Patankar, 1980
cF 0x
F 0
y
G
x
F
t
q
Steady, 1-D
ux/αxF u c ae c/u
Fw Fe
Pee e ee P EPe
u u xF c e Pe
e 1;
PW E
ew x x
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 14
Exponential SchemeCell Peclet number Pe
Pee e ee P EPe
u u xF c e Pe
e 1;
,
,
, . ( )
( )
)
e e P
e e E
Pe 2
e E Pe P E
e
e x x
Pe 1 F u
Pe 1 F u
Pe 1 e 1 Pe 0 5 Pe 1 Pe
u F 1 Pe
Pe x
(Note: F u
Gradual shift to upwind
1D Convective Transport Equation
Uniform Cartesian grid (and ue = uw = u)Pe = Pw = Px = uΔx/α
Exponential Scheme
Linear FVM (Central Difference)
x x
x x
x
x x
P Pe w P E W PP P
P
P W EP P
u uF F e e
e 1 e 1
e 1
e 1 e 1
( ) ( )
x xcell cell
e w P W E
1 0 5R 1 0 5RF F
2 2
. . x
cell xR P
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 15
Exponential Scheme
FeFw
Gn
Gs
Cartesian grids only
x y xx yy
x
y
u v S 0
F u F Glet S 0
G v x y
( )
ew xx
s
n
y
y
0SAΔxGΔxGΔyFΔyFΔxx; ΔyΔy
PPssnnwwee
snew
Pe Pwe we P E w W PPe Pw
Pn Psn sn P N s S PPn Ps
e e w w n n s se w n s
u uF e F e
e 1 e 1 v v
G e G ee 1 e 1u x u x v y v y
P , P , P , P
;
;
yeyw
xn
xs
Exponential SchemeNonuniform Cartesian grids
Uniform Cartesian grid (and ue = uw = u, vn = vs = v)
P P E E W W N N S S P
Pw Pse e w w n n s s
E W N SPe Pw Pn Ps
Pe Pne e w w n n s s
P Pe Pw Pn Ps
C C C C C A S
u y e u y v x e v xC C C C
e 1 e 1 e 1 e 1
e u y u y e v x v xC
e 1 e 1 e 1 e 1
, , ,
; ( , )
, , ,
( ) ( ),
e w x n s y
Px PyE W E N S NPx Py
Px Py
P PPx Py
u x v y P P P P P P constant u v
u y v xC C e C C C e C
e 1 e 1
u y e 1 v x e 1C A x y
e 1 e 1
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 16
FVM: Pure DiffusionUniform Cartesian Grid x y 0 1 S 0. ;
0.25
0.25
0.25 0.25 0.25
0.25
0.25
0.25
Linear Exponential
P E E W W N N S S x y
u x v yC C C C P P; ,
0P ;0P
0v ;0u
yx
FVM: Convection-DiffusionUniform Cartesian Grid x y 0 1 S 0. ;
0.25
0.25
0.2625 0.2375 0.262599
0.249896
0.237609
0.249896
Linear Exponential
1051709.1e ;0P ;1.0P
0v ;1u ;1 Px
yx
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 17
FVM: Convection-DiffusionUniform Cartesian Grid x y 0 1 S 0. ;
0.25
0.25
0.375 0.125 0.379922
0.240156
0.139765
0.240156
Linear Exponential
718281828.2e ;0P ;1P
0v ;1u ;1.0 Px
yx
FVM: Convection-DominantUniform Cartesian Grid x y 0 1 S 0. ;
0.25
0.25
1.5 -1.0 0.833308
0.083327
0.000038
0.083327
Linear Exponential
466.22026e ;0P ;10P
0v ;1u ;01.0 Px
yx
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 18
FVM: Convection-DominantUniform Cartesian Grid x y 0 1 S 0. ;
0.25
0.25
12.75 -12.25 1.000000
0.000000
0.000000
0.000000
Linear Exponential
10688.2e ;0P ;100P
0v ;1u ;001.0 43Px
yx
FVM: Skewed UpwindUniform Cartesian Grid x y 0 1 S 0. ;
0.2625
0.2375
0.2625 0.2375 0.262490
0.237510
0.237510
0.262490
Linear Exponential
1051709.1ee ;1.0P ;1.0P
1v ;1u ;1 PyPx
yx
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 19
FVM: Skewed UpwindUniform Cartesian Grid x y 0 1 S 0. ;
0.375
0.125
0.375 0.125 0.365529
0.134471
0.134471
0.365529
Linear Exponential
718281828.2ee ;1P ;1P
1v ;1u ;1.0 PyPx
yx
Finite-Volume MethodUniform Cartesian Grid x y 0 1 S 0. ;
1.5
-1.0
1.5 -1.0 0.499977
0.000023
0.000023
0.499977
Linear Exponential
466.22026ee ;10P ;10P
1v ;1u ;01.0 PyPx
yx
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 20
FVM: Convection-DominantUniform Cartesian Grid x y 0 1 S 0. ;
12.75
-12.25
12.75 -12.25 0.500000
0.000000
0.000000
0.500000
Linear Exponential
10688.2ee ;100P ;100P
1v ;1u ;001.0 43PyPx
yx
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 21
Error in textbook
AJM
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 22
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 23
Successive Overrelaxation(SOR)
HC Chen 2/19/2020
Chapter 5B: Finite-Volume Method 24
Grid Refinement