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Setting  and  Usage  of  
OpenFOAM  multiphase  
solver(S-‐‑‒CLSVOF)
Graduate  school  of  Engineering  Science  
Osaka  Univ.  
D1
Takuya  Yamamoto
30th	
  OpenCAE	
  study	
  mee2ng	
  @	
  Kansai,	
  Japan	
  
2014/05/31	
Ver.	
  3	
  
updated	
  in	
  2015/7/20
•  Improved	
  solver	
  of	
  OpenFOAM	
  interFoam(VOF)	
  
•  Improved	
  surface	
  tension	
  model(CSF	
  model)	
  by	
  
using	
  re-­‐ini2aliza2on	
  equa2on	
  (Level-­‐Set	
  func2on)	
  
•  Please	
  refer	
  the	
  previous	
  presenta2on	
  (In	
  
Japanese)	
  
25th	
  OpenCAE	
  study	
  mee2ng	
  @	
  Kansai,	
  Japan	
26th	
  OpenCAE	
  study	
  mee2ng	
  @	
  Kansai,	
  Japan	
J. U. Brackbill, D. B. Kothe, C. Zemach, J. Comput. Phys. 100 (1992) 335–354.
CSF	
  model	
VOF	
 C. W. Hirt, B. D. Nichols, J. Comput. Phys. 39 (1981) 201–225. 	
S-­‐CLSVOF(Simple	
  Coupled	
  Volume	
  Of	
  Fluid	
  with	
  Level	
  Set)	
  method	
What  is  S-‐‑‒CLSVOF  solver  
(sclsVOFFoam)?
Generally	
Level-­‐Set	
  method	
  
•  low	
  volume	
  preserva2ve	
  quality	
  
•  Normal	
  unit	
  vector	
  (high	
  accuracy)	
VOF	
  method	
  
•  high	
  volume	
  preserva2ve	
  quality	
  
•  Normal	
  unit	
  vector	
  (low	
  accuracy)	
  
M. Sussman, P. Smereka, S. Osher, J. Comput. Phys. 114 (1994) 146–159.	
CLSVOF(Coupled	
  Volume	
  Of	
  Fluid	
  with	
  Level	
  Set)	
  method	
S-­‐CLSVOF(Simple	
  Coupled	
  Volume	
  Of	
  Fluid	
  with	
  Level	
  Set)	
  method	
Simple	
  coupling	
High	
  accuracy,	
  however,	
  slightly-­‐low	
  volume	
  preserva2ve	
  quality	
BeYer	
  than	
  VOF	
  method,	
  High	
  volume	
  preserving	
  quality	
What  is  S-‐‑‒CLSVOF  solver  
(sclsVOFFoam)?
Specifically	
  
	
  
In	
  A.	
  Albadawi	
  et	
  al.,	
  Int.	
  J.	
  
Mul2phase	
  Flow,	
  53,	
  11-­‐28	
  (2013).	
  
Implemented	
  the	
  S-­‐CLSVOF	
  method	
  
What  is  S-‐‑‒CLSVOF  solver  
(sclsVOFFoam)?
0	
 0	
 0	
 0	
 0	
0	
 0	
 0	
 0.1	
 0.3	
0	
 0	
 0.5	
 0.95	
 1.0	
0	
 0.4	
 1.0	
 1.0	
 1.0	
0	
 0.7	
 1.0	
 1.0	
 1.0	
VOF	
What  is  S-‐‑‒CLSVOF  solver  
(sclsVOFFoam)?
re-­‐ini2aliza2on	
  Eq.	
Level-­‐Set	
  func2on
Version  in  OpenFOAM
•  OpenFOAM-‐‑‒2.0.x
•  OpenFOAM-‐‑‒2.1.1
•  OpenFOAM-‐‑‒2.1.x
Validated	
  only	
  above	
  versions	
Released	
  site	
  (solver	
  and	
  tutorials)	
hYps://bitbucket.org/nunuma/public/src
Usage  (Solver  compilation)
1.  Copy	
  sclsVOFFoam	
  solver	
  to	
  applica2ons/
solvers	
  (cp	
  -­‐r	
  sclsVOFFoam	
  applica2ons/
solvers)	
  
2.  Change	
  directory	
  to	
  sclsVOFFoam	
  (cd	
  
sclsVOFFoam)	
  
3.  Compile(wmake)	
  
4.  Finish	
  solver	
  compila2on	
Please	
  type	
  sclsVOFFoam
Usage  (dam  break)
cp	
  -­‐r	
  $FOAM_TUTORIALS/mul2phase/interFoam/laminar/damBreak	
  .	
copy	
  damBreak	
  folder	
edit	
  damBreak	
  folder	
1.  Edit	
  constant/transportProper2es	
  
Add	
  the	
  following	
  commnts	
  in	
  transportProper2es	
  
deltaX	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  deltaX	
  [	
  0	
  0	
  0	
  0	
  0	
  0	
  0	
  ]	
  0.01;	
  
	
  
2.  Add	
  psi(Level-­‐Set	
  func2on)	
  in	
  0	
  folder	
  (ini2al	
  condi2on)	
  
(Based	
  on	
  alpha1)	
  
cp	
  -­‐r	
  0/alpha1	
  0/psi	
  
	
  
	
  
	
  
3.  Execute	
  sclsVOFFoam	
(deltaX	
  value	
  is	
  the	
  cell	
  width	
  
near	
  interface	
  posi2on)	
Edit	
  psi(Non-­‐dimension,	
  Boundary	
  condi2ons	
  are	
  zeroGradient)
Usage  (dam  break)
Change	
  based	
  on	
  interFoam	
  tutorial	
  case	
  
1.  In	
  transportProper2es,	
  you	
  must	
  write	
  grid	
  spacing	
  
(DeltaX).	
  
2.  You	
  must	
  define	
  ini2al	
  condi2ons	
  and	
  boundary	
  
condi2ons	
  of	
  Level-­‐Set	
  func2on(psi).	
Cau;on	
  
•  Boundary	
  condi2on	
  for	
  Level-­‐Set	
  func2on	
  have	
  not	
  
been	
  implemented.	
  
(You	
  can’t	
  use	
  fixed	
  contact	
  angle.	
  )	
  
•  You	
  can	
  use	
  only	
  zero	
  gradient	
  for	
  level	
  set	
  func2on.	
  
Summary
•  Advance  boundary  conditions  of  Level-‐‑‒
Set  function  have  not  been  
implemented.
•  By  changing  a  tutorial  of  interFoam,  
one  can  easily  execute  the  solver.
• If  there  are  something  
wrong,  please  send  e-‐‑‒mail  
to  me.
• Please  correct  my  English!!  
• Please  teach  me!!
tak_1031@hotmail.co.jp	
E-­‐mail	
  address
References
1.  G. Tryggvason, R. Scardovelli and S. Zaleski, Direct
Numerical Simulations of Gas-Liquid Multiphase Flows,
Cambridge University Press, Cambridge 2011.
2.  C. W. Hirt, B. D. Nichols, J. Comput. Phys. 39 (1981) 201–
225.
3.  J. U. Brackbill, D. B. Kothe and C. Zemach, J. Comput.
Phys. 100 (1992) 335–354.
4.  A. Albadawi et al., Int. J. Multiphase Flow 53 (2013) 11-28.
5.  M. Sussman, P. Smereka and S. Osher, J. Comput. Phys. 114
(1994) 146–159.
Setting and Usage of OpenFOAM multiphase solver (S-CLSVOF)
Support  
Documentation
•  Governing  Equations
Navier-‐‑‒Stokes  Eq.
Advection  of  α
interFoam  (VOF)
sk
gP
t
δσ
ρν
σ
σ
nF
Fvvv
v
=
++∇+−∇=∇⋅+
∂
∂ 2
::	
  liquid	
  phase	
  
::	
  interface	
  
::	
  gas	
  phase	
1=α
0=α
10 <<α
Fluid	
  phase	
  
	
  
Gas	
  phase	
( ) 0=⋅∇+
∂
∂
l
t
vα
α
( ) 0=⋅∇+
∂
∂
vα
α
t
( )( ) 01 =−⋅∇+
∂
∂
g
t
vα
α
Subscripts	
  l,	
  g	
  represent	
  liquid	
  and	
  gas	
  phase	
( )
glr
gl
vvv
vvv
−=
−+= αα 1
Defini;on	
ρ =αρl +(1−α)ρg
µ =αµl +(1−α)µg
( ) 0=⋅∇+
∂
∂
vα
α
t
CSF	
  model
sk
gP
t
δσ
ρν
σ
σ
nF
Fvvv
v
=
++∇+−∇=∇⋅+
∂
∂ 2
::	
  liquid	
  phase	
  
::	
  interface	
  
::	
  gas	
  phase	
1=α
0=α
10 <<α
( ) ( )( ) 01 =−⋅∇+⋅∇+
∂
∂
r
t
vv ααα
α
In	
  alphaEqn.H,	
  the	
  defini2on	
  is	
  
wriYen.	
  
∂α
∂t
+ ∇⋅ αv( )= 0 This	
  term	
  works	
  only	
  interface	
  area	
  
because	
  (1-­‐α)α is	
  included.	
ρ =αρl +(1−α)ρg
µ =αµl +(1−α)µg
interFoam  (VOF)
•  Governing  Equations
Navier-‐‑‒Stokes  Eq.
Advection  of  α
S-‐‑‒CLSVOF  method
∂v
∂t
+v⋅∇v = −∇P +ν∇2
v + Fσ + ρg
::	
  liquid	
  phase	
  
::	
  interface	
  
::	
  gas	
  phase	
1=α
0=α
10 <<α
Level-­‐Set	
  func2on	
  φ	
φ0 = (2α −1)⋅Γ
Γ	
  ;	
  non-­‐dimension	
  number	
Γ = 0.75Δx
Δx	
  ;	
  non-­‐dimension	
  number	
∂φ
∂τ
= S(φ0 ) 1− ∇φ( )
φ x,0( )= φ0 x( )
Re-­‐ini2aliza2on	
  equa2on	
∂α
∂t
+ ∇⋅ αv( )= 0
∇φ
Itera2on	
  number	
  φcorr	
φcorr =
ε
Δτ
ε =1.5Δx
Interface	
  width	
  ε	
ρ =αρl +(1−α)ρg
µ =αµl +(1−α)µg
α∇
Schema2c	
•  Governing  Equations
Navier-‐‑‒Stokes  Eq.
Advection  of  α
∂v
∂t
+v⋅∇v = −∇P +ν∇2
v + Fσ + ρg
::	
  liquid	
  phase	
  
::	
  interface	
  
::	
  gas	
  phase	
1=α
0=α
10 <<α
Fσ =σkδ∇φ
CSF	
  model	
k = −∇⋅nf = −∇⋅
∇φ( )f
∇φ( )f
+δs
$
%
&
&
'
(
)
)
∂α
∂t
+ ∇⋅ αv( )= 0
Dirac	
  func;on	
  δ	
δ φ( )= 0
δ φ( )=
1
2ε
1+cos
πφ
ε
!
"
#
$
%
&
!
"
#
$
%
&
φ >ε
φ ≤ε
Heaviside	
  func;on	
  H	
H φ( )= 0
H φ( )=
1
2
1+
φ
ε
+
1
π
sin
πφ
ε
!
"
#
$
%
&
!
"
#
$
%
&
H φ( )=1
Curvature	
ρ =αρl +(1−α)ρg
µ =αµl +(1−α)µg
•  Governing  Equations
Navier-‐‑‒Stokes  Eq.
Advection  of  α
S-‐‑‒CLSVOF  method
•  Governing  Equations
Navier-‐‑‒Stokes  Eq.
Advection  of  α
∂v
∂t
+v⋅∇v = −∇P +ν∇2
v + Fσ + ρg
::	
  liquid	
  phase	
  
::	
  interface	
  
::	
  gas	
  phase	
1=α
0=α
10 <<α
∂α
∂t
+ ∇⋅ αv( )= 0
H φ( )= 0
H φ( )=
1
2
1+
φ
ε
+
1
π
sin
πφ
ε
!
"
#
$
%
&
!
"
#
$
%
&
H φ( )=1
ρ =αρl +(1−α)ρg
µ =αµl +(1−α)µg
ρ = Hρl +(1− H)ρg
µ = Hµl +(1− H)µg
In	
  A. Albadawi et al. (2013),
no	
  physical	
  property	
  is	
  updated.	
φ < −ε
φ ≤ ε
φ > ε
Heaviside	
  func;on	
  H	
S-‐‑‒CLSVOF  method
Ex.1(Bubble  in  Cavity)
0.1	
  m	
0.1	
  m	
0.5	
  m/s	
0.02	
  m	
liquid	
  1	
liquid	
  2	
Physical	
  Proper;es	
  
Dynamic	
  viscosity 1.0	
  x	
  10-­‐3	
  m2/s	
  
Surface	
  tension	
  10	
  mN/m 	
Purpose	
  
Deforma2on	
  by	
  shear	
  stress	
  
(No	
  Buoyancy	
  flow	
  
Same	
  physical	
  proper2es	
  area	
  used	
  in	
  both	
  	
  
liquid	
  1	
  and	
  liquid	
  2)	
Calc.1	
  
interFoam	
  (VOF)	
  
Calc.	
  2	
  
sclsVOFFoam(S-­‐CLSVOF)	
Numerical	
  Grid	
  
200	
  x	
  200	
  (x,	
  y	
  direc2on)	
x	
y
Calc.1(Bubble  in  Cavity)
VOF	
 S-­‐CLSVOF	
Ini;al	
  condi;on
Calc.1(Bubble  in  Cavity)
VOF	
 S-­‐CLSVOF
Calc.  2(Dam  Break)
0.584	
  m	
0.584	
  m	
0.048	
  m	
0.292	
  m	
0.292	
  m	
0.1461	
  m	
phase	
  1	
  
Dynamic	
  viscosity 1	
  x	
  10-­‐6	
  m2/s	
  
Density  1000	
  kg/m3	
phase	
  1	
phase	
  2	
phase	
  2	
  
Dynamic	
  viscosity 1.48	
  x	
  10-­‐5	
  m2/s	
  
Density  1	
  kg/m3	
Surface	
  tension 70	
  mN/m	
  
VOF	
 S-­‐CLSVOF	
Calc.	
  Time	
  about	
  1.3	
  2mes	
  longer	
  in	
  S-­‐CLSVOF	
Calc.  2(Dam  Break)
VOF	
 S-­‐CLSVOF	
0.2	
  s	
 0.2	
  s	
0.3	
  s	
 0.3	
  s	
0.4	
  s	
 0.4	
  s	
0.5	
  s	
 0.5	
  s	
Calc.  2(Dam  Break)
Laplace  Pressure
•  Verification (A. Albadawi et al.(2013))
Laplace  Pressure
Laplace  Pressure  is  shown  as  following  
equation.
Δp =γ
1
R
+
1
R'
!
"
#
$
%
&
Δp = p0
in
− p∞
out p0
in
p∞
out
Pressure	
  in	
  bubble	
Pressure	
  at	
  outside	
  of	
  bubble	
Compare	
  the	
  numerical	
  and	
  analy2cal	
  pressures	
M. M. Francois et al., J. Comput. Phys., 213, 141-173 (2006).
Verification  problem  1
•  Numerical  domain
Δpexact =γ
1
R
+
1
R'
!
"
#
$
%
& = 2
Δp = p0
in
− p∞
out
p0
in
p∞
out
Pressure	
  at	
  the	
  bubble	
  center	
Pressure	
  at	
  wall	
uniform	
  spacing	
  grid	
  
DX	
  =	
  0.001	
  m	
  (Fine)	
  
	
  	
  	
  	
  	
  	
  =	
  0.0005	
  m	
  (Coarse)	
0.05	
  m	
0.05	
  m	
0.01	
  m	
Laplace	
  pressure(Theory)	
Physical	
  Proper;es	
  
γ	
  0.01	
  N/m	
  
Laplace	
  pressure	
  (Calc.)	
ρg	
  1	
  kg/m3	
  
µg	
  10-­‐5	
  kg/(ms)	
  
ρl	
  1000	
  kg/m3	
  
µl	
  10-­‐3	
  kg/(ms)	
  	
gas	
liquid	
zero	
  gravity	
  condi;on	
  
calc.	
  ;me	
  
0.1	
  sec.	
  	
  
(Δt	
  =	
  1x10-­‐5	
  sec.	
  (Coarse))	
  
(Δt	
  =	
  5x10-­‐6	
  sec.	
  (Fine))	
  
rela;ve	
  pressure	
  error	
  E0	
  
E0 =
Δp− Δpexact
Δpexact
Laplace  Pressure  (VOF)  
•  Result  (VOF(Coarse))
black	
  line	
  (alpha	
  =	
  0.5)
•  Result  (VOF(Fine))
Laplace  Pressure  (VOF)  
black	
  line	
  (alpha	
  =	
  0.5)
Results  (E0,  VOF)
CAlpha	
 0	
 1	
 2	
VOF	
  (Coarse)	
 25.17	
 25.23	
 25.38	
VOF	
  (Fine)	
 19.34	
 19.29	
 19.05	
Δpexact =γ
1
R
+
1
R'
!
"
#
$
%
& = 2
Δp = p0
in
− p∞
out
p0
in
p∞
out
E0 =
Δp− Δpexact
Δpexact
E0	
  depending	
  on	
  CAlpha	
Laplace	
  pressure(Theory)	
Laplace	
  pressure	
  (Calc.)	
Pressure	
  at	
  the	
  bubble	
  center	
Pressure	
  at	
  wall	
rela;ve	
  pressure	
  error	
  E0	
  
•  Result  (SCLSVOF(Coarse))
Laplace  Pressure  (S-‐‑‒CLSVOF)  
black	
  line	
  (alpha	
  =	
  0.5)
•  Result  (SCLSVOF(Fine))
Laplace  Pressure  (S-‐‑‒CLSVOF)  
black	
  line	
  (alpha	
  =	
  0.5)
Results  (E0,  S-‐‑‒CLSVOF)
E0	
  depending	
  on	
  CAlpha	
CAalpha	
 0	
 1	
 2	
VOF	
  (Coarse)	
 25.17	
 25.23	
 25.38	
VOF	
  (Fine)	
 19.34	
 19.29	
 19.05	
SCLSVOF	
  (Coarse)	
 1.557	
 0.1749	
 1.752	
SCLSVOF	
  (Fine)	
 1.496	
 1.210	
 0.9390	
Δpexact =γ
1
R
+
1
R'
!
"
#
$
%
& = 2
Δp = p0
in
− p∞
out
p0
in
p∞
out
E0 =
Δp− Δpexact
Δpexact
Laplace	
  pressure(Theory)	
Laplace	
  pressure	
  (Calc.)	
Pressure	
  at	
  the	
  bubble	
  center	
Pressure	
  at	
  wall	
rela;ve	
  pressure	
  error	
  E0	
  

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Setting and Usage of OpenFOAM multiphase solver (S-CLSVOF)

  • 1. Setting  and  Usage  of   OpenFOAM  multiphase   solver(S-‐‑‒CLSVOF) Graduate  school  of  Engineering  Science   Osaka  Univ.   D1 Takuya  Yamamoto 30th  OpenCAE  study  mee2ng  @  Kansai,  Japan   2014/05/31 Ver.  3   updated  in  2015/7/20
  • 2. •  Improved  solver  of  OpenFOAM  interFoam(VOF)   •  Improved  surface  tension  model(CSF  model)  by   using  re-­‐ini2aliza2on  equa2on  (Level-­‐Set  func2on)   •  Please  refer  the  previous  presenta2on  (In   Japanese)   25th  OpenCAE  study  mee2ng  @  Kansai,  Japan 26th  OpenCAE  study  mee2ng  @  Kansai,  Japan J. U. Brackbill, D. B. Kothe, C. Zemach, J. Comput. Phys. 100 (1992) 335–354. CSF  model VOF C. W. Hirt, B. D. Nichols, J. Comput. Phys. 39 (1981) 201–225. S-­‐CLSVOF(Simple  Coupled  Volume  Of  Fluid  with  Level  Set)  method What  is  S-‐‑‒CLSVOF  solver   (sclsVOFFoam)?
  • 3. Generally Level-­‐Set  method   •  low  volume  preserva2ve  quality   •  Normal  unit  vector  (high  accuracy) VOF  method   •  high  volume  preserva2ve  quality   •  Normal  unit  vector  (low  accuracy)   M. Sussman, P. Smereka, S. Osher, J. Comput. Phys. 114 (1994) 146–159. CLSVOF(Coupled  Volume  Of  Fluid  with  Level  Set)  method S-­‐CLSVOF(Simple  Coupled  Volume  Of  Fluid  with  Level  Set)  method Simple  coupling High  accuracy,  however,  slightly-­‐low  volume  preserva2ve  quality BeYer  than  VOF  method,  High  volume  preserving  quality What  is  S-‐‑‒CLSVOF  solver   (sclsVOFFoam)?
  • 4. Specifically     In  A.  Albadawi  et  al.,  Int.  J.   Mul2phase  Flow,  53,  11-­‐28  (2013).   Implemented  the  S-­‐CLSVOF  method   What  is  S-‐‑‒CLSVOF  solver   (sclsVOFFoam)?
  • 5. 0 0 0 0 0 0 0 0 0.1 0.3 0 0 0.5 0.95 1.0 0 0.4 1.0 1.0 1.0 0 0.7 1.0 1.0 1.0 VOF What  is  S-‐‑‒CLSVOF  solver   (sclsVOFFoam)? re-­‐ini2aliza2on  Eq. Level-­‐Set  func2on
  • 6. Version  in  OpenFOAM •  OpenFOAM-‐‑‒2.0.x •  OpenFOAM-‐‑‒2.1.1 •  OpenFOAM-‐‑‒2.1.x Validated  only  above  versions Released  site  (solver  and  tutorials) hYps://bitbucket.org/nunuma/public/src
  • 7. Usage  (Solver  compilation) 1.  Copy  sclsVOFFoam  solver  to  applica2ons/ solvers  (cp  -­‐r  sclsVOFFoam  applica2ons/ solvers)   2.  Change  directory  to  sclsVOFFoam  (cd   sclsVOFFoam)   3.  Compile(wmake)   4.  Finish  solver  compila2on Please  type  sclsVOFFoam
  • 8. Usage  (dam  break) cp  -­‐r  $FOAM_TUTORIALS/mul2phase/interFoam/laminar/damBreak  . copy  damBreak  folder edit  damBreak  folder 1.  Edit  constant/transportProper2es   Add  the  following  commnts  in  transportProper2es   deltaX                    deltaX  [  0  0  0  0  0  0  0  ]  0.01;     2.  Add  psi(Level-­‐Set  func2on)  in  0  folder  (ini2al  condi2on)   (Based  on  alpha1)   cp  -­‐r  0/alpha1  0/psi         3.  Execute  sclsVOFFoam (deltaX  value  is  the  cell  width   near  interface  posi2on) Edit  psi(Non-­‐dimension,  Boundary  condi2ons  are  zeroGradient)
  • 9. Usage  (dam  break) Change  based  on  interFoam  tutorial  case   1.  In  transportProper2es,  you  must  write  grid  spacing   (DeltaX).   2.  You  must  define  ini2al  condi2ons  and  boundary   condi2ons  of  Level-­‐Set  func2on(psi). Cau;on   •  Boundary  condi2on  for  Level-­‐Set  func2on  have  not   been  implemented.   (You  can’t  use  fixed  contact  angle.  )   •  You  can  use  only  zero  gradient  for  level  set  func2on.  
  • 10. Summary •  Advance  boundary  conditions  of  Level-‐‑‒ Set  function  have  not  been   implemented. •  By  changing  a  tutorial  of  interFoam,   one  can  easily  execute  the  solver.
  • 11. • If  there  are  something   wrong,  please  send  e-‐‑‒mail   to  me. • Please  correct  my  English!!   • Please  teach  me!! tak_1031@hotmail.co.jp E-­‐mail  address
  • 12. References 1.  G. Tryggvason, R. Scardovelli and S. Zaleski, Direct Numerical Simulations of Gas-Liquid Multiphase Flows, Cambridge University Press, Cambridge 2011. 2.  C. W. Hirt, B. D. Nichols, J. Comput. Phys. 39 (1981) 201– 225. 3.  J. U. Brackbill, D. B. Kothe and C. Zemach, J. Comput. Phys. 100 (1992) 335–354. 4.  A. Albadawi et al., Int. J. Multiphase Flow 53 (2013) 11-28. 5.  M. Sussman, P. Smereka and S. Osher, J. Comput. Phys. 114 (1994) 146–159.
  • 15. •  Governing  Equations Navier-‐‑‒Stokes  Eq. Advection  of  α interFoam  (VOF) sk gP t δσ ρν σ σ nF Fvvv v = ++∇+−∇=∇⋅+ ∂ ∂ 2 ::  liquid  phase   ::  interface   ::  gas  phase 1=α 0=α 10 <<α Fluid  phase     Gas  phase ( ) 0=⋅∇+ ∂ ∂ l t vα α ( ) 0=⋅∇+ ∂ ∂ vα α t ( )( ) 01 =−⋅∇+ ∂ ∂ g t vα α Subscripts  l,  g  represent  liquid  and  gas  phase ( ) glr gl vvv vvv −= −+= αα 1 Defini;on ρ =αρl +(1−α)ρg µ =αµl +(1−α)µg ( ) 0=⋅∇+ ∂ ∂ vα α t CSF  model
  • 16. sk gP t δσ ρν σ σ nF Fvvv v = ++∇+−∇=∇⋅+ ∂ ∂ 2 ::  liquid  phase   ::  interface   ::  gas  phase 1=α 0=α 10 <<α ( ) ( )( ) 01 =−⋅∇+⋅∇+ ∂ ∂ r t vv ααα α In  alphaEqn.H,  the  defini2on  is   wriYen.   ∂α ∂t + ∇⋅ αv( )= 0 This  term  works  only  interface  area   because  (1-­‐α)α is  included. ρ =αρl +(1−α)ρg µ =αµl +(1−α)µg interFoam  (VOF) •  Governing  Equations Navier-‐‑‒Stokes  Eq. Advection  of  α
  • 17. S-‐‑‒CLSVOF  method ∂v ∂t +v⋅∇v = −∇P +ν∇2 v + Fσ + ρg ::  liquid  phase   ::  interface   ::  gas  phase 1=α 0=α 10 <<α Level-­‐Set  func2on  φ φ0 = (2α −1)⋅Γ Γ  ;  non-­‐dimension  number Γ = 0.75Δx Δx  ;  non-­‐dimension  number ∂φ ∂τ = S(φ0 ) 1− ∇φ( ) φ x,0( )= φ0 x( ) Re-­‐ini2aliza2on  equa2on ∂α ∂t + ∇⋅ αv( )= 0 ∇φ Itera2on  number  φcorr φcorr = ε Δτ ε =1.5Δx Interface  width  ε ρ =αρl +(1−α)ρg µ =αµl +(1−α)µg α∇ Schema2c •  Governing  Equations Navier-‐‑‒Stokes  Eq. Advection  of  α
  • 18. ∂v ∂t +v⋅∇v = −∇P +ν∇2 v + Fσ + ρg ::  liquid  phase   ::  interface   ::  gas  phase 1=α 0=α 10 <<α Fσ =σkδ∇φ CSF  model k = −∇⋅nf = −∇⋅ ∇φ( )f ∇φ( )f +δs $ % & & ' ( ) ) ∂α ∂t + ∇⋅ αv( )= 0 Dirac  func;on  δ δ φ( )= 0 δ φ( )= 1 2ε 1+cos πφ ε ! " # $ % & ! " # $ % & φ >ε φ ≤ε Heaviside  func;on  H H φ( )= 0 H φ( )= 1 2 1+ φ ε + 1 π sin πφ ε ! " # $ % & ! " # $ % & H φ( )=1 Curvature ρ =αρl +(1−α)ρg µ =αµl +(1−α)µg •  Governing  Equations Navier-‐‑‒Stokes  Eq. Advection  of  α S-‐‑‒CLSVOF  method
  • 19. •  Governing  Equations Navier-‐‑‒Stokes  Eq. Advection  of  α ∂v ∂t +v⋅∇v = −∇P +ν∇2 v + Fσ + ρg ::  liquid  phase   ::  interface   ::  gas  phase 1=α 0=α 10 <<α ∂α ∂t + ∇⋅ αv( )= 0 H φ( )= 0 H φ( )= 1 2 1+ φ ε + 1 π sin πφ ε ! " # $ % & ! " # $ % & H φ( )=1 ρ =αρl +(1−α)ρg µ =αµl +(1−α)µg ρ = Hρl +(1− H)ρg µ = Hµl +(1− H)µg In  A. Albadawi et al. (2013), no  physical  property  is  updated. φ < −ε φ ≤ ε φ > ε Heaviside  func;on  H S-‐‑‒CLSVOF  method
  • 20. Ex.1(Bubble  in  Cavity) 0.1  m 0.1  m 0.5  m/s 0.02  m liquid  1 liquid  2 Physical  Proper;es   Dynamic  viscosity 1.0  x  10-­‐3  m2/s   Surface  tension  10  mN/m  Purpose   Deforma2on  by  shear  stress   (No  Buoyancy  flow   Same  physical  proper2es  area  used  in  both     liquid  1  and  liquid  2) Calc.1   interFoam  (VOF)   Calc.  2   sclsVOFFoam(S-­‐CLSVOF) Numerical  Grid   200  x  200  (x,  y  direc2on) x y
  • 21. Calc.1(Bubble  in  Cavity) VOF S-­‐CLSVOF Ini;al  condi;on
  • 23. Calc.  2(Dam  Break) 0.584  m 0.584  m 0.048  m 0.292  m 0.292  m 0.1461  m phase  1   Dynamic  viscosity 1  x  10-­‐6  m2/s   Density  1000  kg/m3 phase  1 phase  2 phase  2   Dynamic  viscosity 1.48  x  10-­‐5  m2/s   Density  1  kg/m3 Surface  tension 70  mN/m  
  • 24. VOF S-­‐CLSVOF Calc.  Time  about  1.3  2mes  longer  in  S-­‐CLSVOF Calc.  2(Dam  Break)
  • 25. VOF S-­‐CLSVOF 0.2  s 0.2  s 0.3  s 0.3  s 0.4  s 0.4  s 0.5  s 0.5  s Calc.  2(Dam  Break)
  • 26. Laplace  Pressure •  Verification (A. Albadawi et al.(2013)) Laplace  Pressure Laplace  Pressure  is  shown  as  following   equation. Δp =γ 1 R + 1 R' ! " # $ % & Δp = p0 in − p∞ out p0 in p∞ out Pressure  in  bubble Pressure  at  outside  of  bubble Compare  the  numerical  and  analy2cal  pressures M. M. Francois et al., J. Comput. Phys., 213, 141-173 (2006).
  • 27. Verification  problem  1 •  Numerical  domain Δpexact =γ 1 R + 1 R' ! " # $ % & = 2 Δp = p0 in − p∞ out p0 in p∞ out Pressure  at  the  bubble  center Pressure  at  wall uniform  spacing  grid   DX  =  0.001  m  (Fine)              =  0.0005  m  (Coarse) 0.05  m 0.05  m 0.01  m Laplace  pressure(Theory) Physical  Proper;es   γ  0.01  N/m   Laplace  pressure  (Calc.) ρg  1  kg/m3   µg  10-­‐5  kg/(ms)   ρl  1000  kg/m3   µl  10-­‐3  kg/(ms)   gas liquid zero  gravity  condi;on   calc.  ;me   0.1  sec.     (Δt  =  1x10-­‐5  sec.  (Coarse))   (Δt  =  5x10-­‐6  sec.  (Fine))   rela;ve  pressure  error  E0   E0 = Δp− Δpexact Δpexact
  • 28. Laplace  Pressure  (VOF)   •  Result  (VOF(Coarse)) black  line  (alpha  =  0.5)
  • 29. •  Result  (VOF(Fine)) Laplace  Pressure  (VOF)   black  line  (alpha  =  0.5)
  • 30. Results  (E0,  VOF) CAlpha 0 1 2 VOF  (Coarse) 25.17 25.23 25.38 VOF  (Fine) 19.34 19.29 19.05 Δpexact =γ 1 R + 1 R' ! " # $ % & = 2 Δp = p0 in − p∞ out p0 in p∞ out E0 = Δp− Δpexact Δpexact E0  depending  on  CAlpha Laplace  pressure(Theory) Laplace  pressure  (Calc.) Pressure  at  the  bubble  center Pressure  at  wall rela;ve  pressure  error  E0  
  • 31. •  Result  (SCLSVOF(Coarse)) Laplace  Pressure  (S-‐‑‒CLSVOF)   black  line  (alpha  =  0.5)
  • 32. •  Result  (SCLSVOF(Fine)) Laplace  Pressure  (S-‐‑‒CLSVOF)   black  line  (alpha  =  0.5)
  • 33. Results  (E0,  S-‐‑‒CLSVOF) E0  depending  on  CAlpha CAalpha 0 1 2 VOF  (Coarse) 25.17 25.23 25.38 VOF  (Fine) 19.34 19.29 19.05 SCLSVOF  (Coarse) 1.557 0.1749 1.752 SCLSVOF  (Fine) 1.496 1.210 0.9390 Δpexact =γ 1 R + 1 R' ! " # $ % & = 2 Δp = p0 in − p∞ out p0 in p∞ out E0 = Δp− Δpexact Δpexact Laplace  pressure(Theory) Laplace  pressure  (Calc.) Pressure  at  the  bubble  center Pressure  at  wall rela;ve  pressure  error  E0