Diffuse interfaces with compressible fluids, phase ...
Transcript of Diffuse interfaces with compressible fluids, phase ...
![Page 1: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/1.jpg)
Diffuse interfaces with compressible fluids, phase transition and capillarity
Richard SaurelAix Marseille University, M2P2, CNRS 7340
RS2N SAS, Saint-Zacharie
![Page 2: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/2.jpg)
Aims of the “diffuse interface” approach
- Solve liquid and gas flows with real density jumps at arbitrary speeds, separated by interfaces.
- Consider compressibility of both phases.
- Consider surface tension, heat diffusion and phase transition.
- Address waves propagation (possibly shocks).
![Page 3: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/3.jpg)
Petitpas, Massoni, Saurel and Lapébie, 2009, Int. J. Mult. Flows
Example: Underwater supercavitating missile (400 km/h)2 interfaces are present: liquid-vapor and gas-vapor
![Page 4: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/4.jpg)
• The ‘diffuse interfaces’ under consideration come from numerical
diffusion (not capillary nor viscous regularization),
• These interfaces are captured with an hyperbolic flow solver.
• The numerical diffusion can be reduced (1-2 points – work in progress).
• All fluids are considered compressible with a convex EOS.
• The ‘Noble-Abel-stiffened-gas’ EOS is a good prototype example:
(Le Metayer-Saurel, POF, 2016)
It is both an improved SG formulation and simplified van der
Waals formulation
� well defined sound speed
• The liquid and vapour constants are coupled through the phase diagram
(and saturation curves).
( ) k k kk 1 k k
k k
(e q )p 1 p
1 b ∞ρ −= γ − − γ
− ρk =1,2
Preliminary remarks
k k(p cst. instead of p a ²)∞ ∞= = ρ
![Page 5: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/5.jpg)
How to compute the pressure in this artificial mixture zone?
The EOS are discontinuous with restricted domain of validity.
G
L
L
G
�Mixture cell
Diffuse interfaces of simple contact (with compressible fluids)
![Page 6: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/6.jpg)
Starting point of the diffuse interfaces method:Consider mixture cells through a total non-equilibrium two-phase model with 2 velocities, 2 temperatures, 2 pressures
)pp(x
ut
211
i1 −µ=
∂∂α+
∂∂α
0x
u
t11111 =
∂ρ∂α
+∂
ρ∂α
)uu(x
px
)p²u(
t
u12
1i
11111111 −λ+∂
∂α=∂
α+ρα∂+∂ρ∂α
)uu(u)pp(p x
upx
)pE(u
t
E12
'
i21'i
kii
111111111 −λ+−µ−∂
∂α=∂
α+ρα∂+∂ρ∂α
Each phase evolves in its own sub-volume.
The pressure equilibrium condition is replaced by a PDE with relaxation � hyperbolicity is preserved.
3 symmetric equations are used for the second phase.
![Page 7: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/7.jpg)
Fulfill interface conditions of equal pressures and equal velocities � stiff relaxation
ε−<α<ε 1k
‘bubbles’ expansion or contraction forces mechanical equilibrium locally:
)pp(x
ut
211
i1 −µ=
∂∂α+
∂∂α
Similar relaxation is done for velocities (Saurel and Abgrall, JCP, 1999)
This trick enables fulfilment of interface conditions.
µ tends to infinity
![Page 8: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/8.jpg)
In other words…
Solve this ODE system in the limit
This ODE system is replaced by two algebraic systems.
)pp(t
211 −µ=
∂∂α
0t
11 =∂
ρ∂α
)uu(t
u12
111 −λ=∂ρ∂α
)uu(u)pp(p t
E12
'
i21
'
i111 −λ+−µ−=
∂ρ∂α
This method deals with 2 steps:
-Solve the 7 equations model without relaxation terms during a
time step � 2 velocities and 2 pressures are computed in the mixture zone.
-Relax (reset) the 2 velocities and pressures to their equilibrium values:
+∞→µ+∞→λ
![Page 9: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/9.jpg)
Example with velocity relaxation
**2
*1 uuu ==
We have to solve:
At relaxed state
0t
22 =∂
ρ∂α
0t
11 =∂
ρ∂α
)uu(t
u12
111 −λ=∂ρ∂α
)uu(t
u12
222 −λ−=∂ρ∂α
Summing the two momentum equations results in:
( )0
t
uu 222111 =∂
ρα+ρα∂
Integrating this equation between the end of the propagation step and the relaxed step results in:
( ) ( )0
222111
*
222111 uuuu ρα+ρα=ρα+ρα
Thus: ( )( )
( )( )0
2211
0
222111*
2211
0
222111* uuuuu
ρα+ραρα+ρα=
ρα+ραρα+ρα=
The same type of procedure is done for pressure relaxation.
t
u
![Page 10: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/10.jpg)
Computational example: Impact of copper projectile at 5 km/s on a copper tank filled with water, surrounded by air.
Very robust but quite complicated if interface only are under consideration.
Helpful when velocity disequilibrium effects are under consideration.
![Page 11: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/11.jpg)
Reduced model with single pressure and single velocity but two temperatures
+∞→ε=µλ 1,
• Faster computations
• Add physical effects easily (surface tension, phase transition)…
1o fff ε+=
Relaxation coefficients are assumed stiff.
Each flow variable evolves with small perturbations around an equilibrium state.
![Page 12: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/12.jpg)
( )x
u
cc
cc)pp(
2
222
1
211
2
11
2
2221 ∂
∂
αρ
+α
ρρ−ρ
→−µ
Consequence
The pressure relaxation term becomes a differential one:
)pp(x
ut
211
i1 −µ=
∂∂α+
∂∂α
![Page 13: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/13.jpg)
Kapila et al., POF, 2001
2211 ρα+ρα=ρ
( ) 1 1 1 1 1 2 2 2 2 21 1 2 2
1 21 1
1 1 1 2 2 2
1 2
(1 b ) (1 b )e (Y q Y q )
,Y(1 b ) (1 b )
p p1 1
p p( ,e, )
1 1
∞ ∞
−ρ −ρ− +
−ρ −ρ
α γ α γρ − +γ − γ −= ρ α = α α+
γ − γ −
x
u
cc
)cc(
xu
t
2
2
22
1
2
11
2
11
2
2211
∂∂
αρ+
αρ
ρ−ρ=
∂∂α
+∂
∂α
0x
u
t1111 =
∂ρ∂α
+∂
ρ∂α
0x
)p²u(tu =∂
+ρ∂+∂∂ρ
0x
)pE(utE =∂
+ρ∂+∂∂ρ
0x
u
t2222 =
∂ρ∂α
+∂
ρ∂α
A mixture EOS is obtained from the mixture energy definition and pressure equilibrium:
Two-phase shock relations are given in Saurel et al., Shock Waves, 2007.
1 1 1 2 2 2E E Eρ = α ρ + α ρ
1 22 2
1 1 2 2
1c² c c
+ρ ρ ρ
α α=
![Page 14: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/14.jpg)
2 22 2 1 1
2 21 1 2 2
1 2
(ρ c ρ c )αu. α div(u) 0
ρ c ρ ctα α
ρYdiv( Yu) 0
tρ
div(ρu) 0t
ρu Y Ydiv PI ρu u σ Y I 0
t Y
σ Y 1ρ(e u²)
σ Y 1 Y Yρ 2div (ρ(e u²) P)u σ Y I
t ρ 2 Y
−∂ + ∇ − =∂ +
∂ + ρ =∂
∂ + =∂
∂ ∇ ⊗∇ + + ⊗ − ∇ − = ∂ ∇
∇∂ + + ∇ ∇ ⊗∇
+ + + + − ∇ −∂ ∇
��
�
�
� ��
�� �
�
�
��
� �
�� �
� .u 0
=
�
Capillary effects can be added (Perigaud and Saurel, JCP, 2005)
- Compressibilty is considered as well as surface tension.
- Thermodynamics and capillarity are decoupled: no need to enlarge the interface.
- Two Gibbs identities are used: and
- There is no capillary lenght to resolve: Jump conditions deal with capillarity as they deal with shocks (genuine property of the conservative formulation),
k k k kde T ds pdv= − de dSσ =σ
![Page 15: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/15.jpg)
Falling droplet from the roof
(computations done in 2004)
Qualitative comparisons (2D computations) Quantitative comparisons (2D axi)
![Page 16: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/16.jpg)
Phase transition
![Page 17: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/17.jpg)
Examination of the mixture entropy production suggests (Saurel et al., JFM, 2008)
( ) ( )
( ) ( )122222
121111
ggudivt
ggudivt
−ρν−=ρα+∂
ρα∂
−ρν=ρα+∂
ρα∂
�
�
2
2
22
1
2
11
2
2
1
1
12
2
2
22
1
2
11
2
22
1
21
12
2
2
22
1
2
11
2
11
2
221
1
cc)TT(H
cc
cc
)gg()u(divcc
)cc()(grad.u
t
αρ+
αρ
αΓ
+αΓ
−+
αρ+
αρ
α+
α−ρν+
αρ+
αρ
ρ−ρ=α+
∂∂α ��
Pressure relaxation Phase change Heat exchange
( )
( ) 0)upE(divt
E
0)p(graduudivt
u
=+ρ+∂ρ∂
=+⊗ρ+∂ρ∂
Phase transition is modeled as a Gibbs free energy relaxation process.
But two kinetic parameters are now present.
![Page 18: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/18.jpg)
The model involves two entropies
02 <c
Phase transition connects the two isentropes through a kinetic path (mass source terms).Sound speed is defined everywhere:
• Very different of the van der Waals (and variants) approaches
With VdW phase transition is modeled as a kinetic path …. and sound speed is undefined
� Ill posed model.
Mixture
Mixture
cstes =1
cstes =2
2 2
s
pc v 0
v
∂ =− <∂
liquid
liquid
vapor
vapor
1 22 2
1 1 2 2
1c² c c
+ρ ρ ρ
α α=
![Page 19: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/19.jpg)
Thermodynamic closureEach fluid is governed by the NASG EOS (Le Metayer and Saurel, Phys.
Fluids, 2016)
With the help of Gibbs identity and Maxwell relations:
Vh(P,T) C T bP q=γ + +
v V 1
Tg(P,T) ( C q')T C Tln bP q
(p p )
γ
γ−∞
= γ − − + ++
V
P pe(P,T) C T q
P p∞
∞
+ γ= ++
( )�
agitation
k k kk k k k k k
k k attraction
shortdistancesrepulsion
(e q )p ( ,e ) 1 p
1 b ∞ρ −ρ = γ − − γ
− ρ
�����
����
V( 1)C Tv(P,T) b
P P∞
γ −= ++
![Page 20: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/20.jpg)
Each fluid EOS requires 6 parameters:
V,P ,C ,b,q,q'∞γ
EOS parameters
k,VkCγ
k,Vk, C,P∞
)T(h\\ sat,k
)T(v\\ sat,k
kh (P,T)
The liquid and gas saturation curves are used
v(P,T)
VL (T) )T(L\\ sat,V lg q,q
lglglg gg,pp,TT ===,l
Bln(p) A ClnT Dln(p p )
T ∞= + + + + )T(P\\ sat lg 'q,'q
![Page 21: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/21.jpg)
Psat
hl
vl
Lv
hg
vg
![Page 22: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/22.jpg)
Data for water
![Page 23: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/23.jpg)
Kinetic parameters ?
The model requires two relaxation parameters H and ν (heat exchange and evaporation kinetics).
Evaporation front observations in expansion tubes (Simoes Moreira and
Shepherd, JFM, 1999)
Existence of 4 waves: left expansion, evaporation front, contact discontinuity, right shock wave.
![Page 24: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/24.jpg)
Local thermodynamic equilibrium is assumed at interfaces
� Thermodynamic equilibrium is forced locally.
� Waves propagate on both sides of the front and produce metastable states that relax to equilibrium at interfaces only.
� When heat diffusion is present metastable states result of heat transfer as well.
ε−<α<ε∞+
=ν otherwise 0
1 ifH,
k
εαε −<< 1k
Mixture cellSame idea as with contact interfaces:
![Page 25: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/25.jpg)
Stiff thermodynamic relaxation
g g g Le Y e (T,P) (1 Y )e (T,P)= + −
The asymptotic solution when H and ν tend to infinity corresponds to the solution of the algebraic system:
g g g Lv Y v (T,P) (1 Y )v (T,P)= + −
g LT T T= =
g LP P P= =
g L satg (T,P) g (T,P) T T (P)= ⇔ =
Unknowns (Psat,Yg) � equilibriumgY
Non-linear system quite delicate to solve in particular when the states range from two-phase to single phase : Allaire, Faccanoni and Kokh, CRAS 2007
Le Metayer, Massoni and Saurel, ESAIM, 2013
g g sat g L sate Y e (P ) (1 Y )e (P )= + −
g g sat g L satv Y v (P ) (1 Y )v (P )= + −
![Page 26: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/26.jpg)
Simple and fast equilibrium solver (Chiapolino, Boivin, Saurel, IJNMF, 2016)
eql sat lIf (Y and T T )then Y (overheated vapor)≤ ε > = ε
Single phase bounds:
eql sat lIf (Y 1 and T T )then Y 1 (subcooled liquid)≥ − ε < = − ε
Otherwise,
Compute:
eqlY 1ε < < − ε
from the mixture mass definition,
from the mixture energy definition.
These two estimates are equal only when p=psatwhich is unknown.
![Page 27: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/27.jpg)
Minmod
Then compute the product
If ( ) then,Take the minimum variation
Otherwise,Set the variation to zero
Update the mass fraction at equilibrium:
m ml l lY Y Yδ = −
e el l lY Y Yδ = −
compute the variations
m el lY . Yδ δ
m el lY . Y 0δ δ >
m el l lY Min( Y , Y )δ = δ δ
lY 0δ =
equl l lY Y Y= + δ
![Page 28: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/28.jpg)
Example: Shock tube computations with two liquid-gas mixtures and phase change
Symbols: Equilibrium computed with the Newton method.Lines: Equilibrium computed with MinmodDotted lines: Solution in temperature and pressure equilibrium without phase transition.
![Page 29: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/29.jpg)
Liquid
ρliquid = 500 kg/m3
pliquid = 1000 Bar
Vapor
ρvapor = 2 kg/m3
pvapor = 1 Barx
Contact discontinuity
shock
Evaporation front
expansion
29
Back to the Simoes-Moreira Shepherd experiment
4 waves appear
![Page 30: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/30.jpg)
Flashing front in an expansion tube (Moreira and Shepherd, JFM, 1999)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
180 200 220 240 260 280 300Température de surchauffe (°C)
Pression derrière le front (Bar)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
180 200 220 240 260 280 300Température de surchauffe (°C)
Vitesse du front de transition (m/s)Phase transition front speed (m/s) Pressure jump across the front (atm)
Experiments: blue
Computations: redSimilar results were obtained by Zein, Hantke and Warnecke, JCP, 2010Saurel et al., JFM, 2008
![Page 31: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/31.jpg)
Reduced model when heat diffusion is
considered � single temperature
31
![Page 32: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/32.jpg)
==
21
21
TT
gg
Tqcond ∇λ−=�
�
q
g
σ
Example: boiling flows
32
![Page 33: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/33.jpg)
Reduced model in both mechanical and thermal
equilibrium
( )
( )
2 111
(g g )Ydiv Y u
0t
div u 0t
ρν −∂ρ + ρ = ∂
∂ρ + ρ =∂
�
�
( )
( )( )
udiv u u PI 0
tE
div E P u 0t
∂ρ + ρ ⊗ + =∂
∂ρ + ρ + =∂
�
� �
�
The phases evolve with same pressure and same temperature but have
different chemical potentials and evolve in their own subvolumes (very
different of the Dalton law).
Conservative system ( ), comfortable for numerical
resolution.
1 2
1 2
1 2
1 2
u u
p p
T T
g g
===≠
33
U F0
t x
∂ ∂+ =∂ ∂
![Page 34: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/34.jpg)
The model is reminiscent of the reactive Euler equations
but the EOS is very different
( )( ) ( )( )2
1 2 ,1 ,2 2 1 ,2 ,1 1 2
1 1P A A P P A A P P A A
2 4∞ ∞ ∞ ∞= + − + + − − − +
( )( ) ( ) ( )( )k k vk
k ,k1 1 v1 2 2 v,2
Y 1 CA e q P
Y 1 C Y 1 C ∞
γ −= ρ − −
γ − + γ −
34
1 2
1 1 2 2e Y e (T,p) Y e (T,p)= +
1 1 2 2v Y v (T,p) Y v (T,p)= +
1 2
1 2
p p p
T T T
= == =
![Page 35: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/35.jpg)
The sound speed in lower than the previous one
T P W o o dc c≤
35
∑ ρα
=ρ k
2kk
k2w cc
1
![Page 36: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/36.jpg)
Inserting heat diffusion and surface tension
( )11 2 1
Ydiv Y u (g g )
t
∂ρ + ρ = ρν −∂
�
( ) 1 1 2 2
u m mdiv u u PI m I g
t m
E m m mdiv E P m u m I u q q g u
t m
∂ρ ⊗+ ρ ⊗ + − σ − = ρ ∂
∂ρ + σ ⊗+ ρ + + σ − σ − • + α + α = ρ • ∂
��
� �
�� � �
( )div u 0t
∂ρ + ρ =∂
�
36
1m Y= ∇�
Le Martelot, Saurel, Nkonga, IJMF, 2014
![Page 37: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/37.jpg)
11vap
11liq
2
1
K.m.W025.0
K.m.W68.0
s.m81.9g
m.mN73
−−
−−
−
−
=λ
=λ==σ
37
Computations done with an implicit low Mach hyperbolic solver.
Liquid initially at saturation temperature.
uniform heat flux
![Page 38: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/38.jpg)
11vap
11liq
2
1
K.m.W025.0
K.m.W68.0
s.m81.9g
m.mN73
−−
−−
−
−
=λ
=λ==σ
38
Same computations without nucleation sites.
![Page 39: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/39.jpg)
Another example: Atomization of O2 liquid jet surrounded by H2 vapour
![Page 40: Diffuse interfaces with compressible fluids, phase ...](https://reader035.fdocuments.us/reader035/viewer/2022062222/62a4aa8484989419bc19f9f8/html5/thumbnails/40.jpg)
40
Thank you!
Ongoing work:- Extend the Minmod method for mass transfer to the presence of non condensable
gas (liquid suspended in air for example): Chiapolino, Boivin, Saurel, Comp & Fluids, submitted
- Sharpen diffuse interfaces on unstructured meshes (Chiapolino and Saurel, SIAM, in preparation)
Superbee New limiter