A low diffusive Lagrange+Remap scheme for the
simulation of violent air
air-water
water free
free-surface
surface flows
flows.
Applications to dam-break and sloshing events.
1,2, Florian De Vuyst
Aude Bernard-Champmartin
p
y 1
1: CMLA, ENS Cachan/ LRC ENS Cachan - CEA DAM
2: Present adress: INRIA Nice and LRC MESO ENS Cachan - CEA DAM
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Introduction
• Simulation
Simulationoffreesurfaceflows
of free surface flows
• Dam‐breakevents,Offshorepipes,Sloshing events,Effects oftidal
bore…
• Allthese examples imply alargedeformation oftheinterfacebetween the
two fluids:anaccurate method tofollow theinterfaceis required.
We focuson:
• Air/watersimulations:two fluids with ahigh ratioofdensity
• Physical effects that havetobe taken into account:
• Effects ofthecompressibility oftheair
• Since l /  g  1000 ,Archimède’s buoyancy effects
• Airpocket formation
• We neglect:
• Surfacetension
• Viscosity ofthefluids
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Choice of the modelisation
• Ch
Choice
i ofthenumerical
f th numerical method
• Particle methods (mesh‐less)
• Simulationusingg SPH((Smoothed Particle Hydrodynamics
y
y
))techniquesor
q
MPS(Moving particle semi‐implicit)
Without numerical diffusionat theinterface
Need anhigh
an high number ofparticles
of particles
high computational cost
• Numerical methods with amesh
• With interfacecapturing orinterfacetracking
Abletofollow largedeformation oftheinterface
Diffusion at theinterface.
Diffusionat
the interface
 Treatment oftheliquid which is quasi‐incompressible
 asacompressiblefluid with astiff equation ofstate
Less computational cost (than solving thePoissonequation)
High celerity ofsound
Highcelerity
of sound which leads toareduction
to a reduction ofthetimestep
of the time step
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Choice of the EOS for each fluid
• Fortheair,we useanisentropic perfect gas law:
 g 
pg (  g )  P0 
  
 g0 
g
 g  1.4,  g  1.28, P0  105
0
• Forthewater:
• We useanisentropic
use an isentropic modified Taït EOSfortheliquid
EOS for the liquid

0 2
    l


cs
l
l
Bulk modulus
pl ( l )  P0 1  K   0   1 
K
  l 

P0 l




with forinstance:  l  7, l0  1000, c s  350 m.s 1 , (K  175)
P
2
Weaklycompressible:if
P  0 ,   999.59
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System to solve
A four equation volume
Afourequation
volume‐averaged
averaged air
air‐water
watermodelwith
model with pressureequilibrium
pressure equilibrium
closure
 t       u   0
:
 t   cg      cg u   0
where
 t   u     u  u   p   g
mean density
u : mean velocity
cg : concentrationofgas
it
g : gravity
volumefractionofgas
with    g  (1   ) l
density ofgas
density ofliquid
+
pressureequilibrium assumption to
l
get  andthen  g and:
p  Pg (  g )  Pl ( l )
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solved
 g mg

  0,1

m
mg   gV ,
and cg 
ml  (1   ) lV
with aNewtonalgorithm
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Numerical scheme
• Solved with aLagrange
a Lagrange‐remap
remap scheme:BBC
scheme: BBC
[WC84] P. Woodward and P. Colella J. Comp.Phys., 54, 115 (1984).
L
Lagrangian
i phase
h
 Dt
1

 u
 Dt u  p
Dt cg  0
+
ProjectionontheEulerian
cartesian grid with alow‐
difusive procedure to
keep athin interface
• Staggered mesh
Velocities defined on the
edges (
) and other
quantities on the center ( ).
)
Thelagrangian phase:
• Usemultistep intime(predictor‐correctorscheme)
• Centered inspace
p
• Useofapseudo‐viscosity tostabilize thescheme
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LOW‐DIFFUSIVE PROJECTION
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Projection of the quantities
• Attheendofthelagrangianphase:deformedmesh
1/2, L
n 1/2, L
n 1/ 2, L
n 1/2, L
Newvolumeofacell: Vi ,nj1, L  Vi ,nj  t y  uin1/2,
j  ui 1/ 2, j   t x  vi , j 1/2  vi , j 1/2 
• Projectionofthequantitiesonthecartesiancellofvolumexy.
Projection of the quantities on the cartesian cell of volume xy
• Firststep:projectionalongthexdirection:ontheintermediate
1/2, L
n 1/2, L
volume:
Vi ,nj1,* Vi ,nj1,*  Vi ,nj1, L  t y  uin1/2,
j  ui 1/2, j 
• Secondstep:projectionalongtheydirection:
1*
1/2 L
n 1/2,
1/2 L
Vi ,nj1  Vi ,nj1,*
 t x  vin, j 1/2,

v
1/ 2
i , j 1/2   xy
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Projection of the masses at first order
• We focusonthefirststep oftheprojection(thesecondis similar):
Vi ,nj1,*
Massofgasintheintermediatevolume:
g
1/2, L n 1, L
n 1, L
n 1/2, L n 1, L
n 1, L
 (1)
mg in,j1,*  mg in, j  t y uin1/2,
j  i 1/ 2, j  g i 1/2, j  ui 1/2, j  i 1/2, j  g i 1/2, j 
writtenin
conservative
form
Massofliquidintheintermediatevolume:
Mass
of liquid in the intermediate volume Vi ,nj1,* :
1/2, L
n 1, L
n 1, L
n 1/2, L
n 1, L
n 1, L

ml in,j1,*  ml in, j  t y uin1/2,
(1

)

u
(1

)




j
i 1/ 2, j
l i 1/2, j
i 1/2, j
i 1/2, j
l i 1/2, j  (2)
,L
n 1/2,, L
where: Vi ,nj1,*,  Vi ,nj  t x  vin, j 1/2,
1/2  vi , j 1/2 
1, L
• Howtocalculate thevaluesat theedges  in1/2,
  g /l i1/2, j
j and?
n 1, L
Firstchoice:
upwind ofthe
quantities.
1 L ,upw
1,
upw
n 1,
1 L ,upw
upw
 in1/2,

j
g i 1/2, j
1/2, L
 in, j 1, L  g in,j1, L if uin1/2,
j 0
  n 1, L n 1, L
n 1/2, L


if
u
 i 1, j g i 1, j
i 1/2, j  0
stablebutdiffusive.
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Low diffusive projection
• Oth
Other choice
h i forthefluxes:we
f th fl
applied
li d alow‐diffusiveprocedure
l
diff i
d
F. Lagoutière, PhD Thesis (2000)
initially developped fortheadvectionofaquantity
andextended foratwo‐fluid sytems
S.Kokh,F.Lagoutière J.C.P2010
At theinterface:
Continuetoupwind  g and l
 Make
M k a«best»choice
b t h i for
f
low diffusive
tolimit the
numerical
diffusion
1  
1, L
n 1, L
LowDiff
n 1, L ,upw
 in1/2,




j g i 1/2, j
i 1/2, j
g i 1/2, j
n 1, L
i 1/2, j

n 1, L
l i 1/2, j
n 1, L ,upw
 1   iLowDiff


1/2, j
l i 1/2, j
1, L
if uin1/2
0
 iup1/2
1, L
if uin1/2
0
 idown
1/2
stable
not
diffusive
diffusive
unstable
• Idea:take theadvantage ofboth upwind anddownwind scheme:
up
down
up
choose  LowDiff   min( idown
p
 tobe less diffusiveaspossible
,

),
)
max(

,


1/2
i

1/2
i

1/2
i
1/2 ) 

while being stable.
i 1/2
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Quantities defining the interface
• Theinterfaceis
Th i t f
i followed
f ll
d thanks
th k to2quantities:
t 2
titi
Vol of gas
  0,1 volumefractionofgaz
Vol total
mg
c

  0,1
 g
massfractionofgaz
mtot
 
   g , l , cg  
cg  l
cg l  1  cg   g
cg   g ,  l ,   
 g
 g  1    l
• Thequantity
Th
tit cg is
i advected
d t d  t cg  u 
c g  0
respectamaximumprinciple
n 1,*
c
• Sotheprojected valuehastorespectadiscrete
maximumprinciple
gi , j
 in, j 1,*   0,1
• Theprojected valuethanks
tothepressureequilibrium
mg in,j1,*
ml in,j1,*
assumption using theprojected partialmassesand.
mg in,j1,*
ml in,j1,*  0.)
(Ifand
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How to choose the flux i+1/2
1) Construct
C
t t aninterval
i t
l I(trustinterval),
I (t
ti t
l) I     hi 1/2 , H i 1/2 
with hi 1/2  min  i ,  i 1   min  iup1/2 ,  idown
1/ 2  and H i 1/2  max  i ,  i 1 
Maximumprinciple
p
p for
1, L
cg in1/2,
j
cg in,j1,*
 I
If,we
ensure consistanceand
stability properties oftheprojection
of the projection
scheme fortheconcentrationofgas cg
i 1/2
I  I1  I 2 asanintersectionof2conditions
2) Low‐diffusiveprocedure:
L Diff
 i 1/ 2   iLowDiff
1/2
Choose  i 1/2  I less diffusiveas
Low Diff
possible:take
ibl
k  i 1/2 theclosest
h l
down
 i 1/2
valuetobutstaying
inI.
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Low‐diffusive projection
Theorem: WehaveconstructatrustintervalIwhichensuresgood
propertiesforourprojectionscheme:
Trustinterval I:
1/2, L
n 1/2, L
uin1/2,
j  0 & ui 1/2, j  0
if or
,
I : i 1/2 , i 1/2  
1/2 L
1/2,
n 1/2,
1/2 L
uin1/2,

0
&
u
j
i  3/2, j  0
I
1
consistance
it
for
f cg

I
2
stability
t bilit for
f cg
ensuring amaximumprinciple for: theedges theprojected
value
l
value
l
min(x, y )

t

0.5
Under CFL condition
UnderCFLconditionandif:
and if c  2 max  u , v :
max(( u , v , c)
1/2, L
 in 1, L if uin1/2,
j 0

  n 1, L
I
thetrustinterval I  i 1/2 , i 1/2    ()andtake
else

 i 1
upw
i 1/2, j
 iLowDiff
t1/2, j  I
th positi it ofmassesofeach
thepositivity
of masses f
h phases.
h
tensures
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SIMULATIONS
WITH THE CODE ODYSSEY
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Simulations
• Collapseofaliquid
C ll
f li id column
l
with
ith anobstacle
b t l
O.Ubbink Numerical prediction oftwo‐fluid systems
with sharp interfaces,PhD thesis (1997)
a  0.146 m, d  0.024 m
Nx  Ny  150
 g  1.4,  l  7
 0g  1.28 kg.m 3 ,  0l  1000 kg.m 3
P0  105 , csound  350 m.s 1
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Simulations‐ Collapse with an obstacle
Num.Result.
Ubbink
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Num.Result.
with Odyssey
Num.Result.
Ubbink
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Num.Result.
with Odyssey
Page 16
Simulations‐ Collapse with an obstacle
Evolutionofwith thecodeOdyssey with diffmethod ofprojection
t=0 6s
t=0.6s
1st order with
Projection: upwindd
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2nd order with
upwind
d
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1st order with
Low‐Diff.
ff
Page 17
Simulations
• Collapseofaliquid
C ll
f li id column
l
with
ith anobstacle
b t l
D.M.Greeves Simulationofviscous watercolumn
collapseusing adapting hierarchical grids J.Num.
Meth.Fluids 2006
Endofthe
experimental
opentank
a  0.25,
0 25 d  0.04
0 04
Lx  Ly  1 m
Nx  Ny  600
 g  1.4,  l  7
06
0.6m
 0g  1. kg.m 3 ,  0l  1000 kg.m 3
P0  105 , csound  350 m.s 1
Exp: from
Koshizuka etalAparticlemethod..Comp.FluidMech.1995
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Simulations of a Dam‐break
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Simulations of a Dam‐break
Sim.Greeves
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Exp.Koshizuka
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Sim.Odyssey
Page 20
Simulations of a Dam‐break
Sim.Greeves
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Exp.Koshizuka
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Sim.Odyssey
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Gas mass fraction
• Numerical
N
i l diffusionappears
diff i
d t fil
duetofilamentation/fragmentation
t ti /f
t ti
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Velocity field
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Sloshing test cases – pitch motion
• Sloshing
Sl hi duetothepitchmotionofarectangular
d t th it h
ti
f
t
l tank:
t k
J.R.Shao etal .Animproved SPHmethod formodeling
liquid sloshing dynamics.Comp.Fluids 2012
L  0.64m, H  0.14m, hw  0.03m
Nx  300, Ny  67
Thetankis oscillating asa
pendulum accordingg to:
p
 (t )   0 sin r t 
Simulationareperformed
Simulation
are performed inthe
in the
frameofreference ofthetank.

with  0  6 , r  4.34 rad/s T  1.45 s 
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Sloshing test cases – pitch motion
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Sloshing test cases – pitch motion
J.R.Shao etal .Animproved SPHmethod formodeling
We superimpose
p
p
theprofileofthearticle:
p
li id sloshing
liquid
l hi dynamics.Comp.Fluids
d
i
C
Fl id 2012
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Sloshing test cases – surge motion
• Sloshing
Sl hi duetothesurge
d t th
motionofarectangular
ti
f
t
l tank:
t k
J.R.Shao etal .Animproved SPHmethod formodeling
liquid sloshing dynamics.Comp.Fluids
dynamics. Comp. Fluids 2012
L  1.73m, H  1.15m, hw  0.6m
d  0.05
0 05 m
Nx  173, Ny  115
Thetankis moving horizontally according to:
 2 t 
x(t )  A cos 

 T 
with A  0.032 m, T  1.3 s ( forced =4.83 rad/s).
Firstnatural frequency ofthe
fluid inthebox
in the box


fluid  g tanh( hw )  3.77 rad/s
L
L
fluid
 forced
Two frequencies areactingand
Experimentals results areavailable:
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O.M.Faltinsen etal .Multidimensional modal
analysis...J.Fluid.Mechanics 2000
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Sloshing test cases – surge motion
Freesurfaceelevation
f
l
ofwaterat
f
theprobe
h
b
with Odyssey
Scanned
experimental
results of
O.M.Faltinsen etal .
Multidimensional modal
modal
analysis...J.Fluid.Mechanics
2000
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Sloshing test cases – surge motion
We search tofind
to find afitofour
a fit of our curve with afunction
a function asasuperpositionoftwo
as a superposition of two signals
f (t )  A1 sin(1t  1 )  A2 sin(2t   2 ) ,we get:1  3.74  0.01 rad/s, 2  4.83  0.01 rad/s
very closetofluid  3.77 rad/s and  forced =4.83 rad/s
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Free fall of liquid and impact with liquid at rest
Lx  Ly  0.584 m
H  Ly / 4  0.146 m
d  H  0.146
0 146 m
Nx  Ny  350
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Free fall of liquid and impact with liquid at rest
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Free fall of liquid and impact with liquid at rest
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Free fall of liquid and impact with liquid at rest
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Conclusion
• Si
Simulationsoffree‐surfaceflowswithanEulerian
l ti
ff
f
fl
ith E l i numericalmethod
i l
th d
• Lagrange‐Remapscheme
• Ananti
An anti‐diffusive
diffusiveapproachonthegasmassfraction
approach on the gas mass fraction
• Thelow‐diffusiveprojectionallowsustokeepathininterfacebetweenthe
two‐fluids
• Wecanobservetheformationofair‐pocket,formationofblast,ejection
of droplets, fragmentation…
ofdroplets,fragmentation…
• Thetestcasesshowagoodagreementbetweenexperiencesandother
codes:dambreak,sloshingevents.
• Some videos ofsimulationwith thecodeOdyssey areavailable onYou
Tube:
http://www.youtube.com/user/audeBChampmartin
http://www.youtube.com/user/floriandevuyst
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Perspectives
• All
Allowingmoreboundaryconditionsinthecode(onlywallboundary
i
b
d
diti
i th
d ( l
ll b
d
conditionuptonow)
• Takingintoaccountphysicalviscosity,surfacetension.
g
p y
y
• Parallelizationofthecode.
• Publications
• “A
AlowdiffusiveLagrange‐remapschemeforthesimulationof
low diffusive Lagrange‐remap scheme for the simulation of
violentairwaterfree‐surfaceflows.”A.Bernard‐Champmartin and
F.DeVuyst.
submittedtoJournalofComputationalPhysics
http://hal.archives‐ouvertes.fr/hal‐00798783
• “A
AlowdiffusiveLagrange‐remapschemeforthesimulationof
low diffusive Lagrange‐remap scheme for the simulation of
violentairwaterfree‐surfaceflows.Applicationstodam‐breakand
sloshingevents.”Inpreparation.
LRC ENS Cachan – CEA DAM
Gdr EGRIN – 4th April
Page 35