Laurent Boudin Laurent Desvillettes
Lab.J.-L.Lions-CNRS UMR7598 CMLA -CNRSUMR8536
UniversiteParis-VI ENSdeCahan
Bo^teourrier187 61avenueduPresidentWilson
75252ParisCEDEX05 94235CahanCEDEX
FRANCE FRANCE
boudinann.jussieu.fr
desvillemla.ens-ahan.fr
RenaudMotte
Centred'etudesdeBruyeres-le-Ch^atel
CEA/DAM
BP12
91680Bruyeres-le-Ch^atel
FRANCE
renaud.motteea.fr
Abstrat. Inthiswork,weareinterestedinaomplexuid-kinetimodelthataimstotake
intoaounttheompressibilityofthedropletsofthespray. Theambientuidisdesribed
byEuler-likeequations,inwhihthetransferofmomentumandenergyfromthedropletsis
takenintoaount,whilethesprayisrepresentedbyaprobabilitydensityfuntionsatisfying
aVlasov-likeequation. Impliittermsropupbeauseoftheompressibilityofthedroplets.
Afterhavingderivedthemodel,weprovethatglobalonservationsaresatised. Thenwe
presenttwonumerialtests. Therstoneenablestovalidatethenumerialode,whilethe
seondoneisperformedinaphysiallyrealistisituation.
Contents
1. Introdution 1
2. Presentation ofthemodel 2
2.1. The unknownsof theproblem 2
2.2. Equation ofstate 2
2.3. The ompressiblemodel 3
3. Numerial tests 7
3.1. Numerial method 7
3.2. Dropletsinthermalimbalanewiththe uid 8
3.3. Hollow-one spray 10
Appendix: value ofthe physialoeÆients 12
Referenes 12
1. Introdution
Complex two-phase uidsan be modelled inmanydierent ways. The modelknown as
(fully)Eulerian gives a desriptionof bothphasesbyphysialquantitiesonlydependingon
the variables t and x (typially, for example, the density, ow veloity and energy of eah
phase [RavSa ℄).
We are here interested in a mixed uid-kinetimodeling (a.k.a.partile-gas or Eulerian-
Lagrangian),wherethepartiles(ordroplets)aredesribedbyaprobabilitydensityfuntion
(PDF) solution of a kineti equation in a phase spae (inluding at least the variables t,
x and the veloities u
p
of the partiles), whereas the ontinuous uid satises traditional
uid equations. This type of model was introdued by Williams[Wi℄ (see also [Ce℄). It is
partiularlysuitedtopolydispersedows,i.e.ows inwhihthesizeofthedropletsanvary
in a wide range. The same framework was used, for instane, by O'Rourke [ORo℄ (and his
teamintheLosAlamosNationalLaboratory),todeveloptheKivaode[Kiva2℄,[Kiva3℄,and
bymanyotherauthors(forexample,Sainsaulieu[Sa℄,Domelevo[Do ℄orMassotandVilledieu
[MasVi ℄).
This work is an attempt to take into aount the ompressibility of the droplets in a
uid-kinetimodel at both modeling and numerial levels. This question has arisen in the
framework of the study of a spray, in the ontext of the Frenh military nulear ageny
(CEA-DAM), after Motte's rst approah [Mo ℄. The droplets remain spherial but their
radii an vary. Note that, in some papers, e.g.[Kiva2℄, the assumption of spheriity is not
systematiallymade. Moreover, we do nottake into aount theexhanges ofmassbetween
a partile and the ambient medium(vaporization, hemial reations, et.) or between two
(or several) partiles (ollisions, oalesene, breakup [Ba ℄...). That implies, in partiular,
that eah partile has a onstant mass. Moreover, at a thermodynami level, we shall use
thedensity andthespeiinternalenergyeas statevariables, asin[Mo ℄.
Note thatwe shallnottakle theproblemofboundaryonditionsinthiswork.
In setion 2 we derive the equations of our model and give some elements showing its
oherene. Then, in setion 3, we briey disuss the numerial method and present some
numerialresults.
2. Presentation of the model
2.1. The unknowns of the problem. We onsider a omplex two-phase ow made up
of a uid and droplets (or partiles). The unknowns for the uid are the volume fration
(t;x), thedensity
g
(t;x), the ow veloity u
g
(t;x), themass internalenergy e
g
(t;x), the
temperatureT
g
(t;x)and theuidpressurep(t;x).
The partiles are desribed by the PDF f. The value f(t;x;u
p
;e
p
;m
p
) is the number
density of partiles of massm
p
loated at oordinate x at time t, movingwith the veloity
u
p
and having the internal energy e
p
. Afterwards, sine m
p
is a parameter whih does
not hange, we shall write f(t;x;u
p
;e
p
) instead of f(t;x;u
p
;e
p
;m
p
). We also assume the
equality of pressures inside and outside a droplet, so that we do not introdue an extra
quantity p
p
. This assumption is lassial(see [LanLi℄ p.57 or[Dr ℄): the pressures between
two phasesreahing an equilibriumis a very fast (mehanial) phenomenon withrespetto
the(thermodynamial)phenomenonoftemperaturesequilibrium. Thatimpliesinpartiular
thattheradiusr
p
ofa dropletand itsdensity
p
arenotvariablesofthePDF,asin[Kiva2℄,
butarefuntions ofe
p ,m
p
andthepressurep(t;x). Itisasigniant diereneomparedto
other already existingmodels(e.g. [Kiva2℄,[Sa ℄), inwhihone onsiders masstransfers, but
nottheompressibilityof thepartiles,and wherer
p
is a variable ofthe PDF.
2.2. Equation of state. The twoequationsof state fortheuidmake itpossibleto obtain
two algebrairelations betweenp,
g ,e
g and T
g
, thatisto say
p(t;x) = P
1 (
g
(t;x);e
g (t;x));
(2.1)
T (t;x) = T ( (t;x);e (t;x)):
(2.2)
Moreover, one an dene
p (t;x;e
p
) bytheformula
p(t;x)=P
2 (
p (t;x;e
p );e
p );
(2.3)
whih is the pressure equation inside the droplet. Finally, one denes the temperature T
p
inside thedropletby
T
p (t;x;e
p )=T
2 (
p (t;x;e
p );e
p ):
(2.4)
That implies that
p
does not depend on the variables u
p
and m
p
. Moreover, sine the
massofeahdropletisinvariant,r
p
isobviouslyafuntionoft,x,e
p andm
p
. Morepreisely,
wehave
m
p
p
= 4
3 r
p 3
: (2.5)
2.3. The ompressible model. The modelonsidered hereis an extension of thealready
existing model with inompressible droplets used by Motte in [Mo ℄, in whih we add the
terms dueto theompressibilityof thepartiles.
2.3.1. Equations of the model. We propose the followingsystem losed by the equations of
state (2.1){(2.4):
t (
g )+r
x
(
g u
g )=0;
(2.6)
t (
g u
g )+r
x
(
g u
g u
g )+r
x p=
Z
u
p
;e
p m
p f;
(2.7)
t (
g e
g )+r
x
(
g e
g u
g
)+p[
t +r
x (u
g )℄
(2.8)
= Z
up;ep
(m
p +
m
p
p r
x p)(u
g u
p
) 4r
p Nu(T
g T
p )
f;
=1 Z
up;ep m
p
p f;
(2.9)
m
p
= m
p
p r
x
p D
p (u
p u
g )+
g C
a m
p
p 2
d
p
dt (u
p u
g );
(2.10)
m
p
=4r
p Nu(T
g T
p )+
pm
p
p 2
d
p
dt
; (2.11)
t f+u
p r
x f +r
up
(f )+
ep
(f)=0;
(2.12)
where
d
p
dt
=
p
t +u
p r
x
p +
p
e
p :
Let usomment thehoiesarried outin(2.6){(2.12).
Equations (2.6){(2.8) ome from theloalonservation of mass, momentum and internal
energy for the uid. The right-hand sides model the feedbak (transfer of momentum and
energy) of the droplets on the uid (we shall later detail their meaning whileommenting
equations (2.10){(2.11)) and their form is usual[ORo℄, [Kiva2℄, [Mo ℄. One an reognize in
the left hand sides of the equations the usual terms of the two-phase ow equations. Note
that (2.7) an berewritten, thanksto (2.9){(2.10), underthemore usualform
t (
g u
g )+r
x
(
g u
g u
g
)+r
x p=
Z
up;ep
D
p (u
p u
g
)
g C
a m
p
p 2
d
p
dt (u
p u
g )
f:
Equation(2.9)expressesthefatthatthetotalvolumeisthesumofthevolumeoftheuid
f 0(sinef isinitiallynonnegative andsatisesa Vlasovequation), itislearthat1.
Althoughit isnotobviousto mathematiallyprovethat 0,suhmodelsareassumedto
bereliable onlywhen remainsloseto 1 (the volumeof the dropletsremains negligibleto
thevolume oupiedbytheuid). In theomputations shownlater, thisisalways veried.
Equations(2.6){(2.9)togetherwith(2.12)aremoreorlessstandard(f.[ORo℄forexample).
TheVlasovequation(2.12) isobtainedbywritingdowntheonservationofthenumberof
partilesinavolumeofthephasespae(int,x,u
p ande
p
). Notethatonedoesnottakeinto
aount theabrasionof thedropletsandthemasstransfersbetweenbothphases,whereasit
isdone intheKivaode [Kiva2℄, forinstane.
Theformofrelations(2.10) and(2.11) isspeito theaseofompressibledroplets. The
quantities and appear as the variations of u
p and e
p
in the phase spae, and are thus
respetively given by the fundamental relation of dynamis (2.10) and by the equation of
energytransfer (2.11) applied to agiven partile.
In (2.10), the rst two terms already appear when the dropletsare inompressible: they
representthe pressurefore andthedrag fore. Thelasttermisspei to theompressible
ase, and models an eet of mass addition. As a matter of fat, the volume variation of
the droplet impliessome loal movements of the uid whih are very similar to a lassial
addedmassfore. Themainterm m
p
p 2
d
p
dt
representsthevolumevariationof thedropletsdue
to theirompressibilityand theremaining termsareobtained byanalogy withthestandard
addedmassfore [ClGrWe ℄, [RanMa ℄. Note thatinthe omputations shownhere,thisterm
isverysmallwith respet to thedragfore.
Intheequationofenergytransfer (2.11),theterm4r
p Nu(T
g T
p
) isalreadytakeninto
aount when the dropletsareinompressible. It represents the heattransfer between both
phases. The termwithd
p
=dtmodelsthemehanialwork dueto theompressibilityofthe
partiles.
Eventually,the variousphysialoeÆients(Nu, D
p
,et.) whihappear here aredened
intheappendix. They dependonthe various quantitiesof theproblem,suhasu
p ,u
g ,et.
Remark 1. The system(2.1){(2.12) isimpliit inthe sensethat there are timederivatives
(of
p
) inright handsides of some equations. Those an bewritten interms of
t
p beause
of(2.3),thenintermsof
t
g ,
t e
g
beauseof(2.1),of
t
beause of(2.6){(2.8), andnally
of
t
p
itselfbeause of (2.9) (and(2.10){(2.11)). Thusone an provethat
t
p (t;x;e~
p )=
Z
u
p
;e
p
F(t;x;u
p
;e
p
;e~
p )
t
p (t;x;e
p
) f(t;x;u
p
;e
p ) du
p de
p
+G(t;x;e~
p );
(2.13)
wherethere are notimederivativesintheexpressions ofF and G.
WeonlydetailtheexpressionofF intheone-dimensionalase(theexpressionofG iseven
moreintriatethantheone ofF,sowehoosenotto writeitdown). Weintroduethestate
equationsderivativeoeÆients
A
g
(t;x)=
p
uid (
g
(t;x);e
g
(t;x)); B
g
(t;x)=
p
e
uid (
g
(t;x);e
g (t;x));
A
p (t;x;e
p )=
p
drop
(p(t;x);e
p
); B
p (t;x;e
p )=
e
drop
(p(t;x);e
p );
(2.14)
and thefollowingoeÆient(whih equals when thedropletsonstitutean idealgas)
p (t;x;e
p )=
1
p(t;x)
(t;x;e ) 2
B
p (t;x;e
p )
1
:
Forthesake ofsimpliity,wedo notwritethedependeneofthefuntionsinthevariables
t andx. We an thenwritedown
F(u
p
;e
p
;e~
p )=
A
p (e~
p )
m
p
p (e
p )
p (e
p )
2
A
g
g +B
g
p
g +C
a ju
p u
g j
2
:
Weshall explaininSetion3 how thisdiÆultyistakled at thenumerial level.
2.3.2. Global onservations. Firstnotethat thesystem (2.1){(2.12) is losed. Likefor most
of thesystems desribingomplexuids,we verifythatthepropertiesofonservationholds.
Proposition 1. Thetotal energy,momentum and mass of the system(2.1){(2.12) areon-
served.
Proof. Fortheonveniene oftheproof,we set
H
p
= pm
p
p 2
d
p
dt
;
sothat equation (2.11) writes
m
p
=4r
p Nu(T
g T
p )+H
p : (2.15)
We suessively verify the onservations of the total energy, momentum and mass of the
system. We rst notiethat
g
t u
g
=
g (r
x u
g )u
g r
x p
Z
up;ep m
p f (2.16)
by using (2.6) and (2.7) together. Moreover, by using (2.12) and (2.14), the derivation of
(2.9) with respet to tgives
t
=
Z
up;ep m
p
t
p
p 2
f 1
p
t f
= Z
u
p
;e
p m
p
t
p
p 2
f + 1
p u
p r
x f +r
up
(f )+
ep (f)
= Z
up;ep m
p
t
p
p 2
f+ Z
up;ep m
p
p u
p r
x f+
Z
up;ep m
p
p 2
B
p f:
(2.17)
Nowlet usomputethe variations ofthe total energyE of thesystem
dE
dt
= d
dt Z
x
g
e
g +
ju
g j
2
2
+ d
dt Z
x;up;ep m
p
e
p +
ju
p j
2
2
f
= Z
x
t (
g e
g )+
1
2 Z
x
t (
g u
g )u
g +
1
2 Z
x
g u
g
t u
g
+ Z
x;up;ep m
p
ju
p j
2
2 +e
p
t f:
Byusing(2.8),(2.7),(2.16)and(2.12)andbydiretlyeliminatingtheintegratedonservative
terms,one obtains
dE
dt
= Z
x p
t
Z
x pr
x (u
g )
+ Z
x;up;ep m
p (u
g u
p )f +
Z
x;up;ep m
p
p r
x p(u
g u
p )f
Z
x;u
p
;e
p 4r
p Nu(T
g T
p )f
+ 1
2 Z
x u
g
"
r
x
(
g u
g u
g
) r
x p
Z
u
p
;e
p m
p f
#
+ 1
2 Z
x u
g
g (r
x u
g )u
g r
x p
Z
up;ep m
p f
!
Z
x;u
p
;e
p m
p
ju
p j
2
2 +e
p
(r
up
(f )+
ep (f)):
With(2.16){(2.17), we nd
dE
dt
= Z
x;up;ep p
m
p
p 2
t
p f
Z
x;up;ep p
m
p
p u
p r
x f
Z
x;up;ep p
m
p
p 2
p
e
p
f
+ Z
x u
g r
x p+
Z
x;up;ep m
p (u
g u
p )f
+ Z
x;u
p
;e
p m
p
p r
x p(u
g u
p )f
Z
x;u
p
;e
p 4r
p Nu(T
g T
p )f
Z
x u
g r
x p
Z
x;up;ep m
p u
g f +
Z
x;up;ep m
p u
p f +
Z
x;up;ep m
p f:
Thanks to anintegration bypartsinthe seondintegral, andusing(2.9) and (2.15), we get
dE
dt
= Z
x;up;ep p
m
p
p 2
t
p f+
Z
x;up;ep m
p u
p r
x
p
p
f
Z
x;up;ep p
p 2
m
p
p
e
p
f Z
x;up;ep m
p
p r
x pu
g f
+ Z
x;up;ep m
p
p r
x p(u
g u
p )f +
Z
x;up;ep H
p f;
hene
dE
dt
= Z
x;up;ep p
m
p
p 2
t
p f +
Z
x;up;ep m
p
p u
p r
x pf
Z
x;up;ep p
m
p
p 2
u
p r
x
p f
Z
x;up;ep p
p 2
m
p
p
e
p
f
Z
x;up;ep m
p
p u
p
x pf+
Z
x;up;ep H
p f
The total massonservation holds
d
dt Z
x
g +
d
dt Z
x;up;ep m
p f
= Z
x
t (
g )+
Z
x;up;ep m
p
t f
= Z
x r
x
(
g u
g )
Z
x;u
p
;e
p m
p (u
p r
x f+r
u
p
(f )+
e
p (f))
=0;
and sodoesthetotal momentumonservation
d
dt Z
x
g u
g +
d
dt Z
x;u
p
;e
p m
p u
p f
= Z
x
t (
g u
g )+
Z
x;up;ep m
p u
p
t f
= Z
r
x p+r
x
(
g u
g u
g )
Z
x;u
p
;e
p m
p f
Z
x;up;ep (m
p (u
p r
x f)u
p +m
p
e
p
(f))+ Z
x;up;ep m
p f
=0:
That ends theproofof Proposition1.
3. Numerial tests
3.1. Numerialmethod. Ourmethodislosetothatof[Mo ℄andwasoriginallydevelopped
for a simulation ode in the ontext of thenulear industry. We use a splitting of the uid
part (2.6){(2.8) of the modeland of its kineti part (2.10){(2.12). We refer to [BDM-pro ℄
formore detailsaboutthesheme.
3.1.1. Solvingthe uid equations. Thenumerialmethod forsolving(2.6){(2.8) istime-split
into two steps, respetively alled Lagrange and \re-map" (or transport). In the rst step
(Lagrange), we solve asystemwhihisquite similarto theLagrangereformulationof(2.6){
(2.8) (see [GoRav℄): that meansthat we solve these equations ina referential whih follows
the uid ow. The seond step (re-map) gets usbak from the Lagrange referential to the
xed one. The reader will nd the study of a similar sheme for the ompressible Euler
equations(without soureterm)in[DeLag℄.
We here use a nite volume method to disretize this system. This method is spae-
split itself: every uid quantity is omputed at the enter of eah volume, exept the uid
veloities,whihareomputed at thebordersof eah volume.
3.1.2. Solving the kineti equation. We use a partile method to solve the Vlasov equation
(2.12): thePDF f is disretizedbya sum of Diramasses. Consequently, when we need to
ompute some quantity at point x, we are led to use ontrol volumes. The whole physial
spaeisdividedintosuhontrolvolumesthatdonotdependonthedropletsbutareapriori
dened. For thesakeof simpliity,inthisode, ontrolvolumes arehosen to oinide with
thenitevolumes used inthesolvingofthe uidequations.
Forinstane,ifwe wantto omputetheuidvolumefrationat timetandpositionx,we
onsider the ontrol volume V whih ontains x. Then we replae the Dira masses (with
pressure(10 6
Pa) temperature(K) density(kg.m 3
)
uid 1:00 1000 0:388
droplet 1:00 100 12:1
Table 1. Thermodynamistate ofthe uidand thedroplets
volumefration at timetand positionx writes
V
(t)=1 X
p m
p
p jVj
1
1
fxp2Vg
;
whereone sumsoverall thedropletsinside theomputationdomain.
3.1.3. Impliit in time featureof the equations. Aswe already pointedoutinRemark1,the
system(2.1){(2.12)istime-impliit. Wenotiedthatwehadtorstsolvetheimpliitintegral
equation (2.13). In theory,(2.13) shouldbesolved by axed point method: more preisely,
at eah time step, this xed point method should be implemented. However, sine we are
only interested in weakly ompressible droplets, we are led to replae this(very expensive)
proedure bythesimpleapproximation
t
p (t
n+1
;x;e~
p )=
Z
up;ep F(t
n
;x;u
p
;e
p
;e~
p )
t
p (t
n
;x;e
p ) f(t
n
;x;u
p
;e
p ) du
p de
p +G(t
n
;x;e~
p ):
3.2. Droplets inthermal imbalane with the uid. In orderto test thevalidityofour
ode, we onsidera situation in whih the initial datum does notdepend on the variable x
(uniformmedium): insideeahmesh(thevolumeofwhihequalstoV =0:025m 3
),thereare
n=5000 droplets. Both phasesare supposedto be ideal gases. The thermodynamistates
of theuid and eah dropletare given intable 1. Note thatthe uidand the dropletshave
thesame pressurebeause oftheisobari equilibriumassumption.
In this partiular ase, there exists a semi-expliit solution. As a matter of fat, rst,
the uniformitywith respetto x impliesthat there is no dependene on x for every studied
quantity. Next,sinethereisinitiallynomotioninsideboththesprayandtheuid,itislear
that we an take u
g
=0 and that the spray hasonly one elerity u
p
=0. Then f now only
depends on t and e
p
,so thatwe an hoose f as a Diramass: f(t;e
p ) =Æ
e
p
=e
p (t)
. Finally,
we have to note that the equations of state are very simple beause both phases are ideal
gases. Consequently,allthequantitiesarealgebraifuntions ofe
g ande
p
,whihsatisfythe
followingsystemof ordinarydierentialequations
e 0
g
= (
g 1)
0
e
g 1
M
g n
V 4r
p Nu
e
g
vg e
p
vp
;
m
p
p e
0
p
= 4r
p Nu
e
g
v
g e
p
v
p
+(
p 1)m
p e
p
e 0
g
e
g
0
;
where
v
g
(resp.
v
p
)is theuid(resp. droplets)isohori(thatis,whenthevolumeisxed)
spei heat,
g
(resp.
p
) the ratio of the uid (resp. droplets) spei heats, M
g is the
onstant valueof ,and where and r aregiven by
0 2e-05 4e-05 time (s)
0 200 400 600 800 1000
Temperatures (K) Droplets
Fluid
Figure 1. Towards theequilibriumof temperatures.
=
1+ n
V
p 1
g 1
m
p
M
g e
p
e
g
1
;
r
p
=
3
4
p 1
g 1
m
p
M
g e
p
e
g
1=3
:
0 2e-05 4e-05
time (s) 0
200 400 600 800 1000
temperatures (K)
CEA Maple
0 4e-06 8e-06 1.2e-05
time (s) 0
0.4 0.8
Density (kg/m3)
CEA Maple
Figure 2. Temperatureand densityof adroplet.
Two physialphenomenahappen almost simultaneously:
to balanetheinreasingofthetemperature, eahdropletdensitydereasestoreahan
equilibriumvalue(see g.2).
Fig.2 also shows that the numerial results obtained thanks to our ode t very well to
thesemi-expliitsolutions (omputed byMaple).
Thistestwasvery auratetogetbothmathematialand programmingvalidationsofthe
model. Still the thermodynamis leads to very important and unrealisti variations of the
densities ofeah droplet. The followingomputationhas beenperformed inamore realisti
thermodynamial setting.
3.3. Hollow-one spray. This seond numerial test has been set in [Du℄ and allows us
to ompare the numerial results for both ompressible and inompressible droplets. The
droplets are injeted with the same veloity 100m:s 1
in a uid following the border of a
hollow-one(with angle45 Æ
). Theuidisinitiallynotmovingandhasa highertemperature
than the droplets (500K for the uid, 300K for the spray). Both phases satisfy realisti
equationsof state (stiened gasfortheuid,standardtin forthespray).
Figure 3. Evolution oftheuid temperature dueto thedroplets.
Fig.3 desribes the ourring phenomenon: the droplets make the uid move and ool
down. We here propose the evolution of a sample droplet: densities (see g.4) and tem-
peratures (see g.5). In g.4, the density of the inompressible droplet remains onstant,
Figure 4. Density ofthe sampledroplet.
Figure 5. Temperatureof thesample droplet.
temperature. The analysis of g.5 is a bit more intriate: the urve for the ompressible
droplet is behind theone forthe inompressible droplet. As a matter of fat, sine there is
rst themehanial reationof theompressibledroplet(inreasingradius),its temperature
does notinreaseas fast asthe temperature of theinompressible droplet. And we already
pointedoutinsubsetion2.1thatmehanialphenomenaourfasterthanthermodynamial
ones. But eventually, we note that for bothgures, there is approximately a 1% dierene
betweenthe inompressibleand ompressibleurves.
In fat, we notied in other numerial tests that, in the ontext of the nulear industry,
the density of a given droplet typially varies of 1% during the evolution of the spray, so
that we an state thattheeets of theompressibilityof thedropletsseemto be smallfor
a realistithermodynamis. Therefore,theintriatemodelstudiedhere shouldbe usedonly
when very preise results are needed. On the other hand, the numerial omparison is not
muh more expensive when the ompressibility is taken into aount, beause of thi small
variation of the density of a droplet. As a matter of fat, one solves the impliit equations
Appendix: value of the physial oeffiients
To ompleteour modeling, we give the value of every oeÆient whih appears in(2.8){
(2.11). Thephysialjustiationsof ourhoiesan befoundin[RanMa ℄, forinstane.
First of all, let us give some preisions about the drag oeÆient C
d
, the added mass
oeÆient C
a
and theNusseltnumberNu.
We empiriallyxthe oeÆient C
a
=0:5.
TowriteC
d
andNu,letusbegintoreallthedenitionofsomestandardoeÆients, suh
astheReynolds, Mah and Prandtlnumbers
Re= 2r
p
g ju
p u
g j
; Ma= ju
p u
g j
; Pr= C
; (3.1)
where theuid visosity is a given onstant, is the loal sound veloity inthe uid and
theuidspeiheat C and thermalondutivityaregiven.
We thentake forC
d
theformula
C
d
=
~
C
d (Re)
~
(Ma );
(3.2)
with
~
C
d
= 24
Re (
2:65
+ 1
6 Re
2=3
1:78
) si Re<Re
:=1000;
~
C
d
= 24
Re
(
2:65
+ 1
6 Re
2=3
1:78
) si Re>Re
;
and
~
= 1 si Ma<0:5;
= 2:22 si Ma>1;
= 2:44Ma 0:22 si 0:5<Ma<1:
We also take forthe Nusseltnumbertheformula
Nu= 7=4
+0:3Re 1=2
Pr 1=3
1=2
:
Finally,thetermD
p
inthe expressionof thedrag foreis xedbytheexpression
D
p
= 1
2 r
2
p
g C
d ju
p u
g j:
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