ESAIM: M2AN 44 (2010) 693–713 ESAIM: Ma hema ical Modelling and Nume ical Analysis
DOI: 10.1051/m2an/2010015 www.esaim-m2an.o g
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS
TURBULENCE MODEL ∗
Tomas Chac´
on Rebollo1,S
´
ephane Del Pino2and D iss Yakoubi3
Abs ac . This pape in oduces a scheme o he nume ical app oxima ion o a model o wo u bu-
len lows wi h coupling a an in e ace. We conside he a ia ional o mula ion o he coupled model,
whe e he u bulen kine ic ene gy equa ion is o mula ed by ansposi ion. We p o e he con e gence
o he app oxima ion o his o mula ion o 3D lows o la ge u bulen iscosi ies and smoo h enough
lows, whene e bounded in W1,p Sobole no ms o pla ge enough. Unde he same assump ions, we
show ha he limi is a solu ion o he ini ial p oblem. Finally, we gi e some nume ical expe imen s
o enligh en he heo e ical wo k.
Ma hema ics Subjec Classifica ion. 63N30, 76M10.
Recei ed Ma ch 29, 2008. Re ised Augus 1s , 2009.
Published online Feb ua y 23, 2010.
1. In oduc ion
In his con ibu ion we ocus ou a en ion on he modelling o he su ace laye be ween he a mosphe e and
he ocean. We a e in e es ed in designing effec i e p ocedu es o sol e he ollowing coupled model:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−∇·(αi(ki)∇ui)+g ad pi= iin Ωi,
∇·ui=0inΩ
i,
−∇·(γi(ki)∇ki)=αi(ki)|∇ui|2in Ωi,
ui=0on Γi,
ki=0onΓ
i,
αi(ki)∂niui−pini+κ(ui−uj)|ui−uj|=0on Γ,1≤i=j≤2,
ki=λ|u1−u2|2on Γ.
(1.1)
Whe eeach iple(ui,p
i,k
i) is defined in he domain Ωi,1≥i≥2. The gene ic poin in R2, esp. inR3,is
deno ed by x=(x, z), esp. x=(x, y, z).
Keywo ds and ph ases. Ocean-a mosphe e coupling, u bulen lows, con e gence analysis, i e a i e me hod, spec al me hod.
∗T. Chac´on Rebollo was pa ially unded by EU Ma ie-Cu ie Fellowship P og amme, by Spanish Go e nmen G an MTM2006-
01275 and by Jun a de Andaluc´ıa G an P07-FQM-02538.
1Depa amen o de Ecuaciones Di e enciales y Analisis Nume ico, Uni e sidad de Se illa, Spain.
2CEA, DAM, DIF, 91297 A pajon, F ance.
3Labo a oi e Jacques-Louis Lions, Uni e si ´e Pie e e Ma ie Cu ie, 4 place Jussieu, 75005 Pa is Cedex, F ance.
A icle published by EDP Sciences c
EDP Sciences, SMAI 2010
694 T. CHAC ´
ON REBOLLO ET AL.
Sys em (1.1) is a simplified model o wo s a iona y u bulen flows in adjacen domains, coupled by bounda y
condi ions on he in e ace, such as he sys em a mosphe e-ocean. Indeed, i is a simplified ma hema ical
o mula ion o he RANS (Reynolds A e aged Na ie -S okes) model o o de 1 used o simula e a s a iona y
mean flow when con ec ion is neglec ed. This kind o modelling is o en used in enginee ing o geophysics,
see o ins ance Be na di e al. [6], Launde and Spalding [16], Mohammadi and Pi onneau [21], Pique [22],
Wilcox [23].
In wha ollows, Ωi(i=1,2) a e bounded domains o Rd,d=2,3, which a e ei he con ex o o class C1,1,
wi h bounda ies ∂Ωi=Γ
i∪Γ,Γ=Ω1∩Ω1being he in e ace be ween he wo fluids. Γ is assumed o be fla .
Indeed, we assume ha he so-called “ igid lid hypo hesis” (in oduced by B yan in [9]) holds, an hypo hesis
which is s anda d in geophysics and oceanog aphy. Each o he wo u bulen fluids is modeled by a simplified
one-equa ion u bulence model whose unknowns a e he eloci y uiand he u bulen kine ic ene gy (TKE) ki.
In he fi s equa ion we model he gene a ion o eddy iscosi y in flow iby he e m
−αi(ki)∇ui.
The (posi i e) quan i y αi(ki) is he eddy iscosi y. This is a simplifica ion o he usual modelling o Reynolds
S ess Tenso by
Ri≃−αi(ki)∇ui+∇ ui.
We p e e he fi s exp ession o simplici y o ma hema ical analysis, al hough ou analysis s ill holds o he
second one. We also neglec anspo effec s, we in end o analyze hem in a o hcoming pape . The fluids
a e assumed o be incomp essible (second equa ion). In he hi d equa ion we model he gene a ion o TKE by
means o a p oduc ion sou ce e m
αi(ki)|∇ui|2,
al hough he physical one should be
αi(ki)|∇ui+∇ ui|2.
Again, we p e e he fi s exp ession o simplici y o ma hema ical ea men . Also he u bulen diffusion
o TKE is he unc ion γi(ki). We neglec he iscous dissipa ion effec s, o a oid o manage an addi ional
s a is ic o he u bulence (a mixing leng h o he u bulen dissipa ion ε, o ins ance). We assume non-
slipping bounda y condi ions in he bounda y pa s Γi o simplici y ( ou h and fi h equa ions). These in
p ac ice a e eplaced by wall-laws o simula e he gene a ion o u bulence on solid bounda ies. The six h
equa ion globally models he in e ac ion o he wo bounda y laye s on one and ano he side o he in e ace Γ
as ic ion effec s, by means o a se o bounda y condi ions simila o Manning’s law. Finally, he las equa ion
models he p oduc ion o TKE in he in e ace. The coefficien s κiand λa e posi i e.
We assume ha he u bulen diffusions αiand γibelong o W1,∞(R)and e i yαi≥ν, γi≥ν, o some
ν>0. The eddy diffusions usually a e unbounded unc ions o he TKE o he o m a+b√k,asweusein
he nume ical simula ions epo ed in Sec ion 5(see o ins ance [8,17,21]). Bu his ende s he analysis much
mo e complex e en o a one-fluid u bulence model (see [18]). So we conside a simplified model, ha s ill
includes se e al ealis ic non-linea in e ac ions.
Sys em (1.1) was s udied in [3] whe e exis ence and uniqueness o small smoo h solu ions we e p o ed.
Spec al and Fini e Elemen disc e iza ions we e s udied in subsequen pape s by he same au ho s and co-
wo ke s (see [4,5]). In hese pape s, he abili y o hese disc e iza ion echniques o app oach he solu ion o
model (1.1) was p o ed. Howe e , in bo h cases he disc e iza ions achie ed consis ed in ully non-linea se s
o algeb aic equa ions. Ou pu pose he e is o de i e i e a i e p ocedu es o sol e sys em (1.1) ha decouples
he in e ac ion o he p oblem, leading o mildly non-linea p oblems.
Le us in oduce he unc ion spaces
Xi= i∈H1(Ωi); i=0on Γi,
L2
0(Ωi)=qi∈L2(Ωi); Ωi
qi=0
.(1.2)
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 695
Conside also wo conjuga e posi i e eal numbe s and i.e.
1
+1
=1,such ha >d.
We in oduce he ollowing i e a i e p ocedu e o sol e (1.1): once known un
i∈Xi,pn
i∈L2
0(Ωi), ki∈W1, (Ωi),
i=1,2, sol e:
P oblem 1. Ob ain un+1
i∈Xi,i=1,2, such ha
Ωi
αi(kn
i)∇un+1
i:∇ idx−Ωi
(∇· i)pn+1
idx+κiΓ|un+1
i−un+1
j|(un+1
i−un+1
j)· idτ=Ωi
i· idτ,
∀ i∈Xi,and
Ωi
(∇·un+1
i)qidx=0,∀qi∈L2
0(Ωi),
and
P oblem 2. Ob ain kn+1
i∈W1, (Ωi),i=1,2, such ha
kn+1
i=0 onΓ
i,k
n+1
i=λ|un+1
1−un+1
2|2on Γ,
Ωi
γi(kn
i)kn+1
iϕidx=Ωi
αi(kn
i)|∇un+1
i|2ϕidx,∀ϕi∈W1,
0(Ωi).
Rema k 1.1. We ake >d o gi e a sense o he equa ion o he kiin model (1.1). Indeed, he e m
αi(kn
i)|∇un+1
i|2belongs o L1(Ωi), and i ollows om he Sobole Imbedding Theo em ha he es unc ion ϕi
in P oblem 2 belongs o L∞(Ωi), so ha he igh -hand membe is well defined.
Obse e ha P oblem 1 is in eali y non-linea due o he p esence o he Manning-like sou ce e m. This
is a mild non-linea i y due o he mono onic na u e o his e m, ha may be made explici in p ac ice i
mass-lumping echniques a e used.
Ou main esul s a es ha i he sequences (un
i)nand (kn
i)na e espec i ely bounded in W1,3+ε(Ωi)dand
W1,3(Ωi), hen, o small enough da a (in a con enien sense), he i e a i e scheme is con ac ing. This egula i y
is ealis ic, as i is no a om he W1,2 egula i y ha has been p o ed o p oblem (1.1) o gene al da a.
The main ing edien s o show he con e gence o ou scheme a e he con enien choices o es unc ions, and
he use o he ha monic li ings Rio Di ichle bounda y condi ions on Γ on he Ωi(see he p oo o Thm. 3.4).
Ou pape is o ganised as ollows. In Sec ion 2we in oduce a weak o mula ion o he abo e i e a i e
p ocedu e. Sec ion 3is de o ed o p o e he con ac i eness o he TKE sequence. Due o he p oduc ion e m
o he TKE on in e ace Γ: ki=λ|u1−u2|2, i is necessa y o es ima e he exp ession
|un+1
1−un+1
2|2−|un
1−un
2|2H
1
2
00(Γ) ,(1.3)
whe e he special space H
1
2
00(Γ) is he subspace o H1
2(Γ) whose ex ension by ze o o ∂Ω1( o ins ance, i could
be also o ∂Ω2) belongs o H1
2(∂Ω1). An in insic scala p oduc on H
1
2
00(Γ) is defined as
((u, ))H
1
2
00 (Γ) =Γ
u(x) (x)dx+ΓΓ
(u(x)−u(y)) ( (x)− (y))
|x−y|ddxdy+Γ
u(x) (x)
d(x, ∂Γ) ,(1.4)
whe e he fi s wo summands define he H1
2(Γ) scala p oduc (see Adams and Fou nie [1], Thm. 7.48). I s
exp ession in ol es he dis ance d(x, ∂Γ) o he bounda y o ∂Γ. I comes om he es ic ion o H
1
2
00(Γ) o he
scala p oduc in H1
2(∂Ω1) o ins ance. I is gi en by Lions and Magenes in [20], Chap e 1, Theo em 11.7.
696 T. CHAC ´
ON REBOLLO ET AL.
The es ima ion o (1.3) is done using a G is a d’s esul (Lem. 3.5 in he pape a hand, see [14] o he
o iginal e e ence) on es ima es in Ws,p o p oduc s o unc ions o Wsj,pj, o some eal numbe s s, sjand
some non-nega i e in ege s p, pj,j=1,2. Conce ning he sequence o p essu e i e a es (pn
i)n, we use a specific
in -sup condi ion (see Co . 3.7) o show ha i is a Cauchy sequence, see Theo em 3.8.
The con e gence analysis is pe o med in Sec ion 4.InTheo em4.1 we p o e ha he iple (un
i,kn
i,p
n
i)n
has a unique limi , which is a solu ion o he a ia ional o mula ion (2.3)–(2.4).
We finally p esen some nume ical es s in Sec ion 5. These es s a e ealized wi h he so wa e F eeFEM3D
(see [12]) in meaning ul si ua ions, ha ag ee wi h he expec a ions o ou esul .
2. I e a i e scheme
We shall a fi s desc ibe he weak o mula ion o p oblem (1.1). We assume ha αiand γia e bounded
unc ions om he se o nonnega i e eal numbe s R+on o R, and belong o W1,∞(Ωi), which sa is y
∀∈R+,δ
1≥αi()≥νand δ1≥γi()≥ν, (2.1)
and
∀∈R+,|α
i()|≤δ2and |γ
i()|≤δ2,(2.2)
whe e δ1,δ
2and νa e posi i e cons an s.
Sys em (1.1) admi s he ollowing a ia ional o mula ion:
Find (ui,p
i,k
i)∈Xi×L2(Ωi)×W1, (Ωi) such ha , o all ( i,q
i,ϕ
i)∈Xi×L2(Ωi)×W1,
0(Ωi),
ai(ki;ui, i)+bi( i,p
i)+κiΓ|ui−uj|(ui−uj)· idτ=Ωi
i· idx
bi(ui,q
i)=0,(2.3)
and,
ki=0onΓ
i,k
i=λ|ui−uj|2on Γ,and
Ci(ki;ki,ϕ
i)=Ωi
αi(ki)|∇ui|2ϕidx,(2.4)
whe e he o ms ai(·;·,·),b
i(·,·)andCi(·;·,·) a e defined by
ai(i;ui, i)=Ωi
α(i)∇ui:∇ idx,
bi( i,q
i)=−Ωi
qi∇· idx,
Ci(i;ki,ϕ
)=Ωi
γi(i)∇ki·∇ϕidx.
No e ha he bilinea o ms aiand Ciin (2.3) depend on ki.
Rema k 2.1. Since ui∈Xi hen i s ace on Γ belongs o H
1
2
00(Γ). Thus by using defini ion o his space ui|Γ
belongs o H1
2(Γ) and applying he Sobole embedding om H1
2(Γ) in o L3(Γ)d, we conclude ha he in eg al
Γ|ui−uj|(ui−uj)· idτis well defined.
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 697
This o mula ion makes sense, as αi(ki)|∇ui|2∈W1, (Ωi)whenui∈Xi. In Lewandowski [19]i isp o ed
ha his o mula ion admi s a leas a solu ion.
We shall conside he ollowing i e a i e p ocedu e:
Gi en (un
i,p
n
i,kn
i)∈Xi×L2(Ωi)×W1, (Ωi),i=1,2, ob ain (un+1
i,p
n+1
i,kn+1
i)∈Xi×L2(Ωi)×W1, (Ωi),
such ha ∀( i,q
i,ϕ
i)∈Xi×L2(Ωi)×W1,
0(Ωi),
ai(kn
i;un+1
i,∇ i)+bi( i,p
n+1
i)+κiΓ|un+1
i−un+1
j|(un+1
i−un+1
j)· idτ=Ωi
i· idτ, (2.5)
and bi(un+1
i,q
i)=0,(2.6)
and
kn+1
i=0onΓ
i,(2.7)
kn+1
i=λ|un+1
1−un+1
2|2on Γ,(2.8)
Ci(kn
i;kn+1
i,ϕ
i)=Ωi
αi(kn
i)|∇un+1
i|2ϕidx.(2.9)
3. Con ac i eness
In his sec ion we p o e ha he sequence o TKE (kn
i)nis con ac ing, and ha consequen ly he sequences
o eloci ies (un
i)nalso is con ac ing, in he sense ha
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
2

i=1 ∇(un+1
i−un
i)2
L2(Ωi)≤K
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi),and
2

i=1 ∇(kn+1
i−kn
i)2
L2(Ωi)≤K
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi).
(3.1)
We may in e p e hese inequali ies in he sense ha he sequence o pai s (un
i,kn
i)nis con ac ing in he Hilbe
space Xi×L2(Ωi). Howe e , o simpli y ou de i a ion, we shall no explici ly use his space. Finally, we show
ha he p essu es (pn
i)nis a Cauchy sequence.
We suppose om now on ha he sequences (un
i)nand (kn
i)n e i y he ollowing hypo hesis.
Hypo hesis 3.1. ∀n∈N,un
i∈W1,3+ε(Ωi)dand kn
i∈W1,3(Ωi), and one has
un
iW1,3+ε(Ωi)d≤M, kn
iW1,3(Ωi)≤M,
whe e Mand εa e wo ixed posi i e numbe s.
Rema k 3.2. No e ha he na u al es ima es o eloci ies in model (1.1)a einH1no m, no in W1,3no m.
Indeed, choosing iequal o 1
κiun+1
i∈Xiin equa ion (2.5), and summing upon i=1,2gi es
2

i=1
1
κiΩi
αi(kn
i)|∇un+1
i|2dx+Γ|un+1
1−un+1
2|3dτ=
2

i=1
1
κiΩi
iun+1
idx.
Since he in eg a ed e m on Γ is nonnega i e and hanks o (2.1), we deduce
ν
cM
2

i=1 ∇un+1
i2
L2(Ωi)d≤1
cm
2

i=1 Ωi
iun+1
idx.

698 T. CHAC ´
ON REBOLLO ET AL.
Using he Cauchy-Schwa z and Poinca ´e-F ied ichs inequali ies, we ob ain
2

i=1 ∇un+1
i2
L2(Ωi)d≤c
ν2
2

i=1  i2
L2(Ωi)d,(3.2)
whe e cis a posi i e cons an , depending only on he domains Ωiand he ic ion coefficien s κi.
We nex p o e ha he con ac i eness o he TKE implies ha o he eloci ies.
Lemma 3.3. Assume ha Hypo hesis 3.1 holds and ha i∈L2(Ωi)d,i=1,2. Then he e exis s a posi i e
cons an c, depending only on Ωi, such ha
2

i=1 ∇(un+1
i−un
i)2
L2(Ωi)d≤cδ2
2M2
ν2
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi)d.(3.3)
P oo . Le us ake i=1
κi
(un+1
i−un
i)∈Xias a es unc ion in (2.5)a i e a ionsnand n+ 1. Then,
calcula ing he diffe ence be ween bo h ob ained equa ions, and summing on i=1,2, yields
2

i=1
1
κiΩi
αi(kn)∇un+1
i−un
i:∇un+1
i−un
idx
+
2

i=1
1
κiΩiαi(kn)−αikn−1
i∇un
i:∇un+1
i−un
idx
+Γun+1
1−un+1
2un+1
1−un+1
2−|un
1−un
2|(un
1−un
2)·un+1
1−un+1
2−(un
1−un
2)dτ=0.
The ollowing inequali y holds o all ec o s a,b∈Rd,
(|b|b−|a|a)·(b−a)≥0.(3.4)
To p o e i , conside he unc ion J:Rd→ Rdefined by J(a)=2
3|a|3.Jis con ex and diffe en iable. Thus,
(∇J(a)−∇J(b)) ·(b−a)≥0,∀a,b∈Rd.
Then, (3.4) ollows as ∇J(a)=|a|a·b.
We deduce ha
2

i=1
1
κiΩi
αi(kn
i)|∇(un+1
i−un
i)|2dx+
2

i=1
1
κiΩi
(αi(kn
i)−αi(kn−1
i))∇un
i·∇(un+1
i−un
i)dx≤0.(3.5)
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 699
I comes om Hypo hesis 3.1 ha ∇un
ibelongs o L3(Ωi)dand ha ∇un
iL3(Ωi)d≤M. Fu he mo e, acco ding
o he ela ion (2.2) and he canonical injec ion om H1(Ωi) oL6(Ωi)andH¨olde inequali y, we ob ain
ν
2

i=1 ∇(un+1
i−un
i)2
L2(Ωi)d≤δ2
2

i=1 Ωi|kn
i−kn−1
i||∇un
i||∇(un+1
i−un
i)|dx
≤δ2
2

i=1 kn
i−kn−1
iL6(Ωi)∇un
iL3(Ωi)d∇(un+1
i−un
i)L2(Ωi)d,
≤δ2
2M2
2ν
2

i=1 kn
i−kn−1
i2
L6(Ωi)+ν
2
2

i=1 ∇(un+1
i−un
i)2
L2(Ωi)d.
F om his es ima e we conclude ela ion (3.3). 
We nex p o e he con ac i eness o he sequence o TKE (kn
i)n.
Theo em 3.4. Assume ha Hypo hesis 3.1 holds and ha i∈L2(Ωi)d. Then he e exis s a posi i e cons an c,
depending only on Ωiand on he da a κiand λ, such ha o all n∈N∗,
2

i=1 ∇(kn+1
i−kn
i)2L2(Ωi)≤c(δ2
1+1)δ2
2
ν3M
2

i=1 ∇(kn
i−kn−1
i)2L2(Ωi).(3.6)
P oo . The p oo o his heo em is made in se e al s eps.
Fi s s ep. Choice o he es unc ion
We fi s choose a pa icula es unc ion ϕiin he equa ions (2.7)–(2.9). Fo ha pu pose, we need o
in oduce he special space H
1
2
00(Γ) (see [20], Chap. 1, Thm. 11.7 o ins ance). We also need o in oduce he
ollowing ope a o . Le Ribe a con inuous ha monic li ing ope a o om H
1
2
00(Γ) o H1(Ωi), defined as ollows.
Fo any ηin H
1
2
00(Γ), Riηbelongs o H1(Ωi), and sa isfies
⎧
⎨
⎩
−ΔRiη=0inΩ
i,
Riη=ηon Γ,and
Riη=0onΓ
i.
Mo eo e , one has
∀η∈H
1
2
00(Γ),RiηH1(Ωi)≤cRηH
1
2
00(Γ),(3.7)
whe e cR>0 depends only on Ωi.
Acco ding o Hypo hesis 3.1,∀n∈N∗,k
n
i∈W1,3(Ωi), hen i s ace on Γ belongs o W2
3,3(Γ). Thus, by
Sobole ’s injec ions, i belongs o H1
2(Γ). Fu he mo e, kn
i=0onΓ
i, henkn
i|Γbelongs o H
1
2
00(Γ).
The idea consis s in choosing he es unc ion ϕiequal o (kn+1
i−kn
i)−Ri(kn+1
i−kn
i)inequa ion(2.9)a
s eps nand n+ 1. Then, we make he diffe ence be ween bo h ob ained equa ions, and sum upon i=1,2. We
find
ν
2

i=1 ∇(kn+1
i−kn
i)2
0,Ωi≤
7

j=1
Ij,(3.8)
700 T. CHAC ´
ON REBOLLO ET AL.
whe e
I1=
2

i=1Ωi
αi(kn
i)(|∇un+1
i|2−|∇un
i|2)(kn+1
i−kn
i)dx
,
I2=
2

i=1Ωi
(αi(kn
i)−αi(kn−1
i))|∇un
i|2(kn+1
i−kn
i)dx
,
I3=
2

i=1Ωi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇(kn+1
i−kn
i)dx
,
I4=
2

i=1Ωi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇Ri(kn+1
i−kn
i)dx
,
I5=
2

i=1Ωi
αi(kn
i)(|∇un+1
i|2−|∇un
i|2)Ri(kn+1
i−kn
i)dx
,
I6=
2

i=1Ωi
(αi(kn
i)−αi(kn−1
i))|∇un
i|2Ri(kn+1
i−kn
i)dx
,and
I7=
2

i=1Ωi
γi(kn
i)∇(kn+1
i−kn
i)·∇Ri(kn+1
i−kn
i)dx
.
Second s ep. Es ima es o Ij,1≤j≤7
Es ima ion o I1.We w i e |∇un+1
i|2−|∇un
i|2=∇un+1
i−un
i·∇un+1
i+un
i, and use Hypo hesis 3.1,and
ela ion (2.1). Thanks o he Sobole embedding o H1(Ωi)in oL6(Ωi)and omH¨olde and Poinca ´e-F ied ichs
inequali ies, we ob ain
Ωi
αi(kn
i)(|∇un+1
i|2−|∇un
i|2)(kn+1
i−kn
i)dx
≤δ1∇(un+1
i−un
i)L2(Ωi)dun+1
iW1,3(Ωi)d+un
iW1,3(Ωi)dkn+1
i−kn
iL6(Ω)
≤Mcδ1∇(un+1
i−un
i)L2(Ωi)d∇(kn+1
i−kn
i)L2(Ωi),
whe e cis a posi i e cons an , depending only on domains Ωi. To simpli y he calcula ions, we in oduce a
posi i e numbe βwhich we shall fix la e .
Acco ding o Young’s inequali y
1
βa2+βb2≥2ab, ∀a, b ∈R,and ∀β>0,(3.9)
we ob ain
Ωi
αi(kn
i)(|∇un+1
i|2−|∇un
i|2)(kn+1
i−kn
i)dx≤ν
β∇(kn+1
i−kn
i)2
L2(Ωi)+βM2δ2
1c2
ν∇(un+1
i−un
i)2
L2(Ωi)d.
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 701
Summing upon i=1,2, and due o he ela ion (3.3) om Lemma 3.3, he e exis s a posi i e cons an c1,
depending only on Ωi,αiand M, such ha
I1≤c1βδ2
1δ2
2M2
ν3
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi)+ν
β
2

i=1 ∇(kn+1
i−kn
i)2
L2(Ωi).(3.10)
Es ima ion o I2and I3.Using he same a gumen s we used o es ima ion o I1, he e exis s wo posi i e
cons an s, depending only on Ωi,γ
iand M, such ha
I2≤c2βδ2
2M2
ν
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi)+ν
β
2

i=1 ∇(kn+1
i−kn
i)2
L2(Ωi),(3.11)
and
I3≤c3βδ2
2M2
ν
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi)+ν
β
2

i=1 ∇(kn+1
i−kn
i)2
L2(Ωi).(3.12)
Es ima ion o I4.We ecall ha
I4=
2

i=1Γi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇Ri(kn+1
i−kn
i)dx
.
Le us apply he Mean Value Theo em o he unc ion γi,use ela ion(2.2)andH¨olde inequali y. We find
Γi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇Ri(kn+1
i−kn
i)dx≤δ2kn
i−kn−1
iL6(Ωi)∇kn
iL3(Ωi)∇Ri(kn+1
i−kn
i)L2(Ωi).
The con inui y o he li ing ope a o (3.7)andHypo hesis3.1 imply
Γi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇Ri(kn+1
i−kn
i)dx≤cRδ2Mkn
i−kn−1
iL6(Ωi)kn+1
i−kn
iH
1
2
00(Γ).
Acco ding o he con inui y o he canonical injec ion om H1(Ωi) oL6(Ωi), he con inui y o he ace ope a o
om Xi o H
1
2
00(Γ), and using Young’s inequali y (3.9), he e exis s a posi i e cons an c4>0, depending only
on Ωi,γ
i,and M, such ha
Γi
(γi(kn
i)−γi(kn−1
i))∇kn
i·∇Ri(kn+1
i−kn
i)dx≤c4βδ2
2M2
ν∇(kn
i−kn−1
i)2
L2(Ωi)+ν
β∇(kn+1
i−kn
i)2
L2(Ωi).
Summing on i=1,2, we find
I4≤c4βδ2
2M2
ν
2

i=1 ∇(kn
i−kn−1
i)2
L2(Ωi)+ν
β
2

i=1 ∇(kn+1
i−kn
i)2
L2(Ωi).(3.13)
Es ima ion o I5.We ha e
I5=
2

i=1Ωi
αi(kn
i)(|∇un+1
i|2−|∇un
i|2)Ri(kn+1
i−kn
i)dx
.
708 T. CHAC ´
ON REBOLLO ET AL.
embedding om W1,
0(Ωi) oH1
0(Ω), we can w i e o m≥m0
ϕm
i−ϕiH1
0(Ωi)≤η
3·(4.3)
Thanks o he iangula inequali y, we ob ain
Ωi
γi(kn
i)∇kn+1
i·∇ϕidx−Ωi
γi(ki)∇ki·∇ϕidx≤Ωi
γi(kn
i)∇kn+1
i·∇ϕidx−Ωi
γi(kn
i)∇kn+1
i·∇ϕm
idx
+Ωi
γi(kn
i)∇kn+1
i·∇ϕm
idx−Ωi
γi(ki)∇(ki)·∇ϕm
idx
+Ωi
γi(ki)∇(ki)·∇ϕm
idx−Ωi
γi(ki)∇(ki)·∇ϕidx
.
(4.4)
H¨olde inequali y implies
Ωi
γi(kn
i)∇kn+1
i·∇ϕidx−Ωi
γi(ki)∇ki·∇ϕidx≤δ1∇kn+1
iL2(Ωi)∇(ϕm
i−ϕi)L2(Ωi)
+Ωi
γi(kn
i)∇kn+1
i·∇ϕm
idx−Ωi
γi(ki)∇(ki)·∇ϕm
idx
+δ1∇ki0∇(ϕm
i−ϕi)L2(Ωi).
Using ela ions (4.2)–(4.3), and he s ong con e gence o he sequence (kn
i)nin H1(Ωi), we deduce ha o any
unc ion ϕi∈W1,
0(Ωi), we ha e
lim
n→∞ Ωi
γi(kn
i)∇(kn+1
i)·∇ϕidx=Ωi
γi(ki)∇(ki)·∇ϕidx.
To show he second equa ion o he ela ion (4.1), we w i e
Ωi
αi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕidx
≤Ωi
αi(kn
i)|∇un+1
i|2−|∇ui|2|ϕi|dx+Ωi|αi(kn
i)−αi(ki)||∇ui|2|ϕi|dx,
hanks o Hypo hesis 3.1, he Sobole embedding om H1(Ωi) oL6(Ωi)andH¨olde inequali y, we can w i e
o all ϕi∈D(Ωi),
Ωi
αi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕidx
≤δ1ϕi∞∇un+1
iL2(Ωi)d∇uiL2(Ωi)d∇(un+1
i−ui)L2(Ωi)d+cM2δ2ϕi∞kn
i−kiH1(Ωi),
whe e cis a posi i e cons an , depending only on he Ωi. Due o he s ong con e gence in H1o he sequences
(un
i)nand (kn
i)n o uiand ki,
Ωiαi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕidx≤η
3·

AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 709
Le now ϕi∈W1,
0(Ωi). The densi y o D(Ωi)inW1,
0(Ωi), implies ha he e exis s a sequence (ϕm
i)min D(Ωi),
such ha
ϕm
i−ϕiL3(Ωi)≤c(Ωi)ϕm
i−ϕiW1, (Ωi)≤η
3,
whe e c(Ωi) only depends on Ωi. Finally, using he iangula inequali y, we ob ain
Ωiαi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕidx≤Ωiαi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕm
idx
+Ωi
αi(kn
i)|∇un+1
i|2(ϕm
i−ϕi)dx
+Ωi
αi(ki)|∇ui|2(ϕm
i−ϕi)dx
.
Thus,
Ωiαi(kn
i)|∇un+1
i|2−αi(ki)|∇ui|2ϕidx≤αi∞∇un+1
i2
L3(Ωi)dϕm
i−ϕiL3(Ωi)
+ε
3+αi∞∇ui2
L3(Ωi)dϕm
i−ϕiL3(Ωi).
We deduce ha ∀ϕi∈W1,
0(Ωi),lim
n→∞ Ωi
αi(kn
i)|∇un+1
i|2ϕidx=Ωi
αi(ki)|∇ui|2ϕidx.
Thi d s ep. Bounda y condi ions o TKE on Γ
In his s ep, we show ha ki=λ|u1−u2|2on Γ.
Conside he ollowing iangula inequali y
λ|u1−u2|2−kiH1
2(Γ) ≤λ|un
1−un
2|2−kiH1
2(Γ) +λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ).
Due o he s ong con e gence o (kn
i)n o kiin H1(Ωi), and hanks o he con inui y o he ace ope a o om
H1(Ωi) oH1
2(Γ), he sequence kn
icon e ges s ongly o kiin H1
2(Γ). Fu he mo e, kn
i=λ|un+1
1−un+1
2|2
on Γ. Thus,
lim
n→∞ λ|un+1
1−un+1
2|2=ki,
s ongly in H1
2(Γ).
Le us now p o e ha
lim
n→∞ λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ) =0.
Using he iden i y a2−b2=(a−b)(a+b) and Lemma 3.5,weob ain
λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ) ≤λ||(un
1−u1)−(un
2−u2)||H1
2(Γ) || (un
1+u1)−(un
2+u2)||W1−1
3+ε,3+ε(Γ)d,
so,
λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ)
≤λun
1−u1H1
2(Γ) +un
2−u2H1
2(Γ)un
1+u1W1−1
3+ε,3+ε(Γ)d+un
2+u2W1−1
3+ε,3+ε(Γ)d.
Due o he con inui y o he ace ope a o s om W1,3+ε(Ω1)d o W1−1
3+ε,3+ε(Γ)dand om H1(Ωi) oH1
2(Γ),
and by Hypo hesis 3.1, he e exis s a posi i e cons an c, ha only depends on he domains Ωi, such ha
λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ) ≤cMλ un
1−u1H1(Ω1)+un
2−u2H1(Ω2).
710 T. CHAC ´
ON REBOLLO ET AL.
Finally, due o he s ong con e gence o (un
i)n o uiin H1(Ωi), we deduce ha
lim
n→∞ λ|un
1−un
2|2−λ|u1−u2|2H1
2(Γ) =0.
Consequen ly, ki=λ|u1−u2|2on Γ. This, finishes he p oo o his heo em. 
5. Nume ical expe imen s
To conclude his pape , we use he algo i hm in oduced o sol e he in e ac ion o he ocean and he
a mosphe e in a simplified geome y. The disc e iza ion is pe o med using a Spec al me hod based on Legend e
polynomials (see [2,10]o [11] o ins ance) ha we ha e implemen ed in F eeFEM3D4.
The algo i hm p esen ed in his pape is nonlinea , hus i canno be used “as is”. Va ious s a egies could
be employed o ea his nonlinea i y (New on o fixed poin algo i hms, e c.). Acco ding o he mono onic
na u e o he nonlinea ic ion bounda y condi ion on he bounda y Γ, we choose o linea ize he e m
Γ|un+1
i−un+1
j|(un+1
i−un+1
j)· idτ
o P oblem 1. We eplace i by
Γ|un
i−un
j|(un+1
i−un+1
j)· idτ.
This e m is linea and s ill mono onic in he unknowns (un+1
1,un+1
2), due o p ope y (3.4). The p oblems o
(un+1
1,p
n+1
1)and(un+1
2,p
n+1
2) s ill a e coupled bu he o e all p oblem is linea and admi s a unique solu ion:
Ob ain (un+1
i,p
n+1
i,kn+1
i)∈Xi×L2(Ωi)×W1, (Ωi), such ha ∀( i,q
i,ϕ
i)∈Xi×L2(Ωi)×W1,
0(Ωi),
ai(kn
i;un+1
i,∇ i)+bi( i,p
n+1
i)+κiΓ|un
i−un
j|(un+1
i−un+1
j)· idτ=Ωi
i· idτ, (5.1)
and bi(un+1
i,q
i)=0.(5.2)
To pe o m a somewha ealis ic compu a ion, we conside u bulen iscosi ies αiand γiwi h he s uc u e
ν +√k. We conside he da a epo ed in [5]:
•Geome y:
–Ω1=]0,5[ ×]0,1[ ×]0,1[ desc ibes he a mosphe e;
–Ω1=]0,5[ ×]0,−1[ ×]0,1[ is he ocean.
•Physical da a ( aken om [5]):
–γ1(k1)=3×10−3+0.277 ×10−4√k1;
–γ2(k2)=3×10−2+0.185 ×10−5√k2;
–αi(·)=γi(·), o i=1,2.
•F ic ion coefficien s (coming also om [5]):
–κi=10
−3, o i=1,2; and
–λ=5×10−2.
These da a co espond o an ai -sea flow, each modeled by a simplified TKE-mixing laye u bulence model.
The mixing leng hs a e calcula ed by wall laws. The ic ion coefficien s, in hei u n, a e jus en a i e.
The physical uni s a e in MKS sys em. The eloci y bounda y condi ions imposes ha u1=0on Γ1 ˜
Γ1,
u1=(1,0,0) on ˜
Γ1and u2=0on Γ2,whe e˜
Γ1is he uppe ace (y=1)o Ω
1. These se ings a e chosen in
o de o c ea e a d i en ca i y-like flow in Ω1. One expec s o gene a e ano he d i en ca i y-like flow in Ω2
o a ing in he opposi e sense.
4h p://www. ee em.o g/ 3d/.
AN ITERATIVE PROCEDURE TO SOLVE A COUPLED TWO-FLUIDS TURBULENCE MODEL 711
(a) u1on he plane z=0.5(b)u1on he plane x=2.5
(c) u2on he plane z=0.5(d)u2on he plane x=2.5
Figu e 1. Veloci y fields on cu ing planes.
(a) k1on he plane z=0.5(b)k2on he plane z=0.5
Figu e 2. Tu bulen Kine ic Ene gy on cu ing plane.
The esul s a e ob ained using a (PN)3×PN−2disc e iza ion o (ui,p
i) o a oid spu ious modes o he S okes
p oblem. The TKE is disc e ized using a PNspace. Fo his pa icula simula ion we ha e chosen he ollowing
deg ees o uiand ki: 28 in di ec ion xand 8 in di ec ions yand z.
Compu ed eloci y fields uiand he TKE kia e ep esen ed in Figu es 1and 2. The esul s a e quan i a i ely
co ec : The a mosphe e flow gene a es a d i en-lid like flow in he ocean, due o he bounda y condi ions on
he eloci y a z= 0. Also, he e is gene a ion o TKE a z= 0, again due o he TKE gene a ion bounda y
condi ions.
712 T. CHAC ´
ON REBOLLO ET AL.
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
0 1 2 3 4 5 6 7 8 9
u1
u2
k1
k2
Figu e 3. Con e gence his o y: compu ed L2no m o he diffe ence o successi e i e a es.
Figu e 3shows he expec ed exponen ial con e gence a e o he algo i hm due o i s con ac i eness. No e
ha his does no depend on he ype o disc e iza ion (see [24] whe e Fini e Elemen app oxima ions ha e also
been used).
6. Conclusion
In his pape , we ha e p esen ed and analyzed a nume ical scheme o he app oxima ion o a model o wo
s eady u bulen fluids wi h coupling a he in e ace. This is a simplified model o he a mosphe e-ocean
in e ac ion, whe e we ha e neglec ed Co iolis o ces and buoyancy effec s, bu ha e kep se e al non-linea
in e ac ions ac oss he common bounda y.
The p oposed scheme is mainly linea , a mono one nonlinea i y being jus kep a he in e ace be ween
he fluids. We showed he con e gence o he iple (un
i,p
n
i,kn
i), o easonable hypo hesis on he egula i y
o he eloci y and he u bulen kine ic ene gy. This con ibu ion ends wi h some nume ical esul s, in good
ag eemen wi h he heo e ical expec a ions. No ably, he exponen ial con e gence ha appea ed h ough he
con ac i eness o he sequences (un
i)nand (kn
i)nis ound in ou es s.
Se e al ex ensions o his wo k can be conside ed. To name a ew, aking in o conside a ion aniso opic
diffusion and Co iolis o ces (mo e ealis ic) a e s aigh o wa d gene aliza ions o he p esen analysis. Also,
swi ching o uns eady incomp essible flows should be possible using ou app oach. Taking in o accoun buoyancy
effec s is mo e echnically in ol ed, bu i should also be possible, simila ly o he ex ension o he s anda d
analysis o incomp essible Na ie –S okes o buoyancy effec s.
Acknowledgemen s. The au ho s a e g a e ul o Ch is ine Be na di o he help and encou agemen s.
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