An iterative procedure to solve a coupled two-fluids turbulence model

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. 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