Abstract
When particle-laden freshwater is placed above clear saltwater, **double**-**diffusive** sedimentation can arise. Navier-Stokes direct numerical simulations by Burns and Meiburg showed that this process can be dominated by either Rayleigh-Taylor or **double**-**diffusive** fingering instabilities. Based on two- dimensional simulations, those authors identify a single dimensionless parameter that can be em- ployed to distinguish between these regimes. Here we develop a high-performance semi-Lagrangian computational approach that enables us to extend these high Schmidt number simulations to three dimensions, and to confirm the validity of their proposed scaling law for three-dimensional flows.

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In the present work, we study the onset of **double** **diffusive** convection in vertical enclosures with equal and opposing buoyancy forces due to horizontal thermal and concentration gradients 共in the case Gr S /Gr T ⫽⫺1, where Gr S and Gr T are, respectively, the solutal and thermal Grashof
numbers 兲. We demonstrate that the equilibrium solution is linearly stable until the parameter Ra T 兩Le ⫺1兩 reaches a critical value, which depends on the aspect ratio of the cell, A. For the square

1 Introduction
The role of the moisture on the dynamics of atmospheric air has been studied several years ago, from various points of view. For instance, Dudis (1972), Einaudi and Lalas (1973), Durran and Klemp (1982a, b; 1983), considered its influence on the Brunt-V¨ais¨al¨a frequency and inviscid flows (such as mountain lee waves). Einaudi and Lalas and later, Durran and Klemp, focused their attention on media with a non- constant distribution of moisture with respect to the altitude. Deardorff (1976, 1980), and Betts (1982) constructed re- duced schemes using the so-called “liquid-water moist po- tential temperature”: such schemes were used by Bougeault (1981a, b), in order to study the moist atmospheric turbu- lence. From another point of view, Kuo (1961, 1965), Ogura (1963) studied convective instability in dissipative media. More recently, Bretherton and Smolarkiewicz (1987, 1988, 1989) considered the motion, the appearance and the disap- pearance of clouds under moist convection effects. Some consequences of the diffusion taking into account the mi- crostructure of the medium have also been exhibited by Kambe and Takaki (1975), and Merceret (1977). In these works, the **double** **diffusive** characteristics of the medium are either absent (when dissipation is neglected in the medium), or not really taken into account.

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conditions. An analytical and numerical study of natural convection heat and mass transfer through a vertical porous layer subjected to a concentration difference and a temperature difference in the horizontal direction has been studied by Trevisan and Bejan [27]. Many physical systems were modelled as a two-dimensional cavity with the vertical walls held at fixed but different temperatures or concentrations and the connecting horizontal walls considered as adiabatic or impermeable Angirasa et al. [28] were reported a numerical study of combined heat and mass transfer by natural convection adjacent to vertical surfaces situated in fluid saturated porous media. Akbal and Baytas [29] have investigated a radioactive gas transfer depending on the decay of the gas, Schmidt and concentration Grashof numbers by natural convection in a fluid saturated porous medium. Merkin and Mahmood [30] have investigated a model for the convective flow in a fluid-saturated porous medium containing a reactive component. Goyeau et al. [31] have studied the **double** **diffusive** natural convection using Darcy—Brinkman formulation in a porous cavity with impermeable boundaries. Bahloul et al. [32] have investigated the **double** **diffusive** convection in a long vertical cavity heated from the below and imposed concentration gradient from the sides both analytically and numerically.

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Forced convection, where the annular space is partially or completely ﬁlled with a ﬂuid saturated porous medium, has been signiﬁcantly analyzed ( Mohamad, 2003 ; Pavel and Mohamad, 2004 ; Yang and Hwang, 2009 ). Mohamad (2003) investigated heat transfer enhancement for a ﬂow in a pipe fully or partially ﬁlled with porous medium. Forced laminar ﬂow was assumed, and the effects of porous layer thickness on the rate of heat transfer and pressure drop have been presented. The Darcy number was varied in the range of 10 6 to 10. The results show that the plug ﬂow assumption is not valid for Da > 10 4 , where the permeability of the medium is high. Inertia term has signi ﬁcant effect on the Nusselt number ( Pavel and Mohamad, 2004 ). The effects of porosity, porous material diameter and thermal conductivity, as well as Reynolds number on the heat transfer rate and pressure drop, were studied experimentally and numerically. The pipe is subjected to a constant and uniform heat ﬂux. The results obtained show that higher heat transfer rates can be achieved using porous inserts whose diameters approach the diameter of the pipe. Yang and Hwang (2009) investigated numerically the ﬂuid ﬂow behavior and heat transfer enhancements in a pipe fully or partially ﬁlled with porous medium inserted at the core of the conduit. The numerical results reveal that heat transfer can be enhanced by using high thermal conductivity porous inserts. Moreover, the turbulent ﬂow in the conduit without porous medium can dissipate the heat from the heating wall because of the strong forced convectional effect, and the local distributions of the Nusselt number along the ﬂow direction increase with the increase of the Reynolds number and the thickness of the porous layer but increase with the decreasing of the Darcy number. Alkam and Al-Nimr (1998) , presented the transient forced convection in the developing region of a cylindrical channel partially ﬁlled with a porous substrate. The ﬂow within the porous domain is modeled by the Brinkman –Forchheimer-extended Darcy model. Results of this model show that the existence of the porous substrate may improve the Nusselt number at the fully developed region by a factor of 8. However, there is an optimum thickness of the porous substrate beyond which no signi ﬁcant improvement in the Nusselt number is achieved. Teamah and Shoukri (1995) treated a similar phenomenon to determine the effect of radius ratio, aspect ratio and buoyancy ratio on the **double** **diffusive** natural convection in vertical annulus enclosures. The ranges of the

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In the present work, we study the onset of **double** **diffusive** convection in vertical enclosures with equal and opposing buoyancy forces due to horizontal thermal and concentration gradients 共in the case Gr S /Gr T ⫽⫺1, where Gr S and Gr T are, respectively, the solutal and thermal Grashof
numbers 兲. We demonstrate that the equilibrium solution is linearly stable until the parameter Ra T 兩Le ⫺1兩 reaches a critical value, which depends on the aspect ratio of the cell, A. For the square

Abstract. Bifurcation phenomena in a square enclosure, submitted to horiwntal temperature and concentration gradients, is studied when the opposing buoyancy forces due to horizon tal thennal and concentration gradients are equal. We perfonn the linear, weakly non-linear and finite amplitude stability analysis of the equilibrium solution. We verif.y that the onset of **double** **diffusive** convection corresponds to a transcritical bifurcation point. The subcritical solutions are strong attractors beyond a particular value of the thermal Rayleigh number which corresponds to the location of turning point. The structure of subcritical and transcritical steady solutions has been studied. © Académie des Sciences/Elsevier, Paris

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The above studies used large Lewis number binary ﬂu- ids. A large amount of work on **double** **diffusive** instabilities with moderate Lewis numbers ~gas media! has also appeared in the literature. Weaver and Viskanta 17 experimentally and numerically investigated this situation in a square cavity for 0.59,Le,2 and 29.42,N,0.55. They observed good agreement with the experimental results in the cooperating case. But due to the unsteadiness of the ﬂow, they did not obtain agreement in the opposing case. Be´ghein et al. 18 nu- merically obtained some correlations concerning mass and

The above studies used large Lewis number binary ﬂu- ids. A large amount of work on **double** **diffusive** instabilities with moderate Lewis numbers ~gas media! has also appeared in the literature. Weaver and Viskanta 17 experimentally and numerically investigated this situation in a square cavity for 0.59,Le,2 and 29.42,N,0.55. They observed good agreement with the experimental results in the cooperating case. But due to the unsteadiness of the ﬂow, they did not obtain agreement in the opposing case. Be´ghein et al. 18 nu- merically obtained some correlations concerning mass and

Aside from the aspect ratio of the cavity\ **double** di}us! ive convection involves four dimensionless parameters] the Prandtl number Pr\ the Schmidt number Sc\ and the solutal and thermal Grashof numbers Gr S and Gr T \ respectively "all of the parameters will be de_ned in sec! tion 1#[ However\ the qualitative behavior of the **double**! di}usive induced ~ow depends mainly on the buoyancy ratio N Gr S :Gr T "b C DC#:"b T DT# and the Lewis num! ber Le Sc:Pr x:D where D\ x\ b C \ b T \ DC\ and DT are the solutal and thermal di}usivities\ the coe.cients of solutal and thermal expansion\ and the di}erences of concentrations and temperatures between the vertical side!walls\ respectively[ The question of the e}ect of the Lewis number and the buoyancy ratio on the ~ow struc! ture when the temperature and the concentration are imposed along the vertical side!walls has been widely discussed by Bergeon et al[ ð6Ł\ Ghorayeb ð10Ł\ and Bennacer ð11Ł[

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c U[M[R[\ 4491 IMFT!CNRS!UPS\ U[F[R[\ M[I[G[\ 007 route de Narbonne\ 20951 Toulouse Cedex\ France
Abstract
A numerical study is presented of unsteady **double**!di}usive convection in a square cavity with equal but opposing horizontal temperature and concentration gradients[ The boundary conditions along the vertical side!walls are imposed in such a way that the buoyancy ratio N Gr S :Gr T is equal to −0\ where Gr S and Gr T are the solutal and thermal Grashof numbers\ respectively[ In this situation\ steady!state convective ~ow is stable up to a threshold value Gr c0 of the thermal Grashof number which depends on the Lewis number Le[ Beyond Gr c0 \ oscillatory convective ~ows occur[ Here we study the transition\ steady!state ~owÐoscillatory ~ow\ as a function of the Lewis number[ The Lewis number varies between 1 and 34[ Depending on the values of the Lewis number\ the oscillatory ~ow occurring for Gr T slightly larger than Gr c0 is either centro!symmetric " for Le − 06# or asymmetric single frequency ~ow " for Le ¾ 06#[ For larger values of the thermal Grashof number\ the two regimes occur for _xed values of Gr T and Le[ Furthermore\ computations show that Gr c0 reaches a minimum equal to 3[64×09 3 for Le ¼ 6[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

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La réalisation de la maquette de démonstration a été effectuée en respectant les critères relatifs aux noeuds mais avec une solution existante dans le commerce (système DUO). Nous avons pu ainsi construire un système à **double** nappe et **double** courbure et constater la faisabilité de la solution que nous proposions. Il a été nécessaire d'adapter le noeud utilisé. On ne peut utiliser le prototype ainsi construit pour multiplier les modifications géométriques en raison de la "faiblesse" du procédé: des glissements internes et des non coïncidences de fibres moyennes ne le permettent pas. Néammoins on peut considérer que le but recherché est atteint. Il conviendra de construire un prototype mécaniquement résistant avec la solution du noeud en commande numérique pour pousser plus avant les investigations sur le comportement des systèmes ce qui constitue un problème ne rentrant pas dans le cadre de cette étude.

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the **diffusive** limit. It is interesting to see that the thermal conductivity gives a different temperature dependence simply by changing L for the same sample quality of a crystal. The L-dependent thermal conductivity at low temperature is evidence of the crossover of thermal conductivity from **diffusive** to ballistic thermal conductivity which should be observed in the experiment. For a temper- ature larger than 300 K, on the other hand, since the MFP for each phonon mode is smaller than L, the L-dependent thermal conductivity does not appear anymore and the thermal conductivity is expressed only by **diffusive** thermal conductivity, in which phonon-phonon scattering is a dominant contribution to thermal resistivity.

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2) linearity, provided a well-posedness condition is ful- filled by measure µβ (see e.g. [14]).
B. Adjoints of Fractional Derivatives
An extension of these results to fractional derivatives can be done, but care must be taken that they are no more bounded (even compact in fact); the unboundedness of the fractional derivative operators gives rise to a specific **diffusive** formulation, see again [6, ch. 2] for the questions of domains. The key ingredients are:

Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web[r]

DOI: 10.1103/PhysRevFluids.2.093501
I. INTRODUCTION
Natural doubly **diffusive** convection is of considerable importance in a variety of applications, ranging from oceanography [ 1 ] to crystal growth [ 2 ]. This type of convection arises in systems driven by an imposed horizontal temperature difference or imposed horizontal heat flux when this is opposed by a compensating concentration difference or concentration flux. With fixed temperature and concentration boundary conditions this system has been studied in both two [ 3 – 5 ] and three dimensions [ 6 ]. For typical parameter values the onset of convection leads to stationary but subcritical states [ 4 , 5 ] that rapidly evolve into a state of corotating rolls, whose sense of rotation is determined by the Lewis number of the fluid. Early work has focused on the two-dimensional problem, and in particular on the onset of convection and small amplitude behavior near onset [ 3 ]. This work has demonstrated that unless the horizontal gradients are balanced exactly convection will take the form of a large cell with upflow along one sidewall and downflow along the other, provided that the system is not too extended in the horizontal. In particular, no conduction state is present, and normal convection develops as a secondary instability of this base flow. In three dimensions the spatial organization of the resulting flow depends crucially on the aspect ratio of the cavity [ 6 , 7 ]. Complex time dependence can result when the primary branch of steady convection fails to restabilize owing to the presence of instability with respect to convection with a different orientation, a situation that leads to strongly nonlinear oscillations near the onset of primary instability [ 6 ].

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almost surely finite. The local result implies more : that the norm of gradient may blow up only on a closed and negligeable set, of 4/5-capicity zero.....
In the next section the **diffusive** operator of ( 1.1) is rigourously derived from a linear Vlasov-Fokker-Planck eq:gyroFP2D eq:gyroFP2D equation in the limit of large magnetic field. In the third section, some useful lemmas are established, proving regularizing properties of the gyro-average, global preservation of some weighted norm of f , the short time preservation of the u(1 + u 2 )-moment of ∇

2.5.1.2 Problèmes d'assemblage
La simple juxtaposition de modules autocontraints ayant subi une modification symétrique des noeuds 5,6,7 et 8 n’est géométriquement pas possible: une surface à **double** courbure positive ne peut être maillée par des carrés. Seul un maillage composé de losanges peut satisfaire les conditions géométriques d'une surface à **double** courbure. La réalisation d'une maquette élémentaire a permis de montrer qu'en fait les losanges ont une forme très proche de celle des carrés. Une étude théorique plus précise reste toutefois nécessaire d'autant qu'il faut simultanément s'assurer que les conditions d'équilibre d’autocontrainte sont satisfaites.

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modifications of the mean flows and of me an heat and mass transfers are obtained in the range of low frequencies, leading to an intensification of heat and mass transfers in the cavity.[r]

5. CONCLUSION
In this paper, the controllability problem for a wide class of Volterra (scalar) systems has been studied. By considering the **diffusive** representation approach, the result is that any **diffusive** Volterra system x = H(∂ t )u is approximately controllable. This result can be trivially extended to gen- eral **diffusive** non linear and/or non t-invariant Volterra systems of the form x = H(t, ∂ t )f (t, x, v) with f an invertible function 4 , simply by replacing µ(ξ) by µ(t, ξ). In that sense, this controllability result can be viewed as a consistent extension of controllability of scalar differential systems ∂ t x = f (t, x, v), x(0) = 0. It seems now to be interesting to tackle the controllability problem for vec-

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