Nonlinear doubly diffusive convection in vertical enclosures

Bardan, Gérald; Bergeon, Alain; Knobloch, Edgar; Mojtabi, Abdelkader

doi:10.1016/s0167-2789(99)00195-5

Cited by 22 publications

(24 citation statements)

References 44 publications

(94 reference statements)

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“…The supercritical branch of the bifurcating solution is only stable up to a pitchfork bifurcation at Ra = 542.4, following a scenario similar to that described in Ref. [12]. The interesting behavior in this system originates from the subcritical branch.…”

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confidence: 61%

“…In this work we numerically study this latter configuration for a small Prandtl number binary mixture. We consider the case when thermal and solutal buoyancy forces exactly compensate each other, which allows the existence of a quiescent (conductive) state [10,11,12,13]. We have found that in this system there exists a large range of Rayleigh numbers in which the only stable solution is an orbit that features a low-frequency spiking behavior.…”

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confidence: 99%

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Blue Sky Catastrophe in Double-Diffusive Convection

Meca¹,

Mercader²,

Batiste³

et al. 2004

Phys. Rev. Lett.

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A global bifurcation of the blue sky catastrophe type has been found in a small Prandtl number binary mixture contained in a laterally heated cavity. The system has been studied numerically applying the tools of bifurcation theory. The catastrophe corresponds to the destruction of an orbit which, for a large range of Rayleigh numbers, is the only stable solution. This orbit is born in a global saddle-loop bifurcation and becomes chaotic in a period doubling cascade just before its disappearance at the blue sky catastrophe.PACS numbers: 47.27. Te, 47.20.Ky, 44.25.+f Bifurcation theory has long been a very helpful tool in the analysis of complex dynamics of nonlinear systems [1,2]. Whereas different devised scenarios have been found in theoretical models with a few variables, there is a growing interest both in relating real systems with that kind of models (e.g. projecting their dynamics to some relevant degrees of freedom [3]) and in directly analyzing the behavior of these systems in terms of dynamical systems theory (by studying them either experimentally or by realistic models). In this context a great deal of work has been devoted to convection in fluids. Qualitative changes in the dynamics of fluxes maintained out of equilibrium by imposed thermal gradients have provided examples of most of the known bifurcations, and have become a main subject in the area of nonlinear dynamics.In this letter we will show the occurrence of a blue sky catastrophe [BSC] in double diffusive convection. The BSC is a codimension-1 bifurcation that consists in the destruction of a stable periodic orbit as its length and period tend to infinity, while the cycle remains bounded and located at a finite distance from all the equilibrium solutions [1,4]. This destruction is caused by the collision with a non-hyperbolic cycle that appears at the bifurcation point. While approaching the bifurcation the orbit increasingly coils in the zone where the new cycle will appear, which originates the divergence in both period and length. In that point the original cycle becomes an orbit homoclinic to the new cycle. This type of bifurcation is relatively exotic, but can easily be found in slow-fast (i.e. singularly perturbed) systems with at least two fast variables [5].We are interested in double-diffusive fluxes that occur when convection is driven by simultaneous thermal and concentration gradients in a binary mixture [6]. Doublediffusive convection in cavities with imposed vertical gradients exhibits very rich dynamics, and has been used as a system to study pattern formation [7] and transition to chaos [8]. The case of horizontal gradients, which arises naturally in applications such as crystal growth [9] or oceanography [6], has received less attention. In this work we numerically study this latter configuration for a small Prandtl number binary mixture. We consider the case when thermal and solutal buoyancy forces exactly compensate each other, which allows the existence of a quiescent (conductive) state [10,11,12,13]. We have found th...

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confidence: 61%

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confidence: 99%

Blue Sky Catastrophe in Double-Diffusive Convection

Meca¹,

Mercader²,

Batiste³

et al. 2004

Phys. Rev. Lett.

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“…They numerically showed that the primary instability of the equilibrium corresponds to a transcritical bifurcation point. The onset of oscillatory flow 7 and nonlinear bifurcation analysis 8,9 have been done and extensions of the configuration to an inclined cavity 10 and to a tilted porous cavity 11 have also been considered.…”

Section: Introductionmentioning

confidence: 99%

Double-diffusive Marangoni convection in a rectangular cavity: Onset of convection

Chen

Zhan

2010

Physics of Fluids

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Double-diffusive Marangoni convection in a rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal Marangoni effects are of equal magnitude ͑solutal to thermal Marangoni number ratio R =−1͒. In this case a no-flow equilibrium solution exists and can remain stable up to a critical thermal Marangoni number. Linear stability analysis and direct numerical simulation show that this critical value corresponds to a supercritical Hopf bifurcation point, which leads the quiescent fluid directly into the oscillatory flow regime. Influences of the Lewis number Le, Prandtl number Pr, and the cavity aspect ratio A ͑height/length͒ on the onset of instability are systematically investigated and different modes of oscillation are obtained. The first mode is first destabilized and then stabilized. Sometimes it never gets onset. A physical illustration is provided to demonstrate the instability mechanism and to explain why the oscillatory flow after the onset of instability corresponds to countersense rotating vortices traveling from right to left in the present configuration, as obtained by direct numerical simulation. Finally the simultaneous existence of both steady and oscillatory flow regimes is shown. While the oscillatory flow arises from small disturbances, the steady flow, which has been described in the literature, is induced by finite amplitude disturbances.

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“…They showed that the primary instability of the equilibrium corresponds to a transcritical bifurcation point that is determined by Ra c ͑Le− 1͒, where Ra c is the critical Rayleigh number and Le is the Lewis number. The onset of oscillatory flow as a function of Le was studied by Ghorayeb et al 7 Nonlinear bifurcation analysis 8,9 has been done and extensions of the configuration to an inclined cavity 10 and to porous cavities 11,12 have also been considered. More recently, the situation in a three-dimensional enclosure was studied by Bergeon and Knobloch.…”

Section: Introductionmentioning

confidence: 99%

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Chen

Zhan

et al. 2010

Physics of Fluids

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Double diffusive convection in a porous layer saturated with viscoelastic fluid using a thermal non-equilibrium model Phys. Fluids 23, 054101 (2011) Influence of vibrations on thermodiffusion in binary mixture: A benchmark of numerical solutions Phys. Fluids 19, 017111 (2007) Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field Phys. Fluids 18, 073302 (2006) Additional information on Phys. Fluids Double-diffusive buoyancy convection in an open-top rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude ͑buoyancy ratio R =−1͒. In this case, a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr c . Linear stability analysis and direct numerical simulation show that depending on the cavity aspect ratio A, the first primary instability can be oscillatory, while that in a closed cavity is always steady. Near a codimension-two point, the two leading real eigenvalues merge into a complex coalescence that later produces a supercritical Hopf bifurcation. As Gr further increases, this complex coalescence splits into two real eigenvalues again. The oscillatory flow consists of counter-rotating vortices traveling from right to left and there exists a critical aspect ratio below which the onset of convection is always oscillatory. Neutral stability curves showing the influences of A, Lewis number Le, and Prandtl number Pr are obtained. While the number of vortices increases as A decreases, the flow structure of the eigenfunction does not change qualitatively when Le or Pr is varied. The supercritical oscillatory flow later undergoes a period-doubling bifurcation and the new oscillatory flow soon becomes unstable at larger Gr. Random initial fields are used to start simulations and many different subcritical steady states are found. These steady states correspond to much stronger flows when compared to the oscillatory regime. The influence of Le on the onset of steady flows and the corresponding heat and mass transfer properties are also investigated.

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Nonlinear doubly diffusive convection in vertical enclosures

Cited by 22 publications

References 44 publications

Blue Sky Catastrophe in Double-Diffusive Convection

Blue Sky Catastrophe in Double-Diffusive Convection

Double-diffusive Marangoni convection in a rectangular cavity: Onset of convection

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

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