Flow transitions in laminar Rayleigh–Bénard convection in a cubical cavity at moderate Rayleigh numbers

Pallarès, Jordi; Grau, F.X.; Giralt, Francesc

doi:10.1016/s0017-9310(98)00192-6

Cited by 73 publications

(69 citation statements)

References 17 publications

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“…This fact should be taken into account when experimentally observed or numerically obtained supercritical flows are compared. It was mentioned already that seven distinct branches of supercritical states for the Rayleigh-Bénard convection in a cube were calculated recently in [22]. Multiple patterns of Bénard-Marangoni instability in three-dimensional boxes were also obtained in [18] using a spectral approach similar to one described here.…”

Section: Discussionmentioning

confidence: 73%

“…Note that in the three-dimensional case the streaklines are not necessarily closed curves. As it follows from the following figures, a liquid particle can travel from one convective roll to another (see also [22]). In the following text the word "perturbation" will be used instead of the term "the most dangerous perturbation.…”

Section: Fig 3 Dependence Of the Critical Rayleigh Number On The Asmentioning

confidence: 95%

“…Substitution of (23), (24) in the boundary conditions (e.g., (17)- (20)) defines a set of linear equations for the coefficientsf im ,f im ,g jl ,ĝ jl ,h im ,ĥ im . After these coefficients are determined, the Galerkin series (22) satisfy all the linear homogeneous boundary conditions analytically (for details see [4]). Note also that the projection of the pressure gradient on the divergent-free basis (23), (24) is analytically zero [4], which means that the Galerkin procedure eliminates the perturbation of pressure from (14).…”

Section: Methodsmentioning

confidence: 99%

“…It is obvious that at larger Ra more modes become unstable which can lead to an appearance of multiple steady states. Recently, seven different supercritical steady states in a cubic cavity heated from below were calculated in [22].…”

Section: Problem 4: Square Horizontal Cross Sectionmentioning

confidence: 99%

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Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

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The article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of Rayleigh-Bénard instability in rectangular two-and three-dimensional boxes. The solution is carried out by the spectral Galerkin method with globally defined, three-dimensional, divergent-free basis functions, which satisfy all boundary conditions. The general description of these three-dimensional basis functions, which can be used for a rather wide spectrum of problems, is presented. The results of the parametric calculations are presented as neutral curves showing the dependence of the critical Rayleigh number on the aspect ratio of the cavity. The neutral curves consist of several continuous branches, which belong to different modes of the most dangerous perturbation. The patterns of different perturbations are also reported. The results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two-and three-dimensional models. Some extensions of the considered benchmark problem are discussed.

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Section: Discussionmentioning

confidence: 73%

Section: Fig 3 Dependence Of the Critical Rayleigh Number On The Asmentioning

confidence: 95%

Section: Methodsmentioning

confidence: 99%

Section: Problem 4: Square Horizontal Cross Sectionmentioning

confidence: 99%

See 2 more Smart Citations

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

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“…The second application that we consider is the Oberbeck-Boussinesq approximation to convection which can lead to multiple steady-state regimes (Gelfgat et al (1999)) that are physically realizable (Pallares et al (1999)). The stochastic bifurcation properties for this ‡ow have been studied in Venturi et al (2010); here we are interested to perform a detailed uncertainty quanti…cation analysis of both the transient and steady state regimes.…”

Section: Rayleigh-bénard Convectionmentioning

confidence: 99%

Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

2013

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We provide a new framework for the study of ‡uid ‡ows presenting complex uncertain behavior. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal …eld equations. By numerically solving these equations we evolve in a fully coupled way the mean ‡ow and the statistical and spatial characteristics of the stochastic ‡uctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that reduce considerably the computational cost. We analyze the transformation of energy from stochastic modes to mean dynamics, and vice-versa, by deriving exact expressions that quantify the interaction among di¤erent components of the ‡ow. The developed framework is illustrated through speci…c ‡ows in unstable regimes. In particular we consider the ‡ow behind a disk and the Rayleigh-Bénard convection for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for di¤erent parameters associated with the considered ‡ows and we reveal the low-dimensionality of the underlying stochastic attractor.

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Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: Past history and current status

Lappa¹

2005

Cryst. Res. Technol.

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The paper presents a comparative study of a number of theoretical/experimental/numerical results concerning the dynamics of natural (gravitational), Marangoni and related mixed convection in various geometrical models of widely-used technologies for the production of single-crystalline materials (Horizontal and vertical Bridgman growth, Czochralski method, Floating Zone Technique). Emphasis is given to fundamental knowledge provided over the years by landmark analyses as well as to very recent contributions. Such a knowledge is of paramount importance since it is validating new, more complex models, accelerating the current trend towards predictable and reproducible phenomena and finally providing an adequate scientific foundation to industrial processes which are still conducted on a largely empirical basis. A deductive approach is followed with fluid-dynamic systems of growing complexity being treated as the discussion progresses.

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Flow transitions in laminar Rayleigh–Bénard convection in a cubical cavity at moderate Rayleigh numbers

Cited by 73 publications

References 17 publications

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: Past history and current status

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