Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

This is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.     

Infinite Prandtl number limit of Rayleigh‐Bénard convection

We rigorously justify the infinite Prandtl number model of convection as the limit of the Boussinesq approximation to the Rayleigh‐Bénard convection as the Prandtl number approaches infinity. This is a singular limit problem involving an initial layer. © 2003 Wiley Periodicals, Inc.     

Analysis of turbulent diffusion in the temperature variance dissipation rate equation

Results of a new direct numerical simulation (DNS) for the Rayleigh‐Bénard convection at Prandtl number Pr = 0.025 and Rayleigh number Ra = 100,000 are used to analyse the turbulent diffusion term in the transport equation for the temperature variance dissipation rate. These DNS results are also used to investigate the performance of statistical models for this turbulent diffusion term. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The destabilizing effect of boundary slip on Bénard convection

We investigate the influence of slip boundary conditions on the onset of Bénard convection in an infinite fluid layer. It is shown that the critical Rayleigh number is a decreasing function of the slip length, and therefore boundary slip is seen to have a destabilizing effect. Chebyshev‐tau and compound matrix formulations for solving the eigenvalue problem are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary layers associated with the 3-D Boussinesq system for Rayleigh–Bénard convection

We investigate the boundary layer effects of the 3-D incompressible Boussinesq system for Rayleigh–Bénard convection with vanishing diffusivity limit. By adopting the multi-scale analysis and the asymptotic expansion methods of singular perturbation theory, we construct an exact approximating solution for the viscous and diffusive Boussinesq system with well-prepared initial data. In addition, we obtain the convergence result of the vanishing diffusivity limit.     

Pattern formations of 2D Rayleigh–Bénard convection with no‐slip boundary conditions for the velocity at the critical length scales

We study the Rayleigh–Bénard convection in a 2D rectangular domain with no‐slip boundary conditions for the velocity. The main mathematical challenge is due to the no‐slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free‐slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold R_c, the system bifurcates to an attractor, which is an (m ? 1)‐dimensional sphere, where m is the number of eigenvalues, which cross zero as R crosses R_c. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when m = 2. More precisely, we rigorously prove that when m = 2, the bifurcated attractor is homeomorphic to a one‐dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent viscosity

The classical problem of thermal-convection involving the classical Navier–Stokes fluid with a constant or temperature dependent viscosity, within the context of the Oberbeck–Boussinesq approximation, is one of the most intensely studied problems in fluid mechanics. In this paper, we study thermal-convection in a fluid with a viscosity that depends on both the temperature and pressure, within the context of a generalization of the Oberbeck–Boussinesq approximation. Assuming that the viscosity is an analytic function of the temperature and pressure we study the linear as well as the non-linear stability of the problem of Rayleigh–Bénard convection. We show that the principle of exchange of stability holds and the Rayleigh numbers for the linear and non-linear stability coincide.     

Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent viscosity

The classical problem of thermal-convection involving the classical Navier–Stokes fluid with a constant or temperature dependent viscosity, within the context of the Oberbeck–Boussinesq approximation, is one of the most intensely studied problems in fluid mechanics. In this paper, we study thermal-convection in a fluid with a viscosity that depends on both the temperature and pressure, within the context of a generalization of the Oberbeck–Boussinesq approximation. Assuming that the viscosity is an analytic function of the temperature and pressure we study the linear as well as the non-linear stability of the problem of Rayleigh–Bénard convection. We show that the principle of exchange of stability holds and the Rayleigh numbers for the linear and non-linear stability coincide.     

Thermosolutal convection at infinite Prandtl number: initial layer and infinite Prandtl number limit

We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes, Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate of O(?).     

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.     

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

In this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.     

On Thermal Convection in a Large Box

Daniels [4] has shown that the presence of imperfectly insulating sidewalls transforms the onset of Bénard convection in a layer of fluid confined by one pair of lateral boundaries from a bifurcation to a continuous transition as the Rayleigh number increases through the critical value. In this paper these ideas are extended to accommodate rectangular containers with large, horizontal dimensions. The appropriate boundary conditions for the amplitude equations are derived, and a limited discussion of the properties of the solutions is undertaken. In particular it is shown that, in agreement with earlier theories and experiments, there is a preference for rolls parallel to the shorter side.     

The effect of boundary conditions on the onset of chaos in Rayleigh–Bénard convection using energy-conserving Lorenz models

The influence of 16 boundary conditions on linear and nonlinear stability analyses of Rayleigh–Bénard system is reported. A Stuart–Landau amplitude equation for the Rayleigh–Bénard system between stress-free, isothermal boundary conditions is derived and the procedure used in this derivation serves as guidance for constructing an appropriate Fourier–Galerkin expansion for the other 15 boundary conditions to derive a generalized Lorenz model. The influence of the boundary conditions comes within the coefficients of the generalized Lorenz model. It is shown that the obtained generalized Lorenz model is energy conserving and for certain boundary conditions it retains features of the classical Lorenz model. Further, the principle of exchange of stabilities is shown to be valid for the present problem and hence it is the steady-state, linearized version of the generalized Lorenz model which yields an analytical expression for the Rayleigh number. On minimizing this expression with respect to wave number the critical Rayleigh number at which the onset of regular convective motion occurs in the form of rolls is determined for all 16 boundary conditions. It is found that these values are in good agreement with those of previous investigations leading to the conclusion that the chosen minimal Fourier–Galerkin expansion is a valid one. Exhibition of chaotic motion in the generalized Lorenz system at the Hopf Rayleigh number is studied. The phase-space plots which indicate a clear-cut visualization of the transition from regular convective motion to chaotic motion in the generalized Lorenz system are presented. Further, existence of a developing region for chaos (mildly chaotic motion) and windows of periodicity are captured using the bifurcation diagrams. It is concluded from the phase-space plots and the bifurcation diagrams that the generalized Lorenz model for certain sets of boundary conditions retains all the features of the classical Lorenz model. Such a conclusion cannot be made by a linear stability analysis and the need thus for a nonlinear analysis is highlighted in the paper.     

On the principle of exchange of stabilities in the magnetohydrodynamic simple Bénard problem

The present paper establishes a new result which was long sought after in the field of magnetoconvection, namely, the validity of the principle of exchange of stabilities for the magnetohydrodynamic simple Bénard problem in the regime Qσ₁ ? π² where Q is the Chandrasekhar number and σ₁ is the magnetic Prandtl number. This result is applicable for quite general boundary conditions and provides a natural extension of A. Pellew and R. V. Southwell's (Proc. Roy. Soc. London Ser. A176 (1940), 312–343) result for the simple Bénard problem. A corresponding result for rotatory magnetohydrodynamic simple Bénard problem is also given.     

The (3,4) Mode Interaction in the GeoFlow Framework

The problem of convection in a self‐gravitating spherical shell of fluid is commonly encountered in sciences like astrophysics and geophysics (earth's liquid core). The GEOFLOW‐experiment is a project of the European Space Agency in order to perform the spherical Rayleigh‐Bénard convection problem on the International Space Station in a micro‐gravity environment: the central force field is simulated by a dielectrophoretic one. Beyond a critical Rayleigh number Ra_c, generically an unique spherical ℓ mode becomes unstable and only stationary or travelling waves solutions are expected near the onset. But, for a critical aspect ratio η_c two consecutive modes (ℓ, ℓ + 1) are unstable. The (1,2) and (2,3) interactions have showed a rich bifurcation diagram, in particular, we have found heteroclinic cycles predicted by the theoretical study. Because of the experiment requirements, only the (3,4) one is possible. So, this paper purposes to analyse this bifurcation in non‐rotating case in the GEOFLOWframework using the theory of bifurcation with the spherical symmetry.     

Multichannel type-I intermittency in two models of Rayleigh–Bénard convection

We report on the first observation of transitions to deterministic chaos via type-I intermittency with two channels of re-injection in two equivariant autonomous dynamical systems. First, we consider the standard Lorenz system which is equivariant under the action of the rotation of π around the z-axis. We also consider the same phenomenon in a nine-dimensional model of the Rayleigh–Bénard convection which is equivariant under the action of the Klein four group, of all isometries mapping a rectangle, which is not a square, on itself.     

The Perturbed Problem on the Boussinesq System of Rayleigh-Bénard Convection

In this paper,the infinite Prandtl number limit of Rayleigh-B′enard convection is studied.For well prepared initial data,the convergence of solutions in L∞(0,t;H2(G)) is rigorously justified by analysis of asymptotic expansions.     

Stability analysis of Rayleigh�CB��nard convection in a porous medium

In this paper we study the problem of Rayleigh?CBénard convection in a porous medium. Assuming that the viscosity depends on both the temperature and pressure and that it is analytic in these variables we show that the Rayleigh?CBénard equations for flow in a porous media satisfy the idea of exchange of stabilities. We also show that the static conduction solution is linearly stable if and only if the Rayleigh number is less than or equal to a critical Rayleigh number. Finally, we show that a measure of the thermal energy of the fluid decays exponentially which in turn implies that the L² norm of the perturbed temperature and velocity also decay exponentially.     

Fronts in reactive convection: Bounds,stability, and instability

This paper examines a simplified active combustion model in which the reaction influences the flow. We consider front propagation in a reactive Boussinesq system in an infinite vertical strip. Nonlinear stability of planar fronts is established for narrow domains when the Rayleigh number is not too large. Planar fronts are shown to be linearly unstable with respect to long‐wavelength perturbations if the Rayleigh number is sufficiently large. We also prove uniform bounds on the bulk burning rate and the Nusselt number in the KPP reaction case. © 2003 Wiley Periodicals, Inc.     

Pullback attractors for nonautonomous 2D Bénard problem in some unbounded domains

In this paper, we study the 2D Bénard problem, a system with the Navier–Stokes equations for the velocity field coupled with a convection–diffusion equation for the temperature, in an arbitrary domain (bounded or unbounded) satisfying the Poincaré inequality with nonhomogeneous boundary conditions and nonautonomous external force and heat source. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite‐dimensional pullback D_σ‐attractor for the process associated to the problem. Copyright © 2013 John Wiley & Sons, Ltd.