Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number

We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for Rayleigh‐Bénard convection converge to that of the infinite‐Prandtl‐number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite‐Prandtl‐number model for convection as a valid simplified model for convection at large Prandtl number even in the long‐time regime. © 2006 Wiley Periodicals, Inc.     

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.     

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(?).     

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.     

Solution of the boussinesq equations by the finite element method

A finite element procedure is presented for the calculation of two-dimensional transient convective/conductive heat transfer in a fluid region. The governing equations are expressed in terms of the primitive variables; the flow is assumed to be laminar, and the fluid incompressible within the Boussinesq approximation. Three typical problems are examined: flow through a sudden enlargement, natural convection in rectangular enclosures, and natural convection between horizontal concentric cylinders. An assessment of the characteristics of the flow regime is made in association with varying dimensionless Prandtl and Rayleigh numbers, as well as cavity aspects ratios. The upper limit for the Rayleigh number in the present paper is 10⁷. Wherever possible, the results are compared with existing solutions obtained by other numerical methods.     

侧向局部加热对流的周期性

通过流体力学方程组的数值模拟,研究了侧向局部加热条件下Prandtl数Pr=0.0272时流体对流的周期性.结果表明:随着Grashof数Gr的增加,对流按稳态对流、单局部周期对流、双局部周期对流、准周期对流的顺序发展.当Gr<3.6×103时,对流为稳态;在3.6×103相似文献   

De la convection naturelle instationnaire sur une plaque verticale infinie

Summary This paper considers the unsteady laminar free convection produced by heating an infinite vertical flat plate. Analytical solutions in closed forms for the velocity and temperature profiles, when the surface temperature varies stepwise and as a power of time, are obtained for Prandtl numbers close to unity.Results are reported for several values of the Prandtl number in case of a stepwise varying plate temperature.     

Solitary Roll Vortices in the Mixed Convection of a Vertical Fluid Layer

Some earlier experimental and numerical findings from convection in vertical channels suggest that localized convection rolls could play a role in the transition process of free or mixed convection. In the present work solitary convection roll vortices for a vertical fluid layer with stable stratification, differential shear and differentially heated side-walls have been obtained numerically for a fluid of unit Prandtl number. The solutions appear through saddle-node bifurcations and in certain parameter ranges they do also exist for linearly stable basic flow. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of rotation on the onset of double diffusive convection in a Darcy porous medium saturated with a couple stress fluid

The effect of rotation on the onset of double diffusive convection in a horizontal couple stress fluid-saturated porous layer, which is heated and salted from below, is studied analytically using both linear and weak nonlinear stability analyses. The extended Darcy model, which includes the time derivative and Coriolis terms, has been employed in the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. The effect of Taylor number, couple stress parameter, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the rotation, couple stress parameter and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The Lewis number has a stabilizing effect in the case of stationary and finite amplitude modes, with a destabilizing effect in the case of oscillatory convection. The Darcy–Prandtl number and normalized porosity advances the onset of oscillatory convection. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. The transient behavior of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge–Kutta method.     

Influence of surface mass transfer on mixed convection flows over non-isothermal horizontal flat plates

Non-similar solution of a steady mixed convection flow over a horizontal flat plate in the presence of surface mass transfer (suction or injection) is obtained when there is power-law variation in surface temperature. The surface temperature is assumed to vary as a power of the axial coordinate measured from the leading edge of the plate. A non-similar mixed convection parameter is considered which covers the whole convection regime, namely from pure free convection to pure forced convection. Numerical results are reported here to account the effects of Prandtl number, surface temperature, surface mass transfer parameter (suction or injection) on velocity and temperature profiles, and skin friction and heat transfer coefficients.     

Banerjeeet al’s characterization theorem in thermohaline convection and its magnetorotatory extensions

The paper mathematically establishes that magnetorotatory thermohaline convection of the Veronis [7] type cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermohaline Rayleigh numberR _s, the Lewis number τ, the thermal Prandtl number σ, the magnetic Prandtl number σ₁, the Chandrasekhar numberQ and the Taylor numberT satisfy the inequality \(R_s 4\pi ^4 \left[ {1 + \frac{\tau }{{\sigma \pi ^2 }}\left\{ {\pi ^2 - \left( {O\sigma _1 + \frac{T}{{\pi ^2 }}} \right)} \right\}} \right]\) when both the boundary surfaces are rigid thus achieving magnetorotatory extension of an important characterization theorem of Banerjeeet al (1992) on the corresponding hydrodynamic problem. A similar characterization theorem is mathematically established in the context of the magnetorotatory thermohaline convection of the Stern (1960) type.     

Nonlinear Stability Problem of a Rotating Doubly Diffusive Porous Layer

Lyapunov direct method is applied to study the non-linear conditional stability problem of a rotating doubly diffusive convection in a sparsely packed porous layer. For a Darcy number greater than or equal to 1000, and for any Prandtl number, Taylor number, and solute Rayleigh number it is found that the non-linear stability bound coincides with linear instability bound. For a Darcy number less than 1000, for a Prandtl number greater than or equal to one, and for a certain range of Taylor number, a coincidence between the linear and nonlinear (energy) stability thermal Rayleigh number values is still maintained. However, it is noted that for a Darcy number less than 1000, as the value of the solute Rayleigh number or the Taylor number increases, the coincidence domain between the two theories decreases quickly.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

In this paper, the initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating solution with expansions up to any order are given and the convergence rates O(ɛ ^m+1/2) and the optimal convergence rates O(ɛ ^m+1) are obtained respectively. This improves the result of J.G. SHI.     

侧向加热腔体中的多圈型对流斑图

基于流体力学方程组的数值模拟,研究了倾角θ=90°时侧向加热的大高宽比腔体中的对流斑图.对于Prandtl数Pr=6.99的流体,在相对Rayleigh数2≤Ra r≤25的范围内,腔体中发生的是单圈型对流斑图.对于Pr=0.0272的流体,取Ra r=13.9,随着计算时间的发展,腔体中由最初的单圈型对流斑图过渡到多圈型对流斑图,这是出现在侧向加热大高宽比腔体中的新型对流斑图.对不同Ra r情况的计算结果表明,Ra r对对流斑图的形成存在明显的影响.当Ra r≤4.4时是单圈型对流滚动;当Ra r=8.9~11.1时是过渡状态;当Ra r≥13.9时是多圈型对流滚动.对流最大振幅和Nusselt数Nu随着相对Rayleigh数的增加而增加.该对流斑图与Pr=6.99时对流斑图的比较说明,对流斑图的形成依赖于Prandtl数.     

Instability mechanisms in buoyant-thermocapillary liquid pools

A cylindrical volume of fluid, with a free surface on top, is heated by a parabolic heat flux from above. Two physical effects drive a flow: thermocapillary effects due to free-surface temperature gradients introduced by the non-uniform heat flux and buoyancy forces due to gravity. The basic axisymmetric flow is computed by finite volumes and its stability is investigated by a linear-stability analysis. It is found that the critical stability boundaries and modes are similar to those known from the half-zone model of crystal growth. For low Prandtl numbers the critical mode is steady and three-dimensional. We find an asymptotic critical value in the limit of vanishing Prandtl number. For increasing Prandtl number the critical Reynolds number increases. Near unit Prandtl number no threshold could be found with the present computational limitations. For Prandtl numbers larger than unity, the critical mode is oscillatory and the critical Reynolds number decreases with the Prandtl number. We present evidence that the low- and high-Prandtl-number instabilities are essentially centrifugal respectively due to the hydrothermal-wave mechanism. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oscillatory free convection flow of a second order fluid from a vertical plate

This paper investigates the problem of free convection flow of a second order liquid in the boundary layer from a semi-infinite vertical plate in which the mean surface temperature varies as a function of the distance from the leading edge of the plate. Study of the oscillatory flow is restricted to small amplitudesε only. Several graphs have been drawn and tables have been presented to depict the effect of elasticity of the liquid and Prandtl number on the velocity and temperature distributions and Nusselt number.     

Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux

Effect of free convection on the visco-elastic fluid (Walter - B’ type) flow past an infinite vertical plate accelerating in its own plane with constant heat flux is examined analytically. It is found that for given values of Grashof number, Prandtl number and Newtonian parameter; flow velocity at any point increases with the increase in time and non-Newtonian parameter, however, it decreases with both, the heating and cooling of the plate.     

Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate

A numerical study of magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate is presented. The onset position characterized by the local Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The Coriolis force and the Lonertz force have significant effects on the formation of longitudinal vortices and the associated instability. Positive rotation stabilizes the flow on the rotating flat surface. On the contrary, a negative rotation destabilizes the flow. The flow is found more stable as the value of Hartmann number increases. The numerical data show reasonable agreement with the experimental results with the case of thermal instability in natural convection over a flat plate heated from below.     

Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.