Finite Amplitude Convection Research Articles

NATURAL CONVECTION OF NANOFLUIDS IN A CAVITY INCLUDING THE SORET EFFECT

Convection of a binary mixture in a cavity is studied numerically. The flow is driven by a buoyancy force due to an externally applied constant temperature difference on the vertical wall of the cavity, while the horizontal surfaces are impermeable and adiabatic. A nanofluid is used and the effects of the cross phenomenon “Soret effect” were considered in the analysis. The flows are found to be dependent on the particle concentration `, the Rayleigh number RaT, the Lewis number Le, the solutal to thermal buoyancy ratio N, and the thermal boundary conditions. Numerical results for finite amplitude convection, obtained by solving numerically the full governing equations, are found to be in good agreement with the analytical solution based on the scale analysis approach. We have proposed a modified formulation of the conservation equations governing the flow and heat transfer of nanofluids, taking into account important changes of nanofluid thermal conductivity and viscosity as well as the spatial change of the particle concentration that is induced by the Soret effect. Results have shown that such an effect increases nanofluid heat transfer. The optimal particle volume concentration, which maximizes heat transfer, is estimated to be 2%. The increase of natural convection with nanoparticle concentration is weak in comparison to that found in forced convection.

An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq fluid, which is heated and salted from below in the presence of Soret coefficient is studied analytically using both linear and nonlinear stability analyses. The normal mode technique is used in the linear stability analysis while a weak nonlinear analysis based on a minimal representation of double Fourier series method is used in the nonlinear analysis. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameters, solute Rayleigh number, Soret parameter and Lewis number on the stationary, oscillatory, finite amplitude convection and heat and mass transfer are shown graphically.

The onset of double diffusive convection in a sparsely packed porous layer using a thermal non-equilibrium model

We examine the effect of local thermal non-equilibrium on double diffusive convection in a fluid-saturated sparsely packed porous layer heated from below and cooled from above, using both linear and nonlinear stability analyses. The Brinkman model is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that a small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusion that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, ratio of diffusivities, Vadasz number and Darcy number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Natural convection in a shallow cavity containing two superposed layers of immiscible binary liquids

This paper reports an analytical study of the natural convection in a shallow rectangular cavity containing two superposed layers of binary immiscible fluids. A uniform heat flux is applied to the horizontal walls of the enclosure while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection) or by a temperature gradient (Soret effects). An analytical solution of the steady form of the governing equations is derived using a parallel flow approximation. The critical Rayleigh numbers for the onset of critical, $${R_{\rm TC}^{\rm sup}}$$ , or subcritical, $${R_{\rm TC}^{\rm sub}}$$ , convection are predicted by the present theory. For finite amplitude convection the present model yields explicitly the flow patterns, Nusselt and Sherwood numbers in terms of the governing parameters of the problem. In the limit of a single fluid layer the present theory is found to be in agreement with the results reported in the literature.

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Double diffusive convection in a porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and nonlinear stability analyses. The Darcy model with time derivative term is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusion that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, ratio of diffusivities and Vadasz number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.

Darcy-Bénard Convection using a Non-Fourier Thermal Model

In the paper the effect of replacing the classical Fourier thermal model by the non-classical non-Fourier thermal model on Darcy-Benard convection in a fluid-saturated densely-packed porous medium is studied and, in particular, how the Nusselt numbers of the solid and fluid phases are affected by the adoption of such a model is considered. A weighted average Nusselt number is also defined and analyzed to study the overall heat transfer in the case of steady finite-amplitude convection which is amenable to analytical treatment. Sub-critical instability is ruled out in mono-diffusive convection but the inclusion of a second diffusing component or rotation is shown to facilitate its manifestation.

An analytical study of linear and non-linear double diffusive convection with Soret effect in couple stress liquids

The double diffusive convection in a two-component couple stress liquid layer with Soret effect is studied using both linear and non-linear stability analyses. The linear theory is based on normal mode technique and the non-linear analysis is based on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret parameter, the solute Rayleigh number, the Prandtl number and the diffusivity ratio on the stationary, oscillatory and finite amplitude convection are shown graphically. It is found that the effects of couple stress are quite large and the positive Soret number enhances the stability while the negative Soret number enhances the instability. The non-linear theory predicts that, finite amplitude motions are possible only for negative Soret parameter. The transient behaviour of thermal and solute Nusselt numbers has been investigated by solving numerically a fifth order Lorenz model using Runge–Kutta method.

Flows in a fluid layer induced by the combined action of a shear stress and the Soret effect

Convection in a horizontal fluid layer of a binary mixture is studied analytically and numerically. In the formulation of the problem, use is made of the Boussinesq approximation. Neumman boundary conditions are specified for the temperature on all walls of the cavity. In addition of the Soret contribution, a shear stress, τ, is applied on the upper free surface of the layer. The flows are found to be dependent of the Darcy–Rayleigh number, R T, the Lewis number, Le, the solutal to thermal buoyancy ratio, ϕ, the shear stress, τ and the thermal boundary conditions. Numerical results for finite amplitude convection, obtained by solving numerically the full governing equations, are found to be in good agreement with the analytical solution based on the parallel flow approach. For given sets of the control parameters, the occurrence of multiple steady state solutions is demonstrated. The existence of subcritical bifurcations for both stabilizing and destabilizing mass flux is also demonstrated.

Onset of convection and finite amplitude flow due to Soret effect within a horizontal sparsely packed porous enclosure heated from below

The Soret effect on thermal natural convection within a horizontal porous enclosure heated uniformly from below by a constant heat flux is studied analytically and numerically using the Brinkman–extended Darcy model along with the Boussinesq approximation. The governing parameters of the problem are the Darcy–Rayleigh number, R T, the Lewis number, Le, the separation parameter, ϕ, the Darcy number, Da, and the aspect ratio of the enclosure, A r. In the limit of a shallow enclosure, an analytical closed form solution is obtained on the basis of a parallel flow approximation. A numerical study is also conducted to validate the results of the analytical predictions. The thresholds for the onset of stationary and finite amplitude convection are determined explicitly as function of Le, ϕ and Da. The obtained results show that in the ϕ– Le plane there exists three regions that correspond to different parallel flow regimes, namely the subcritical and supercritical flows. The separation parameter has a strong effect on the threshold of instabilities and on the characteristics of heat and mass transfer. The threshold for Hopf bifurcation is obtained on the basis of the linear stability analysis. The wavenumber and the oscillation frequency at the onset of instabilities are null for an infinite extent layer.

Influence of grain size evolution on convective instability

Grain size, one of the most important microstructural properties of materials, evolves during creep deformation to minimize the free energy of polycrystalline aggregates. We apply a model of grain size evolution to the study of convective instability of cooling boundary layers. The grain size evolution model is coupled to a composite rheology where the deformation rate is the sum of that due to dislocation creep and grain size sensitive diffusion creep. The onset of convection is sensitive to grain growth rates and the initial grain size. The formation of convective instabilities is enhanced by stresses induced by plate motions; therefore small‐scale convection is more likely to occur beneath fast‐moving plates. In finite amplitude convection, grain size evolution leads to high viscosity in regions where convective stresses are low and can induce viscosity contrasts exceeding one order of magnitude. Such viscosity contrasts are sufficient to influence the dynamics of convection, often leading to domains which remain isolated from the well‐mixed convecting fluid. A composite viscosity including diffusion creep, which has a lower activation energy than dislocation creep, reduces the effective temperature dependence of viscosity.

Nonlinear Convection in Porous Media: A Review

The review article deals mainly with nonlinear convection (NLC) in porous media and discusses analytical and numerical techniques to handle them. A section on experimental heat transfer in porous media relates to the state of the art on the subject. Five techniques for studying NLC are presented in a logical order of traversal. The first technique, due to Lyapunov, is useful to obtain energy bounds and in conjunction with a variation technique can be used to investigate for possible subcritical instabilities. The power integral and the spectral methods concern steady finite-amplitude convection. The former method is useful in obtaining information on physically preferred cell patterns, whereas the latter can handle cross-interaction of different modes in addition to estimating heat transfer. The Fourier decomposition for unsteady large-amplitude convection is capable of predicting chaos and quantifying heat transfer. The finite-difference method, or any numerical method, when guided by the results of the existing analytical methods and experiment, can be used effectively to handle a more general problem with realistic boundary conditions. The results of the experimental and theoretical study are meant to mutually ratify the respective findings. The present scenario on heat transfer in porous media is such that not all observed aspects can be covered in a theoretical study and also not all results predicted by the theory are experimentally realizable. It thus calls for concerted effort from various quarters. It is on this ground that the review puts together many aspects of NLC in porous media, taking essential excerpts from previous works, with an unavoidable lean on the authors' own works.

INFLUENCE OF THE SORET EFFECT ON CONVECTION IN A BINARY FLUID LAYER WITH A FREE UPPER SURFACE

We investigate natural convection in a binary fluid filling a horizontal cavity with an undeformable free upper surface. The cavity is heated from below and cooled from above by a constant heat flux while the vertical walls are maintained adiabatic. The fluid is subject to the Soret effect. In the limit of a shallow cavity, an analytical model is derived on the basis of a parallel flow approximation. The critical Rayleigh numbers for the onset of supercritical and subcritical convection are predicted in terms of the governing parameters of the problem. Also, results are obtained for finite amplitude convection. The study is completed by a numerical solution of the full governing equations

A systematic experimental study of rapidly rotating spherical convection in water and liquid gallium

Results of finite-amplitude convection experiments in a rotating spherical shell are presented. Water (Prandtl number P=7) and liquid gallium ( P=0.027) have been used as working fluids. In both liquids, convective velocities could be measured in the equatorial plane using an ultrasonic Doppler velocimetry technique. The parameter space has been systematically explored, for values of the Ekman and Rayleigh numbers E>7×10 −7 and Ra<5×10 9. Both measured convective velocity and zonal circulation are much higher in liquid gallium than in water. A scaling analysis is formulated, which shows that higher convective velocities are an effect of the low Prandtl number in liquid gallium, and that higher zonal flows can be explained through a Reynolds stress mechanism. The Reynolds numbers in gallium ( Re=250–2000) are higher indeed than in water ( Re=25–250). An inertial regime sets up at high Re, in which kinetic energy does not dissipate at the scale of convective eddies and is transferred up to the scale of the container, where it is dissipated through Ekman friction of zonal flow. This upwards energy transfer can be seen as an effect of quasigeostrophic (QG) turbulence. Applying the scaling relations to an hypothetic non-magnetic flow in the Earth’s core yields Reynolds numbers of the order of 10 8, in fair agreement with values required for dynamo action, convective velocities of order 10 −3 m/s, zonal flow of similar amplitude, and eddy scales as low as 10 km.

On numerical stability analysis of double-diffusive convection in confined enclosures

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

Finite-amplitude convection in the presence of finitely conducting boundaries

Finite-amplitude convection in the form of rolls and their stability with respect to infinitesimal disturbances is investigated in the case of boundaries of the horizontal fluid layer which exhibit a thermal conductivity comparable to that of the fluid. It is found that even when rolls represent the preferred mode at the onset of convection a transition to square cells may occur at slightly supercritical Rayleigh numbers. The phenomenon of inertial convection in low Prandtl number fluids appears to become more pronounced as the conductivity of the boundaries is reduced. Modulated convection rolls have also been found as solutions of the problem. But they appear to be unstable in general. Experimental observations have been made and are found in general agreement with the theoretical predictions.

Finite-amplitude double-component convection due to different boundary conditions for two compensating horizontal gradients

Finite-amplitude convective steady flows that do not bifurcate from the respective conduction state are discovered. They arise as the compensating horizontal gradients of two density-affecting components with equal diffusivities but different boundary conditions are applied to the Boussinesq fluid at rest with and without stable vertical stratification. These flows emanate from convection in a laterally heated stably stratified slot. Their relevance to convective states in a horizontal slot with two vertical gradients, emphasizing universality of the underlying type of convection, is discussed.

Onset of convection in an anisotropic porous layer of finite lateral extent

A linear stability analysis is performed to study the onset of convection in a horizontal saturated porous layer of finite aspect ratio and anisotropic in permeability. The critical Rayleigh number is established as a function of the permeability ratio, the inclination angle of the principal axes and the enclosure aspect ratio. A linear stability analysis is conducted through a numerical procedure. Also, for the weak finite amplitude convection, results are obtained by solving numerically the full governing equations.

Convection in a low Prandtl number fluid layer rotating about a vertical axis

Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found.

Onset of double-diffusive convection in a rectangular porous cavity subject to mixed boundary conditions

The onset of double-diffusive convection in a horizontal porous cavity is studied numerically using linear stability analysis. In the formulation of the problem, use is made of the Darcy model with the Boussinesq approximation. Mixed boundary conditions for heat and solute are specified on the horizontal walls of the enclosure while the two vertical ones are impermeable and adiabatic. The Galerkin and the finite element methods are used to solve the perturbation equations. The onset of convection is found to be dependent of the aspect ratio of the cavity, A, normalized porosity, ε, Lewis number, Le, solutal to thermal buoyancy ratio, N, and the thermal and solutal boundary conditions. For a confined enclosure, it is shown that there exists a supercritical Rayleigh number, R sup TC, for the onset of the supercritical convection and an overstable Rayleigh number, R over TC, at which overstability may arise. Furthermore, the overstable regime is shown to exist up to a critical Rayleigh number, R osc TC, at which the transition from the oscillatory to direct mode convection occurs. However, for an infinite layer ( A→∞) the results indicate the absence of an overstable regime. Numerical results for finite amplitude convection, obtained by solving numerically the full governing equations, demonstrate that subcritical convection is possible.