Stability Of Natural Convection Research Articles

Stability of natural convection in a vertical layer of Navier-Stokes-Voigt fluid

The stability of natural convection in a vertical layer of viscoelastic fluid confining between rigid-isothermal side walls held at different temperatures is investigated numerically using the Chebyshev collocation method. The viscoelastic behavior is modelled by means of a constitutive equation encompassing the Navier-Stokes-Voigt fluid or the Kelvin-Voigt fluid of order zero. The onset of convective instability is examined by linearizing the governing equations for the perturbations and an appropriate extension of Squire's theorem is given making a case to consider only two-dimensional perturbation stability equations. The numerical solution of the stability eigenvalue problem leads to the determination of the neutral stability condition. The dependence of the Kelvin-Voigt parameter on the critical stability parameters and also on the Prandtl number at which the point of transition from stationary to travelling-wave mode occurs is thoroughly analysed. The Kelvin-Voigt parameter shows an important role on the travelling-wave mode instability where it inducts both stabilizing and destabilizing effects on the base flow depending on the values of Prandtl number, while its impact on the stationary mode is found to be very weak. Furthermore, the streamlines and isotherms of the perturbation modes presented herein demonstrate the development of complex dynamics at the critical state. The results of a Newtonian fluid are obtained as a particular case.

Stability of natural convection in a heated vertical corner

The spatial stability of three-dimensional non-parallel natural convection in a uniformly-heated vertical corner is examined for the first time. Two self-similar steady base flows are determined, which are distinguished by their behaviour far from the corner. The first solution (A) reduces to the flow induced by a heated semi-infinite vertical plate, with a crossflow component that is driven by the entrainment into the corner. The second solution (B) is new and in contrast has a crossflow that increases linearly with the spanwise coordinate. In both solutions, the crossflow is directed towards the corner, being driven by the so-called `chimney effect’. By adopting a PSE approach, we show that the computed base flows become unstable to viscous modes at sufficiently large Grashof numbers. In the case of solution A, the eigenmodes that initially dominate are well approximated by the two-dimensional stability problem for the boundary layer over a heated vertical plate. However, further downstream these modes are later overcome by modes that are corner-specific (and cannot be captured by a simpler planar far-field stability problem).

The role of a second diffusing component on the Gill–Rees stability problem

The stability of natural convection in a vertical porous layer using a local thermal nonequilibrium model was first studied by Rees (Transp Porous Med 87:459–464, 2011) following the proof of Gill (J Fluid Mech 35:545–547, 1969), called the Gill–Rees stability problem. The aim of the present study is to investigate the implication of an additional solute concentration field on the Gill–Rees problem. The stability eigenvalue problem is solved numerically and some novel results not observed in the studies of double-diffusive natural convection in vertical porous (local thermal equilibrium case) and non-porous layers are disclosed. The possibility of natural convection parallel flow in the basic state becoming unstable due to the addition of an extra diffusing component is established. In some cases, the neutral stability curves of stationary and travelling-wave modes are connected to form a loop within which the flow is unstable indicating the requirement of two thermal Darcy–Rayleigh numbers to specify the stability/instability criteria. Moreover, the change in the mode of instability is recognized in some parametric space. The results for the extreme cases of the scaled interphase heat transfer coefficient are discussed.

Energy stability of thermally modulated inclined fluid layer

The stability of natural convection in thermally modulated inclined fluid layer is analyzed using linear instability analysis and generalized energy stability theory. A sufficient condition for the global stability of the fluid layer is obtained. The stability boundaries are found in terms of the Rayleigh number. Shooting method is used to find the stability limits numerically. Uncertain stability region is observed between the linear and the nonlinear stability boundaries. The onset of instability depends upon the frequency and the amplitude of modulation.

Finite Darcy–Prandtl Number and Maximum Density Effects on Gill's Stability Problem

Abstract The simultaneous effect of a time-dependent velocity term in the momentum equation and a maximum density property on the stability of natural convection in a vertical layer of Darcy porous medium is investigated. The density is assumed to vary quadratically with temperature and as a result, the basic velocity distribution becomes asymmetric. The problem has been analyzed separately with (case 1) and without (case 2) time-dependent velocity term. It is established that Gill's proof of linear stability effective for case 2 but found to be ineffective for case 1. Due to the lack of Gill's proof for case1, the stability eigenvalue problem is solved numerically and observed that the instability sets in always via traveling-wave mode when the Darcy–Prandtl number is not larger than 7.08. The neutral stability curves and isolines are presented for different governing parameters. The critical values of Darcy–Rayleigh number corresponding to quadratic density variation with respect to temperature, critical wave number, and the critical wave speed are computed for different values of governing parameters. It is found that the system becomes more stable with increasing Darcy–Rayleigh number corresponding to linear density variation with respect to temperature and the Darcy–Prandtl number.

Impact of Thermal Non-equilibrium on the Stability of Natural Convection in an Oldroyd-B Fluid-Saturated Vertical Porous Layer with Internal Heat Sources

The stability of natural convection in a vertical layer of heat-generating Darcy porous medium saturated with an Oldroyd-B fluid using a local thermal non-equilibrium (LTNE) model has been investigated numerically. The impermeable vertical walls of the porous layer are maintained at different uniform temperatures. As a consequence of LTNE model, two temperature equations representing the fluid and solid phases separately are used for the heat transport equation. A uniform volumetric heating in both fluid and solid phases is considered and the transfer of heat between the phases is considered in the basic state. The internal heating introduced asymmetry in the basic flow which led to the existence of competing modes. The intricacies of internal heat source strength in the fluid and the solid phases are clearly discerned on the stability of the system. The stress relaxation parameter $$\Lambda_{1}$$ , fluid-heat generation parameter $$Q_{f}$$ , solid-heat generation parameter $$Q_{s}$$ and the porosity-modified conductivities ratio $$\gamma$$ were found to exhibit destabilizing effect on the system, while the strain retardation parameter $$\Lambda_{2}$$ shows an opposite trend even in the presence of internal heating.

Investigation on the stability of natural convection in an annular cavity with non-isothermal walls

The onset and nature of unsteady natural convection in a tall, closed annular cavity with non-isothermal walls is investigated experimentally. In the first of a sequence of experiments, thermal boundary conditions imposed on the inner and outer cylinders promote steady bi-cellular natural convection airflow inside the cavity. As the buoyant potential of the upper convection cell is increased, steady flow inside the cavity first bifurcates into mono-periodic flow and then follows a quasi-periodic route to chaotic flow. Similarly, when the buoyant potential of the lower convection cell is increased, steady flow bifurcates into mono-periodic flow and then follows an intermittent route to chaos. All of the observed unsteady airflows inside the annular cavity were three-dimensional, with all of the periodic flows exhibiting a gentle swirling motion in the azimuthal direction.

Stability of natural convection in a vertical non-Newtonian fluid layer with an imposed magnetic field

A numerical study has been conducted to analyze the influence of a uniform horizontal magnetic field on the stability of buoyancy driven parallel shear flow in a differentially heated vertical layer of an electrically conducting couple stress fluid; a type of non-Newtonian fluid. Within the framework of linear stability theory, the resulting complex generalized eigenvalue problem is solved numerically using the Chebyshev collocation method with QZ algorithm. The critical Grashof number $$G_{c}$$ and the corresponding wave number $$\alpha_{c}$$ and wave speed $$c_{c}$$ are computed for a wide range of couple stress parameter $$\varLambda_{c}$$ , Prandtl number $$Pr$$ and Hartmann number $$M$$ . It is found that the value of $$Pr$$ at which the instability switches over from stationary to travelling-wave mode increases with increasing $$M$$ and decreasing $$\varLambda_{c}$$ . The effect of magnetic field is to delay the onset of instability while an opposite kind of behavior is observed with increasing $$\varLambda_{c}$$ . The streamlines presented herein demonstrate the development of complex dynamics at the transition mode.

Boundary and inertia effects on the stability of natural convection in a vertical layer of an anisotropic Lapwood–Brinkman porous medium

The stability of natural convection in a fluid-saturated vertical anisotropic porous layer is investigated. The vertical rigid walls of the porous layer are maintained at different constant temperatures, and anisotropy in both permeability and thermal diffusivity is considered. The flow in the porous medium is described by the Lapwood–Brinkman model, and the stability of the basic flow is analysed numerically using Chebyshev collocation method. The presence of inertia is to inflict instability on the system and in the absence of which the system is always found to be stable. The mechanical and thermal anisotropies exhibit opposing contributions on the stability characteristics of the system. The mode of instability is interdependent on the values of Prandtl number and thermal anisotropy parameter, while it remains unaltered with the mechanical anisotropy parameter. The effect of increasing Prandtl and Darcy numbers shows a destabilizing effect on the system. Besides, simulations of secondary flow and energy spectrum have been analysed for various values of physical parameters at the critical state.

Effect of Local Thermal Nonequilibrium on the Stability of Natural Convection in an Oldroyd-B Fluid Saturated Vertical Porous Layer

The effect of local thermal nonequilibrium (LTNE) on the stability of natural convection in a vertical porous slab saturated by an Oldroyd-B fluid is investigated. The vertical walls of the slab are impermeable and maintained at constant but different temperatures. A two-field model that represents the fluid and solid phase temperature fields separately is used for heat transport equation. The resulting stability eigenvalue problem is solved numerically using Chebyshev collocation method as the energy stability analysis becomes ineffective in deciding the stability of the system. Despite the basic state remains the same for Newtonian and viscoelastic fluids, it is observed that the base flow is unstable for viscoelastic fluids and this result is qualitatively different from Newtonian fluids. The results for Maxwell fluid are delineated as a particular case from the present study. It is found that the viscoelasticity has both stabilizing and destabilizing influence on the flow. Increase in the value of interphase heat transfer coefficient Ht, strain retardation parameter Λ2 and diffusivity ratio α portray stabilizing influence on the system while increasing stress relaxation parameter Λ1 and porosity-modified conductivity ratio γ exhibit an opposite trend.

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter $$\Lambda _2 $$ portrays stabilizing influence on the system while increasing stress relaxation parameter $$\Lambda _1$$ displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.

Stability of natural convection in a vertical layer of Brinkman porous medium

A classical linear stability theory is applied to emphasize the effect of inertia on the stability of buoyancy-driven parallel shear flow in a vertical layer of porous medium. The Lapwood–Brinkman model with fluid viscosity different from effective viscosity is used to describe the flow in a porous medium. The resulting eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Darcy–Rayleigh number \(R_\mathrm{Dc} \), the critical wave number \(a_\mathrm{c}\) and the critical wave speed \(c_\mathrm{c}\) are computed over a wide range of values of the Darcy–Prandtl number \(Pr_\mathrm{D}\) and the Darcy number \({\tilde{D}}a\). Depending on the choice of physical parameters, instability occurs due to the presence of inertia. The value of \(Pr_\mathrm{D}\) at which the transition from stationary to traveling-wave mode instability takes place increases with decreasing \({\tilde{D}}a\). Besides, the effect of decreasing \({\tilde{D}}a\) shows destabilizing effect if the instability is via stationary mode, and on the contrary, it exhibits a dual behavior if the instability is through traveling-wave mode. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state. In the energy spectrum, transition of instability from one type to another is found to take place as a function of \(Pr_\mathrm{D}\). The disturbance kinetic energy due to surface drag and viscous force plays no significant role in the stability of flow throughout the domain of \(Pr_\mathrm{D}\) considered.

Stability of natural convection in a vertical dielectric couple stress fluid layer in the presence of a horizontal AC electric field

The combined effect of couple stresses and a uniform horizontal AC electric field on the stability of buoyancy-driven parallel shear flow of a vertical dielectric fluid between vertical surfaces maintained at constant but different temperatures is investigated. Applying linear stability theory, stability equations are derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number Gc, the critical wave number ac and the critical wave speed cc are computed for wide ranges of couple stress parameter Λc, AC electric Rayleigh number Rea and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. The value of Prandtl number at which the transition from stationary to travelling-wave mode takes place is found to be independent of AC electric Rayleigh number even in the presence of couple stresses but increases significantly with increasing Λc. Moreover, the effect of increasing Rea is to instill instability, while the couple stress parameter shows destabilizing effect in the stationary mode but it exhibits a dual behavior if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.

Effect of Horizontal Alternating Current Electric Field on the Stability of Natural Convection in a Dielectric Fluid Saturated Vertical Porous Layer

The stability of natural convection in a dielectric fluid-saturated vertical porous layer in the presence of a uniform horizontal AC electric field is investigated. The flow in the porous medium is governed by Brinkman–Wooding-extended-Darcy equation with fluid viscosity different from effective viscosity. The resulting generalized eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Grashof number Gc, the critical wave number ac, and the critical wave speed cc are computed for a wide range of Prandtl number Pr, Darcy number Da, the ratio of effective viscosity to the fluid viscosity Λ, and AC electric Rayleigh number Rea. Interestingly, the value of Prandtl number at which the transition from stationary to traveling-wave mode takes place is found to be independent of Rea. The interconnectedness of the Darcy number and the Prandtl number on the nature of modes of instability is clearly delineated and found that increasing in Da and Rea is to destabilize the system. The ratio of viscosities Λ shows stabilizing effect on the system at the stationary mode, but to the contrary, it exhibits a dual behavior once the instability is via traveling-wave mode. Besides, the value of Pr at which transition occurs from stationary to traveling-wave mode instability increases with decreasing Λ. The behavior of secondary flows is discussed in detail for values of physical parameters at which transition from stationary to traveling-wave mode takes place.

Stability of natural convection in a vertical couple stress fluid layer

The stability of buoyancy-driven parallel shear flow of a couple stress fluid confined between vertical plates is investigated by performing a classical linear stability analysis. The plates are maintained at constant but different temperatures. A modified Orr–Sommerfeld equation is derived and solved numerically using the Galerkin method with wave speed as the eigenvalue. The critical Grashof number Gc, critical wave number ac and critical wave speed cc are computed for wide ranges of couple stress parameter Λc and the Prandtl number Pr. Based on these parameters, the stability characteristics of the system are discussed in detail. The value of Prandtl number, at which the transition from stationary to travelling-wave mode takes place, increases with increasing Λc. The couple stress parameter shows destabilising effect on the convective flow against stationary mode, while it exhibits a dual behaviour if the instability is via travelling-wave mode. The streamlines and isotherms presented demonstrate the development of complex dynamics at the critical state.

Stochastic bifurcation analysis of Rayleigh–Bénard convection

Stochastic bifurcations and stability of natural convection within two-dimensional square enclosures are investigated by different stochastic modelling approaches. Deterministic stability analysis is carried out first to obtain steady-state solutions and primary bifurcations. It is found that multiple stable steady states coexist, in agreement with recent results, within specific ranges of Rayleigh number. Stochastic simulations are then conducted around bifurcation points and transitional regimes. The influence of random initial flow states on the development of supercritical convection patterns is also investigated. It is found that a multi-element polynomial chaos method captures accurately the onset of convective instability as well as multiple convection patterns corresponding to random initial flow states.

Stability of natural convection in superposed fluid and porous layers: Equivalence of the one- and two-domain approaches

Stability analyses of thermal and/or solutal natural convection in a configuration composed by a fluid layer overlying a homogeneous porous medium have been performed using different modeling approaches, especially for the treatment of the interfacial region. Comparisons between the one-domain approach and the two-domain formulation have shown important discrepancies of the marginal stability curves. This note shows that, according to Kataoka [I. Kataoka, Local instant formulation of two-phase flow, Int. J. Multiphase Flow 12(5) (1986) 745–758.], the differentiation of the macroscopic properties of the porous layer at the interface (porosity, permeability, thermal effective diffusivity) must be considered in the meaning of distributions. In that case, the one- and the two-domain approaches are shown to be equivalent and very good agreement is indeed found when comparing the results obtained with both approaches.

Stability of natural convection in superposed fluid and porous layers: Influence of the interfacial jump boundary condition

Macroscopic modeling of momentum transport at a fluid-porous interface has been improved by the derivation of a stress jump boundary condition related to the spatial variations of the effective properties of the porous medium at the interfacial region. This Communication concerns the influence of this jump condition on the onset of thermal natural convection in fluid/porous stratified layers. A linear stability analysis shows that, for small depth ratio, the effective jump coefficient strongly influences the bimodal marginal stability curves. At large wave numbers, the “fluid mode” (the convective flow is confined in the fluid layer) is found to be more unstable while the porous mode, corresponding to small wave numbers, remains unchanged. The influence of the thickness and the dimensionless permeability of the porous layer is also presented.

Stability of Natural Convection in Superposed Fluid and Porous Layers Using Integral Transforms

A stability analysis of thermal natural convection in superposed fluid and porous layers is carried out. The two-layer system is described using a one-domain formulation, and the eigenvalue problem resulting from the stability analysis is solved using the generalized integral transform technique (GITT). The numerical results confirm that the onset of convection can have a bimodal nature depending on the depth ratio. The influence of the dimensionless permeability and thermal diffusivity ratio are investigated.

Stability of natural convection in an inclined square duct with perfectly conducting side walls

We have performed three-dimensional linear stability analyses for natural convection in an inclined square duct. The duct is heated from the bottom, while the lateral walls are assumed to be perfectly thermal conducting. Three-dimensional transverse rolls whose axes are normal to the axis of the duct occur from the motionless state when the Rayleigh number exceeds a critical value and the duct is placed horizontally ( θ = 0°). However, it is found that when the duct is placed inclined ( θ = 0.01°), a two-dimensional longitudinal roll which is unchanged in the axis of the duct occurs and is stable if the Rayleigh number is small.