Onset Of Double-diffusive Convection Research Articles

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Double-diffusive buoyancy convection in an open-top rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude (buoyancy ratio Rρ=−1). In this case, a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Grc. Linear stability analysis and direct numerical simulation show that depending on the cavity aspect ratio A, the first primary instability can be oscillatory, while that in a closed cavity is always steady. Near a codimension-two point, the two leading real eigenvalues merge into a complex coalescence that later produces a supercritical Hopf bifurcation. As Gr further increases, this complex coalescence splits into two real eigenvalues again. The oscillatory flow consists of counter-rotating vortices traveling from right to left and there exists a critical aspect ratio below which the onset of convection is always oscillatory. Neutral stability curves showing the influences of A, Lewis number Le, and Prandtl number Pr are obtained. While the number of vortices increases as A decreases, the flow structure of the eigenfunction does not change qualitatively when Le or Pr is varied. The supercritical oscillatory flow later undergoes a period-doubling bifurcation and the new oscillatory flow soon becomes unstable at larger Gr. Random initial fields are used to start simulations and many different subcritical steady states are found. These steady states correspond to much stronger flows when compared to the oscillatory regime. The influence of Le on the onset of steady flows and the corresponding heat and mass transfer properties are also investigated.

The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer

The double diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer, which is heated and salted from below, is studied analytically. The modified Darcy equation that includes the time derivative term is used to model 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 parameter, solute Rayleigh number, Lewis number, couple stress parameter, and Vadasz number on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the thermal anisotropy parameter, couple stress parameter, and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The mechanical anisotropy parameter has destabilizing effect on stationary, oscillatory, and finite amplitude convection. The Lewis number has stabilizing effect in the case of stationary and finite amplitude modes, with dual effect in the case of oscillatory convection. Vadasz number advances the onset of oscillatory convection. The heat and mass transfer decrease with an increase in the values of couple stress parameter, while both increase with an increase in the value of solute Rayleigh number and mechanical anisotropy parameter. The thermal anisotropy parameter and Lewis number have contrasting effect on the heat mass transfer.

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.

Double-diffusive convection in a Darcy porous medium saturated with a couple-stress fluid

The onset of double-diffusive convection in a couple-stress fluid-saturated horizontal porous layer is studied using linear and weak nonlinear stability analyses. The modified Darcy equation that includes the time derivative term and the inertia term is used to model the momentum equation. The expressions for stationary, oscillatory and finite-amplitude Rayleigh number are obtained as a function of the governing parameters. The effect of couple-stress parameter, solute Rayleigh number, Vadasz number and diffusivity ratio on stationary, oscillatory and finite-amplitude convection is shown graphically. It is found that the couple-stress parameter and the solute Rayleigh number have a stabilizing effect on stationary, oscillatory and finite-amplitude convection. The diffusivity ratio has a destabilizing effect in the case of stationary and finite-amplitude modes, with a dual effect in the case of oscillatory convection. The Vadasz number advances the onset of oscillatory convection. The heat and mass transfer decreases with an increase in the values of couple-stress parameter and diffusivity ratio, while both increase with an increase in the value of the solute Rayleigh number.

The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer

The onset of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer is studied using a linear and a weak nonlinear stability analyses. The modified Darcy law for the viscoelastic fluid of the Oldroyd type is used to model the momentum equation. The onset criterion for stationary and oscillatory convection is derived analytically. There is a competition between the processes of thermal, solute diffusions, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary. The effect of anisotropy parameters, Darcy–Prandtl number, relaxation, and retardation parameters on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method is used to find the transient heat and mass transfer. The effect of various parameters on heat and mass transfer is also brought out.

Onset of Double Diffusive Convection in a Thermally Modulated Fluid-Saturated Porous Medium

The onset of double diffusive convection in a sparsely packed porous medium was studied under modulated temperature at the boundaries, and a linear stability analysis has been made. The primary temperature field between the walls of the porous layer consisted of a steady part and a timedependent periodic part and the Galerkin method and the Floquet were used. The critical Rayleigh number was found to be a function of frequency and amplitude of modulation, Prandtl number, porous parameter, diffusivity ratio and solute Rayleigh number.

The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer

The effect of rotation and anisotropy on the onset of double diffusive convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. The effect of rotation, mechanical and thermal anisotropy parameters, and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The finite amplitude analysis is performed to find the thermal and solute Nusselt numbers. The effect of various parameters on heat and mass transfer is also investigated.

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.

Double Diffusive Convection in a Rotating Porous Layer with Temperature Modulation on the Boundaries

Linear stability analysis is performed to find the effect of temperature modulation on the onset of double diffusive convection in a rotating horizontal layer of a fluid-saturated porous medium using Darcy-Lapwood-Brinkman's model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability, and temperature modulation on the onset of double diffusive convection in a porous medium has been investigated using Galerkin method. The value of the critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Prandtl number, Taylor number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. Furthermore, it is found that both rotation and the porous medium have stabilizing influences on the system. The effect of other parameters on the stability of the system is also discussed.

Temperature Modulation of Double Diffusive Convection in a Horizontal Fluid Layer

Linear stability analysis is performed for the onset of thermosolutal convection in a horizontal fluid layer with rigid-rigid boundaries. The temperature field between the walls of the fluid layer consists of two parts: a steady part and a time-dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of temperature modulation on the onset of thermosolutal convection has been studied using the Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Prandtl number, diffusivity ratio and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of the diffusivity ratio and solute Rayleigh number on the stability of the system are also discussed.

The effect of stable thermal stratification on the onset of double-diffusive convection in electrochemical systems

The onset time of double-diffusive convection in time-dependent, nonlinear concentration fields was investigated theoretically and experimentally. The stability analysis was conducted on the basis of the propagation theory. Under the linear stability theory, the linearized perturbation equations were transformed similarly by using the concentration penetration depth as a length-scaling factor. The newly derived stability equations were solved numerically. Also the onset time was determined experimentally by employing an electrochemical technique where 0.03–0.3 M CuSO 4 + 1.5 M H 2SO 4 aqueous solutions were adopted as electrolyte. The onset time of double-diffusive convection was delayed or shortened depending on the degree of stable stratification.

Effect of thermal modulation on the onset of double diffusive convection in a horizontal fluid layer

The effect of time periodic boundary temperatures on the onset of double diffusive convection in a horizontal two component fluid layer is studied using a linear stability analysis. The perturbation method is used to compute the critical thermal Rayleigh number and the corresponding wave number for small amplitude temperature modulation. The correction thermal Rayleigh number is calculated as a function of frequency of the modulation, Prandtl number, solute Rayleigh number, and the diffusivity ratio. It is found that the thermal modulation may stabilize an unstable system or destabilize a stable system. In particular it is found that low frequency symmetric modulation is destabilizing whereas the asymmetric modulation and lower wall temperature modulation are stabilizing. The effect of the solute Rayleigh number, ratio of the diffusivities and the Prandtl number are also reported. It is also found that the effect of modulation disappears for large frequency.

Effect of time-periodic boundary temperatures on the onset of double diffusive convection in a horizontal anisotropic porous layer

The effect of time-periodic boundary temperatures on the onset of double diffusive convection in a fluid-saturated anisotropic porous medium is studied by making a linear stability analysis. A perturbation method based on small amplitude of the imposed temperature modulation is used to compute the critical values of thermal Rayleigh number and wave number. The correction thermal Rayleigh number is calculated as a function of frequency of modulation, viscosity ratio, anisotropy parameter, porous parameter, Prandtl number, diffusivity ratio and solute Rayleigh number. The effect of various physical parameters is found to be significant at moderate values of the frequency. We found that it is possible to advance or delay the onset of double diffusive convection by proper tuning of the frequency of modulation of the wall temperature. The effect of various parameters on the stability of the system is brought out.

The onset of penetrative double-diffusive convection

The onset of double-diffusive convection in a horizontal fluid layer is studied. The density is assumed to depend quadratically on the temperature and linearly on the solute concentration. Under the Boussinesq approximation, the linear stability of the conduction state is investigated with respect to the oscillatory and steady convection modes. For steady onset, the critical thermal Rayleigh number is found to be a double-valued function of the solutal Rayleigh number as long as the relative maximum of the density profile exists within the fluid layer. Driving mechanisms of the steady convections are discussed.

THE EFFECTS OF STABLE THERMAL STRATIFICATION ON THE ONSET OF DOUBLE-DIFFUSIVE CONVECTION

The time of the onset of double-diffusive convection in time-dependent, nonlinear concentration fields is investigated theoretically. When the initially quiescent horizontal fluid layer with a uniform temperature gradient experiences a sudden concentration change from below, the concentration field could invoke a convective motion. The related stability analysis, with considering Soret effect, is conducted on the basis of the propagation theory. Under the linear stability theory the concentration penetration depth is used as a length scaling factor and the linearized perturbation equations of similarity transform are solved numerically. The resulting critical time to mark the onset of regular cells is obtained as a function of the thermal Rayleigh number, the solute Rayleigh number, and the Soret effect coefficient

The Effect of AnisotropicThermoconvective Currents on the Onset of Double-Diffusive Convection in a Sparsely Packed Porous Medium

The Effect of AnisotropicThermoconvective Currents on the Onset of Double-Diffusive Convection in a Sparsely Packed Porous Medium

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.

Onset of double-diffusive convection in a saturated porous layer with time-periodic surface heating

The stability of a fluid saturated, horizontal porous layer in the presence of a solute concentration gradient and time-periodic thermal gradient is examined. The modulated gradient is the result of a sinusoidal upper surface temperature which models the effect of variable solar radiation heating of the layer. Darcy's law and the Boussinesq approximation are employed, and we assume an equation of state linear in temperature and concentration. A linear stability analysis is carried out to obtain predictions for the onset of convection and critical wavenumbers for the system. The critical conditions are obtained via the Galerkin method and Floquet theory. The effects of variable concentration gradient, temperature modulation amplitude and frequency are examined, and compared with the results obtained analytically from the corresponding unmodulated problem. It is shown that instabilities can occur as convective motions which are synchronous or subharmonic with the surface heating, or can be identified via complex conjugate Floquet exponents. The neutral stability curves at the transitions between instabilities are found to be bimodal when the temperature is time-periodic, and are characterized by jumps in the critical wavenumbers.

Stabilité des écoulements de convection thermosolutale en cavité carrée

Bifurcation phenomena in a square enclosure, submitted to horizontal temperature and concentration gradients, is studied when the opposing buoyancy forces due to horizontal thermal and concentration gradients are equal. We perform the linear, weakly non-linear and finite amplitude stability analysis of the equilibrium solution. We verify that the onset of double diffusive convection corresponds to a transcritical bifurcation point. The subcritical solutions are strong attractors beyond a particular value of the thermal Rayleigh number which corresponds to the location of turning point. The structure of subcritical and transcritical steady solutions has been studied.