Onset Of Oscillatory Convection Research Articles

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.

Hydrothermal instability of thermocapillary convection in large-Prandtl-number liquid bridges under microgravity

Linear stability analysis was performed to study the mechanism of transition of thermocapillary convection in liquid bridges with liquid volume ratios ranging from 0.4 to 1.2, aspect ratio of 0.75 and Prandtl number of 100. 2-D governing equations were solved to obtain the steady axi-symmetric basic flow and temperature distributions. 3-D perturbation equations were discretized at the collocation grid points using the Chebyshev-collocation method. Eigenvalues and eigenfunctions were obtained by using the Q–R method. The predicted critical Marangoni numbers and critical frequencies were compared with data from space experiments. The disturbance of the temperature distribution on the free surface causes the onset of oscillatory convection. It is shown that the origin of instability is related to the hydrothermal origin for convections in large-Prandtl-number liquid bridges.

Rayleigh–Benard convection in rotating fluids

Linear and weakly nonlinear properties of Rayleigh–Benard convection in rotating fluids are investigated. Linear stability analysis is studied to investigate analytically the effect of Coriolis force on gravity-driven convection for idealised stress-free boundary conditions. We have derived a nonlinear one-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation. A coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at the supercritical Hopf bifurcation are derived and discussed the stability regions of travelling and standing waves.

Nonlinear thermohaline convection in rotating fluids

Linear and weakly nonlinear properties of thermohaline convection in rotating fluids are investigated. Linear stability analysis is studied by plotting graphs for different values of physical parameters relevant to the Earth’s outer core and oceans. We have derived a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation and shown the occurrence of Eckhaus and zigzag instabilities. We have studied heat transfer by using Nusselt number which is obtained from Landau–Ginzburg equation at the onset of stationary convection for the steady case. A coupled two-dimensional Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection are derived and the stability regions of travelling and standing waves discussed.

The effect of rotation on the onset of convection in a horizontal anisotropic porous layer

The effect of rotation and anisotropy on the onset of 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. A modified energy equation including the thermal anisotropy is used. 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 steady finite amplitude analysis is performed using truncated Fourier series to find the Nusselt number. The effect of various parameters on heat transfer is investigated.

Nonlinear Convection in a Sparsely Packed Porous Medium Due to Compositional and Thermal Buoyancy

Soret-driven convection in a two-component fluid layer subject to vertical temperature and concentration gradients is investigated analytically and numerically. The Darcy-Lapwood-Brinkman model for the momentum equation and the Boussinesq approximation is used to study the linear and weakly nonlinear properties of convection in two-component fluid in a sparsely packed porous medium due to compositional and thermal buoyancy. We have derived a nonlinear twodimensional Landau-Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation. We have studied the Nusselt number contribution from Landau-Ginzburg equation at the onset of stationary convection. Two coupled nonlinear one-dimensional Landau-Ginzburg type equations with complex coefficients near the onset of oscillatory convection at the supercritical Hopf bifurcation are derived and the stability regions of travelling and standing waves are discussed.

The Onset of Oscillatory Convection in a Horizontal Porous Layer Saturated with Viscoelastic Liquid

The onset of thermal convection in an isothermally heated, horizontal porous layer saturated with viscoelastic liquid was analyzed analytically under the linear theory. An existing constitutive model, which is rather simple, was employed to examine the effects of relaxation times. It is shown clearly that oscillatory instabilities can set in before stationary modes are exhibited. The peculiar behavior of the frequency at the critical state was discussed in connection to polymeric liquids.

Various flow patterns in thermocapillary convection in half-zone liquid bridge of high prandtl number fluid

Various flow patterns induced by a thermocapillary-driven convection in a half-zone liquid bridge of a high Prandtl number fluid (Pr = 0(10)) far beyond the critical condition were investigated experimentally. After the onset of oscillatory convection, one can observe several types of flow patterns with increasing a temperature difference between the both end surfaces of the bridge. The flow patterns were categorized through flow visualization, measurement of surface temperature variation and reconstruction of the pseudo-phase space.

Transition to oscillatory natural convection of water near its density maximum in a tall enclosure

Numerical simulations have been performed for three‐dimensional natural convection of water near its maximum‐density (cold water) inside rectangular enclosures with differential heating at the vertical (left and right) walls. The horizontal (top and bottom) walls and the lateral (front and rear) walls are taken as insulated. Computations are performed for the buoyancy‐driven convection of cold water with density inversion parameter θm = 0.5 in the enclosures with aspect ratio (height/width) Ay = 8 and depth ratios (depth/width) Az = 0.5, 1, and 2. The influence of the depth ratio on the onset of oscillatory convection in a cold‐water‐filled enclosure is investigated. The presence of the lateral walls tends to suppress the onset of unsteadiness in the convective flow. The main features of the oscillatory convection flow and temperature fields as well as the instability mechanism in the three‐dimensional enclosure were similar to those found in the two‐dimensional model. However, there exists a strong three‐dimensionality in the spatial distribution of the fluctuation amplitude. With the decrease of the depth ratio, the damping effect of the lateral walls becomes increasingly pronounced, leading to a reduced heat transfer rate.

Influence of Variable Viscosity on the Onset of Hydrothermal Waves in Thermocapillary Liquid Bridges

The float-zone crystal-growth process is investigated in the framework of the half-zone model. The influence of a temperature-dependent viscosity on the axisymmetric steady thermocapillary flow and its stability with respect to nonaxisymmetric perturbations is examined by means of a linear-stability analysis. The onset of oscillatory convection is studied numerically by a mixed Chebyshev-collocation finite-difference method. Detailed calculations have been carried out for PR = 4 and unit aspect ratio. It is found that the temperature-dependent viscosity decreases the critical Reynolds number for the onset of three-dimensional flow. Since the threshold shift is a function of the azimuthal wavenumber of the neutral mode, the critical mode may depend on the magnitude of the viscosity variation.

Influence of Variable Viscosity on the Onset of Hydrothermal Waves in Thermocapillary Liquid Bridges

The float-zone erystal-growth process is investigated in the framework of the half-zone model. The influence of a temperature-dependent viscosity on the axisymmetric steady thermocapillary flow and its stability with respect to nonaxisymmetric perturbations is examined by means of a linear-stability analysis. The onset of oscillatory convection is studied numerically by a mixed Chebyshev-collocation finite-difference method. Detailed calculations have been carried out for Pr=4 and unit aspect ratio. It is found that the temperature-dependent viscosity decreases the critical Reynolds number for the onset of three-dimensional flow. Since the threshold shift is a function of the azimuthal wavenumber of the neutral mode, the critical mode may depend on the magnitude of the viscosity variation.

Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis

In the present work, we study the onset of double-diffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless double-diffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case Le⩾1 and for ε Le2<1 the first primary bifurcation is always a Hopf bifurcation for any A and φ except for φ=+90°. For ε Le2>1 only stationary primary bifurcations exist. In the case Le<1, zones where stationary and oscillatory primary bifurcations exist are separated by a curve depending on A and φ. The last part of this work consists of a series of numerical simulations. The onset of stationary and oscillatory convection is obtained numerically at the critical Rayleigh number predicted by linear analysis. We also verified the frequency of oscillations for several sets of dimensionless parameters. The numerical simulations show multiple subcritical solutions.

Onset of oscillatory binary fluid convection in finite containers.

The onset of oscillatory convection in binary fluid mixtures in a two-dimensional domain with realistic boundary conditions on all boundaries is determined as a function of the fluid parameters and the aspect ratio Gamma of the container. The first unstable mode has either odd or even parity under left-right reflection. Depending on Gamma and the separation ratio S, this mode has the form of a standing wave, or a "chevron," consisting of a pair of waves propagating outwards from the cell center (or, in some cases, inwards towards it). Codimension--two points at which odd and even parity modes are simultaneously marginally stable are determined, as are various Takens-Bogdanov points. For fixed S<S(TB), all mode interactions among modes of like parity, arising as Gamma varies, are of the nonresonant Hopf-Hopf type; however, the details of the modal interchange are organized by resonant Hopf bifurcations with 1:1 resonance. Particular attention is paid to the asymptotic mode structure as Gamma-->infinity, and to the gap (in Rayleigh number and oscillation frequency) between successively unstable modes. The results quantify the parameter regime in which the weakly nonlinear dynamics of the system can be described in terms of the interaction of the first odd and even parity oscillatory modes.

The onset of oscillatory Marangoni convection in a semi-infinitely deep layer of fluid

In this paper we use classical linear stability theory to analyse the onset of thermocapillary-driven Marangoni convection in a semi-infinitely deep layer of quiescent fluid. We use a combination of analytical and numerical techniques to obtain for the first time a detailed description of the marginal stability curves for the onset of oscillatory convection. We perform a comprehensive asymptotic analysis of the marginal curves in the limits of both long-wave and short-wave perturbations and present the results of numerical calculations which illustrate the effect of varying the problem parameters on the marginal curves and hence on the critical Marangoni number, frequency of oscillation and wavenumber for the onset of convection.

Geometric effects on convective coupling and interfacial structures in bilayer convection

The effects of boundaries on bilayer convection in a cylinder with idealized boundary conditions are studied. Using as examples two bilayer systems, a plot of the Rayleigh number versus aspect ratio (radius-height) is calculated. For certain systems, the type of convection coupling, either thermal or viscous, will change as the width of the container changes, even as the height is fixed. Additionally, the interfacial structure is calculated to help identify the driving force for the convection. Sudden changes in the convection coupling and the dominant driving force of convection are found as the container radius increased. The oscillatory onset of convection and the possibility of oscillating nonlinear dynamics are also identified.

Experimental techniques in natural convection

Experimental techniques in natural convection heat transfer employed in the author's laboratory are introduced. The techniques are mostly related to visualization of flow, temperature field, and heat flux distribution in fluids. Three topics are presented, the first being natural convection in a horizontal rectangular liquid layer driven by surface tension and buoyancy. The patterns of flow were visualized by suspending fine aluminum flakes in the liquid. At the same time, the distribution of the temperature gradient in the liquid was visualized by an optical method making use of the refraction of light. The second topic is the onset of oscillatory convection in the Czochralski growth melt. In this case a forced flow due to rotation of the crystal and the vessel is superimposed on the buoyancy convection, resulting in an oscillatory flow under certain circumstances. The flow pattern and the temperature distribution in the liquid were visualized simultaneously by suspending in the liquid a microencapsulated temperature-sensitive liquid crystal. Periodical oscillation of the flow and the temperature was clearly recognized. The third topic is the rollover of double liquid layers that were stratified stably due to a density difference. A small-scale experiment was carried out to clarify the basic mechanism of rollover. The tracer method was used to visualize boundary layer flow along the vertical side wall and the shadowgraph technique to visualize the density distribution in the liquid layers. The article emphasizes the importance of visual observation in the investigation of natural convection phenomena.

Onset of oscillatory convection in binary mixtures with Sorét effects and solidification

The oscillatory instability of convection in a binary liquid-mixture with negative Sorét effects coupling Bénard convection and solidification is investigated when the temperature of the upper boundary is below the freezing point of the two-component system. Complete rejection of the lean component from the solid into the liquid phase is assumed. The Sorét coefficient and the coefficients of thermal and solutal expansion depend strongly on the local concentration in the liquid and, consequently, on the thickness of the solidified layer. Linear stability analysis for typical ethanol-water mixtures demonstrates that the variation of concentration is a dominant effect in the destabilization of the conductive state.

Travelling and standing waves in magnetoconvection

The problem of Boussinesq magnetoconvection with periodic boundary conditions is studied using standard perturbation techniques. It is found that either travelling waves or standing waves can be stable at the onset of oscillatory convection, depending on the parameters of the problem. When travelling waves occur, a steady shearing flow is present that is quadratic in the amplitude of the convective flow. The weakly nonlinear predictions are confirmed by comparison with numerical solutions of the full partial differential equations at Rayleigh numbers 10% above critical. Modulated waves (through which stability is transferred between travelling and standing waves) are found near the boundary between the regions in parameter space where travelling waves and standing waves are preferred.

EXPERIMENTAL STUDY ON OSCILLATORY THERMOCAPILLARY CONVECTION

The transition processes from steady flow into oscillatory flow in a liquid bridge of the half floating zone are studied experimentally. Two methods of noncontacted diagnoses are developed to measure the distribution of critical Marangoni numbers described by the onset of the oscillation at the free surface of the liquid bridge. The experimental results obtained for both cases of the upper rod heated and the lower rod heated agree with the prediction by Rayleigh's instability theory. The sensitive relations between the relatively fat or slender liquid bridge and the onset of oscillatory convection are also discussed to reveal the insight of the pressure distribution near the free surface. The experiments have been performed in a small liquid bridge, where the Bond number is much smaller than 1, alld the results can be used to simulate the experiment in the microgravity environment.

The influence of buoyancy on the onset of oscillatory convection in a half floating zone

The critical Marangoni number is determined by the optical method, a new parameter describing the free surface shape is sensitive to the onset of oscillation. The experiments are performed for half floating-zone with either a upper rod heated (URH) or a lower rod heated (LRH). The results agree with the instability analyses of buoyancy influence (see, for examples, /1–3/).