Plane Horizontal Layer Research Articles

Stability of convective rolls in a horizontal layer rotating about an inclined axis

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity Ω = ( Ω 1 , Ω 2 , Ω 3 ) . (i) We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary Ω , and by the reduced problem the set of equations for the angular velocity ( Ω 1 , 0 , Ω 3 ) . Here x,y are horizontal Cartesian coordinates in the layer and z is the vertical one. We prove that a y-independent solution to one of the two problems is also a solution to the second one. (ii) We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let β be the angle between the horizontal projection of Ω and the rolls axes. We show that β = 0 for the least stable mode. When the axis of rotation is horizontal, this is proven analytically, and for Ω 3 ≠ 0 , the result is obtained numerically. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of Ω 1 and Ω 2 and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. (iii) We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for Ω = ( 0 , Ω 2 , Ω 3 ) and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as Ω 2 increases. Namely, on increasing Ω 2 , the interval of the Rayleigh numbers for which convective rolls are stable increases.

Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid

We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary.

Thermal Vibrational Convection in a Rotating Plane Layer

The influence of rotation on the thermal vibrational convection of liquid in a horizontal plane layer is studied experimentally. The convection in the layer is excited by the vibrations of circular polarization in the horizontal plane. The research is carried out in a liquid layer heated from above, under conditions of a strong stabilizing effect of gravity. In this case, the vibrational convection is excited at a relatively high intensity of vibrations and manifests itself in the development of short-wavelength convective structures in the form of beads. It is found that the rotation has a stabilizing effect and increase the vibrational convection excitation threshold. A map of convective stability is plotted on the plane of dimensionless parameters: vibrational parameter, and dimensionless velocity of rotation, for different values of the gravitational Rayleigh number. The results are compared with the classic case of gravitational convection in a rotating layer. It is shown that, similarly to the case of gravitational convection, rotation has a stabilizing effect on vibroconvective stability; the threshold value of the vibration parameter increases with the dimensionless velocity. It is found that at large negative values of the gravitational Rayleigh number, when cellular convective structures of vibroconvective nature are characterized by large wave numbers, in the studied area of the dimensionless frequency, rotation has little effect on the wave number of spatial convective structures, but changes their shape. (Clinical trials: no any clinical trials were performed).

Convective instability in multicomponent mixtures with Soret effect

We study the two-dimensional modes of gravity-dependent convective instability arising in a ternary mixture in a plane horizontal layer. The gravity force is oriented perpendicular to the layer heated from below. It is assumed that the system is in the state of mechanical equilibrium of a mixture considered as the base state of the system, which can become unstable under certain conditions. The mathematical model uses two-dimensional Navier-Stokes equations, transfer equations for describing mixture components with the Soret effect, and a heat transfer equation. Under the influence of a concentration gradient, the heat transfer symmetric to the Soret effect is neglected because it is usually small in water-based mixtures. The possible concentration-dependent diffusion and cross-diffusion of dissolved components are also neglected. We investigate a convective stability using linear approximation and under finite-amplitude convection regimes. The stability analysis of the base state after linearization of the governing equations with respect to the state of mechanical equilibrium predicts various convection modes. For each case, a neutral curve and the dependences of the disturbance growth decrements on both the Rayleigh number and the wave number are plotted. We show that in the case of the layer being heated from below, there are ranges of governing parameters where both long-wave and short-wave modes of oscillatory instability occur. The numerical analysis of a nonlinear coupling between these modes has revealed that such instabilities develop in the ternary mixtures whose components are arranged (due to the Soret effect) in different directions with respect to the temperature gradient. The effect of the Rayleigh number on the flow structure and heat mass transfer is demonstrated.

Effect of rotation on thermal convection in horizontal plane layer subject to circular vibration

The stability of quasi-equilibrium of a horizontal liquid plane layer with isothermal boundaries of different temperatures, subject to circular vibrations in the horizontal plane, is experimentally studied. The rotation of the cavity around the vertical axis is set independently. In order to exclude from consideration the destabilizing effect of the gravity field, the layer is heated from the top. In the absence of rotation, the circular vibrations lead to the threshold excitation of vibrational thermal convection in the layer of nonisothermal liquid, in spite of its stable stratified in the gravity field. At given vibration amplitude and temperature difference at the layer boundaries, the threshold is determined by sharp increase in heat transfer with a monotonic increase in the vibration frequency. The size of the convective spatial structures is determined by the layer thickness. In the case of high-frequency vibrations, convection is determined by dimensionless parameters: gravitational Rayleigh number Ra and vibrational parameter R v. The thresholds of vibrational convection excitation on the plane of these parameters in the case of circular vibrations coincide with the theoretically predicted stability boundary for linear vibrations. It is shown that rotation has a stabilizing effect on vibroconvective stability, similar to the case of gravitational convection. The threshold value of the vibration parameter R v grows with an increase in the dimensionless rotation velocity ω rot. Under conditions of the performed experiment, the structure of convective cells and the wave number of structures in the supercritical region do not change in comparison with ω rot = 0.

Exact solution for stable convective concentration flows of a Couette type

The paper presents an exact solution describing a steady-state diffusion layered Couette-type flow of a viscous incompressible fluid binary mixture, induced by specifying a parabolic wind at one of the boundaries of the flow region. The flow is modeled using the system of equations of concentration convection, which consists of the Navier-Stokes equations (in the Boussinesq approximation), the incompressibility equation, and the equation for changing the concentration of the light phase of a binary mixture. The solution of this nonlinear overdetermined system of equations is sought within the framework of the Lin-Sidorov-Aristov class. The solvability of the reduced system of equations with respect to the components of the velocity field, concentration and pressure fields is shown. A plane horizontal infinite layer of constant thickness is considered as the flow region. A distinctive feature of the presented exact solution is taking into account the impermeability property of a solid hydrophilic surface, which limits the fluid layer under consideration from below. In the course of analyzing the exact solution describing the distribution of the velocity field, it was shown that in some cases the flow can be reduced to a unidirectional one. In the general case, each of the nonzero components of the velocity vector can have at most one zero point inside the layer under consideration, and they will coincide only if the flow is reducible to a unidirectional one. It is shown that the corresponding shear stress field can be stratified into two zones, in each of which the stress retains its sign and changes it upon passing to another zone. The investigation of the concentration field revealed the fundamental possibility of stratification of the background concentration field into two zones compared to the reference value. At the same time, the background pressure field can be divided into three zones in relation to the reference level. Thus, the solution proposed can describe return flows localized (for certain combinations of edge parameters and physical characteristics of the fluid) near the boundaries of the considered fluid layer.

Nonlinear regimes of Soret-driven convection of ternary fluid with fixed vertical heat flux at the boundaries.

Nonlinear regimes of the Soret-induced convection of a ternary mixture in a rectangular cavity with the vertical heat flux at the boundaries are studied numerically. Linear stability analysis performed earlier for a plane horizontal layer has shown that in a wide range of parameters the longwave instability is dominant. Nonlinear calculations performed in the present work for the parameter ranges where the monotonic longwave instability is dominant according to the linear stability analysis, show that at large supercriticalities the multi-vortex stationary flows could be realized. With the decrease of absolute value of the Rayleigh number Ra the series of flow structure transformations takes place. This transformation is accompanied by the decrease of a number of the vortices such that near the threshold the flow becomes two-vortex, i.e. the flow with maximum possible wavelength (minimum possible wave number) is realized. Thus, the nonlinear calculations confirm the conclusions of the linear stability analysis that the longwave instability is dominant. For the parameter ranges where according to the linear stability analysis the oscillatory longwave instability is dominant, the nonlinear calculations show that at small values of the net separation ratio the oscillatory regimes could be observed just in a narrow range of Ra close to the convection threshold and with the growth of net separation ratio the range of Ra where they can be observed is extended.

Vibroconvective stability of liquid in horizontal plane layer subject to circular translational vibrations

Thermal vibrational convection of liquid in a horizontal plane layer heated from above and subject to circular translational vibrations in horizontal plane is experimentally investigated. It is shown that circular vibrations of sufficient intensity lead to a threshold excitation of the vibrational convection of a fluid, which is stably stratified in a gravity field. The thickness of the liquid layer and the vibrational parameters varied. The results of an experimental study of the stability of a quasi-equilibrium in the area of large negative values of the gravitational Rayleigh number agree well with results of linear stability theory. It is found that thermal vibrational convection in a plane layer heated from above develops in the form of two-dimensional convective rolls, the size of which is determined by the layer thickness. The wavenumber of the convective structures is in satisfactory agreement with the theoretically predicted.

Effects of anisotropic diffusion on onset of rotating magnetoconvection in plane layer; stationary modes

ABSTRACTTurbulence in the Earth's outer core not only increases all diffusive coefficients, but it can lead to their anisotropic properties. Therefore, the model of rotating magnetoconvection in horizontal plane layer rotating about vertical axis and permeated by homogeneous horizontal magnetic field, influenced by anisotropic diffusivities, viscosity and thermal diffusivity, is advanced by considering the magnetic diffusivity as anisotropic too. The case of full anisotropy, i.e. all coefficients anisotropic, is compared with both the case possessing isotropic diffusion coefficients and the case of partial anisotropy, i.e. mixed case with isotropic and anisotropic diffusive coefficients (viscosity and thermal diffusivity anisotropic and magnetic diffusivity isotropic). The existence and preference of instabilities is sensitive to all non-dimensional parameters, as well as on anisotropic parameter, the ratio of horizontal and vertical diffusivities. Two types of anisotropy, BM (introduced by Braginsky and Meytlis) and SA (stratification anisotropy) are studied. BM as well as SA were applied by Šoltis and Brestenský to the study of the partial anisotropy; this study is extended, in this paper, to full anisotropy cases (full SA and full BM) and it is shown that the style of convection given by the onset of stationary modes is more affected by anisotropic diffusivities in BM than in SA anisotropy. The important influence of strong anisotropies in the Earth's core dynamics is stressed.

Rayleigh-Taylor instability of a miscible interface in a confined domain

On the basis of the phase-field approach, we investigate the simultaneous diffusive and convective evolution of an isothermal binary mixture of two slowly miscible liquids that are confined in a horizontal plane layer. We assume that two miscible liquids are brought into contact, so the binary system is thermodynamically unstable and the heavier liquid is placed on top of the lighter liquid, so the system is gravitationally unstable. Our model takes into account the non-Fickian nature of the interfacial diffusion and the dynamic interfacial stresses at a boundary separating two miscible liquids. The numerical results demonstrate that the classical growth rates that characterise the initial development of the Rayleigh-Taylor instability can be retrieved in the limit of the higher Peclet numbers (weaker diffusion) and thinner interfaces. The further nonlinear development of the Rayleigh-Taylor instability, characterised, e.g., by the size of the mixing zone, is however limited by the height of the plane layer. On a longer time scale, the binary system approaches the state of thermodynamic and hydrodynamic equilibrium. In addition, a novel effect is found. It is commonly accepted that the interface between the miscible liquids slowly smears in time due to diffusion. We however found that when the binary system is subject to hydrodynamic transformations the interface boundary stretches, so its thickness changes (the interface becomes thinner) on a much faster convective time scale. The thickness of the interface is inversely proportional to the surface tension, and the stronger surface tension limits the development of the Rayleigh-Taylor instability.

Plane-Parallel Advective Flow in a Horizontal Layer of an Incompressible Fluid with an Internal Linear Heat Source

We present a new exact solution of the Navier-Stokes equations in the Oberbeck-Boussinesq approximation describing a plane-parallel advective flow in a plane horizontal layer of an incompressible fluid with solid boundaries. At the boundaries, a linear temperature distribution is defined in the presence of an internal heat source that is linear with respect to the horizontal coordinate. Examples of such solutions are given. The possibility of an analytical determination of the velocity and temperature of such flows is demonstrated. The velocity profile has not a cubic profile, which is usual for advective flows, but a more complex form depending on the source type.

On the Long-Range Influence of Earthquake Rupture Zones

This paper proposes a new approach to estimating the long-range influence of crustal volumes that involve anomalous stress behavior. It is shown that the consideration of this problem within elasticity theory underestimates the long-range action, determining a rapid decrease in the influence of the volume involving anomalous stresses on the state of stress in the earth crust, as one goes away from the source of the anomaly. In these approaches the influence of the anomaly at great distances (100 radii of the anomalous volume) is due to a reduction of strain by a factor of 106 compared with the deformation in the anomalous volume. It is proposed in estimating the long-range influence of anomalous volumes to take account of the fact that there are extensive regions in a supercritical state at different crustal horizons. Such regions are fault zones in the upper crust and layers of high fluid pressure in the middle crust (waveguides). The presence of regions of supercritical deformation where the earth behaves inelastically (pseudoplastic and quasiplastic behavior) produces a different regime of long-range influence due to incipient anomalies of stress, different from the purely elastic response. We showed that in the 2-D case (a plane horizontal layer) and the classical Coulomb–Mohr criterion, the reduction in the stress disturbance due to an anomalous inclusion can be given by the law 1/r (r is the lateral distance relative the center of the inclusion when normalized by the size of the inclusion). The expressions derived in this study show that the level of disturbed strain at great distances is reduced by a factor of only 100 when under horizontal compression or tension. This is lower by a factor of 104 compared with what would be expected in an elastic medium. Consequently, the change in the level of stress due to an anomaly in a medium that is experiencing irreversible deformation occurs at a substantially lower rate in relation to increasing distance than in a purely elastic medium, where the decay of stress behaves like 1/r 2.

Effects of variable thermal diffusivity on the structure of convection

The structure of multiscale convection in a thermally stratified plane horizontal fluid layer is investigated by means of numerical simulations. The thermal diffusivity is assumed to produce a thin boundary sublayer convectively much more unstable than the bulk of the layer. The simulated flow is a superposition of cellular structures with three different characteristic scales. In contrast to the largest convection cells, the smaller ones are localised in the upper portion of the layer. The smallest cells are advected by the larger-scale convective flows. The simulated flow pattern qualitatively resembles that observed on the Sun.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДВУМЕРНОЙ КОНВЕКЦИИ РЭЛЕЯ–БЕНАРА

Рассматривается конвекция Рэлея–Бенара (естественная конвекция). Это течение, которое образуется в вязкой среде при нагреве снизу и охлаждении сверху. В результате происходит образование вихрей (конвективных ячеек). Данный процесс описывается системой нелинейных дифференциальных уравнений в приближении Обербека–Буссинеска. В качестве управляющих параметров, определяющих режимы конвекции, выбраны число Рэлея и число Прандтля. Система решена методом конечных элементов с помощью вычислительного пакета FEniCS. Получены численные результаты при различных числах Рэлея. Исследована интегральная характеристика (число Нуссельта) в зависимости от числа Рэлея.

On Compression of a Heavy Compressible Layer of an Elastoplastic or Elastoviscoplastic Medium

The problem of deformation of a horizontal plane layer of a compressible material is solved in the framework of the theory of small strains. The upper boundary of the layer is under the action of shear and compressing loads, and the no-slip condition is satisfied on the lower boundary of the layer. The loads increase in absolute value with time, then become constant, and then decrease to zero.Various plasticity conditions are consideredwith regard to the material compressibility, namely, the Coulomb–Mohr plasticity condition, the von Mises–Schleicher plasticity condition, and the same conditions with the viscous properties of the material taken into account. To solve the system of partial differential equations for the components of irreversible strains, a finite-difference scheme is developed for a spatial domain increasing with time. The laws of motion of elastoplastic boundaries are presented, the stresses, strains, rates of strain, and displacements are calculated, and the residual stresses and strains are found.

Heat transfer in an infinite layer with fractal distribution of heating elements

The effect of inhomogeneous temperature distribution at boundary on convection in an infinite horizontal plane layer is investigated. Direct numerical simulation of the compressible non-isothermal flow in a cubic cell with rigid horizontal walls is performed under periodic vertical boundary conditions. Laminar and weakly non-linear flow regimes are determined. The heated area on the cell bottom obeys regular or fractal distributions. The intensities of heat flux through the layer are compared for different heterogeneous distributions of heating elements at a specific temperature gradient in the case when the area of heated surface remains constant. Fractal geometry shows the appearance of the multiscale structure of the flow and the enhancement of heat transfer.

Thermal convection in a layer of magnetic colloid based on a single-component fluid

Experiments on studying thermal gravitational convection in an undecane-based magnetic colloid layer are carried out. Undecane is a single-component carrier fluid. We use a cylindrical cavity with a diameter of 58mm and a height of 2.4mm to model a horizontal plane layer. Convection in the magnetic colloid layer heated from below is observed by means of a thermocouple system and a thermal imager. Several series of thermocouple measurements for a heat flux through the layer and thermal imaging survey for temperature fields at the magnetic colloid surface are performed at various average colloid temperatures. The average temperature increases from 20 to 55°C in increments of 5°C. A regime in the form of convective patterns consisting of stable downward flows in their centers and unstable upward flows along the edges is found in experiments. The Rayleigh number range for the regime shrinks as the average temperature increases. It can be seen from the convection regime map constructed in our study. We propose the hypothesis, according to which shrinkage of the Rayleigh number range and the instability of upward flows for this regime is due to the effect of aggregate sedimentation on convection in a horizontal magnetic colloid layer. Aggregate sizes decrease as the average colloid temperature rises.

Acoustic wave effect on a stability of convective flow in a horizontal channel subjected to the horizontal temperature gradient

The effect of acoustic wave propagating in longitudinal direction on stationary convective flow in a horizontal channel of rectangular cross-section in the presence of horizontal temperature gradient and its stability is studied. It is shown that in the presence of sufficiently intensive acoustic wave stationary flow is of coaxial character, the fluid moves in the direction of temperature gradient near sidewalls and in the opposite direction in the interior of the channel. At Pr< ∼0.2 hydrodynamic shear instability mode is most dangerous. This mode is strongly stabilized by acoustic wave at low Prandtl numbers. The range of Prandtl number values where acoustic wave can stabilize the basic flow becomes wider with the channel width growth. At high Prandtl number values the spiral instability modes are most dangerous. The oscillatory spiral mode is slightly stabilized by acoustics. The monotonic spiral mode is destabilized by acoustic wave, especially in the wide channels. In general, for all instability modes the sidewalls suppress the development of perturbations and stationary flow in the channel is more stable than that in plane horizontal layer. In the channel of square cross-section the oscillatory spiral mode is completely suppressed.

Convection of colloidal suspensions stratified by thermodiffusion and gravity.

The convective stability thresholds and nonlinear evolution of convective rolls are numerically investigated in a plane horizontal layer of a colloidal suspension with positive separation ratio in the case of no-slip, impermeable horizontal boundaries. The characteristics of the steady and oscillatory patterns are analyzed under heating and gravity stratification. The standing and traveling waves are found as stable solutions within certain domains of parameters (on the plane of the Rayleigh and the Boltzmann numbers). Complex bifurcation and spatiotemporal properties are caused by the interaction of gravity sedimentation, Soret-induced gradients, and convective mixing of the fluid.

Stratification-induced scale splitting in convection

The coexistence of motions on various scales is a remarkable feature of solar convection, which should be taken into account in analyses of the dynamics of magnetic fields. Therefore, it is important to investigate the factors responsible for the observed multiscale structure of solar convection. In this study, an attempt is made to understand how the scales of convective motions are affected by the particularities of the static temperature stratification of a fluid layer. To this end, simple models are considered. The equations of two-dimensional thermal convection are solved numerically for a plane horizontal fluid layer heated from below, in an extended Boussinesq approximation that admits thermal-diffusivity variations. These variations specify the stratification of the layer. The static temperature gradient in a thin sublayer near the upper surface of the layer is assumed to be many times larger than in the remainder of the layer. In some cases, distributed heat sinks are assumed to produce a stably stratified region overlying the convective layer. Manifestations of the scale-splitting effect are noted, which depend on the boundary conditions and stratification; it becomes more pronounced with the increase of the Rayleigh number. Small-scale convection cells are advected by larger-scale flows. In particular, the phase trajectories of fluid particles indicate the presence of complex attractors, which reflect the multiscale structure of the flow. The effect of the stably stratified upper sublayer on the flow scales is also considered.