Thermal Gravitational Convection Research Articles

Rayleigh — Benard problem for Polymer Solution

There are three mathematical models describing the motion of aqueous solutions of polymers: the second grade fluid model (Rivlin — Eriksen), the hereditary model (Voitkunsky — Amfilokhiev — Pavlovsky), and its asymptotic simplification (Pavlovsky). This work considers the problem of fluid equilibrium stability in a horizontal fluid layer heated from below or from above. Also, equations of thermal gravitational convection for all three models are derived. Three types of boundary conditions are considered: two solid boundaries; the lower solid boundary and the upper free boundary; two free boundaries (the Rayleigh problem). For the case of heating from below, the principle of perturbation monotonicity is established that ensures the spectral problem eigenvalues to be of real type. This greatly simplifies the determination of the critical Rayleigh numbers. It turned out that these numbers coincide with the critical Rayleigh numbers in the classical Rayleigh — Benard problem. In the case of heating from above at large temperature gradients, the perturbation decrements become complex, but their real parts are negative. The conclusion that the relaxation properties of a second grade fluid and an aqueous solution of polymers do not lead to a change in the critical Rayleigh number may seem strange at first glance. According to our assumption, it is explained by the base state of the liquid being a state of rest.

Thermocapillary deformations of a two-layer system of liquids under laser beam heating

Heat and mass transfer in a bilayer system of viscous incompressible liquids with interfaces have been studied theoretically and experimentally. Thermocapillary convection and interface deformations have been induced by an intense thermal exposure on the free boundary due to the action of a laser beam. The evolution of the interfaces and liquid behavior have been experimentally investigated. The static profile of the liquid – liquid interface and free surface have been measured using the laser-sheet technique. The scheme of the experiment and parameters of the physical system have been presented. The theoretical study of the system dynamics includes the development of a correct mathematical model in the framework of continuum mechanics and creation of an effective numerical algorithm, validated with the help of new experimental results. An outline of the numerical method has been given. The proposed approach allows one to describe the characteristics of the arising flows, to predict the shape of thermocapillary deformations of the liquid interfaces, and to analyze the interaction of the thermal gravitational and thermocapillary convection. The mechanisms causing the occurrence of the decaying oscillations that were first discovered by the authors have been discussed. The results of the numerical study for a two-layer system of the “silicone oil – glycerin” type are in good agreement with the experimental data, and enable to estimate the influence of the upper layer thickness on the features of fluid motion and to forecast the rupture time of the top liquid under local heating at the free boundary. The rupture time depends on the rate at which the threshold temperature difference in the top layer sets in. For systems with relatively large thickness of the upper layer the appearance of a stable profile of the thermocapillary flexure on the free surface without layer breakdown is characteristic.

Natural convection of a hybrid nanofluid affected by an inclined periodic magnetic field within a porous medium

Heat transfer enhancement for various engineering systems can be achieved by the inclusion of metal nanoparticles inside the heat transfer liquid. Such an effect can be improved by considering the hybrid nanofluid when nanoparticles of different materials are added to the base fluid. The present study is devoted to computational analysis of thermal gravitational convection within a porous chamber under the impact of tilted periodic magnetic force. Governing equations formulated employing the single-phase nanoliquid approach, Brinkman-extended Darcy model for transport processes within the porous layer with the local thermal non-equilibrium approach, Boussinesq approximation for the description of the buoyancy force and magnetic field, have been resolved using the Galerkin finite element method. Impacts of the Darcy number, Hartmann number, Rayleigh number, periodicity of the magnetic field, magnetic field inclination angle, thermal conductivity ratio, and medium porosity on flow and thermal patterns have been examined. It has been found that parameters of the periodic magnetic field have the non-monotonic influence of the heat transfer performance.

The effect of porosity and permeability variations on convective stability of a two-layer system under longitudinal vibration in zero gravity

The effect of the porosity and permeability variations on the onset of average convection in a single-component fluid within an non-uniformly heated horizontal layer partially filled with a porous zone under zero gravity conditions is studied. The system of the fluid and porous layers oscillates as a whole with high frequency and small amplitude in the longitudinal direction. Permeability is uniform within the porous zone and related to porosity by the Carman–Kozeny formula. The method of constructing a fundamental system of solutions as well as the orthogonalization of vectors for partial solutions are applied to simulate a linear stability problem with respect to the fluid quasi-equilibrium state in the layers numerically. The instability threshold relative to the short-wave and long-wave convective rolls is found. An abrupt change in the type of instability from the short-wave to long-wave ones occurs with the growth of the porosity from 0.3 to 0.8. A similar variation also presents in the case of thermal gravitational convection in layered fluid systems with porous zones under Earth conditions. Convection in weightlessness distinguishes from that in the gravitational field by a non-monotonic behavior of the instability threshold with increasing porosity at different fixed frequencies of vibration. The onset of convection delays at low frequencies and, on the contrary, speeds up at high enough frequencies, if the porosity belongs to the interval from 0.3 to 0.8. In addition, one studies the effect of two types of boundary conditions for tangential velocities near the interface between the layers on the instability threshold in the system with a low permeable porous zone.

Modeling hydrodynamics and heat exchange in response to liquid metal buoyant flow in a rectangular channel in a coplanar magnetic field

The hydrodynamics and heat exchange in response to liquid metal (LM) buoyant flow in a rectangular channel with approximately 3/1 aspect ratio in a coplanar magnetic field (MF) on the assumption of one-sided channel heating have been studied. The flow configuration is approximated to the heat exchange channel of the cooling system of a TOKAMAK fusion reactor LM blanket. Experiments were performed on the mercury magneto-hydrodynamic (MHD) test stand as part of MPEI – JIHT of RAS facility. Under the conditions of buoyant flow, the measurable effect of thermal gravitational convection (TGC) interacting with the external MF in some environments was detected, it causes occurrence and development of layered flow variabilities. Averaged velocity, channel wall temperature profiles are shown. The MF causes suppression of turbulent transfer resulting in an increase in heated wall temperature. A numerical model of target values under experimental conditions is proposed. In the process of numerical simulation, the differential equation system: the continuity equation, the motion equation for three velocity components and the energy equation, was solved. To compute turbulence, the Reihardt model adapted for the rectangular channel was used. Computational data agree satisfactorily with experimental data.

Behavior of the Melt in the Vertical Bridgman Method with a Low Axial Temperature Gradient

The thermal conditions for the growth of Ge crystals with a diameter of 50 mm by the vertical Bridgman method in the case of low thermal-gravitational convection are studied using model experiments. Distilled water being hydrodynamically similar to the Ge melt is used as the model liquid. When modelling by means of the light cut method, it is established that mixture particles move along the heat flow direction from top to bottom. It is shown that an axial temperature gradient of 2 K/cm or more increases the contribution of thermal diffusion to mass transfer at a vertical flow rate of 0.09 mm/s or more. The numerical simulation of thermal convection in the Oberbeck–Boussinesq approximation confirms the absence of convection under the given thermal conditions. However, the deviation of the container axis from the vertical by 0.5° during the process of crystal growth contributes to the increase in the flow rates in the liquid phase up to 0.55 mm/s.

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.

A mathematical and numerical framework for the analysis of compressible thermal convection in gases at very high temperatures

The relevance of non-equilibrium phenomena, nonlinear behavior, gravitational effects and fluid compressibility in a wide range of problems related to high-temperature gas-dynamics, especially in thermal, mechanical and nuclear engineering, calls for a concerted approach using the tools of the kinetic theory of gases, statistical physics, quantum mechanics, thermodynamics and mathematical modeling in synergy with advanced numerical strategies for the solution of the Navier-Stokes equations. The reason behind such a need is that in many instances of relevance in this field one witnesses a departure from canonical models and the resulting inadequacy of standard CFD approaches, especially those traditionally used to deal with thermal (buoyancy) convection problems. Starting from microscopic considerations and typical concepts of molecular dynamics, passing through the Boltzmann equation and its known solutions, we show how it is possible to remove past assumptions and elaborate an algorithm capable of targeting the broadest range of applications. Moving beyond the Boussinesq approximation, the Sutherland law and the principle of energy equipartition, the resulting method allows most of the fluid properties (density, viscosity, thermal conductivity, heat capacity and diffusivity, etc.) to be derived in a rational and natural way while keeping empirical contamination to the minimum. Special attention is deserved as well to the well-known pressure issue. With the application of the socalled multiple pressure variables concept and a projection-like numerical approach, difficulties with such a term in the momentum equation are circumvented by allowing the hydrodynamic pressure to decouple from its thermodynamic counterpart. The final result is a flexible and modular framework that on the one hand is able to account for all the molecule (translational, rotational and vibrational) degrees of freedom and their effective excitation, and on the other hand can guarantee adequate interplay between molecular and macroscopic-level entities and processes. Performances are demonstrated by computing some incompressible and compressible benchmark test cases for thermal (gravitational) convection, which are then extended to the high-temperature regime taking advantage of the newly developed features.

Mathematical simulation of nonstationary regimes of natural convection in a cubical enclosure with finite-thickness heat-conducting walls

Nonstationary regimes of conjugate thermal-gravitational convection in a cubical enclosure in the conditions of horizontal temperature difference is numerically analyzed. The external surfaces of two opposite walls were at constant different temperatures, the rest external edges were considered adiabatic. The mathematical model based on Oberbeck-Bussinesq equations is formulated in dimensionless natural velocity-pressure variables. Typical temperature and velocity fields, which represent the effect of the nonstationarity factor, Prandtl number, and thermophysical characteristics of the enclosure solid walls on the flow and heat transfer, are obtained.

Thermal gravitational convection of a side-heated supercritical fluid with variable physical properties

Thermal gravitational convection in a supercritical medium is studied at parameter values higher than those corresponding the liquid-gas phase transition point. Steady convective flow and heat transfer are modeled in an extended region with a square cross-section, the temperatures on the lateral boundaries being constant and different. Both Boussinesq flows, in which spatial variations of the density and the thermodynamic parameters are small, and non-Boussinesq flows with considerably spatially-variable physical properties are considered. The effect of the temperature inhomogeneity parameter on the distinctive features of the flow and the heat transfer is determined.

Determination of the similarity criteria for thermal gravitational convection in a supercritical fluid

Natural-convection flows in continua are studied at the parameter values beyond the critical point of the liquid-gas phase transition. The theory of similarity of supercritical convective flows is generalized by taking account for the dependence of the thermodynamic properties of a medium on the temperature and the density. In the case of an arbitrary two-parameter medium calibration relations, which serve for determining the similarity criteria of convection, are derived. In the fluid with the Van der Waals equation of state the calibration relations are reduced to the form of functions of the dimensionless temperature and density.

Three-dimensional convective interactions in an inclined longitudinal air layer

The results of a three-dimensional numerical simulation of thermal-gravitational convection for an arbitrary direction of the gravity force are presented. The calculations were carried out for a rectangular domain with the solid-wall aspect ratio 4:0.5:1. The cavity filled with air was heated from below. The convection regime was laminar with Rayleigh numbers Ra = 103–105. The cavity was gradually turned about the short side in different directions in the range 0–90° to study how the initial conditions affect the convection. The average heat transfer characteristics and the spatial flow patterns are presented. A comparison with two-dimensional calculations is given. The conditions of the occurrence of a hysteresis in the steady state of convection are found, a regime map is constructed, and the specific features of the interaction of heat fluxes are described.

Instabilities of thermal gravitational convection and heat transfer in the Czochralski model at different Prandtl numbers

The results of the calculations of critical Grash of numbers, at which flowfield and temperature fluctuations originate in the axisymmetric and three-dimensional models of crystal growth by pulling from a melt, are presented. The salient features of the convection and heat transfer structure in the zones of stabilization and changeover of dangerous modes are studied over a wide Prandtl number range under different boundary conditions on the melt surface and compared with the experimental data.

The influence of elastic compressibility of the mantle on thermal-gravitational convection: The convective instability of the gravitational stress pattern

The set of equations for the problem of thermal-gravitational convection accounts for compressibility of solid bodies, which changes for elementary volumes moving during convection process in fields of the initial temperature and the initial gravitational stress pattern for rheology of an elastic-viscous Maxwell body. It was shown that equations of momentum conservation in the vertical direction and heat transfer for steady convection differ from the equation for incompressible liquid by terms containing the rate of elastic volume change and the connected rate of heat change. It was established that an additional term in the momentum conservation equation defines a new class of the instable state of a solid body, which is able to form huge deformations at the expense of plasticity and creep at large segments of time—flow in the field of gravity force—instability of the gravitational stress pattern of elastic-plastic body. Analysis of different boundary conditions for which this instability can be realized in the form of convective cells showed that the convection rate is totally defined by reconstruction processes of vertical stresses on horizontal boundaries close to the initial gravitation pattern. Alignment process of these stresses can be provided not only by erosion and denudation processes occurring on the Earth’s surface, but also by processes on the inner boundaries of the tectonosphere which provide isostasy in the mantle.

Modeling convective and wave processes and heat transfer in near-supercritical media. An overview

The results of mathematical modeling and experimental investigation of flows and heat transfer in supercritical media in the vicinity of the critical thermodynamic point under microgravity and terrestrial conditions are considered. The effects of thermal gravitational convection and thermoacoustics complicated by the adiabatic compression effect, as well as the special features of two- and three-dimensional supercritical structures, are discussed. The experimental results obtained aboard the space station Mir are interpreted. The projects of experiments aboard the International Space Station, together with their terrestrial applications, are discussed.

Methods for modeling convective and wave processes and heat transfer in near-critical media. An overview

The paper outlines the theoretical studies of flows and heat transfer in near-critical media in the vicinity of the critical thermodynamic point under microgravity and terrestrial conditions. The criteria determining the characteristics of thermal gravitational convection and thermal acoustics complicated by the adiabatic compression effect, the distinctive features of convection generation, and the hierarchy of non-Boussinesq mathematical models and the their calibration on the basis of the Boussinesq model are discussed.

Patterns and heat transfer in Rayleigh-Bénard thermal gravitational convection in helium (3He) near the critical point

Thermal gravitational convection in a bottom-heated layer of near-critical 3He is considered. The range of criteria determining the convection parameters beyond the stability threshold is discussed. The specific features of 2D and 3D supercritical structures, the adiabatic compression effect, and heat transfer are considered.

Thermal and hydrodynamic relationships between the melt in the thermochemical plume conduit and the horizontal mantle flow

When modeling the formation and dynamics of thermochemical plumes, geologists and geophysicists of the world use equations of free convection in the Boussinesq approximation with addition of the mass transfer equation [1]. It is assumed that the base of the mantle corresponds to a dense chemical layer that was formed owing primarily to accretion of the oceanic crust in the lower mantle. The plume is forming in the thermal boundary zone in the presence of this dense chemical layer at the base of the mantle. The ascending migration of the plume under thermal and concentration gravitational convection is stimulated by the high temperature gradient between the core‐mantle interface and the surrounding mantle (750 K). The ascending movement of the plume is accompanied by the capture of material from the dense chemical layer. Our model of the thermochemical plume [2] differs from that accepted in [1]. Its difference from the latter is discussed in detail elsewhere [3]. According to our model, thermochemical plumes form at the core‐mantle interface under the influence of the heat flow from the outer core and local influx of a chemical dope that reduces the melting point of the lower mantle substance, which results in mantle substrate melting and upward plume migration [2, 3]. The heat and mass transfer due to interaction between horizontal free convection mantle flows and the conduit of the thermochemical plume located under the oceanic plate away from the mid-oceanic ridge (MAR) is considered in [4] with presentation of the main equations for calculating the heat and mass transfer by the thermochemical plume interacting with the horizontal convective mantle flow and quantitative correlation between the dope concentration at the plume base and the Lewis number. The purpose of this study is calculation of the heat exchange between the conduit of the thermochemical plume ascending from the core‐mantle interface and the horizontal mantle flow. We present the model of the plume conduit interacting with horizontal free convection flows in the upper and lower mantle and determine local coefficients of the heat transfer at the interface between the melt and the surrounding mantle (plume conduit boundary), as well as the diameter of the Hawaiian plume interacting with horizontal mantle flows. Based on the calculated heat exchange between the plume and the horizontal lower mantle flow, we also estimate the main parameters of the lower mantle. Let us consider the possible structure of the conduit and head part of the Hawaiian plume based on experimental modeling and seismic studies of the upper mantle beneath the Island of Hawaii. The thermal power of the Hawaiian plume is estimated to be 3 · 10 11 W, and the Raleigh number (Ra) for the free convection flow in the conduit of the Hawaiian plume is

Laser-supported combustion wave in the field of gravity

Numerical simulation is used for studying the propagation of slow laser-supported combustion waves (LCW) in atmospheric air. The gasdynamic structure is investigated of the flow field which accompanies the movement of laser-supported combustion wave. A special feature of the problem being solved is the inclusion of gravity; this results in the emergence of convective motions in the thermal wake of LCW. It is demonstrated that different modes of thermal gravitational convection are observed depending on the conditions of LCW movement. A two-dimensional radiation-gasdynamic model is given, which includes the continuity and Navier-Stokes equations, the energy equation, and the equations of laser and thermal radiation transport in a multi-group approximation. The real thermal and spectral optical properties of air in the temperature range from 300 to 20 000 K are taken into account.

Charge dissolution and heat and mass transfer during hydrothermal crystal growth with heat field rotation

Analytical modeling of laminar thermal gravitational convection in a vertical porous cylinder of radius R and height H (R is much smaller than H) with a low-compressed fluid (a model of the charge-filled bottom part of autoclave) is performed by the Poincare method. Boundary thermal conditions of the first kind are set at the cylinder boundary. Along with the main thermal field (stationary, homogeneous along the azimuthal coordinate, and linear along the vertical; the temperature gradient is set at the top), an additional cyclically rotating azimuthal inhomogeneous thermal field is applied to the cylinder wall. Asymptotic representations of the temperature, velocity, and concentration fields in the porous layer are obtained in the criteria form. It is shown that applying an additional temperature field increases the charge dissolution rate without increasing the average temperature gradient.