INSTABILITY OF COUPLED THERMO-SOLUTE CAPILLARY CONVECTION IN LIQUID BRIDGE AND CONTROL BY ROTATING MAGNETIC FIELD

浮区法因具有无坩埚接触污染的生长优点而成为生长高完整性和高均匀性单晶材料的重要技术.但熔体中存在的毛细对流会给浮区法晶体生长带来极大挑战,这是由于对流的不稳定会导致晶体微观瑕疵的产生和宏观条纹等缺陷的形成.为了提高浮区法生长单晶材料的品质,研究浮区法晶体生长中毛细对流特性及如何控制其不稳定性显得尤为重要.本文采用数值模拟的方法对半浮区液桥内Si_xGe_1-x体系中存在的热质毛细对流展开研究并施加旋转磁场对其进行控制.结果表明：纯溶质毛细对流表现为二维轴对称模式,温度场主要由热扩散作用决定,而浓度场则由对流和溶质扩散共同支配;纯热毛细对流呈现三维稳态非轴对称流动,浓度分布与熔体内热毛细对流的流向密切相关,等温线在对流较大的区域发生弯曲;耦合溶质与热毛细对流则为三维周期性旋转振荡流.施加旋转磁场后,熔体周向速度沿径向向外增大,熔体内浓度场和流场均呈现二维轴对称分布.     

半浮区液桥热毛细振荡流

采用非定常、三维直接数值模拟方法研究大Pr数半浮区液桥热毛细对流从定常流向振荡流的过渡过程．文中详细描述了热毛细振荡流的起振和振荡特征,给出了液桥横截面上振荡流的流场和温度分布．在地面引力场条件下计算的结果与地面实验的结果进行比较,得出液桥水平截面上的流场和温度分布图样以一定的速度旋转,自由表面固定点处流体的环向流速正、负交替变化的一致结论．     

Regime diagram and three-dimensional effects of convective interactions in the hydrodynamic Czochralski model

The generalized diagram of the critical Grashof numbers as functions of the Prandtl number is presented. The diagram shows the zones of occurrence of flow field and temperature fluctuations in the axisymmetric and three-dimensional formulations of the crystal growth model using the pulling from a melt. The structure of thermals at high Prandtl numbers and the distinctive features of the three-dimensional convection structure in the zones of stabilization and hazardous mode changeover at different Prandtl numbers are discussed. The effect of crystal rotation on the flow and temperature field patterns is estimated.     

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.     

Branching of an axially symmetric convective flow

The stability of an axisymmetric convective gas flow with respect to finite disturbances of the bottom temperature is studied numerically using a finite-difference method. The convective flow considered approximately describes the free convection developing in the atmosphere due to heating of the substrate surface. The temperature disturbance used increases the intensity of one of two possible flows and suppresses the other flow, with the opposite signs of the vortices. Using the methods of numerical experiment, the corresponding problem of branching of the solution is examined. It is shown that the transformation of one flow into the other far from the threshold of the onset of convection requires substantial disturbances of the temperature.     

Convection of a heat-generating fluid in a rotating horizontal cylinder

Thermal convection of a fluid in a horizontal cylinder rotating about its own axis with uniformly volume-distributed internal heat sources is experimentally investigated. The enclosure boundary temperature was kept constant. The threshold of the excitation of convective flows and their structure are studied as functions of the heat-release intensity and the rotation velocity. The experiments are performed with water and water-glycerin solutions. It is shown that rapidly rotating fluid is in a stable quasiequilibrium state, namely, the temperature distribution is axisymmetric and has a maximum at the center of the enclosure. It is found that with decrease in the rotation velocity a convective flow arises thresholdwise, in the form of vortex cells periodically arranged along the axis. The thermal convection in the rotating enclosure is shown to be determined by the effects of two different mechanisms. One of these is due to the centrifugal force of inertia and plays the stabilizing role, while the other, thermovibrational mechanism is connected with nonisothermal fluid oscillations under the action of gravity in the enclosure-fitted reference frame and is responsible for the occurrence of mean thermal convection. The boundaries of the convection generation are plotted in the plane of the governing dimensionless parameters and the heat transfer in the supercritical region is studied.     

Convective heat transfer from a rotating or nonrotating axisymmetric body

An analysis of steady laminar mixed-convection heat transfer from a rotating or nonrotating axisymmetric body is presented. A mixed-convection parameter is proposed to serve as a controlling parameter that determines the relative importance of the forced and the free convection. In addition, a rotation parameter is introduced to indicate the relative contributions of the flow forced convection and the rotational forced convection. The values of both these two parameters lie between 0 and 1. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection from the forced-convection limit (rotating or nonrotating bodies) to the pure free-convection limit (non-rotating bodies) and the entire regime of forced convection from the pure flow forced-convection limit (nonrotating bodies) to pure rotational forced-convection limit (rotating bodies). The effects of mixed-convection intensity, body rotation, fluid suction or injection, and fluid Prandtl number on the velocity profiles, the temperature profiles, the skin-friction parameter, and heat transfer parameter are clearly illustrated for both cases of buoyancy assisting and opposing flow conditions.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Combined forced and free convection in an axisymmetric stagnation flow on a vertical non-isothermal cylinder

Boundary layer solutions are presented to study the effects of buoyancy on forced convection in an axisymmetric stagnation flow over a vertical cylinder with arbitrary surface heat flux variations. Numerical solutions are given for the governing momentum and energy equations. Two flow regions, namely, the buoyancy-assisted and buoyancy-opposed cases are analyzed. It is observed that the wall shear stress and surface heat transfer rate increase or decrease with the buoyancy force parameter depending upon the flow regime being buoyancy-assisted or buoyancy-opposed.     

高径比对GaAs熔体液桥热毛细对流失稳的影响

采用基于谱元法线性稳定性分析方法,研究了高径比对GaAs熔体(Pr=0.068)液桥热毛细对流失稳的影响,同时结合能量分析揭示了热毛细对流的失稳机制.研究结果表明:与典型低普朗特数(例如Pr=0.011)熔体静态失稳模式和典型高普朗特数(例如Pr>1)熔体振荡失稳模式不同,GaAs熔体热毛细对流失稳模式依赖于液桥高径比...     

Unsteady mixed convection over two-dimensional and axisymmetric bodies

The unsteady laminar incompressible mixed convection flow over a two-dimensional body (cylinder) and an axisymmetric body (sphere) has been studied when the buboyancy forces arise from both thermal and mass diffusion and the unsteadiness in the flow field is introduced by the time dependent free stream velocity. The nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The results indicate that for the thermally assisting flow the local skin friction, heat transfer and mass diffusion are enhanced when the buoyancy force from mass diffusion assists the thermal buoyancy force. But this trend is opposite for the thermally opposing flow. The point of zero skin friction moves upstream due to unsteadiness. No singularity is observed at the point of zero skin friction for unsteady flow unlike steady flow. The flow reversal is observed after a certain instant of time. The velocity overshoot occurs for assisting flows.     

Convection experiments in an inclined narrow cavity

The liquid flow behaviour in a small vertical gap with a heated and a cooled sidewall was studied experimentally in a former work as far as heat and mass transfer are concerned [Heiland et al. in Heat Mass Transf 43:863–870, 2007]. Following this, the study of thermal convection in a narrow cavity with variable inclination angle has been performed with liquid crystal techniques. Velocity and temperature fields of the flow have been measured. The results show that the strongest convection intensity arises in a vertical cavity.     

Experimental study of the stability of long axisymmetric liquid bridges between solid supports at different temperatures

The stability of axisymmetric, long liquid bridges held captive between two coaxial, circular solid disks kept at different temperatures is considered. Because of the temperature difference between the supporting disks, a thermally-induced surface tension gradient and its associated flow (Marangoni convection) appear in the liquid column, modifying (decreasing) the capillary stability of the bridge. The influence on the stability limits of long, axisymmetric liquid bridges of the combined effect of gravity acceleration and thermally induced surface tension gradients was experimentally analyzed by using very small size liquid bridges (between disks 1 mm in diameter). Experimental results are compared with available analytical results.     

导电流体热对流中磁场的致稳作用

本文导出了在磁场作用下导电流体热对流流动的方程组及其定解条件,用数值方法模拟了由磁场控制的单晶生长热对流问题,计算结果说明磁场可以有效地抑制流动在壁面处的分离、单胞对流变为多胞对流以及速度和温度的振荡等热不稳定现象,说明了磁场对不稳定热对流有明显的致稳作用。     

Petrov-Galerkin methods for natural convection in directional solidification of binary alloys

A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.     

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.     

Free convection boundary layer flow of a micropolar fluid past slender cones

Using the theory of micropolar fluids developed by Eringen, the transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past slender vertical cones are investigated. The case of constant surface heat flux is considered in this paper. Using perturbation techniques, the governing equations for momentum, angular momentum and energy have been solved numerically. Graphical representations for the velocity, angular velocity and thermal functions are presented for various physical and fluid property parameters.     

Hybrid lattice Boltzmann finite‐difference simulation of axisymmetric swirling and rotating flows

The axisymmetric flows with swirl or rotation were solved by a hybrid scheme with lattice Boltzmann method for the axial and radial velocities and finite‐difference method for the azimuthal (or swirl) velocity and the temperature. An incompressible axisymmetric lattice Boltzmann D2Q9 model was proposed to solve the axial and radial velocities through inserting source terms into the two‐dimensional lattice Boltzmann equation. Present hybrid scheme was firstly validated by simulations of Taylor–Couette flows between two concentric cylinders. Then the benchmark problems of melt flow in Czochralski crystal growth were studied and accurate results were obtained. Numerical experiment demonstrated that present axisymmetric D2Q9 model is more stable than the previous axisymmetric D2Q9 model (J. Comp. Phys. 2003; 186 (1):295–307). Hence, compared with the previous model, present numerical method provides a significant advantage in simulation melt flow cases with high Reynolds number and high Grashof number. Copyright © 2006 John Wiley & Sons, Ltd.     

Flow and temperature field measurements of thermal convection in a small vertical gap using liquid crystals

Thermal convection in a small vertical gap is studied experimentally applying digital particle image velocimetry/thermometry. This optical method enables the simultaneous measurement of two-dimensional flow and temperature fields in a liquid. The principle is based on seeding the liquid flow medium with thermochromic liquid crystal particles. The temperature is measured by the crystal particles which change their reflected colour as function of temperature. The flow velocity is measured by using the same particles as flow tracers. The investigation shall contribute to the understanding of the fluid mechanical behaviour of biological liquids within micro reactor systems. However, the problem is also of fundamental interest as far as heat and mass transfer is concerned. Measured temperature and flow velocity fields are presented and discussed. Presented in part at the 4th Chemnitz/Hamburger Colloquium (CHC) on Microflows, Hamburg, Germany, November 2004.     

Generalized lattice‐BGK concept for thermal and chemically reacting flows at low Mach numbers

The lattice‐BGK method has been extended by introducing additional, free parameters in the original formulation of the lattice‐BGK methods. The relationship between these parameters and the macroscopic moment equations is analysed by Taylor series and Chapman–Enskog expansion. The parameters are determined from the macroscopic moment equations by comparisons with the governing equations to be modelled. Extensions are presented for the Navier–Stokes equations at low Mach numbers in Cartesian or axisymmetric coordinates with constant or variable density, for scalar convection–diffusion equations and for equations of Poisson type. The generalized lattice‐BGK concept is demonstrated by two applications of chemical engineering. These are the computation of chemically reacting flow through an axisymmetric reactor and of the transport and deposition of particles to filters under the action of different forces. Copyright © 2006 John Wiley & Sons, Ltd.