A Schur complement formulation for solving free-boundary,Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm–Newton approach, which employs Newton’s method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt–solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.     

Semianalytical analysis of thermal effect in LD double-side-pumped rectangular laser crystal

A semianalytical method to analyze the thermal effect in a LD double-side-pumped rectangular laser crystal is put forward. Through the analysis of working characteristics of the laser crystal, a thermal model that matches actual situations of the laser crystal is established. General expressions of temperature field and thermal distortion field in the laser crystal can be obtained by a novel method to solve the heat conduction equation of orthotropic material. This semianalytical method can be used to calculate the temperature field and thermal distortion field in other LD double-side-pumped laser crystals and is applied to Nd:YVO₄ crystal in detail in this paper, and two methods of effectively reducing thermal distortion in the laser crystal are offered. Results show that a maximum temperature rise of 362.2 °C and a maximum thermal distortion of 5.55 μm are obtained in Nd:YVO₄ crystal when the output power of the two laser diodes are both 30 W. When the off-center distance is 0.6 mm, the maximum thermal distortion is reduced by 37.7%; when the thickness of the crystal is reduced from 2.0 mm to 1.4 mm, the maximum thermal distortion is reduced by 31.7%. Results in this paper can offer theoretical base for better solving thermal problems in laser system.     

The effect of two-dimensional shear flow on the stability of a crystal interface in a supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of twodimensional shear flow on the stability of a crystal interface in the supercooled melt of a pure substance.Similar to the three-dimensional shear flow as described in our previous paper,the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude.However,compared with the case of the Laplace equation for a steady-state thermal diffusion field,due to the existence of time partial derivatives of the temperature fields in the diffusion equation the absolute value of the gradients of the temperature fields increases,therefore destabilizing the interface.The circular interface is more unstable than in the case of Laplace equation without time partial derivatives.The critical stability radius of the crystal interface increases with shearing rate increasing.The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

Coupled thermal-optic effects and electrical modulation mechanism of birefringence crystal with Gaussian laser incidence

We study the Gaussian laser transmission in lithium niobate crystal(LiNbO3) by using the finite element method to solve the electromagnetic field's frequency domain equation and energy equation. The heat generated is identified by calculating the transmission loss of the electromagnetic wave in the birefringence crystal, and the calculated value of the heat generated is substituted into the energy equation. The electromagnetic wave's energy losses induced by its multiple refractions and reflections along with the resulting physical property changes of the lithium niobate crystal are considered.Influences of ambient temperature and heat transfer coefficient on refraction and walk-off angles of O-ray and E-ray in the cases of different incident powers and crystal thicknesses are analyzed. The E-ray electrical modulation instances, in which the polarized light waveform is adjusted to the rated condition via an applied electrical field in the cases of different ambient temperatures and heat transfer coefficients, are provided to conclude that there is a correlation between ambient temperature and applied electrical field intensity and a correlation between surface heat transfer coefficient and applied electrical field intensity. The applicable electrical modulation ranges without crystal breakdown are proposed. The study shows that the electrical field-adjustable heat transfer coefficient range becomes narrow as the incident power decreases and wide as the crystal thickness increases. In addition, it is pointed out that controlling the ambient temperature is easier than controlling the heat transfer coefficient. The results of the present study can be used as a quantitative theoretical basis for removing the adverse effects induced by thermal deposition due to linear laser absorption in the crystal, such as depolarization or wave front distortion, and indicate the feasibility of adjusting the refractive index in the window area by changing the heat transfer boundary conditions in a wide-spectrum laser.     

The effect of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt of pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of Laplace equation for steady state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

定向结晶条件下聚乙二醇6000的强动力学效应

在单向温度场条件下, 采用不同抽拉速度实现了聚乙二醇6000的定向生长、界面形貌的实时观测及界面温度的测量, 进而揭示了其生长机制. 实验结果表明, 随着抽拉速度的增大, 界面的温度逐渐减小, 过冷度逐渐增大. 运用高聚物结晶的次级形核理论模型, 对实验数据进行了计算, 得到在界面过冷度为13.5 K左右时, 生长机制发生了由区域Ⅱ向区域Ⅲ的转变. 实验数据与等温结晶数据的比较发现等温结晶方法中获得过冷度相对较大, 是因为其包含了热过冷. 聚乙二醇6000定向结晶过程中需要的最大动力学过冷度为20 K, 说明由于高聚物的二维形核, 其生长主要由界面动力学控制, 具有较强的动力学效应.     

基于OpenFOAM的投影法磁流体求解器开发与验证

在开源计算流体力学C++工具包OpenFOAM环境下开发了低磁雷诺数条件下的磁流体求解器,并进行了验证。采用投影算法求解动量方程和压力泊松方程;采用非结构网格同位相容守恒算法求解电势泊松方程、感应电流和洛伦兹力;采用边界耦合方法求解流固耦合电势场。通过对均匀磁场下导电方管和导电圆管内的完全发展磁流体层流的数值模拟和解析解的对比,对求解器进行了验证。进一步对非均匀强磁场作用下导电方管和导电圆管内完全发展磁流体层流进行了数值模拟,并与ALEX实验结果进行了比较。数值解和实验结果吻合良好。所开发的求解器可用于复杂结构强磁场作用下磁流体的数值模拟研究。     

Full multi grid method for electric field computation in point-to-plane streamer discharge in air at atmospheric pressure

Streamers dynamics are characterized by the fast propagation of ionized shock waves at the nanosecond scale under very sharp space charge variations. The streamer dynamics modelling needs the solution of charged particle transport equations coupled to the elliptic Poisson’s equation. The latter has to be solved at each time step of the streamers evolution in order to follow the propagation of the resulting space charge electric field. In the present paper, a full multi grid (FMG) and a multi grid (MG) methods have been adapted to solve Poisson’s equation for streamer discharge simulations between asymmetric electrodes. The validity of the FMG method for the computation of the potential field is first shown by performing direct comparisons with analytic solution of the Laplacian potential in the case of a point-to-plane geometry. The efficiency of the method is also compared with the classical successive over relaxation method (SOR) and MUltifrontal massively parallel solver (MUMPS). MG method is then applied in the case of the simulation of positive streamer propagation and its efficiency is evaluated from comparisons to SOR and MUMPS methods in the chosen point-to-plane configuration. Very good agreements are obtained between the three methods for all electro-hydrodynamics characteristics of the streamer during its propagation in the inter-electrode gap. However in the case of MG method, the computational time to solve the Poisson’s equation is at least 2 times faster in our simulation conditions.     

A composite grid solver for conjugate heat transfer in fluid–structure systems

We describe a numerical method for modeling temperature-dependent fluid flow coupled to heat transfer in solids. This approach to conjugate heat transfer can be used to compute transient and steady state solutions to a wide range of fluid–solid systems in complex two- and three-dimensional geometry. Fluids are modeled with the temperature-dependent incompressible Navier–Stokes equations using the Boussinesq approximation. Solids with heat transfer are modeled with the heat equation. Appropriate interface equations are applied to couple the solutions across different domains. The computational region is divided into a number of sub-domains corresponding to fluid domains and solid domains. There may be multiple fluid domains and multiple solid domains. Each fluid or solid sub-domain is discretized with an overlapping grid. The entire region is associated with a composite grid which is the union of the overlapping grids for the sub-domains. A different physics solver (fluid solver or solid solver) is associated with each sub-domain. A higher-level multi-domain solver manages the entire solution process.     

高功率半导体激光端面泵浦方形掺Nd3+离子激光晶体热形变研究

为了解决高功率半导体激光器端面泵浦激光晶体引起的热效应问题,激光晶体泵浦端面的热形变必须进行准确的计算.通过对于全固态激光器中激光晶体的工作特点分析,建立了矩形截面激光晶体热分析模型.基于热传导方程,提出了泊松方程的一种新解,并获得了矩形截面激光晶体端面热形变分布的一般解析表达式.同时讨论了半导体激光器偏心泵浦激光晶体给端面热形变带来的影响.与有限元分析方法以及其他数值分析方法相比,解析分析方法不会给计算引入任何的误差.热形变的解析分析为解决激光晶体的热效应问题以及提高激光器的性能提供了理论的依据.     

Theory of First,Second and Zero Sound in One-Dimensional Crystals

The propagation of thermal and mechanical disturbances in a non-conducting crystal is studied on the basis of a generalized transport equation for phonons, derived in a previous paper. This equation agrees with the Peierls-Boltzmann equation for slowly varying inhomogeneities in space and time and is consistent with the perturbation treatment of the phonon self energy for high frequencies and short wave-lengths. A formal solution of the transport equation is obtained, which allows us to study in detail the transition between the low-frequency, collision-dominated regime and the high-frequency, essentially collision-less regime. Using a single relaxation time approximation we discuss the propagation of “sound” waves and obtain explicit expressions for the dispersion of the sound velocity and the damping, valid for arbitrary frequencies. In order to elucidate more clearly the technique used we consider in this paper an one-dimensional crystal, but the essential results can easily be taken over to the threedimensional case. A separate report on this will be published elsewhere.     

水中金属丝电爆炸拉氏磁流体动力学模拟方法

基于拉格朗日描述,建立了水中金属丝电爆炸的单温磁流体动力学模型,并给出一种高阶混合有限元离散求解方法。拉氏可压缩流体方程组中,速度定义在H1连续有限元空间,内能定义在L2间断有限元空间实现物质界面的精确捕捉,存在激波的区域引入张量人工粘性抑制数值振荡。磁扩散方程仅考虑周向磁通量密度,简化为标量方程,使用H1连续有限元方法离散求解。焦耳热和洛伦兹力作为源项引入实现磁流体方程的耦合。数值算例表明:磁扩散求解器能够求解存在不同电导率的多介质磁扩散问题;拉氏流体求解器能够精确追踪物质界面,具有较好的激波分辨能力;耦合RLC电路的磁流体求解器能够复现水中金属丝电爆炸加热相变、冲击波的产生与传播、放电模式转变等物理过程。     

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie–Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.     

High Thermal Rectifications Using Liquid Crystals Confined into a Conical Frustum

In recent years, phononics, that studies thermal analogs of electronic devices, has become an important subject due to the need for better use of energy resources influenced by growing demand. On developing of these analogs, for example, thermal diodes, a successful route is the design of nanostructured materials (e.g., carbon nanotubes). However, these materials entail increased costs due to the use of complex techniques/equipments, while alternative cheaper materials present nearly comparable efficiency. In this work, we investigate how a thermal diode made by an alternative material (nematic liquid crystal), confined in a conical frustum capillary, can be optimized to achieve high rectifications. In such capillary tube, the thermotropic nematic liquid crystal 5CB produces an axially anisotropic defect called escaped radial disclination. With the molecular director field of such structure, we obtain the thermal conductivity tensor of the diode and solve the steady-state regime of Laplace and Fourier equations using the finite element method. We observed the anisotropy of the system with the non-linear temperature dependences of the molecular thermal conductivities that rectify the heat flux at rates up to 1266% at room temperature. Studying the sensitivity of the system with respect to shape and molecular and thermal aspects, we found that the improved thermal diode is suitable to be miniaturized and applied on well-determined areas, and it is robust against variations of the inward pumped heat flux. This work contributes to the usage of liquid crystals in non-display devices, having potential applications on controlling the heat flux through surfaces.     

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.     

Methods for creating the self-regulating mechanisms of passive systems for ensuring thermal regime of devices for space application

The passive systems for ensuring the thermal regime of space objects, which are in current use, can preserve the temperature at the same level only under a certain orientation of spacecraft and at a constant heat release of the equipment. At a variable orientation of the space object, its temperature may vary due to a possible illumination of the radiator-emitter of the system for ensuring the thermal regime by the solar or planet radiation. At a variable heat release the object temperature changes due to an off-design regime of the radiator operation. To compensate for the given effects the active elements are used — the heaters and coolers. This reduces the reliability of thermal regulation system and, consequently, the reliability of the entire spacecraft. The solutions are proposed, which create in the absence of active elements in the thermal regulation system the internal mechanisms enabling a compensation of the variability of heat fluxes both at a reorientation of the spacecraft and at a variation of the internal heat release.     

Semianalytical thermal analysis of rectangle Nd:GGG in heat capacity laser

Based on the theory of anisotropy semianalytical thermal analysis, the temperature field of rectangle Nd:GGG heat capacity laser crystal is investigated. Through an analysis of the working characteristics of the heat capacity laser crystal, a thermal model of heat capacity laser crystal is established. Using a new method for the anisotropic medium heat conduction equation, a temperature field expression of rectangle Nd:GGG heat capacity laser crystal for pumping stage and cooling stage is obtained, respectively. These results show that when using the output power of 300 W LD end-pumped rectangle Nd:GGG crystal for 10 seconds, the maximum temperature rise in the center of the pump face is 180.18°C, and after stopping pumping for 100 seconds, the maximum temperature rise drops to 0.74%. These results from this work provide a theoretical basis for the optimized design of a LD end-pumped heat capacity laser.     

基于改进元胞自动机模型的三元合金枝晶生长的数值模拟

在二元合金元胞自动机模型的基础上,通过耦合多元合金热力学相平衡求解器PanEngine, 建立了三元合金改进的元胞自动机模型,可模拟初生相枝晶的生长过程. 模型考虑了曲率过冷和成分过冷对界面平衡溶质成分的影响,通过不同组元的无量纲溶质过饱和度方程和界面溶质守恒方程之间的耦合来求解界面生长速率,并通过PanEngine计算界面处的液相线温度. 采用本模型模拟了Al-7%Si-xMg三元合金自由枝晶的生长形态, 结果表明Mg含量的增加会抑制枝晶一次臂的生长和二次臂的产生.同时模拟了不同抽拉速度下 Al-7%Si-0.5%Mg合金柱状枝晶的竞争生长过程,随着抽拉速度的增大,柱状枝晶一次枝晶臂间距逐渐减小, 与Hunt理论模型符合较好.     

跨流域高超声速绕流环境Boltzmann模型方程统一算法研究

如何准确可靠地模拟从外层空间高稀薄流到近地面连续流的航天器高超声速绕流环境与复杂流动变化机理是流体物理的前沿基础科学问题. 基于对Boltzmann方程碰撞积分的物理分析与可计算建模, 确立了可描述自由分子流到连续流区各流域不同马赫数复杂流动输运现象统一的Boltzmann模型速度分布函数方程, 发展了适于高、低不同马赫数绕流问题的离散速度坐标法和直接求解分子速度分布函数演化更新的气体动理论数值格式, 建立了模拟复杂飞行器跨流域高超声速飞行热环境绕流问题的气体动理论统一算法. 对稀薄流到连续流不同Knudsen数0.002 ≤Kn_∞ ≤1.618、不同马赫数下可重复使用卫星体再入过程(110–70 km)中高超声速绕流问题进行算法验证分析, 计算结果与典型文献的Monte Carlo直接模拟值及相关理论分析符合得较好. 研究揭示了飞行器跨流域不同高度高超声速复杂流动机理、绕流现象与气动力/热变化规律, 提出了一个通过数值求解介观Boltzmann模型方程, 可靠模拟高稀薄自由分子流到连续流跨流域高超声速气动力/热绕流特性统一算法.     

最近邻弱交换相互作用对纳米管中Blume-Capel模型热学性质的研究

利用有效场理论研究了纳米管上最近邻弱交换相互作用下spin-1纳米管中Blume-Capel模型的内能、比热和自由能,得到了系统的内能、比热和自由能与最近邻弱交换相互作用和晶场的关系。结果表明：最近邻弱交换相互作用和晶场强度等诸多因素相互竞争,使系统表现出比 = = =1时的BC模型更为复杂的热学性质;系统内能随温度的变化曲线表现出不连续性;比热随温度的变化出现奇异性;高温对自由能的影响更加明显。