正交分解法在热对流系统稳定性分析中的应用研究
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摘要
热对流在自然界是普遍存在的，在许多领域，需要对热对流的稳定性加以控制。很多热对流是流场在各种驱动力，如浮力、表面张力以及磁力、离心力等驱动作用下失稳后的一种流动状态。对热对流稳定性的研究主要有线性、非线性稳定性理论分析和里亚普诺夫函数方法。这些研究方法的一个主要内容是求解出流动发生转捩的参数大小及参数变化区间，以及不稳定性流动将向何处发展等问题。但对于某一确定的流动而言，却不能分析是由哪些不稳定性因素导致了这个流动结果，以及这些不稳定性因素对流场不稳定性的贡献比重情况。为此，本研究试图在这方面作初步的尝试和探讨。
正交分解法借助于实验或数值计算结果，可以把一个振荡流场分解为一组线性权重组合的正交函数基(主要模式)的递加，每一个函数基都包含了流场各个点的信息，并且可以定量地给出各个函数基对整个流场的贡献比重。为此，在本研究中，尝试利用正交分解法来分析各不稳定性因素对流场不稳定性的贡献比重情况。
Czochralski(CZ)法是从熔体中生长单晶体的重要方法之一。在单晶的生长过程中，熔体的流动状态是影响单晶质量的主要因素之一。复杂的熔体流动，主要是由浮力不稳定性、毛细力引起的表面张力不稳定性以及由旋转引起的离心力不稳定性等引起的。为了更好的控制熔体的热对流，需要对热对流系统中各个不稳定性因素进行分析探讨。为此我们以CZ法晶体生长系统中坩埚内熔体为研究对象，在轴对称情况下，仅考虑浮力、表面张力、和晶体旋转时，建立物理数学模型。采用有限体积法和HSMAC算法，编写计算程序，对生长炉子内的熔液热对流进行了大量的模拟计算。
首先直接利用数值模拟的流场结果，定性地判断出浮力和表面张力各自对热对流场起主要作用以及作用相当的区域；然后在数值模拟流场结果的基础上，借助于正交分解法，抽取出流场的主要基本模式，获取了浮力不稳定性和表面张力不稳定性所对应主要模式各自的结构特点，如，浮力不稳定性表现为整体和旋转性，表面张力不稳定性表现为局部和移动性。通过对主要基本模式的分析比较，定性地探讨了浮力和表面张力在不同组合时，对熔液热对流不稳定性的贡献比重情况。最后探讨了主要基本模式和流场稳定性的关联，从而在理论上解释了流场主要模式可以用于分析流场不稳定性的原因。通过以上的分析，表明正交分解法可用于分析热对流不稳定性。
最后我们又在两个方面作了扩展尝试：第一、考虑晶体旋转，主要分析晶转为30转／min时，晶转对浮力和表面张力不稳定性的影响。得到了，在浮力较小时，晶体旋转对表面张力起强化作用；在浮力较大时，对表面张力起抑制作用；第二、在定性探讨的基础上，利用流场波动动能，初步构建了一种定量分析浮力和表面张力对熔液热对流不稳定性贡献比重的方法。
通过本文的分析，表明正交分解可以用于分析热对流不稳定性。拓宽了正交分解法的应用领域。
Thermal convection exists universally in nature. In general, thermal convection is needed to control. Thermal convection is mostly generated by flow field losing its stability under the driving forces, such as buoyancy, surface tension, magnetic force, centrifugal force. To study the stability in thermal convection, some stability analysis methods have already been developed and widely used, e.g., Lyapurov V-function, linear and non-linear stability analysis. These methods are mainly to determine critical value at which flow instability occur, the bifurcation and the region of control parameters. However, when more than one instabilities exists, these methods can not reveal the dominant instabilities which lead to the flow and contribution of the instabilities to the flow. Therefore, the present research aims to make the preliminary attempt and discussion in this aspect.
The method of Proper Orthogonal Decomposition(POD) is used to extract spatial information from a set of time series data available on the spatio-temporal domains. These data can be obtained from experiments or numerical simulations. It can also provide quantificational contribution of each dominant to the field. So, POD was used to analysis thermal convection instability in the study.
The Czochralski (CZ) crystal growth is one of the most important methods of producing single crystals from the melt. In single crystal growth, the quality of crystals is closely related to the transport phenomena of melt in the furnace. The observed complicated melt flows are caused by different types of instability such as buoyancy -induced instability, Marangoni instability and vortical instability. In order to control the thermal convection in the furnace better, we need to discuss the mechanism which forms the thermal convection. So as the melt in the furnace of the study object, we build a physically mathematic mold in suppose of symmetry. Melt flow are only caused by buoyancy、surface tension and circumrotation of single crystals. Simulation study to the thermal convection in the furnace is carried out using the finite volume method and HSMAC (Highly Simplified Marker and Cell) arithmetic by writing a mathematical procedure in Fortran.
First direct numerical simulation is used to obtain the flow fields driven by both buoyancy and surface tension and also the flow fields driven by one of these forces alone. And it is gained that different contribution of them to the thermal convection in the furnace by direct numerical simulation. Proper orthogonal decomposition is then employed to extract the basic modes from the flow fields. By comparing the basic modes between these situations, the dominant instability involved in such complicated flows has been revealed. It is found that the basic modes corresponding to buoyancy-induced instability are global and turning and those to Marangoni-induced instability are local and traveling. It is also gained that different contribution of them to the thermal convection in different combination of them. At last we discuss the connection of dominant modes to flow field instability, and explain why dominant modes can analyze the thermal convection instability.
Finally we have an attempt in two aspects. First, Circumrotation of single crystal was considered (30r/min). How circumrotation of single crystal affect to buoyancy and surface tension instability is investigated. It is found that circumrotation of single crystal inhibits to buoyancy when the buoyancy is small, otherwise accelerated. Second, Based On qualitative discussion result, a quantitative analysis method is constructed to get what contribution proportion of buoyancy and the surface tension to thermal convective instability in melts by use of the field undulation kinetic energy.
By the analysis, it offers some reference data to control instability of thermal convection in the furnace. It indicates that the proper orthogonal decomposition be a powerful method in instability analysis. At the same time, it widen the study fields of proper orthogonal decomposition.
正交分解法借助于实验或数值计算结果，可以把一个振荡流场分解为一组线性权重组合的正交函数基(主要模式)的递加，每一个函数基都包含了流场各个点的信息，并且可以定量地给出各个函数基对整个流场的贡献比重。为此，在本研究中，尝试利用正交分解法来分析各不稳定性因素对流场不稳定性的贡献比重情况。
Czochralski(CZ)法是从熔体中生长单晶体的重要方法之一。在单晶的生长过程中，熔体的流动状态是影响单晶质量的主要因素之一。复杂的熔体流动，主要是由浮力不稳定性、毛细力引起的表面张力不稳定性以及由旋转引起的离心力不稳定性等引起的。为了更好的控制熔体的热对流，需要对热对流系统中各个不稳定性因素进行分析探讨。为此我们以CZ法晶体生长系统中坩埚内熔体为研究对象，在轴对称情况下，仅考虑浮力、表面张力、和晶体旋转时，建立物理数学模型。采用有限体积法和HSMAC算法，编写计算程序，对生长炉子内的熔液热对流进行了大量的模拟计算。
首先直接利用数值模拟的流场结果，定性地判断出浮力和表面张力各自对热对流场起主要作用以及作用相当的区域；然后在数值模拟流场结果的基础上，借助于正交分解法，抽取出流场的主要基本模式，获取了浮力不稳定性和表面张力不稳定性所对应主要模式各自的结构特点，如，浮力不稳定性表现为整体和旋转性，表面张力不稳定性表现为局部和移动性。通过对主要基本模式的分析比较，定性地探讨了浮力和表面张力在不同组合时，对熔液热对流不稳定性的贡献比重情况。最后探讨了主要基本模式和流场稳定性的关联，从而在理论上解释了流场主要模式可以用于分析流场不稳定性的原因。通过以上的分析，表明正交分解法可用于分析热对流不稳定性。
最后我们又在两个方面作了扩展尝试：第一、考虑晶体旋转，主要分析晶转为30转／min时，晶转对浮力和表面张力不稳定性的影响。得到了，在浮力较小时，晶体旋转对表面张力起强化作用；在浮力较大时，对表面张力起抑制作用；第二、在定性探讨的基础上，利用流场波动动能，初步构建了一种定量分析浮力和表面张力对熔液热对流不稳定性贡献比重的方法。
通过本文的分析，表明正交分解可以用于分析热对流不稳定性。拓宽了正交分解法的应用领域。
Thermal convection exists universally in nature. In general, thermal convection is needed to control. Thermal convection is mostly generated by flow field losing its stability under the driving forces, such as buoyancy, surface tension, magnetic force, centrifugal force. To study the stability in thermal convection, some stability analysis methods have already been developed and widely used, e.g., Lyapurov V-function, linear and non-linear stability analysis. These methods are mainly to determine critical value at which flow instability occur, the bifurcation and the region of control parameters. However, when more than one instabilities exists, these methods can not reveal the dominant instabilities which lead to the flow and contribution of the instabilities to the flow. Therefore, the present research aims to make the preliminary attempt and discussion in this aspect.
The method of Proper Orthogonal Decomposition(POD) is used to extract spatial information from a set of time series data available on the spatio-temporal domains. These data can be obtained from experiments or numerical simulations. It can also provide quantificational contribution of each dominant to the field. So, POD was used to analysis thermal convection instability in the study.
The Czochralski (CZ) crystal growth is one of the most important methods of producing single crystals from the melt. In single crystal growth, the quality of crystals is closely related to the transport phenomena of melt in the furnace. The observed complicated melt flows are caused by different types of instability such as buoyancy -induced instability, Marangoni instability and vortical instability. In order to control the thermal convection in the furnace better, we need to discuss the mechanism which forms the thermal convection. So as the melt in the furnace of the study object, we build a physically mathematic mold in suppose of symmetry. Melt flow are only caused by buoyancy、surface tension and circumrotation of single crystals. Simulation study to the thermal convection in the furnace is carried out using the finite volume method and HSMAC (Highly Simplified Marker and Cell) arithmetic by writing a mathematical procedure in Fortran.
First direct numerical simulation is used to obtain the flow fields driven by both buoyancy and surface tension and also the flow fields driven by one of these forces alone. And it is gained that different contribution of them to the thermal convection in the furnace by direct numerical simulation. Proper orthogonal decomposition is then employed to extract the basic modes from the flow fields. By comparing the basic modes between these situations, the dominant instability involved in such complicated flows has been revealed. It is found that the basic modes corresponding to buoyancy-induced instability are global and turning and those to Marangoni-induced instability are local and traveling. It is also gained that different contribution of them to the thermal convection in different combination of them. At last we discuss the connection of dominant modes to flow field instability, and explain why dominant modes can analyze the thermal convection instability.
Finally we have an attempt in two aspects. First, Circumrotation of single crystal was considered (30r/min). How circumrotation of single crystal affect to buoyancy and surface tension instability is investigated. It is found that circumrotation of single crystal inhibits to buoyancy when the buoyancy is small, otherwise accelerated. Second, Based On qualitative discussion result, a quantitative analysis method is constructed to get what contribution proportion of buoyancy and the surface tension to thermal convective instability in melts by use of the field undulation kinetic energy.
By the analysis, it offers some reference data to control instability of thermal convection in the furnace. It indicates that the proper orthogonal decomposition be a powerful method in instability analysis. At the same time, it widen the study fields of proper orthogonal decomposition.
引文
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[2] Landau, L.D.,C.R.Acad.Sci.USSR,44(4994),311-314.
[3] Stuart,J.T.,J.EM.,4(1958),1-21.
[4] Stuart,J.T.,J.EM.,9(1960),353-370.
[5] 周恒，尤学一．中国科学．5(1992)．457-504。
[6] 曾丹苓．工程非平衡热动力学．北京：科学出版社．1991年：149-201。
[7] 过增元．热流体学．清华大学出版社．1992：178-179。
[8] Chengjun Jing.Numerical studies of wave pattern in an oxide melt in the Czochralski crystal growth.J.Crystal Growth 265-2004.
[9] Chengjun Jing. Effect of internal radiative heat transfer on spoke pattern on oxide melt surface in Czochralski crystal growth.J.Crystal Growth 259-2003.
[10] Chengjun Jing. Effect of RF coil position on spoke pattern on oxide melt surface in Czochralski crystal growth.J.Crystal Growth 252-2003.
[11] 闵乃本著．晶体生长的物理基础．上海科学技术出版社．1980年：174。
[12] G. Berkooz. Observations on the proper orthogonal decomposition. In T. B. Gatski,S. Sarkar, and C. G. Speziale, editors, Studies in Turbulence, pages 229{247. Spdnger-Vedag, New york, 1991.
[13] G.Berkooz, PJ.Holmes and J. L Lumley,The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25:539-575, 1993.
[14] D.H.Chambers, R.J.Adrian,P.Moin, D.S.Stewart and H.J.Sung, Karhunen-Loeve expansion of Burgers' model of turbulence, Phys, Fluids, 31(9):2573-2582, September 1988.
[15] Philip J. Holmes. Can dynamical systems approach turbulence? In J. L.Lumley, editor, Whther Turbulence? Turbulence at the Crossroads, volume 357 of Lect.Notes Phys.,1991: 195-249.
[16] M. Kirby, J. P. Boris, and L. Sirovich. A proper orthogonal decomposition of a simulated supersonic shear layer. Int. J. Numerical Methods Fluids, 10: 411-428, 1990.
[17] L. Sirovich. Turbulence and the dynamics of coherent structures, Parts Ⅰ-Ⅲ. Quarterly of Applied Mathematics, ⅩLⅤ(3): 561-590, 1987.
[18] L Sirovich, B. W. Knight, and J. D. Rodriguez. Optimal low-dimensional dynamical approximations. Quarterly of Applied Mathematics, ⅩLⅧ(3):535{548, September 1990.
[19] Henk Aling, Suman Banerjee, Anil K. Bangia, Vernon Cole, Jon Ebert, Abbas Emami-Naeini, Klavs F. Jensen, Ioannis G. Kevrekidis, and Stanislav Shvartsman. Nonlinear model reduction for simulation and control of rapid thermal processing. In Proceedings of the American control Conference, volume 4, p2233-2238, 1997.
[20] G.M. Kepler, H.T.Tran, and H.T. Banks. Reduced order model compensator control of species transport in a CVD reactor.CRSC Technical Report CSRC-TR99-25, North Carolina State University, April 1999.
[21] HungV. Ly and Hien T. Tran. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. CRSC Technical Report CRSC-TR98-13, North Carolina State University, 1998. In press, Quart. Appl. Math.
[22] A. Theodoropoulou, R. A. Adomaitis, and E. Za_riou. Model reduction for optimization of rapid thermal chemical vapor deposition systems. IEEE Transactions on Semicon-ductor Manufacturing, 11(1): 85-98, 1998.
[23] Marco Fahl. Computation of PODs for fluid flows with Lanczos methods. Technical Report 99-13, Universitat Trier, 1999.
[24] Hung V. Ly and Hien T. Tran. Modeling and control of physical processes using properorthogonal decomposition. In press, Journal of Mathematical and Computer Modeling, February 1999.
[25] H.T. Banks, R. C. H. del Rosario, and R. C. Smith. Reduced order model feedbackcontrol design: numerical implementation in a thin shell model. CRSC Technical ReportCSRC-TR98-27, North Carolina State University, July 1998.
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