论文总字数：33342字

摘 要

方封闭腔内的自然热对流是计算流体力学（CFD）的经典问题，同时也是验证新算法的基准算例。方腔流模型具有边界条件设置简单、内部流动易于分析比对、与实际工程问题联系紧密等突出特点，传统的CFD方法在模拟方腔流模型方面已有一套完整的体系与方法，为许多实际工程问题提供了解决方法，但其始终存在如下不足：复杂网格模型计算过程耗时过长；复杂边界难以描述；依托于连续性假设。作为一种新兴的流体计算方法，格子玻尔兹曼方法（LBM）凭借其程序简洁、易于并行计算、并不依托于连续性假设等优点，在计算流体力学领域影响日重。本文采用格子玻尔兹曼方法，针对差异加热方腔内的自然热对流问题构筑了双分布函数模型，进行了数值模拟。研究了在恒定普朗特数（）下可变瑞利阻尼数（，，，）与几何纵横比（）对封闭方腔内对流换热情况的影响，以努塞尔数和流线的变化衡量传热形式与速率，结果以等温图、流线图与速度图的形式给出，并与fluent模拟结果进行比对以验证模型准确性。为对计算结果进行进一步分析，适当地调整流体控制参数，多次运行程序以观察比较封闭方腔内流动情况和换热情况的变化趋势。结果表明：瑞利阻尼数低于时，热端壁面无量纲温度沿腔体长度方向线性下降，腔体内的换热以热传导为主，随着瑞利阻尼数的增加，该曲线偏离线性，对流换热逐渐占据主导作用。腔体内流线、等温线始终呈中心对称，腔体中心部随着瑞利阻尼数的增加有出现二次流的趋势，同时边界层会随着瑞利阻尼数增加而减薄。腔体长高比的增加或瑞利阻尼数的增加都会导致努塞尔数的增加，反之亦然。沿壁面的努塞尔数随着腔体高度方向变小，其变小的程度随着瑞利数的增加而增加。热端和冷端壁面的平均努塞尔数从1.143增加到8.454，分别对应瑞利数和。通过把平均努塞尔数与基准解比较，表明本文模拟结果与基准解存在2%-4%的相对误差，再次验证了该模型的准确性与可靠性。

关键词：格子玻尔兹曼方法 自然对流 双分布函数

Numerical Study of Natural Thermal Convection in a Square Cavity Based on Lattice Boltzmann Method

Abstract

Natural thermal convection in a square enclosed cavity is a classic problem of computational fluid dynamics (CFD), and it is also a benchmark example to verify the new algorithm. The square cavity flow model has the outstanding characteristics of simple boundary condition setting, easy analysis and comparison of internal flow, and close connection with actual engineering problems. The traditional CFD method has a complete system and method for simulating the square cavity flow model. The engineering problem provides a solution, but it always has the following shortcomings: the calculation process of the complex grid model takes too long; the complex boundary is difficult to describe; relying on the continuity assumption. As an emerging fluid calculation method, the lattice Boltzmann method (LBM), with its simple program, easy parallel calculation, and does not rely on the continuity assumption, has an increasing influence on the field of fluid computing. In this paper, the lattice Boltzmann method is used to construct a double distribution function model for the problem of natural thermal convection in a differentially heated square cavity, and numerical simulation is performed. The effects of variable Rayleigh damping number () and geometric aspect ratio () on convective heat transfer in a closed square cavity under constant Prandtl number () are studied. The heat transfer form and rate are measured, and the results are given in the form of an isothermal graph, a streamline graph, and a velocity graph, and are compared with fluent simulation results to verify the accuracy of the model. In order to further analyze the calculation results, adjust the fluid control parameters appropriately, and run the program multiple times to observe and compare the changing trends of the flow and heat transfer conditions in the closed square cavity. The results show that when the Rayleigh damping number is lower than , the dimensionless temperature of the hot end wall decreases linearly along the length of the cavity. The heat transfer in the cavity is dominated by heat conduction. As the Rayleigh damping number increases, the curve deviates from linearity. Heat transfer gradually takes the leading role. The streamlines and isotherms in the cavity are always center-symmetric. The center of the cavity has a trend of secondary flow with the increase of the Rayleigh damping number, and the boundary layer will thin as the Rayleigh damping number increases. An increase in the cavity aspect ratio or an increase in the Rayleigh damping number will cause an increase in the Nusselt number, and vice versa. The Nusselt number along the wall surface decreases with the height of the cavity, and the degree of its decrease increases with the increase of the Rayleigh number. The average Nusselt number of the hot and cold walls increased from 1.143 to 8.454, corresponding to the Rayleigh number and .By comparing the average Nusselt number with the benchmark solution, it shows that there is a relative error of 2% -4% between the simulation results and the benchmark solution, the accuracy and reliability of the model are verified again.

Keywords: lattice Boltzmann method; natural convection; double distribution function

目 录

摘 要 I

Abstract II

符号对照表 1

第一章 绪论 3

1.1研究意义 3

1.2国内外研究进展 4

1.3研究内容 7

第二章 数值方法 8

2.1玻尔兹曼分布函数 8

2.2 BGKW近似 9

2.3边界条件 10

2.4自然热对流双分布函数模型 10

2.4.1控制方程 11

2.4.2碰撞过程 12

2.4.3迁移过程 13

2.5宏观流动参数 14

2.6模型验证 15

第三章 结果与讨论 16

3.1几何纵横比 16

3.1.1长高比为0.5 16

3.1.2长高比为1.0 20

3.1.3长高比为2.0 24

3.2努塞尔数 28

第四章 总结与展望 31

4.1总结 31

4.2展望 31

参考文献 32

附录 35

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