自由表面旋涡的数值模拟与理论分析
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摘要
旋涡是自然界中常见的流动现象，如龙卷风，水力设备进水口前的自由表面
旋涡。旋涡流动极易受到外界因素的影响，它的结构和运动机理也十分复杂，吸
气旋涡还包含气液两相间的相互作用。本文对旋转容器底部排水所形成的盆池旋
涡现象进行了理论分析与数值模拟研究，以期再现自由表面旋涡运动规律，建立
临界淹没深度的预测模型，为深入研究水力工程中表面旋涡流动提供理论指导。
主要研究成果如下：
1）采用大涡模拟 LES（Large eddy simulation）与 VOF（Volume of fluid）方
法，模拟再现了吸气旋涡现象，计算所得气芯尺寸与实验结果吻合。当向中心聚
集的涡量大于从出口排除的涡量时，即形成吸气旋涡。LES 方法模拟出了旋转容
器底部的径向射流现象和 Ekman 层，以及吸气旋涡的气泡从气芯脱落的过程，分
析气芯内部流场结构，涡破裂现象是导致气芯失稳近而造成气泡脱落的原因。
2）采用雷诺平均 RANS（Reynolds-averaged Navier–Stokes equations）中的
RNG
k
湍流模型与 VOF（Volume of fluid）方法，模拟出了涡量向中心聚集的过
程，但是涡量的耗散过大，消耗了形成旋涡所必需的能量，未能得到吸气旋涡现
象。
3）采用分段函数表示径向速度分布，改进了 Burgers 涡模型。以最大径向速
度所对应的径向坐标为界，边界外的径向速度公式改写为“点汇”形式，边界内
的中心区域依然沿用 Burgers 涡模型的假设，克服了 Burgers 涡模型中径向速度分
布公式的缺陷，消除了“点汇”在原点处的奇性，且在无穷远处结果符合真实物
理现象。
4）在球坐标系下研究旋涡运动的空间效应，得到了空间轴对称“点汇”与“点
涡”叠加的模型。分析粘性的影响，在球坐标系下旋涡结构中应该还存在有类似
二次流的结构。
5）基于 Burgers 涡模型，综合考虑“汇”的作用，理论推导得到了一个新的
“临界淹没深度”预测公式，采用该公式所得预测值与相应的实验值吻合良好，
对预测临界淹没深度具有一定的指导意义。
关键词：自由表面旋涡；大涡模拟；Burgers 涡；点汇与点涡；临界
淹没深度
ABSTRACT
Vortex flow is very common in nature, such as tornado and free-surface vortex
above intakes of hydraulic systems. However vortex flow is sensitive to external factors,
and its mechanism is very complex, when coming to the air-core vortex, it will
encounter the problem of two-phase interaction. In order to find out the law of
free-surface vortex, and establish a model to estimate the critical submergence, bath-tub
vortex above outlet of a swirling barrel were simulated, and the results were studied
theoretically. The specific works were list as below:
1) Large-eddy Simulation (LES) were used to simulate the bath-tub vortex. LES
successfully got the air-core vortex, and the size of air-core is the same as what
observed in experiment, swirl flow developing to air-core vortex is due to the vortices
accumulation at center area. Meanwhile LES also got the radial jet phenomenon and the
Ekman layer at the bottom of vessel. And LES simulated the process of bubble tip-off
from the air-core as well, to study internal flow structure, vortex breakdown was the
reason to make bubble tip-off from air-core.
2) RNG
k
turbulence model of RANS got the process of vortices
accumulation, but because of its over dissipation, RNG
k
turbulence model cannot
get the result of air-core vortex.
3) To make an improvement of Burgers model, the radial velocity was expressed as
a piece-wise function. The function was separated at the place where the radial velocity
is the largest, the assumption of radial velocity expression in Burgers model was
adopted at the internal domain, and at the outside, radial velocity was expressed as a
―sink‖. Then the singularity at infinite was overcome, and the flaw of radial velocity
expression in Burgers model was fixed.
4) To study the problem of vortex at the spherical coordinate system, a kind of
spacial ―vortex‖ plus ―sink‖ model was got to descript the basic vortex structure. While
taking the viscous effect into consideration, it found that, in the real vortex, the
secondary flow structure should be existed.
5) On the basis of Burgers model and taking the ―sink‖ effect into consideration, a
model to estimate the critical submergence was got, and it was prove to be useful by
validating with others’ experimental data.
Keyword: free-surface vortex, LES, Burgers vortex model, sink plus
vortex, critical submergence
目录
摘要
ABSTRACT
第一章 绪论 .................................................................................................................... 1
1.1 研究背景及意义 .................................................................................................... 1
1.2 国内外研究历史与现状 ........................................................................................ 3
1.2.1 理论研究 .......................................................................................................... 3
1.2.2 实验研究 .......................................................................................................... 5
1.2.3 数值模拟 .......................................................................................................... 6
1.3 本文的主要工作 .................................................................................................... 7
第二章 控制方程与计算方法 ........................................................................................ 9
2.1 控制方程 ................................................................................................................ 9
2.1.1 流动控制方程 .................................................................................................. 9
2.1.2 湍流的控制方程 ............................................................................................ 10
2.2 数值计算 ............................................................................................................. 11
2.2.1 湍流的数值计算 ............................................................................................ 11
2.2.2VOF 方法简介 ................................................................................................ 15
本章小结 .................................................................................................................... 17
第三章 旋涡的数值模拟 .............................................................................................. 18
3.1 模型与网格 .......................................................................................................... 18
3.1.1 计算模型 ........................................................................................................ 18
3.1.2 网格划分 ........................................................................................................ 19
3.1.3 进出口条件 .................................................................................................... 20
3.2 LES 方法与 RNG
k
模型的比较 ................................................................... 20
3.2.1 自由液面的变化 ............................................................................................ 20
3.2.2 切向速度的变化 ............................................................................................ 23
3.2.3 涡量的变化 .................................................................................................... 25
3.2.4 湍流粘性 ........................................................................................................ 26
3.3 盆池涡内部结构的 LES 模拟 ............................................................................. 27
3.3.1 底部径向射流与 Ekman 层........................................................................... 28
3.3.3 气泡脱落 ........................................................................................................ 29
本章小结 .................................................................................................................... 32
第四章 旋涡运动的理论分析 ...................................................................................... 33
4.1 涡的来源 .............................................................................................................. 33
4.1.1 科氏力(Coriolis force) ................................................................................... 33
4.1.2 工程中涡的来源 ............................................................................................ 35
4.2 理论分析 .............................................................................................................. 36
4.2.1 二维平面涡流动 ............................................................................................ 37
4.2.2 具有轴向拉伸作用的轴对称涡 .................................................................... 44
4.2.3 球坐标系下的涡 ............................................................................................ 52
4.3 预测临界淹没深度的预测 .................................................................................. 57
4.3.1 临界汇球面（CSSS） .................................................................................. 58
4.3.2 临界汇球面的重新定义 ................................................................................ 60
4.3.3 涡的作用 ........................................................................................................ 62
4.3.4“汇”与“涡”的综合效应来预测临界淹没深度 ..................................... 64
本章小结 .................................................................................................................... 65
第五章 结论与展望 ...................................................................................................... 67
5.1 结论 ...................................................................................................................... 67
5.2 展望 ...................................................................................................................... 68
主要符号表 .................................................................................................................... 69
参考文献 ........................................................................................................................ 71
在读期间公开发表的论文 ............................................................................................ 75
致谢 ................................................................................................................................ 76
第一章 绪论
1
第一章 绪论
1.1 研究背景及意义
旋涡是自然界中常见的现象，从古至今一直吸引着人们去探索它，研究它，
如风中旋转着飘落的树叶，河流的水面上泛起的小水涡，足以摧毁一切、令人畏
惧的龙卷风，和流传在水手中间能够将他们带往另一个世界的“海漏”。旋涡在人
类的历史中，神秘而又美妙，是力量的象征，又是生命的开始与终结，而人类的
文明也如旋涡般绽放。
在文艺复兴时期，里昂纳多.达芬奇已经意识到涡是一种动量和能量的表现形
式，他通过定性的实验和观察认识到势流涡与刚体旋转的不同，并且首次提出了
湍流这一流动形态，意识到湍流中包含了大量的涡[1]。
图 1.1 老人与旋涡，里昂纳多.达芬奇的自画像
从工业革命至今，科学技术飞速发展，水利设备在生活与生产中得到了广泛
的应用，人类对于流体有了更深刻的认识，对于旋涡的态度也从原始崇拜，转向
理性得去认识。旋涡是涡量聚集的一种表现，毫无疑问旋涡必定与旋转不可分离，
Lugt[1]给出了两种定义：1）旋涡是一群绕公共中心旋转的流体微团；2）微团围绕
某一点做刚体旋转的角速度称为涡量。
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作者：赵德峰
分类：高等教育资料
价格：15积分
属性：79 页
大小：5.53MB
格式：PDF
时间：2025-01-09
