-
Notifications
You must be signed in to change notification settings - Fork 0
/
Solving-dispersion-relation-in-linear-stability-for-clean-jet.html
381 lines (276 loc) · 21.8 KB
/
Solving-dispersion-relation-in-linear-stability-for-clean-jet.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
<!DOCTYPE html>
<html lang="zh-CN">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width">
<meta name="theme-color" content="#ddd"><meta name="generator" content="Hexo 6.2.0">
<link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
<link rel="icon" type="image/png" sizes="32x32" href="/images/logo_32x32.png">
<link rel="icon" type="image/png" sizes="16x16" href="/images/logo_16x16.png">
<link rel="mask-icon" href="/images/logo.png" color="#ddd">
<link rel="stylesheet" href="/css/main.css">
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans+SC:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.1.1/css/all.min.css" integrity="sha256-DfWjNxDkM94fVBWx1H5BMMp0Zq7luBlV8QRcSES7s+0=" crossorigin="anonymous">
<script class="next-config" data-name="main" type="application/json">{"hostname":"example.com","root":"/","images":"/images","scheme":"Gemini","darkmode":false,"version":"8.12.1","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12},"copycode":{"enable":true,"style":null},"bookmark":{"enable":false,"color":"#222","save":"auto"},"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},"stickytabs":false,"motion":{"enable":false,"async":false,"transition":{"post_block":"fadeIn","post_header":"fadeInDown","post_body":"fadeInDown","coll_header":"fadeInLeft","sidebar":"fadeInUp"}},"prism":false,"i18n":{"placeholder":"搜索...","empty":"没有找到任何搜索结果：${query}","hits_time":"找到 ${hits} 个搜索结果（用时 ${time} 毫秒）","hits":"找到 ${hits} 个搜索结果"}}</script><script src="/js/config.js"></script>
<meta name="description" content="色散方程 射流线性稳定性中色散方程（Dispersion relation）是描述射流受到扰动时，时间扰动增长率和空间扰动增长率的关系。假设粘性射流受到轴向波数为 \(k\) 、周向阶数为 \(n\)、时间增长率为 \(\omega\) 的扰动，粘性相关无量纲数为 \(Oh\)，则色散方程的无量纲形式 \(D(Oh;k,n,\omega)=0\) 如下所示 \[ \begin{equati">
<meta property="og:type" content="article">
<meta property="og:title" content="粘性射流线性稳定性色散方程求解">
<meta property="og:url" content="http://example.com/Solving-dispersion-relation-in-linear-stability-for-clean-jet.html">
<meta property="og:site_name" content="Day day up!">
<meta property="og:description" content="色散方程 射流线性稳定性中色散方程（Dispersion relation）是描述射流受到扰动时，时间扰动增长率和空间扰动增长率的关系。假设粘性射流受到轴向波数为 \(k\) 、周向阶数为 \(n\)、时间增长率为 \(\omega\) 的扰动，粘性相关无量纲数为 \(Oh\)，则色散方程的无量纲形式 \(D(Oh;k,n,\omega)=0\) 如下所示 \[ \begin{equati">
<meta property="og:locale" content="zh_CN">
<meta property="article:published_time" content="2020-03-19T16:00:00.000Z">
<meta property="article:modified_time" content="2022-04-21T16:00:00.000Z">
<meta property="article:author" content="Chilton Nieh">
<meta property="article:tag" content="Matlab">
<meta property="article:tag" content="射流">
<meta property="article:tag" content="瑞利不稳定性">
<meta property="article:tag" content="Bessel 函数">
<meta name="twitter:card" content="summary">
<link rel="canonical" href="http://example.com/Solving-dispersion-relation-in-linear-stability-for-clean-jet.html">
<script class="next-config" data-name="page" type="application/json">{"sidebar":"","isHome":false,"isPost":true,"lang":"zh-CN","comments":true,"permalink":"http://example.com/Solving-dispersion-relation-in-linear-stability-for-clean-jet.html","path":"/Solving-dispersion-relation-in-linear-stability-for-clean-jet.html","title":"粘性射流线性稳定性色散方程求解"}</script>
<script class="next-config" data-name="calendar" type="application/json">""</script>
<title>粘性射流线性稳定性色散方程求解 | Day day up!</title>
<noscript>
<link rel="stylesheet" href="/css/noscript.css">
</noscript>
</head>
<body itemscope itemtype="http://schema.org/WebPage">
<div class="headband"></div>
<main class="main">
<header class="header" itemscope itemtype="http://schema.org/WPHeader">
<div class="header-inner"><div class="site-brand-container">
<div class="site-nav-toggle">
<div class="toggle" aria-label="切换导航栏" role="button">
<span class="toggle-line"></span>
<span class="toggle-line"></span>
<span class="toggle-line"></span>
</div>
</div>
<div class="site-meta">
<a href="/" class="brand" rel="start">
<i class="logo-line"></i>
<p class="site-title">Day day up!</p>
<i class="logo-line"></i>
</a>
<p class="site-subtitle" itemprop="description">做人最重要的的是😀</p>
</div>
<div class="site-nav-right">
<div class="toggle popup-trigger">
</div>
</div>
</div>
<nav class="site-nav">
<ul class="main-menu menu"><li class="menu-item menu-item-home"><a href="/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a></li><li class="menu-item menu-item-archives"><a href="/archives/" rel="section"><i class="fa fa-archive fa-fw"></i>归档<span class="badge">38</span></a></li><li class="menu-item menu-item-tags"><a href="/tags/" rel="section"><i class="fa fa-tags fa-fw"></i>标签<span class="badge">41</span></a></li><li class="menu-item menu-item-categories"><a href="/categories/" rel="section"><i class="fa fa-th fa-fw"></i>分类<span class="badge">6</span></a></li><li class="menu-item menu-item-favorites"><a href="/favorites/" rel="section"><i class="fa fa-heart fa-fw"></i>收藏</a></li><li class="menu-item menu-item-about"><a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a></li>
</ul>
</nav>
</div>
<div class="toggle sidebar-toggle" role="button">
<span class="toggle-line"></span>
<span class="toggle-line"></span>
<span class="toggle-line"></span>
</div>
<aside class="sidebar">
<div class="sidebar-inner sidebar-nav-active sidebar-toc-active">
<ul class="sidebar-nav">
<li class="sidebar-nav-toc">
文章目录
</li>
<li class="sidebar-nav-overview">
站点概览
</li>
</ul>
<div class="sidebar-panel-container">
<!--noindex-->
<div class="post-toc-wrap sidebar-panel">
<div class="post-toc animated"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%89%B2%E6%95%A3%E6%96%B9%E7%A8%8B"><span class="nav-number">1.</span> <span class="nav-text">色散方程</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E7%AC%A6%E5%8F%B7%E6%B1%82%E8%A7%A3"><span class="nav-number">2.</span> <span class="nav-text">符号求解</span></a></li></ol></div>
</div>
<!--/noindex-->
<div class="site-overview-wrap sidebar-panel">
<div class="site-author site-overview-item animated" itemprop="author" itemscope itemtype="http://schema.org/Person">
<img class="site-author-image" itemprop="image" alt="Chilton Nieh"
src="/images/logo.png">
<p class="site-author-name" itemprop="name">Chilton Nieh</p>
<div class="site-description" itemprop="description">路漫漫其修远兮，吾将上下而求索</div>
</div>
<div class="site-state-wrap site-overview-item animated">
<nav class="site-state">
<div class="site-state-item site-state-posts">
<a href="/archives/">
<span class="site-state-item-count">38</span>
<span class="site-state-item-name">日志</span>
</a>
</div>
<div class="site-state-item site-state-categories">
<a href="/categories/">
<span class="site-state-item-count">6</span>
<span class="site-state-item-name">分类</span></a>
</div>
<div class="site-state-item site-state-tags">
<a href="/tags/">
<span class="site-state-item-count">41</span>
<span class="site-state-item-name">标签</span></a>
</div>
</nav>
</div>
<div class="links-of-author site-overview-item animated">
<span class="links-of-author-item">
<a href="https://github.com/yourname" title="GitHub → https://github.com/yourname" rel="noopener" target="_blank"><i class="fab fa-github fa-fw"></i>GitHub</a>
</span>
</div>
</div>
</div>
</div>
</aside>
<div class="sidebar-dimmer"></div>
</header>
<div class="back-to-top" role="button" aria-label="返回顶部">
<i class="fa fa-arrow-up"></i>
<span>0%</span>
</div>
<noscript>
<div class="noscript-warning">Theme NexT works best with JavaScript enabled</div>
</noscript>
<div class="main-inner post posts-expand">
<div class="post-block">
<article itemscope itemtype="http://schema.org/Article" class="post-content" lang="zh-CN">
<link itemprop="mainEntityOfPage" href="http://example.com/Solving-dispersion-relation-in-linear-stability-for-clean-jet.html">
<span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
<meta itemprop="image" content="/images/logo.png">
<meta itemprop="name" content="Chilton Nieh">
</span>
<span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
<meta itemprop="name" content="Day day up!">
<meta itemprop="description" content="路漫漫其修远兮，吾将上下而求索">
</span>
<span hidden itemprop="post" itemscope itemtype="http://schema.org/CreativeWork">
<meta itemprop="name" content="粘性射流线性稳定性色散方程求解 | Day day up!">
<meta itemprop="description" content="">
</span>
<header class="post-header">
<h1 class="post-title" itemprop="name headline">
粘性射流线性稳定性色散方程求解
</h1>
<div class="post-meta-container">
<div class="post-meta">
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-calendar"></i>
</span>
<span class="post-meta-item-text">发表于</span>
<time title="创建时间：2020-03-20 00:00:00" itemprop="dateCreated datePublished" datetime="2020-03-20T00:00:00+08:00">2020-03-20</time>
</span>
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-calendar-check"></i>
</span>
<span class="post-meta-item-text">更新于</span>
<time title="修改时间：2022-04-22 00:00:00" itemprop="dateModified" datetime="2022-04-22T00:00:00+08:00">2022-04-22</time>
</span>
<span class="post-meta-item">
<span class="post-meta-item-icon">
<i class="far fa-folder"></i>
</span>
<span class="post-meta-item-text">分类于</span>
<span itemprop="about" itemscope itemtype="http://schema.org/Thing">
<a href="/categories/%E7%A0%94%E7%A9%B6%E7%9B%B8%E5%85%B3/" itemprop="url" rel="index"><span itemprop="name">研究相关</span></a>
</span>
</span>
</div>
</div>
</header>
<div class="post-body" itemprop="articleBody">
<h2 id="色散方程">色散方程</h2>
<p>射流线性稳定性中色散方程（Dispersion
relation）是描述射流受到扰动时，时间扰动增长率和空间扰动增长率的关系。假设粘性射流受到轴向波数为
<span class="math inline">\(k\)</span> 、周向阶数为 <span
class="math inline">\(n\)</span>、时间增长率为 <span
class="math inline">\(\omega\)</span> 的扰动，粘性相关无量纲数为 <span
class="math inline">\(Oh\)</span>，则色散方程的无量纲形式 <span
class="math inline">\(D(Oh;k,n,\omega)=0\)</span> 如下所示 <span
class="math display">\[
\begin{equation}\label{eq1}
\begin{vmatrix}
D_{11} & D_{12} & D_{13} \\
2\mathrm{i}kI_n^\prime(k) & -\frac{l^2}{k^2}I_n^\prime(l)-I_{n+1}(l)
& \frac{l^2}{k^2}I_n^\prime(l)+I_{n-1}(l) \\
2\mathrm{i}n[kI_n^\prime(k)-I_n(k)] & lI_{n+2}(l) & lI_{n-2}(l)
\end{vmatrix}
= 0,
\end{equation}
\]</span> <span id="more"></span></p>
<p>其中 <span class="math display">\[
\begin{equation}
\begin{aligned}
D_{11} &= \omega\left[ 2Ohk^{2}I_{n}^{ \prime \prime }(k)+ \omega
I_{n}(k) \right] -(1-n^{2}-k^{2})kI_{n}^{ \prime }(k), \\
D_{12} &= 2\mathrm{i}\omega OhlI_{n+1}^{ \prime }(l)
-(1-n^{2}-k^{2})\mathrm{i}I_{n+1}(l), \\
D_{13} &= -2\mathrm{i}\omega OhlI_{n-1}^{ \prime }(l) +
(1-n^{2}-k^{2})\mathrm{i}I_{n-1}(l),
\end{aligned}
\end{equation}
\]</span></p>
<p>以及 <span class="math display">\[
l=\sqrt{k^2+\frac{\omega}{Oh}}.
\]</span> 以上方程出现第一类修正 Bessel 函数 <span
class="math inline">\(I_n (x)\)</span>，它是一个无穷级数，求解方程 <span
class="math inline">\(\eqref{eq1}\)</span> 时涉及求解 <span
class="math inline">\(I_n (x)\)</span> 的反函数。</p>
<h2 id="符号求解">符号求解</h2>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><span class="line">clear; close all; clc;</span><br><span class="line"><span class="comment">% 定义初始变量</span></span><br><span class="line">Oh0 = <span class="number">1</span>;</span><br><span class="line">n0 = <span class="number">0</span>;</span><br><span class="line">k0 = <span class="number">1e-2</span>:<span class="number">1e-2</span>:<span class="number">1</span>;</span><br><span class="line"></span><br><span class="line"><span class="comment">% 定义符号量</span></span><br><span class="line">syms Oh k omega n l</span><br><span class="line">assume(in(<span class="built_in">real</span>(omega),<span class="string">'positive'</span>));</span><br><span class="line"></span><br><span class="line"><span class="comment">% 行列式</span></span><br><span class="line">D11 = omega*(<span class="number">2</span>*Oh*k^<span class="number">2</span>*diff(<span class="built_in">besseli</span>(n,k),k,<span class="number">2</span>) + omega*<span class="built_in">besseli</span>(n,k)) ...</span><br><span class="line"> - (<span class="number">1</span> - n^<span class="number">2</span> - k^<span class="number">2</span>)*k*diff(<span class="built_in">besseli</span>(n,k),k);</span><br><span class="line">D12 = <span class="number">2</span><span class="built_in">i</span>*omega*Oh*l*diff(<span class="built_in">besseli</span>(n+<span class="number">1</span>,l),l)...</span><br><span class="line"> - (<span class="number">1</span> - n^<span class="number">2</span> - k^<span class="number">2</span>)*<span class="number">1</span><span class="built_in">i</span>*<span class="built_in">besseli</span>(n+<span class="number">1</span>,l);</span><br><span class="line">D13 = <span class="number">-2</span><span class="built_in">i</span>*omega*Oh*l*diff(<span class="built_in">besseli</span>(n<span class="number">-1</span>,l),l)...</span><br><span class="line"> + (<span class="number">1</span> - n^<span class="number">2</span> - k^<span class="number">2</span>)*<span class="number">1</span><span class="built_in">i</span>*<span class="built_in">besseli</span>(n<span class="number">-1</span>,l);</span><br><span class="line">D21 = <span class="number">2</span><span class="built_in">i</span>*k*diff(<span class="built_in">besseli</span>(n,k),k);</span><br><span class="line">D22 = -l^<span class="number">2</span>/k^<span class="number">2</span>*diff(<span class="built_in">besseli</span>(n,l),l) - <span class="built_in">besseli</span>(n+<span class="number">1</span>,l);</span><br><span class="line">D23 = l^<span class="number">2</span>/k^<span class="number">2</span>*diff(<span class="built_in">besseli</span>(n,l),l) + <span class="built_in">besseli</span>(n<span class="number">-1</span>,l);</span><br><span class="line">D31 = <span class="number">2</span><span class="built_in">i</span>*n*(k*diff(<span class="built_in">besseli</span>(n,k),k) - <span class="built_in">besseli</span>(n,k));</span><br><span class="line">D32 = l*<span class="built_in">besseli</span>(n+<span class="number">2</span>,l);</span><br><span class="line">D33 = l*<span class="built_in">besseli</span>(n<span class="number">-2</span>,l);</span><br><span class="line">D = [D11, D12, D13;...</span><br><span class="line"> D21, D22, D23;...</span><br><span class="line"> D31, D32, D33];</span><br><span class="line">eqn = det(D);</span><br><span class="line">eqn = subs(eqn, l, <span class="built_in">sqrt</span>(k^<span class="number">2</span>+omega/Oh));</span><br><span class="line">eqn = subs(eqn, Oh, Oh0);</span><br><span class="line">eqn = subs(eqn, n, n0);</span><br><span class="line"></span><br><span class="line"><span class="comment">% 求解</span></span><br><span class="line">omega0 = <span class="built_in">zeros</span>(<span class="number">1</span>,<span class="built_in">length</span>(k0));</span><br><span class="line"><span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">1</span>:<span class="built_in">length</span>(k0)</span><br><span class="line"> eqn0 = subs(eqn, k, k0(<span class="built_in">i</span>));</span><br><span class="line"> omega0(<span class="built_in">i</span>) = vpasolve(eqn0, omega, <span class="number">1</span>);</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 保存数据</span></span><br><span class="line">data = [k0', omega0'];</span><br><span class="line">save([<span class="string">'Oh='</span>, num2str(Oh0), <span class="string">'_n0='</span>, num2str(n0), <span class="string">'.txt'</span>], <span class="string">'data'</span>, <span class="string">'-ascii'</span>);</span><br></pre></td></tr></table></figure>
</div>
<footer class="post-footer">
<div class="post-tags">
<a href="/tags/Matlab/" rel="tag"># Matlab</a>
<a href="/tags/%E5%B0%84%E6%B5%81/" rel="tag"># 射流</a>
<a href="/tags/%E7%91%9E%E5%88%A9%E4%B8%8D%E7%A8%B3%E5%AE%9A%E6%80%A7/" rel="tag"># 瑞利不稳定性</a>
<a href="/tags/Bessel-%E5%87%BD%E6%95%B0/" rel="tag"># Bessel 函数</a>
</div>
<div class="post-nav">
<div class="post-nav-item">
<a href="/Gerris-tutorial.html" rel="prev" title="Gerris 入门教程">
<i class="fa fa-chevron-left"></i> Gerris 入门教程
</a>
</div>
<div class="post-nav-item">
<a href="/COMSOL-simulation-of-viscous-jet.html" rel="next" title="粘性射流非线性断裂过程模拟">
粘性射流非线性断裂过程模拟 <i class="fa fa-chevron-right"></i>
</a>
</div>
</div>
</footer>
</article>
</div>
</div>
</main>
<footer class="footer">
<div class="footer-inner">
<div class="copyright">
©
<span itemprop="copyrightYear">2022</span>
<span class="with-love">
<i class="fa fa-heart"></i>
</span>
<span class="author" itemprop="copyrightHolder">Chilton Nieh</span>
</div>
<div class="powered-by">由 <a href="https://hexo.io/" rel="noopener" target="_blank">Hexo</a> & <a href="https://theme-next.js.org/" rel="noopener" target="_blank">NexT.Gemini</a> 强力驱动
</div>
</div>
</footer>
<script src="https://cdnjs.cloudflare.com/ajax/libs/animejs/3.2.1/anime.min.js" integrity="sha256-XL2inqUJaslATFnHdJOi9GfQ60on8Wx1C2H8DYiN1xY=" crossorigin="anonymous"></script>
<script src="/js/comments.js"></script><script src="/js/utils.js"></script><script src="/js/next-boot.js"></script>
<script class="next-config" data-name="mermaid" type="application/json">{"enable":true,"theme":{"light":"default","dark":"dark"},"js":{"url":"https://cdnjs.cloudflare.com/ajax/libs/mermaid/9.1.1/mermaid.min.js","integrity":"sha256-8L3O8tirFUa8Va4NSTAyIbHJeLd6OnlcxgupV9F77e0="}}</script>
<script src="/js/third-party/tags/mermaid.js"></script>
<script class="next-config" data-name="enableMath" type="application/json">true</script><script class="next-config" data-name="mathjax" type="application/json">{"enable":true,"tags":"none","js":{"url":"https://cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.1/es5/tex-mml-chtml.js","integrity":"sha256-hlC2uSQYTmPsrzGZTEQEg9PZ1a/+SV6VBCTclohf2og="}}</script>
<script src="/js/third-party/math/mathjax.js"></script>
</body>
</html>