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<?xml version="1.0" encoding="utf-8"?>
<search>
<entry>
<title>一些燃烧学理论知识</title>
<link href="/2024/10/23/%E4%B8%80%E4%BA%9B%E7%87%83%E7%83%A7%E5%AD%A6%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/"/>
<url>/2024/10/23/%E4%B8%80%E4%BA%9B%E7%87%83%E7%83%A7%E5%AD%A6%E7%90%86%E8%AE%BA%E7%9F%A5%E8%AF%86/</url>
<content type="html"><![CDATA[<h1>火焰高度</h1><p>在roper’s相关的工作中报道，浮力主导的层流共流扩散火焰（圆环喷嘴），火焰高度公式为：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mo>=</mo><mfrac><mover accent="true"><mi>Q</mi><mo>˙</mo></mover><mrow><mn>4</mn><mi>π</mi><mi>D</mi><mi>ln</mi><mo></mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mfrac><mfrac><msub><mi>T</mi><mi>f</mi></msub><msub><mi>T</mi><mn>0</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">H_f=\frac{\dot{Q}}{4 \pi D \ln (1+1 / S)} \frac{T_f}{T_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5332em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5972em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1/</span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">Q</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>这里的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">H_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>为可视火焰高度，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>Q</mi><mo>˙</mo></mover></mrow><annotation encoding="application/x-tex">\dot{Q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1146em;vertical-align:-0.1944em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">Q</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1944em;"><span></span></span></span></span></span></span></span></span>是燃料的体积流率，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>是摩尔扩散系数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>是氧化剂相对燃料的摩尔当量比，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>是环境温度，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">T_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>是平均火焰温度</p><p>从公式中可以看出，在其它变量固定的情况下，燃料侧的体积流率（燃料的速度）和火焰高度成正比</p><figure class="highlight markdown"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs markdown">{% note success %}<br></code></pre></td></tr></table></figure>]]></content>
</entry>
<entry>
<title>在OpenFOAM中构建矩方程模型</title>
<link href="/2024/10/07/%E5%9C%A8OpenFOAM%E4%B8%AD%E6%9E%84%E5%BB%BASE%E5%8F%8C%E6%96%B9%E7%A8%8B%E6%A8%A1%E5%9E%8B/"/>
<url>/2024/10/07/%E5%9C%A8OpenFOAM%E4%B8%AD%E6%9E%84%E5%BB%BASE%E5%8F%8C%E6%96%B9%E7%A8%8B%E6%A8%A1%E5%9E%8B/</url>
<content type="html"><![CDATA[<h1>SE双方程模型</h1><p>SE双方程模型=单分布模型，它是一种简化的模型假设，假定碳烟的粒径分布为单分布。具体的模型假定可以参考<a href="https://ignite.byu.edu/sootlib_documentation/md_models_2_mono.html">SootLib: Monodispersed (byu.edu)</a>。其它矩方程模型也可以在这篇文章中找到。</p><p>值得说明的是，目前的SootLib仍然有很多的BUG</p><h3 id="一些耦合的注意事项">一些耦合的注意事项</h3><p>在求解M0的过程中，由于M0的数量级跨越很大（在0-1e30左右），这会造成在计算时候源项的作用非常“迅速”，如果时间步长给的太大（哪怕是0.001这中数量级），会造成M0的负数情况，进而触发非物理修正，造成结果的不正确。 在22000网格中测试，临界时间步长是5*10^-6，才能捕获M0,不知道有没有其它方法进行修改改进</p><h3 id="线性化源项">线性化源项</h3><p>参考这位大佬的文章<a href="https://marinecfd.xyz/post/openfoam-source-term-treatment/">OpenFOAM 中源项的实现 - OpenFOAM 之道 (marinecfd.xyz)</a></p><p>关于SuSp源项，这个帖子讲的非常详细<a href="https://www.cfd-china.com/topic/2535/%E5%85%B3%E4%BA%8E%E6%BA%90%E9%A1%B9%E5%A4%84%E7%90%86%E6%96%B9%E6%B3%95fvm-su%E5%92%8Cfvm-sp%E7%9A%84%E7%96%91%E9%97%AE/17">关于源项处理方法fvm::su和fvm::sp的疑问 (cfd-china.com)</a><br>看了收获很大</p><p>源项线性化的用途可以保证物理量非负以及增强对角占优。</p><h1>具体实现耦合框架</h1><h2 id="输运方程">输运方程</h2><p>在OpenFOAM中，求解数密度矩的输运方程：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>M</mi><mi>r</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">∂</mi><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>l</mi></msub></mrow></mfrac><mrow><mo fence="true">(</mo><msub><mi>U</mi><mi>l</mi></msub><msub><mi>M</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo>=</mo><mfrac><mi mathvariant="normal">∂</mi><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>l</mi></msub></mrow></mfrac><mrow><mo fence="true">(</mo><mover accent="true"><mi>D</mi><mo>ˉ</mo></mover><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>M</mi><mi>r</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>l</mi></msub></mrow></mfrac><mo fence="true">)</mo></mrow><mo>−</mo><mfrac><mi mathvariant="normal">∂</mi><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mi>l</mi></msub></mrow></mfrac><mrow><mo fence="true">(</mo><msub><mi>U</mi><mrow><mi>T</mi><mo separator="true">,</mo><mi>l</mi></mrow></msub><msub><mi>M</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mover accent="true"><mi>M</mi><mo>˙</mo></mover><mi>r</mi></msub><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>r</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{r}\frac{\partial M_r}{\partial t}+\frac{\partial}{\partial x_l}\left(U_l M_r\right)=\frac{\partial}{\partial x_l}\left(\bar{D} \frac{\partial M_r}{\partial x_l}\right)-\frac{\partial}{\partial x_l}\left(U_{T, l} M_r\right)+\dot{M}_r, \\r=0,1,2, \ldots, \infty,\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4509em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4509em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4509em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4509em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.0556em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.26em;"><span class="pstrut" style="height:3.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∞</span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p><figure class="highlight less"><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></pre></td><td class="code"><pre><code class="hljs less"><span class="hljs-selector-tag">fvScalarMatrix</span> <span class="hljs-selector-tag">MOM0Eqn</span><br><br> ( <br><br> <span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">ddt</span>(rho , M0) +<span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">div</span>(phi, M0 ) \+ <span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">div</span>(phiTh, M0)<span class="hljs-comment">// 添加热泳项 </span><br><br> == <span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">laplacian</span>(diffusionCoeff * rhoGas , M0 ) + SM0 );<br><br> <span class="hljs-selector-tag">fvScalarMatrix</span> <span class="hljs-selector-tag">MOM1Eqn</span><br><br> (<span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">ddt</span>(rho , M1) +<span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">div</span>(phi, M1) \+ <span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">div</span>(phiTh, M1) <span class="hljs-comment">// 添加热泳项</span><br><br> ==<span class="hljs-attribute">fvm</span>::<span class="hljs-built_in">laplacian</span>( diffusionCoeff * rhoGas, M1 ) +SM1 );<br></code></pre></td></tr></table></figure><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mrow><mo fence="true">(</mo><mi>ρ</mi><msub><mi>M</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mrow><mrow><mi mathvariant="normal">∂</mi><mi>t</mi></mrow></mfrac><mo>+</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow><mo fence="true">(</mo><mi>ϕ</mi><msub><mi>M</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow><mo fence="true">(</mo><msub><mi>ϕ</mi><mtext>th </mtext></msub><msub><mi>M</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow><mo fence="true">(</mo><msub><mi>ρ</mi><mrow><mi>g</mi><mi>a</mi><mi>s</mi></mrow></msub><mo>⋅</mo><mi>D</mi><mo>⋅</mo><mi mathvariant="normal">∇</mi><msub><mi>M</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>S</mi><msub><mi>M</mi><mn>0</mn></msub></msub></mrow><annotation encoding="application/x-tex">\frac{\partial\left(\rho M_0\right)}{\partial t}+\nabla \cdot\left(\phi M_0\right)+\nabla \cdot\left(\phi_{\text {th }} M_0\right)=\nabla \cdot\left(\rho_{g a s} \cdot D \cdot \nabla M_0\right)+S_{M_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord mathnormal">ϕ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">th </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9334em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>其中第一项是时间导数项，第二项是对流项，第三项是热泳项，第四项是扩散项，第五项是源项<br>其中，相关文献提到扩散项可以忽略不计，而热泳项在AMREX中并未考虑。具体的建模公式可以参考文献。</p><p>这里我是直接进行求解，有文献（NGA求解器）采用分裂算子的方法进行计算。</p><p>热泳项：<br>volVectorField gradT = fvc::grad(T); // 温度梯度</p><p> olVectorField Vth = - (Ct * mu / rho) * (fvc::grad(T) / T); // 计算热泳速度<br> surfaceScalarField phiTh = (fvc::interpolate(rho * Vth) & mesh.Sf()); // 计算热泳通量</p><p> dimensionedScalar Ct(“Ct”, dimensionSet(0,0,0,0,0,0,0), 0.65); // 热泳系数，通常取值约为0.55<br>扩散项：</p><p>dimensionedScalar Ds(“Ds”, dimensionSet(0, 2, -1, 0, 0), 1e-6); //文献参考54为1e-6</p><p>源项的计算：</p><figure class="highlight css"><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></pre></td><td class="code"><pre><code class="hljs css"> `std::vector<double> yGas{Y_H<span class="hljs-selector-attr">[cellI]</span>, Y_H2<span class="hljs-selector-attr">[cellI]</span>, Y_O<span class="hljs-selector-attr">[cellI]</span>, Y_O2<span class="hljs-selector-attr">[cellI]</span>, Y_OH<span class="hljs-selector-attr">[cellI]</span>,Y_H2O<span class="hljs-selector-attr">[cellI]</span>,Y_CO<span class="hljs-selector-attr">[cellI]</span>, Y_C2H2<span class="hljs-selector-attr">[cellI]</span> };`<br><br> `std::vector<double> Msoot{M0<span class="hljs-selector-attr">[cellI]</span>* rhoGas<span class="hljs-selector-attr">[cellI]</span> , M1<span class="hljs-selector-attr">[cellI]</span>* rhoGas<span class="hljs-selector-attr">[cellI]</span> , M2<span class="hljs-selector-attr">[cellI]</span>* rhoGas<span class="hljs-selector-attr">[cellI]</span>, M3<span class="hljs-selector-attr">[cellI]</span>* rhoGas<span class="hljs-selector-attr">[cellI]</span>}; // soot moment values <span class="hljs-selector-attr">[M0, M1, M2, M3]</span>`<br><br> `S<span class="hljs-selector-class">.setState</span>(T<span class="hljs-selector-attr">[cellI]</span>, <span class="hljs-selector-tag">p</span><span class="hljs-selector-attr">[cellI]</span>, rhoGas<span class="hljs-selector-attr">[cellI]</span>, muGas<span class="hljs-selector-attr">[cellI]</span>, yGas, yPAH, Msoot, nsoot );<br></code></pre></td></tr></table></figure><p>这里大致的求解步骤：<br>给一个小的M1 M0初值，然后与其它所需化学物质温度等变量作为输入进行矩的源项计算，然后进行矩的输运方程计算。</p><h3 id="一些openfaom相关的操作注意事项">一些openfaom相关的操作注意事项</h3><p>如果出现warning：<code>Wedge patch ... is not planar.</code>，一般是由于算例设定的存储精度偏低，可适当调高<code>system/controlDict</code>中的参数<code>writePrecision</code>的设定数值，默认为6，可适当调高到12，甚至更高</p><p><a href="https://zhuanlan.zhihu.com/p/160882863">在OpenFOAM中使用二维轴对称的.msh网格 - 知乎 (zhihu.com)</a></p><h2 id="耦合物种消耗">耦合物种消耗</h2><p>考虑碳烟带来的物质消耗，以02为例</p><p>在物种的输运方程添加一个源项。</p><figure class="highlight asciidoc"><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></pre></td><td class="code"><pre><code class="hljs asciidoc">fvScalarMatrix YiEqn<br>(<br>fvm::ddt(rho, Yi)<br>+<br>(<br>turbName == "laminar"<br>? (mvConvection->fvmDiv(phi, Yi) + mvConvection->fvmDiv(phiUc, Yi))<br>: mvConvection->fvmDiv(phi, Yi)<br><span class="hljs-section">)</span><br><span class="hljs-section">==</span><br>(<br>splitting<br>? fvm::laplacian(DEff(), Yi)<br><span class="hljs-meta">: (fvm::laplacian(DEff(),</span> Yi) + combustion->R(Yi))<br>)<br>);<br>if (Yi.name() == "O2") {<br><br>YiEqn -= O2ConsumptionRate; <br><br>}<br></code></pre></td></tr></table></figure><p>注意的是在openfoam中，并不存在数学意义上的==号，而是将“==”重载为“-”号，所以在添加源项时候应该是减去这一项。这里采用了线性化源项的操作避免带来收敛的问题。<br>openfoam的susp ,sp方法用起来很奇怪，好在YiEqn -= O2ConsumptionRate;这个方法是绝对正确的。</p><h2 id="修正质量守恒方程">修正质量守恒方程</h2><p>同时也要考虑碳烟带来的气相质量损失，修改连续性方程 fvScalarMatrix rhoEqn</p><p>其中，totalConsumptionRate为所有化学物质的总消耗</p><figure class="highlight lisp"><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></pre></td><td class="code"><pre><code class="hljs lisp"> fvScalarMatrix rhoEqn<br><br> (<br><br> fvm:<span class="hljs-symbol">:ddt</span>(<span class="hljs-name">rho</span>)<br><br> \+ fvc:<span class="hljs-symbol">:div</span>(<span class="hljs-name">phi</span>)<br><br> ==<br><br> totalConsumptionRate<br><br> )<span class="hljs-comment">;</span><br></code></pre></td></tr></table></figure><h2 id="修正过氧化和非物理结果">修正过氧化和非物理结果</h2><p>在结束计算后，由于实际物理意义中矩不可能有负数，而计算过程中由于过氧化带来的负数矩要进行修正，用一个较小的数字进行替代。<br>这里使用了对数正态分布对M1和M2进行估计</p><figure class="highlight gcode"><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></pre></td><td class="code"><pre><code class="hljs gcode"><span class="hljs-keyword">if</span> <span class="hljs-comment">(M0[cellI] <= 0)</span><br><br>{<br><br> F_<span class="hljs-name">M0</span>[cellI] =<span class="hljs-number">1</span>;<br><br> F_<span class="hljs-name">M2</span>[cellI] =<span class="hljs-name">M0</span>[cellI];<br><br><span class="hljs-name">M0</span>[cellI] =<span class="hljs-number">1</span>e<span class="hljs-number">+3</span>;<br><br>}<br><br><span class="hljs-keyword">if</span> <span class="hljs-comment">(M1[cellI] <= 0.0)</span><br><br>{ <span class="hljs-name">M1</span>[cellI] = <span class="hljs-name">M0</span>[cellI] * mavg * std::exp<span class="hljs-comment">(0.5 * std::pow(sigL,2.0)</span>); <span class="hljs-comment">// estimate an M1 value based on M0 and lognormal distribution</span><br><br> F_<span class="hljs-name">M1</span>[cellI] =<span class="hljs-number">1</span>;<br><br> }<br><br><span class="hljs-keyword">if</span> <span class="hljs-comment">(M2[cellI] <= 0.0)</span><br><br>{<span class="hljs-name">M2</span>[cellI] =<span class="hljs-name">M0</span>[cellI] * std::pow<span class="hljs-comment">(mavg,2)</span> * std::exp<span class="hljs-comment">(0.5 * 4 * std::pow(sigL,2.0)</span>); <span class="hljs-comment">// estimate an M2 value based on M0 and lognormal distribution</span><br>}<br></code></pre></td></tr></table></figure><p>F_M0 F_M1等变量是为了标记进行非物理结果修正的网格，方便后续debug</p><h2 id="碳烟ppm转化公式">碳烟ppm转化公式</h2><p>求得的M1还不是我们常用的碳烟体积分数，假定碳烟的密度为1850g/cm^3<br>sootfv[cellI] = M1[cellI] * rho[cellI] /1850</p><h2 id="压力边界条件设置">压力边界条件设置</h2><p>不要设置fixValue边界条件，这样改变燃烧室的压力时候会计算报错。正确的压力边界条件设置：</p><p>dimensions [1 -1 -2 0 0 0 0];</p><p>internalField uniform 201325;</p><p>全部改为zeroGradient条件</p><h2 id="温度边界条件">温度边界条件</h2><p>杜博建议：将碰嘴改为绝热边界条件可以减少举升</p><h2 id="光学薄膜假设辐射项">光学薄膜假设辐射项</h2><p>理论基础：<br>参考Maciej Chmielewski等人的文章《Planck Mean Absorption Coefficients of H2O, CO2, CO and NO for radiation numerical modeling in combusting flows》</p><p>所谓光学薄膜假设实际上就是在计算能量方程的右边添加一个辐射源项</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>q</mi><mtext>loss </mtext></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mfrac><msub><mi>x</mi><mi>i</mi></msub><msub><mi>x</mi><mtext>tot </mtext></msub></mfrac><msub><mi>p</mi><mtext>tot </mtext></msub><msub><mi>κ</mi><mi>i</mi></msub><mi>σ</mi><mrow><mo fence="true">(</mo><msup><mi>T</mi><mn>4</mn></msup><mo>−</mo><msubsup><mi>T</mi><mi>b</mi><mn>4</mn></msubsup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">q_{\text {loss }}=\sum_{i=1}^N \frac{x_i}{x_{\text {tot }}} p_{\text {tot }} \kappa_i \sigma\left(T^4-T_b^4\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">loss </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">tot </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">tot </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">κ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></p><p>其中，第一项是需要计算的物种摩尔浓度（无量纲），第二个是总压（atm),第三个是普朗克平均吸收系数（m pa)-1 Tb是环境温度，这里我设置为300K。</p><p>碳烟的拟合是通过：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>Q</mi><mtext>soot </mtext></msub><mo>=</mo><msub><mi>C</mi><mtext>soot </mtext></msub><mo>⋅</mo><msub><mi>f</mi><mi>v</mi></msub><mo>⋅</mo><msup><mi>T</mi><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">Q_{\text {soot }}=C_{\text {soot }} \cdot f_v \cdot T^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">soot </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">soot </span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></p><p>其中Csoot为</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>C</mi><mtext>soot</mtext></msub><mo>=</mo><mn>3.334</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mfrac><mtext>W</mtext><mrow><mo stretchy="false">(</mo><msup><mtext>m</mtext><mn>3</mn></msup><msup><mtext>K</mtext><mn>5</mn></msup><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">C_{\text{soot}} = 3.334 \times 10^{-4} \frac{\text{W}}{(\text{m}^3 \text{K}^5)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">soot</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3.334</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.3877em;vertical-align:-1.0273em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.2227em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">m</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord"><span class="mord text"><span class="mord">K</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8873em;"><span style="top:-3.1362em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">W</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0273em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>fv为碳烟的体积分数</p><h3 id="普朗克平均吸收系数（m-pa-1-的拟合">普朗克平均吸收系数（m pa)-1 的拟合</h3><p>这里通过6次多项式进行拟合，数据来源是PELELMX提供的。</p><p><img src="/images/%E8%BE%90%E5%B0%84%E5%90%B8%E6%94%B6%E7%B3%BB%E6%95%B0%E6%8B%9F%E5%90%88%E6%9B%B2%E7%BA%BF.png" alt="辐射吸收系数拟合曲线"></p><figure class="highlight inform7"><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></pre></td><td class="code"><pre><code class="hljs inform7">double calculate_ap(double T, const std::vector<double>& coeffs) {<br> // T 是温度，coeffs 是存储 c0, c1, c2, c3, c4, c5 的系数组<br> double T_inv = T; // <br> double ap = coeffs<span class="hljs-comment">[0]</span> <br> + coeffs<span class="hljs-comment">[1]</span> * T_inv <br> + coeffs<span class="hljs-comment">[2]</span> * std::pow(T_inv, 2) <br> + coeffs<span class="hljs-comment">[3]</span> * std::pow(T_inv, 3) <br> + coeffs<span class="hljs-comment">[4]</span> * std::pow(T_inv, 4) <br> + coeffs<span class="hljs-comment">[5]</span> * std::pow(T_inv, 5)<br> + coeffs<span class="hljs-comment">[6]</span> * std::pow(T_inv, 6);<br> return ap;<br> }<br> if(radiationButton =1){<br> Qsoot<span class="hljs-comment">[cellI]</span> = Csoot<span class="hljs-comment">[cellI]</span> * pow(T<span class="hljs-comment">[cellI]</span>, 5) * (sootfv<span class="hljs-comment">[cellI]</span>);<br> absorptivity_CO2<span class="hljs-comment">[cellI]</span> = calculate_ap(T<span class="hljs-comment">[cellI]</span>, CO2_coeffs); //注意这里吸收系数算出来单位/(cm*bar)-1<br> absorptivity_CH4<span class="hljs-comment">[cellI]</span> = calculate_ap(T<span class="hljs-comment">[cellI]</span>, CH4_coeffs);<br> absorptivity_CO<span class="hljs-comment">[cellI]</span> = calculate_ap(T<span class="hljs-comment">[cellI]</span>, CO_coeffs);<br> absorptivity_H2O<span class="hljs-comment">[cellI]</span> = calculate_ap(T<span class="hljs-comment">[cellI]</span>, H2O_coeffs);<br> //计算公式：kpl系数*总压（atm)*摩尔分数 *玻尔兹曼常数* T相关 气体的总摩尔质量通过理想气体状态方程进行换算<br> mean_MW<span class="hljs-comment">[cellI]</span> = rho<span class="hljs-comment">[cellI]</span>/ p<span class="hljs-comment">[cellI]</span>* 8.314 * T<span class="hljs-comment">[cellI]</span>;<br> QgasRadLoss<span class="hljs-comment">[cellI]</span> = (p<span class="hljs-comment">[cellI]</span> / 101325) * (<br> (Y_H2O<span class="hljs-comment">[cellI]</span> / 18 * mean_MW<span class="hljs-comment">[cellI]</span> * absorptivity_H2O<span class="hljs-comment">[cellI]</span>) + <br> (Y_CO2<span class="hljs-comment">[cellI]</span> / 44 * mean_MW<span class="hljs-comment">[cellI]</span> * absorptivity_CO2<span class="hljs-comment">[cellI]</span>) + <br> (Y_CO<span class="hljs-comment">[cellI]</span> / 28 * mean_MW<span class="hljs-comment">[cellI]</span> * absorptivity_CO<span class="hljs-comment">[cellI]</span>) + <br> (Y_CH4<span class="hljs-comment">[cellI]</span> / 16 * mean_MW<span class="hljs-comment">[cellI]</span> * absorptivity_CH4<span class="hljs-comment">[cellI]</span>)<br> ) * 100 * (5.67e-8) * (pow4(T<span class="hljs-comment">[cellI]</span>) - pow4(Tb<span class="hljs-comment">[cellI]</span>));}<br><br></code></pre></td></tr></table></figure><h2 id="数值模拟验证">数值模拟验证</h2><table><thead><tr><th>FUEL温度 流速</th><th>AIR温度 流速</th><th>机理</th><th>是否启用rad</th><th>机理</th><th>model</th></tr></thead><tbody><tr><td>433 0.15</td><td>344 0.5</td><td>1668</td><td>on</td><td>mechnism.yaml</td><td>默认</td></tr><tr><td>433 0.15</td><td>344 0.5</td><td>1668</td><td>off</td><td>mechnism.yaml</td><td>默认</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table><p>主要是想验证这里rad会不会影响举升。</p><h3 id="稳态条件">稳态条件</h3><p>参考文献35</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> RelativeError </mtext><mo>=</mo><mfrac><mrow><mo fence="true">∥</mo><msub><mi>V</mi><mi>i</mi></msub><mo>−</mo><msub><mi>V</mi><mrow><mi>i</mi><mo>−</mo><mn>100</mn></mrow></msub><mo fence="true">∥</mo></mrow><mrow><mo fence="true">∥</mo><msub><mi>V</mi><mrow><mi>i</mi><mo>−</mo><mn>100</mn></mrow></msub><mo fence="true">∥</mo></mrow></mfrac><mo>∗</mo><mn>100</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\text { RelativeError }=\frac{\left\|V_i-V_{i-100}\right\|}{\left\|V_{i-100}\right\|} * 100 \%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord"> RelativeError </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;">∥</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">100</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∥</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;">∥</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">100</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">∥</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">100%</span></span></span></span></span></p><p>这里是用100个时间步后的值，误差小于0.5%就认为是达到稳定状态</p><p>在涉及到化学反应的计算中想要达到完全的稳态是很难的</p><p><a href="https://blog.csdn.net/qq_15963745/article/details/134784788">Fluent验证案例49：燃烧器燃烧-完善修改_fluent点火怎么设置-CSDN博客</a></p><p>参考这个人的设置，一般能量的残差下降到1e-6下，其余物种残差下降到1e-5下就可以认为是达到稳态</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext> Residual </mtext><mo>=</mo><mrow><mo fence="true">∣</mo><mfrac><mtext> Current value of equation residual </mtext><mtext> Maximum value of initial guess or solution </mtext></mfrac><mo fence="true">∣</mo></mrow></mrow><annotation encoding="application/x-tex">\text { Residual }=\left|\frac{\text { Current value of equation residual }}{\text { Maximum value of initial guess or solution }}\right|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord"> Residual </span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Maximum value of initial guess or solution </span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Current value of equation residual </span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:4.4em;"></span><span style="width:0.333em;height:2.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="2.400em" viewBox="0 0 333 2400"><path d="M145 15 v585 v1200 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-1200 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z"/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>一个收敛较好的残差曲线图：</p><p><img src="/images/%E6%AE%8B%E5%B7%AE%E5%A5%BD%E5%9B%BE.png" alt="好的残差曲线图"></p><p>在这个残差曲线图中，大部分物种浓度都降到了1e-5以下，而一个收敛不好的残差曲线图如下所示，震荡非常严重，而且没有达到收敛要求</p><p><img src="/images/%E6%AE%8B%E5%B7%AE%E5%9D%8F%E5%9B%BE.png" alt="不好的残差曲线图"></p>]]></content>
</entry>
<entry>
<title>碳烟的HACA机理以及在AMREX中的实现</title>
<link href="/2024/10/06/%E7%A2%B3%E7%83%9F%E7%9A%84HACA%E6%9C%BA%E7%90%86%E4%BB%A5%E5%8F%8A%E5%9C%A8AMREX%E4%B8%AD%E7%9A%84%E5%AE%9E%E7%8E%B0/"/>
<url>/2024/10/06/%E7%A2%B3%E7%83%9F%E7%9A%84HACA%E6%9C%BA%E7%90%86%E4%BB%A5%E5%8F%8A%E5%9C%A8AMREX%E4%B8%AD%E7%9A%84%E5%AE%9E%E7%8E%B0/</url>
<content type="html"><![CDATA[<h1>HACA原理</h1><p>参考<a href="https://ignite.byu.edu/sootlib_documentation/md_models_1_chemistry.html">SootLib: Chemistry and Physical Models (byu.edu)</a>HACA机理由下面六个反应构成</p><table><thead><tr><th></th><th style="text-align:left">Reaction</th><th style="text-align:left">Rate constant</th></tr></thead><tbody><tr><td>(1)</td><td style="text-align:left">C(s)-H + H ↔ C(s) . + H 2</td><td style="text-align:left">k1=4.2×1013e−13/RT</td></tr><tr><td></td><td style="text-align:left"></td><td style="text-align:left">kr1=3.9×1012e−11/RT</td></tr><tr><td>(2)</td><td style="text-align:left">C(s)-H + OH ↔ C(s) . + H 2O</td><td style="text-align:left">k2=1.0×1010T0.734e−1.43/RT</td></tr><tr><td></td><td style="text-align:left"></td><td style="text-align:left">kr2=3.68×108T1.139e−17.1/RT</td></tr><tr><td>(3)</td><td style="text-align:left">C(s) . + H → C(s)-H</td><td style="text-align:left">k3=2.0×1013</td></tr><tr><td>(4)</td><td style="text-align:left">C(s) . + C 2H 2 → C(s)-H + H</td><td style="text-align:left">k4=8.0×107T1.56e−3.8/RT</td></tr><tr><td>(5)</td><td style="text-align:left">C(s) . + O 2 → 2CO + products</td><td style="text-align:left">k5=2.2×1012e−7.5/RT</td></tr><tr><td>(6)</td><td style="text-align:left">C(s)-H + OH → CO + products</td><td style="text-align:left">k6=0.13⋅1290T−1/2</td></tr></tbody></table><p>（1）-（4）代表碳烟的表面增长，（5）（6）代表氧化</p><p>其中值得注意的是：反应（1）-（5）需要进行稳态自由基点位数密度的计算</p><p>碳烟自由基点位由以下公式进行计算：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>χ</mi><msup><mi>C</mi><mo lspace="0em" rspace="0em">−</mo></msup></msub><mo>=</mo><mrow><mo fence="true">[</mo><mfrac><mrow><msub><mi>k</mi><mn>1</mn></msub><mo stretchy="false">[</mo><mi mathvariant="normal">H</mi><mo stretchy="false">]</mo><mo>+</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">H</mi></mrow><mo stretchy="false">]</mo></mrow><mrow><msub><mi>k</mi><mrow><mi>r</mi><mn>1</mn></mrow></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>k</mi><mrow><mi>r</mi><mn>2</mn></mrow></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>k</mi><mn>3</mn></msub><mo stretchy="false">[</mo><mi mathvariant="normal">H</mi><mo stretchy="false">]</mo><mo>+</mo><msub><mi>k</mi><mn>4</mn></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>k</mi><mn>5</mn></msub><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow></mrow></mfrac><mo fence="true">]</mo></mrow><msub><mi>χ</mi><mrow><mi>C</mi><mo>−</mo><mi>H</mi></mrow></msub><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">\chi_{C^{-}} = \left[\frac{k_1[\mathrm{H}] + k_2[\mathrm{OH}]}{k_{r 1}\left[\mathrm{H}_2\right] + k_{r 2}\left[\mathrm{H}_2 \mathrm{O}\right] + k_3[\mathrm{H}] + k_4\left[\mathrm{C}_2 \mathrm{H}_2\right] + k_5\left[\mathrm{O}_2\right]}\right] \chi_{C-H},</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3419em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7027em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">−</span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathrm">O</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathrm">H</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathrm">H</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">OH</span></span><span class="mclose">]</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span></span></p><p>其中C-H表面点位被假定为</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>χ</mi><mrow><mi>C</mi><mo>−</mo><mi>H</mi></mrow></msub><mo>=</mo><mn>2.3</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup><mtext> sites </mtext><mi mathvariant="normal">\</mi><msup><mi>m</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi_{C-H}=2.3 \times 10^{15} \text { sites } \backslash m^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2.3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span></span></span></span></span><span class="mord text"><span class="mord"> sites </span></span><span class="mord">\</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p>而剩下的k则是相关的化学反应速率常数，r代表逆反应。一旦碳烟自由基表面反应点位被计算，碳烟的生长速率可以这样表示</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>R</mi><mrow><mi>g</mi><mi>r</mi><mi>w</mi></mrow></msub><mo>=</mo><msub><mi>k</mi><mn>4</mn></msub><msub><mi>χ</mi><mi>C</mi></msub><mo>⋅</mo><mrow><mo fence="true">[</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">R_{g r w}=k_4 \chi_C \cdot\left[\mathrm{C}_2 \mathrm{H}_2\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathrm">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathrm">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span></span></p><p>碳烟被O2和OH氧化的速率表示为：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="center" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>R</mi><mrow><mi>o</mi><mi>x</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi>k</mi><mn>5</mn></msub><msub><mi>χ</mi><mi>C</mi></msub><mo>⋅</mo><mo stretchy="false">[</mo><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>R</mi><mrow><mi>o</mi><mi>x</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi>k</mi><mn>6</mn></msub><msub><mi>χ</mi><mrow><mi>C</mi><mo>−</mo><mi>H</mi></mrow></msub><msub><mi>P</mi><mrow><mi>O</mi><mi>H</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}R_{o x i} = k_5 \chi_C \cdot [\mathrm{O}_2] \\R_{o x i} = k_6 \chi_{C-H} P_{O H}\end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">o</span><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathrm">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">o</span><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">O</span><span class="mord mathnormal mtight" style="margin-right:0.08125em;">H</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h1>AMREX中的代码功能的实现</h1><p>先吐槽一句，PELELMX中的代码，可能是我的代码水平太低，总感觉可读性太差！！<br>阿伦尼乌斯方程形式：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>⋅</mo><msup><mi>T</mi><mi>n</mi></msup><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>E</mi><mi>a</mi></msub><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi>R</mi><mo>⋅</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">k(T)=A \cdot T^n \cdot e^{-E_a /(R \cdot T)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7144em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1645em;"><span style="top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span><span class="mbin mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>其中A是频率因子，T是温度，n是温度指数，Ea是活化能，R是气体常数</p><p>在PELE中，</p><p>A_f和A_b分别是正向和逆向反应的频率因子<br>n_f和n_b分别是正向和逆向的温度指数<br>ER_f和ER_b分别是正向和反向反应的活化能，这里ER代表活化能除以气体常数的值</p><p>PELE使用的是CGS（厘米-克-秒）单位，注意的是这里和sootlib对应不上是因为单位的原因</p><p><code>static constexpr amrex::Real RU = 8.31446261815324e7;</code></p><p><code>static constexpr amrex::Real Avna = 6.022140857e23;</code><br>此处的RU是标准常数</p>]]></content>
</entry>
<entry>
<title>HMOM方法原理</title>
<link href="/2024/10/06/HMOM%E6%96%B9%E6%B3%95%E5%8E%9F%E7%90%86/"/>
<url>/2024/10/06/HMOM%E6%96%B9%E6%B3%95%E5%8E%9F%E7%90%86/</url>
<content type="html"><![CDATA[<h1>HMOM方法原理</h1><p>HMOM(Hybrid Method of Moments)是一种结合了MOMIC和DQMOM两种方法的矩方法<br>MOMIC:在数值上很容易实现但是不能解释碳烟的双模分布<br>DQMOM：是精确的，但是数值上很难实现</p><h2 id="MOMIC">MOMIC</h2><p>参考Mueller等人的文章<strong>A joint volume-surface model of soot aggregation</strong></p><p><strong>with the method of moments</strong><br>采用Muller等人的体积-表面积（VS)对碳烟颗粒进行描述<br>碳烟直径可表述为：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mi>p</mi></msub><mo>=</mo><mn>6</mn><mi>V</mi><msup><mi>S</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">d_p=6 V S^ { -1 }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">6</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>碳烟数密度可表述为：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>n</mi><mi>p</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>36</mn><mi>π</mi></mrow></mfrac><msup><mi>V</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><msup><mi>S</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">n_p=\frac{1}{36 \pi} V^{-2} S^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">36</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span></p><p>所以对于一个双变量描述的碳烟来说，NDF的矩有两个指标（一个是体积，一个是表面积）。</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>M</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msubsup><mi>V</mi><mi>i</mi><mi>x</mi></msubsup><msubsup><mi>S</mi><mi>i</mi><mi>y</mi></msubsup><msub><mi>N</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">M_{x, y}=\sum_i V_i^x S_i^y N_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3277em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>i是对二维空间上的求和</p><p>另外一个原始版本的矩的定义可以供参考</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>M</mi><mi>r</mi></msub><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><msup><mi>m</mi><mi>r</mi></msup><mi>n</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mi>d</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">M_r=\int_0^{\infty} m^r n(m) d m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">m</span></span></span></span></span></p><p>在MOMIC中，输运方程求解一组矩，然后源项用对数多项式进行已知矩的插值</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext> Joint moments of the soot NDF </mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><menclose notation="left right"><mtable rowspacing="0.16em" columnalign="left left" columnspacing="1em" rowlines="solid none none none none"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Moment </mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Physical quantity </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>M</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Number density </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>M</mi><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Particle volume </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>M</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Particle surface area </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>M</mi><mrow><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Primary particle diameter </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>M</mi><mrow><mo>−</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> Number of primary particles </mtext></mstyle></mtd></mtr></mtable></menclose></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&\text { Joint moments of the soot NDF }\\&\begin{array}{ll}\hline \text { Moment } & \text { Physical quantity } \\\hline M_{0,0} & \text { Number density } \\M_{1,0} & \text { Particle volume } \\M_{0,1} & \text { Particle surface area } \\M_{1,-1} & \text { Primary particle diameter } \\M_{-2,3} & \text { Number of primary particles } \\\hline\end{array}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.04em;vertical-align:-4.27em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.77em;"><span style="top:-9.82em;"><span class="pstrut" style="height:5.89em;"></span><span class="mord"></span></span><span style="top:-5.27em;"><span class="pstrut" style="height:5.89em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.27em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.77em;"><span style="top:-9.82em;"><span class="pstrut" style="height:5.89em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord"> Joint moments of the soot NDF </span></span></span></span><span style="top:-5.27em;"><span class="pstrut" style="height:5.89em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.89em;"><span style="top:-5.85em;"><span class="pstrut" style="height:5.85em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.85em;"><span style="top:-6.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Moment </span></span></span></span><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-0.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.85em;"><span style="top:-6.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Physical quantity </span></span></span></span><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Number density </span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Particle volume </span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Particle surface area </span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Primary particle diameter </span></span></span></span><span style="top:-0.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord"> Number of primary particles </span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:5.85em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-8.5em;"><span class="pstrut" style="height:5.85em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-9.7em;"><span class="pstrut" style="height:5.85em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.35em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.27em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h2 id="QMOM-Quadrature-based-moment">QMOM Quadrature-based moment</h2><p>值得一提的是QMOM方法（1997年提出）在2016年被选为AS&T杰出论文。</p><p><a href="https://www.aaar.org/awards/annual-awards/ast-outstanding-publication-award/">https://www.aaar.org/awards/annual-awards/ast-outstanding-publication-award/</a></p><p>摘要：所谓正交矩，就是利用高斯求积（可以理解为一种插值方法）对已知的矩进行离重构。</p><p>参考资料书：<strong>Computational Models for Polydisperse Particulate andMultiphase Systems</strong></p><p>一些基本的概念：<br>Univariate distributions:单变量分布描述（即NDF只用一个变量描述，一般是体积或者质量（体积*固定密度））</p><p>而双变量描述则是利用两个变量描述如HMOM方法中的V,S，双变量可以描述聚集体的状态<br>矩可以通过下列式子进行定义：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>m</mi><mi>k</mi></msub><mo>=</mo><msub><mo>∫</mo><msub><mi mathvariant="normal">Ω</mi><mi>ξ</mi></msub></msub><mi>n</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msup><mi>ξ</mi><mi>k</mi></msup><mrow><mtext> </mtext><mi mathvariant="normal">d</mi></mrow><mi>ξ</mi><mo separator="true">,</mo><mspace width="1em"/><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">m_k=\int_{\Omega_{\xi}} n(\xi) \xi^k \mathrm{~d} \xi, \quad k=0,1,2, \ldots,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.475em;vertical-align:-1.115em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04601em;">ξ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.115em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mord"><span class="mspace nobreak"> </span><span class="mord mathrm">d</span></span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></p><h3 id="正交多项式">正交多项式</h3><p>定义下式：</p><p>若满足</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><msub><mi mathvariant="normal">Ω</mi><mi>ξ</mi></msub></msub><mi>n</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>β</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>ξ</mi><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> for </mtext><mi>α</mi><mo mathvariant="normal">≠</mo><mi>β</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>></mo><mn>0</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> for </mtext><mi>α</mi><mo>=</mo><mi>β</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\int_{\Omega_{\xi}} n(\xi) P_\alpha(\xi) P_\beta(\xi) \mathrm{d} \xi \begin{cases}=0 & \text { for } \alpha \neq \beta, \\ >0 & \text { for } \alpha=\beta,\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04601em;">ξ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.115em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>其中</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>k</mi><mrow><mi>α</mi><mo separator="true">,</mo><mn>0</mn></mrow></msub><msup><mi>x</mi><mi>α</mi></msup><mo>+</mo><msub><mi>k</mi><mrow><mi>α</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><msup><mi>x</mi><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>k</mi><mrow><mi>α</mi><mo separator="true">,</mo><mi>α</mi></mrow></msub><mtext> </mtext></mrow><annotation encoding="application/x-tex">P_\alpha(\xi)=k_{\alpha, 0} x^\alpha+k_{\alpha, 1} x^{\alpha-1}+\cdots+k_{\alpha, \alpha}\</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0005em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1502em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace"> </span></span></span></span></span></p><p>则称P为n(ξ)的正交多项式，若满足</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><msub><mi mathvariant="normal">Ω</mi><mi>ξ</mi></msub></msub><mi>n</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>β</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>ξ</mi><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> for </mtext><mi>α</mi><mo mathvariant="normal">≠</mo><mi>β</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext> for </mtext><mi>α</mi><mo>=</mo><mi>β</mi><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\int_{\Omega_{\xi}} n(\xi) P_\alpha(\xi) P_\beta(\xi) \mathrm{d} \xi= \begin{cases}0 & \text { for } \alpha \neq \beta, \\ 1 & \text { for } \alpha=\beta .\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.475em;vertical-align:-1.115em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04601em;">ξ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.115em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> for </span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord">.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>则称P为标准正交多项式。P是由n(ξ)和积分域唯一确定的。</p><h3 id="高斯求积">高斯求积</h3><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><msub><mi mathvariant="normal">Ω</mi><mi>ξ</mi></msub></msub><mi>n</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>ξ</mi><mo>≈</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>w</mi><mi>α</mi></msub><mi>g</mi><mrow><mo fence="true">(</mo><msub><mi>ξ</mi><mi>α</mi></msub><mo fence="true">)</mo></mrow><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">\int_{\Omega_{\xi}} n(\xi) g(\xi) \mathrm{d} \xi \approx \sum_{\alpha=1}^N w_\alpha g\left(\xi_\alpha\right),</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.475em;vertical-align:-1.115em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04601em;">ξ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.115em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0954em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.046em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></p><p>在矩输运方程中的任何一项都可以写成左边形式，其中第一项w称为权重函数，第二项称为插值公式的节点，N是节点的总数。不同的插值公式有对应不同的精度，具体可以参考书。这里主要介绍高斯求积（具有2N-1阶精度）。</p><p>对于精度这个概念，我的理解是在M的2N-1矩下，该重构的式子任然可以正确描述。<br>有如下充分且必要定理：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mo>∫</mo><msub><mi mathvariant="normal">Ω</mi><mi>ξ</mi></msub></msub><mi>n</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>ξ</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>g</mi><mrow><mo fence="true">(</mo><msub><mi>ξ</mi><mi>α</mi></msub><mo fence="true">)</mo></mrow><msub><mi>w</mi><mi>α</mi></msub><mo>+</mo><msub><mi>R</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{\Omega_{\xi}} n(\xi) g(\xi) \mathrm{d} \xi=\sum_{\alpha=1}^N g\left(\xi_\alpha\right) w_\alpha+R_N(g)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.475em;vertical-align:-1.115em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.4336em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04601em;">ξ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.115em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathrm">d</span><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0954em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.046em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;">)</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span></span></span></span></span></p><p>对于高斯求积，有2N-1精度，且它的N个节点ξa的值与对应n(ξ)的正交多项式在积分域上的根重合。</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>m</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>w</mi><mi>α</mi></msub><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>m</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>w</mi><mi>α</mi></msub><msub><mi>ξ</mi><mi>α</mi></msub><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>m</mi><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>w</mi><mi>α</mi></msub><msubsup><mi>ξ</mi><mi>α</mi><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>m</mi><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>w</mi><mi>α</mi></msub><msubsup><mi>ξ</mi><mi>α</mi><mrow><mn>2</mn><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}m_0 & =\sum_{\alpha=1}^N w_\alpha, \\m_1 & =\sum_{\alpha=1}^N w_\alpha \xi_\alpha, \\\vdots & \\m_{2 N-2} & =\sum_{\alpha=1}^N w_\alpha \xi_\alpha^{2 N-2}, \\m_{2 N-1} & =\sum_{\alpha=1}^N w_\alpha \xi_\alpha^{2 N-1} .\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:15.7418em;vertical-align:-7.6209em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:8.1209em;"><span style="top:-10.1209em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-6.7254em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.6583em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.17em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:2.2254em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:7.6209em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:8.1209em;"><span style="top:-10.1209em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-6.7254em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.046em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-3.6583em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span></span></span><span style="top:-1.17em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.046em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:2.2254em;"><span class="pstrut" style="height:3.8283em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04601em;">ξ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.046em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:7.6209em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>这里的整数矩是已知量，因为高斯求积有2N-1精度，因此可以列出上面的式子。</p><p>左边有2N个已知的整数矩，右边有N个权重函数w,和N个节点ξ，这个方程是封闭的。<br>但这个不是一个简单的线性方程组，需要用到一些数值解法，有包括PD算法等，详细的解法参考书50页。</p><h2 id="多分布Multivariate-distributions"><strong>多分布Multivariate distributions</strong></h2><p>在前面介绍中，n(m)NDF是作为质量或者体积的单变量分布函数进行计算，考虑多分布的情况。</p>]]></content>
</entry>
<entry>
<title>linux系统常用指令</title>
<link href="/2024/10/05/linux%E7%B3%BB%E7%BB%9F%E5%B8%B8%E7%94%A8%E6%8C%87%E4%BB%A4/"/>
<url>/2024/10/05/linux%E7%B3%BB%E7%BB%9F%E5%B8%B8%E7%94%A8%E6%8C%87%E4%BB%A4/</url>
<content type="html"><![CDATA[<figure class="highlight routeros"><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></pre></td><td class="code"><pre><code class="hljs routeros"><span class="hljs-built_in">find</span> . -name <span class="hljs-string">"*:Zone.Identifier"</span> -type f -delete 清除.Zone文件<br>sudo ldconfig 更新<br><span class="hljs-built_in">find</span> <span class="hljs-variable">$WM_PROJECT_DIR</span> -name probesDi 查找文件名<br><span class="hljs-built_in">find</span> . -name <span class="hljs-string">'*'</span> | xargs grep -1 Spark 寻找当前文件里面带有Spark关键词的文件<br></code></pre></td></tr></table></figure><p>动态库缓存</p><p>在原有基础上进行细化网格</p><figure class="highlight gams"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs gams">refineMesh <span class="hljs-comment">//首先在新的案例文件目录，这个命令会创建一个新的时间文件夹只包含有网格数据</span><br><br><span class="hljs-function"><span class="hljs-title">mapFields</span></span> ../template/ -consistent -sourceTime latestTime <span class="hljs-comment">//映射指令，这里在新的案例文件目录映射template案例文件的最新时间步数据到此案例refineMesh生成的时间文件上</span><br></code></pre></td></tr></table></figure><p>在c++中，1/3代表整除操作，如果想得到一个浮点数，要这样写1.0/3.0</p><h1>github相关</h1><figure class="highlight crmsh"><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></pre></td><td class="code"><pre><code class="hljs crmsh">git pull origin development 更新已经拉取的<br><br>git <span class="hljs-keyword">clone</span> <span class="hljs-title">--recursive</span> URL 连通子模块一起拉取<br><br>git restore examples/simple_example.cc 恢复远程仓库中的最新版本，丢弃本地修改<br><br>git clean -fd 清理所有未被跟踪的文件和文件夹<br><br>git clean -fdn 查看显示即将被删除的文件和目录<br><br>git fetch origin 从远程仓库拉取最新的更改<br><br>git reset --hard origin/<span class="hljs-keyword">master</span> <span class="hljs-title">重置本地分支 ，这会将本地的master</span>分支重置为远程仓库最新提交<br><br>git status 验证同步<br></code></pre></td></tr></table></figure><h1>python相关</h1><figure class="highlight maxima"><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></pre></td><td class="code"><pre><code class="hljs maxima">conda <span class="hljs-built_in">deactivate</span>退出当前环境<br><br>conda <span class="hljs-built_in">activate</span> deepflame 激活deepflame环境<br><br>conda create -n deepflame python=<span class="hljs-number">3.9</span> 激活一个名字为deepflame的新环境<br><br>conda <span class="hljs-built_in">deactivate</span> 退出conda环境<br></code></pre></td></tr></table></figure><h1>cantera</h1><p><code>ck2yaml --input=C2H4-42s-171r.inp --thermo=C2H4-42s-therm.dat --transport=C2H4-42s-trans.dat</code></p><p>转化机理为yaml格式</p>]]></content>
</entry>
<entry>
<title>使用Amrex计算碳烟</title>
<link href="/2024/10/05/%E4%BD%BF%E7%94%A8Amrex%E8%AE%A1%E7%AE%97%E7%A2%B3%E7%83%9F/"/>
<url>/2024/10/05/%E4%BD%BF%E7%94%A8Amrex%E8%AE%A1%E7%AE%97%E7%A2%B3%E7%83%9F/</url>
<content type="html"><![CDATA[<h1>Amrex的安装</h1><p>Amrex主要由三个部分构成：PelePhysics PeleLMeX PeleC ,我们这里计算碳烟主要是用到了前两个，PelePhysics主要提供化学计算，而后者提供流体求解器，具体的说明和安装可以参考<a href="https://amrex-combustion.github.io/">AMReX-Combustion: Suite of adaptive mesh reacting flow codes for reacting flows</a></p><p>在amrex中，辐射计算是通过基于球谐函数法的辐射热传递模型。</p><h1>常见BUG总结</h1><p>（1）</p><p>==============================================================================<br><code><!--Est. time step - Conv: 2.348325225e-06, divu: 0.0007054938829--><!--STEP [30000] - Time: 0.1481225918, dt 2.348325225e-06--><!--0::Assertionamrlev >= 0 && amrlev < this->m_num_amr_levels' failed, file "/home/zhangjiaqian/PelePhysics/Submodules/amrex/Src/LinearSolvers/MLMG/AMReX_MLCellLinOp.H", line 504 !!!--></code><br><code><!--SIGABRT--></code><br>在添加辐射模块的时候，提示遇到断言错误，这是因为mx_level的大小和ref_ratio的数量没有保持一致导致的，如果设置了多层次细化比率，请确保其大小与 <code>amr.max_level</code> 一致。<br><code>amr.max_level = 2 # maximum level number allowed</code><br><code>amr.ref_ratio = 2 2 # refinement ratio</code></p><p>同时因为不知道的bug,这里如果直接启动还是会报错，建议先算几步，然后在input里面指定重新计算</p><h1>燃烧配置</h1><h2 id="1物理建模">1物理建模</h2><p>由于Amrex是使用自适应网格技术，它的一个最大的优点就是不用自己画网格，调节粗细。这里采用我们实验室的燃烧器计算图纸进行建模。</p><p><img src="/images/%E5%AE%9E%E9%AA%8C%E5%AE%A4%E5%9B%BE%E7%BA%B8.jpg" alt="燃烧器模型"></p><p>选择项目提供的案例里最相近的模型进行修改，这里我们选择PeleLMeX\Exec\Production\DiffBunsen2D</p><p>案例文件，从文件名上也可以看出这是一个二维扩散本生灯相关的案例。</p><h3 id="1）CMakeLists-txt">1）CMakeLists.txt</h3><p>先看看CMakeLists.txt(值得说明的是AMREX为大规模并行计算设计的结构非常好，它使用了编译可执行文件+Input文本的方式进行计算，而不依赖其它相关项，这里后面也会讲)</p><p><code>target_include_directories(pelelmex PUBLIC $<BUILD_INTERFACE:${CMAKE_CURRENT_LIST_DIR}>)</code></p><p><code>set(PELE_CHEMISTRY_MODEL drm19 PARENT_SCOPE)</code><br><code>set(PELE_EOS_MODEL Fuego PARENT_SCOPE)</code><br><code>set(PELE_TRANSPORT_MODEL Simple PARENT_SCOPE)</code><br><code>set(PELE_DIM "2" PARENT_SCOPE)</code><br><code>set(PELE_MPI ON PARENT_SCOPE)</code></p><p><code>target_sources(pelelmex</code><br><code>PRIVATE</code><br><code>pelelmex_prob.cpp</code><br><code>pelelmex_prob.H</code><br><code>pelelmex_prob_parm.H</code><br><code>)</code></p><p>第一行不用管，这是告诉系统寻找相关头文件和执行安装当前目录的</p><p><code>set(PELE_CHEMISTRY_MODEL drm19 PARENT_SCOPE)</code> 使用drm19化学机理，amrex使用不同化学机理需要使用yaml格式文件转化（如果只有标准chemkin格式则需要先用ck2转化为yaml），详细的转化可以参考官网给的教程<a href="https://amrex-combustion.github.io/PelePhysics/Ceptr.html#generating-pele-compatible-mechanisms-for-a-single-chemistry">CEPTR: Chemistry Evaluation for Pele Through Recasting — PelePhysics 2022.10 documentation (amrex-combustion.github.io)</a></p><p>剩下的几行分别代表使用的状态模型（EOS），输运模型，计算维数，并行计算，以及目录下面的一些头文件等</p><h3 id="2）GNUmakefile">2）GNUmakefile</h3><p>上面的Cmakelist似乎对后面的编译操作影响不大（我一般是两个都改，但是make操作应该是只针对这个文件的，不懂，后面再看）<br>里面的文件也是进行相关的设置，值得注意的是，这里需要指定Pele的目录，如果你不是在案例目录操作则需要对应更改，推荐使用绝对路径</p><p><code>PELE_HOME ?= ../../..</code><br><code>include $(PELE_HOME)/Exec/Make.PeleLMeX</code></p><h3 id="3）pelelmex-Prob-cpp">3）pelelmex_Prob.cpp</h3><p>这个文件只定义了一个函数，这个函数是用来读取input字典里面的初始条件参数的。</p><p><code>PeleLM::readProbParm()</code></p><p><code>{****</code></p><p><code>}</code></p><h3 id="4）pelelmex-prob-H">4）pelelmex_prob.H</h3><p>这里具体定义了物理模型<br><em>// Find the zone of the cell to identify the inflow, initial conditions and BC</em></p><p><em>// There are 4 zones : 1 = Fuel inflow ; 2 = Obstacle ; 3 = Oxidizer inflow ; 4</em></p><p><em>// = Air inflow Schema : [ 1 | 2 | 3 | 4 ]</em></p><p>这是一个层流同轴扩散火焰的模型，其中1是燃料出口，2是墙壁，三是氧化剂出口，4是空气出口（这个在我的模型里目前没用到）</p><p>具体代码实现</p><p><code>const amrex::Real* prob_lo = geomdata.ProbLo();</code></p><p><code>const amrex::Real* prob_hi = geomdata.ProbHi();</code></p><p><code>amrex::Real split12 = prob_lo[0] + prob_parm.Xf;</code></p><p><code>amrex::Real split23 = prob_lo[0] + prob_parm.Xf + prob_parm.Xe;</code></p><p><code>amrex::Real split34 = prob_lo[0] + prob_parm.Xf + prob_parm.Xe + prob_parm.Xc;</code></p><p>所以如果要更改燃料碰嘴大小等，需要在input里面更改对应的参数</p><p>这里的文件应该是每个网格读取一次，然后根据x的值返回对应zone的标签。接下来的代码包括设置点火，二次函数燃料出口等</p><h3 id="5）pelelmex-prob-parm-H">5）pelelmex_prob_parm.H</h3><p>设置相关的物理模型参数初始条件</p><p>在input中也能改，且input的优先级要高于这个初始文件</p><h3 id="6）input">6）input</h3><h4 id="DOMAIN-DEFINITION">DOMAIN DEFINITION</h4><p><code>#----------------------DOMAIN DEFINITION------------------------</code><br><code>prob.Xmax = 0.0185 #Maximum size in the x direction</code><br><code>prob.Ymax = 0.037 #Maximum size in the y direction</code><br><code>prob.Zmax = 0.0185 #Maximum size in the z direction</code></p><p>展示不清楚这个和下面有什么区别，先不动</p><p>geometry.is_periodic = 0 0 # For each dir, 0: non-perio,periodic<br>geometry.coord_sys = 1 # 0 => cart, 1 => RZ<br>geometry.prob_lo = 0.0 0.0 # x_lo y_lo (z_lo)<br>geometry.prob_hi = 0.0185 0.037 # x_hi y_hi (z_hi)</p><p>第一个设置周期边界条件，第二个设置轴对称模型，第三个设置矩形的对角</p><h4 id="BC-FLAGS">BC FLAGS</h4><p>Interior, Inflow, Outflow, Symmetry,</p><p>SlipWallAdiab, NoSlipWallAdiab, SlipWallIsotherm, NoSlipWallIsotherm</p><p>peleLM.lo_bc = Symmetry Inflow<br>peleLM.hi_bc = SlipWallAdiab Outflow<br>设置对应的边界条件，不用改动</p><h4 id="AMR-CONTROL">AMR CONTROL</h4><p>amr.n_cell = 128 256 32 # Level 0 number of cells in each direction<br>amr.v = 1 # AMR verbose 输出日志的详细程度<br>amr.max_level = 1 # maximum level number allowed 最大细化等级，0代表不使用AMR<br>amr.ref_ratio = 2 2 2 2 # refinement ratio 对应每个方向上细化率，2代表细化一倍，第一个2代表从0level -1level 的细化倍率，以此类推<br>amr.regrid_int = 2 # how often to regrid<br>amr.n_error_buf = 4 2 2 2 # number of buffer cells in error est<br>amr.grid_eff = 0.7 # what constitutes an efficient grid<br>amr.blocking_factor = 16 # block factor in grid generation (min box size)<br>amr.max_grid_size = 64 # max box size</p><p>意思见注释</p><h4 id="Problem">Problem</h4><p>#--------------------------- Problem -------------------------------<br>prob.Xf = 0.0011 # Radius of the fuel pipe<br>prob.Xe = 0.0001 # Thickness of the extruded tube<br>prob.Xc = 0.0113 # Size of the oxidizer fuel</p><p>prob.Yin = 0.003 # Height of the inflow entrance at t=0s<br>prob.V_fu = 0.15 # Speed of the fuel in inflow<br>prob.V_ox = 0.031 # Speed of the oxidizer in inflow<br>prob.V_air = 0.0 # Speed of the air flow around the burner<br>prob.T_fu = 300 # Temperature of the fuel<br>prob.T_ox = 300 # Temperature of the oxidizer<br>prob.T_air = 300 # Temperature of the air<br>prob.T_obst = 450 # Temperature of the fuel pipe (=obstacle)</p><p>prob.do_ignition = 1 # Do you want to do ignition ?<br>prob.ign_rad = 0.001 # Size of the ignition blob<br>prob.ign_T = 1900 # Temperature inside the ignition blob</p><p>prob.P_mean = 100000.0 # Ambient pressure in the chamber</p><p>prob.dilution = 0.7 # Proportion of CH4 in the fuel mix. Ch4 is diluted in N2</p><p>见字面意思，注意这里dilution0.7代表燃料的质量分数</p><h4 id="PeleLM-CONTROL">PeleLM CONTROL</h4><p>#-------------------------PeleLM CONTROL----------------------------<br>peleLM.gravity = 0.0 -9.81 0.0</p><p>peleLM.v = 2 输出日志等级<br>peleLM.incompressible = 0 0代表可压缩流体<br>peleLM.rho = 1.17<br><a href="http://peleLM.mu">peleLM.mu</a> = 0.0<br>peleLM.use_wbar = 1 指定是否使用分子量平均，<code>1</code> 表示开启。<br>peleLM.sdc_iterMax = 2<br>peleLM.floor_species = 0<br>peleLM.num_init_iter = 3<br>peleLM.advection_scheme = “Godunov_PLM” 指定对流的数值方案，这里使用 “Godunov_PLM”（Piecewise Linear Method），通常是一种高分辨率方案<br>peleLM.do_temporals = 1 # use to calculate the integrals<br>peleLM.do_mass_balance = 1 否进行质量守恒计算，<code>1</code> 表示启用。<br>#peleLM.deltaT_verbose = 1 控制时间步调节信息输出的详细程度。<br>peleLM.deltaT_iterMax = 15 调整时间步的最大迭代次数。</p><h4 id="AMR-CONTROL-2">AMR CONTROL</h4><p>#amr.restart = chk05000<br>amr.check_int = 100<br>amr.plot_int = 100<br>amr.max_step = 100<br>amr.dt_shrink = 0.01 时间步缩减因子，初始时用较小的时间步以保证稳定性。<br>#amr.stop_time = 0.001<br>amr.init_dt = 0.000001 初始迭代次数，在时间积分开始前进行的初始化迭代次数。<br>amr.cfl = 0.05<br>amr.derive_plot_vars = avg_pressure mag_vort mass_fractions</p><p>peleLM.chem_integrator = “ReactorCvode” 使用 CVODE 进行化学积分<br>peleLM.use_typ_vals_chem = 1 # Use species/temp typical values in CVODE<br>ode.rtol = 1.0e-7 # Relative tolerance of the chemical solve<br>ode.atol = 1.0e-6 # Absolute tolerance factor applied on typical values<br>cvode.solve_type = denseAJ_direct # CVODE Linear solve type (for Newton direction)<br>cvode.max_order = 4 # CVODE max BDF order.</p><h4 id="REFINEMENT-CONTROL">REFINEMENT CONTROL</h4><p>#--------------------REFINEMENT CONTROL------------------------<br>amr.refinement_indicators = HR <code>HR</code>（Heat Release，即热释放率) 根据热释率进行细化<br>amr.HR.max_level = 1 设置 <code>HR</code> 指标的最大细化层级为 <code>1</code>，这意味着网格在满足该细化条件的区域最多可以细化到 Level 1。<br>amr.HR.value_greater = 5e7 如果某区域的热释放率大于 <code>5e7</code>，则触发网格细化。<br>amr.HR.field_name = HeatRelease 指定用于细化条件的字段名称为 <code>HeatRelease</code>。这通常是由求解器计算出的一个场变量，代表燃烧过程中热释放的速率。</p><p>#amr.refinement_indicators = magVort 涡量幅度（<code>mag_vort</code>）来细化网格。<br>#amr.magVort.max_level = 1<br>#amr.magVort.value_greater = 500.0<br>#amr.magVort.field_name = mag_vort</p><p>#amr.refinement_indicators = boxNoz 这个指标用于根据几何区域（box）的范围来细化网格，通常用于细化某些特定物理区域，例如喷嘴。<br>#amr.boxNoz.max_level = 3 网格细化最大层级为 <code>3</code>，比其他指标的层级更高。<br>#amr.boxNoz.in_box_lo = 0.000 0.0 指定用于细化的几何区域，范围在 <code>in_box_lo</code> 和 <code>in_box_hi</code> 之间。这里区域在二维平面上，限定了细化的空间范围。<br>#amr.boxNoz.in_box_hi = 0.0045 0.0045</p><p>#amrex.fpe_trap_invalid = 1 启用对非法浮点操作的捕获，例如计算未定义的操作（如 <code>NaN</code>）。<br>#amrex.fpe_trap_zero = 1 启用对除零操作的捕获，如果浮点运算中发生了除零错误，将触发异常。<br>#amrex.fpe_trap_overflow = 1 启用对浮点运算溢出的捕获，当浮点数超过最大范围导致溢出时触发异常。</p>]]></content>
</entry>
<entry>
<title>个人博客搭建指南</title>
<link href="/2024/10/04/%E4%B8%AA%E4%BA%BA%E5%8D%9A%E5%AE%A2%E6%90%AD%E5%BB%BA%E6%8C%87%E5%8D%97/"/>
<url>/2024/10/04/%E4%B8%AA%E4%BA%BA%E5%8D%9A%E5%AE%A2%E6%90%AD%E5%BB%BA%E6%8C%87%E5%8D%97/</url>
<content type="html"><![CDATA[<p>我的所有网站博客配置都是跟着这个文章写的<br><a href="https://blog.csdn.net/yaorongke/article/details/119089190">GitHub Pages + Hexo搭建个人博客网站，史上最全教程_hexo博客-CSDN博客</a></p><p>一些常见的运行命令：<br><code>hexo g -d</code></p><p>部署到GitHub上（要提前配置gitHub相关的信息）</p><p><code>hexo s</code></p><p>这两个用来本地启动</p><p><code>hexo new post 个人博客搭建指南</code><br>这个指令用来新建post文</p>]]></content>
</entry>
<entry>
<title>Hello World</title>
<link href="/2024/10/02/hello-world/"/>
<url>/2024/10/02/hello-world/</url>
<content type="html"><![CDATA[<p>Welcome to <a href="https://hexo.io/">Hexo</a>! This is your very first post. Check <a href="https://hexo.io/docs/">documentation</a> for more info. If you get any problems when using Hexo, you can find the answer in <a href="https://hexo.io/docs/troubleshooting.html">troubleshooting</a> or you can ask me on <a href="https://github.com/hexojs/hexo/issues">GitHub</a>.</p><h2 id="Quick-Start">Quick Start</h2><h3 id="Create-a-new-post">Create a new post</h3><figure class="highlight bash"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo new <span class="hljs-string">"My New Post"</span><br></code></pre></td></tr></table></figure><p>More info: <a href="https://hexo.io/docs/writing.html">Writing</a></p><h3 id="Run-server">Run server</h3><figure class="highlight bash"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo server<br></code></pre></td></tr></table></figure><p>More info: <a href="https://hexo.io/docs/server.html">Server</a></p><h3 id="Generate-static-files">Generate static files</h3><figure class="highlight bash"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo generate<br></code></pre></td></tr></table></figure><p>More info: <a href="https://hexo.io/docs/generating.html">Generating</a></p><h3 id="Deploy-to-remote-sites">Deploy to remote sites</h3><figure class="highlight bash"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo deploy<br></code></pre></td></tr></table></figure><p>More info: <a href="https://hexo.io/docs/one-command-deployment.html">Deployment</a></p>]]></content>
</entry>
</search>