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Lists-of-NS-equations.html
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NS 方程索引
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<p>流体流动类型众多，可以按照流体粘性种类划分，也可以按照流体压缩性划分。本文参考了
COMSOL 帮助文档，列出主要常用的各种流体流动方程，以作参考。</p>
</blockquote>
<p><img
src="https://raw.githubusercontent.com/NChilton/Picture/main/img/202111301507193.png" /></p>
<span id="more"></span>
<h2 id="标准-ns-方程">标准 NS 方程</h2>
<p>NS 方程用于描述速度场 <span
class="math inline">\(\boldsymbol{u}\)</span> 和压力场 <span
class="math inline">\(p\)</span> 的关系，可以写作 <span
class="math display">\[
\begin{align}
\underbrace{\rho \left( \frac{\partial \boldsymbol{u}}{\partial t} +
(\boldsymbol{u}\cdot\nabla) \boldsymbol{u} \right)}_\text{惯性力} &
= \underbrace{-\nabla p}_\text{压力} + \underbrace{\nabla\cdot
\boldsymbol{K}}_\text{粘性力} +
\underbrace{\boldsymbol{F}}_\text{体积力}, \label{eq_momentum} \\
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\boldsymbol{u}) &
= 0. \label{eq_mass}
\end{align}
\]</span> 其中 <span class="math inline">\(\rho\)</span>
是流体密度，粘性力项中张量 <span
class="math inline">\(\boldsymbol{K}\)</span> 与动力粘度 <span
class="math inline">\(\mu\)</span> 和速度 <span
class="math inline">\(\boldsymbol{u}\)</span> 相关 <span
class="math display">\[
\boldsymbol{K} = \mu(\nabla
\boldsymbol{u}+(\nabla\boldsymbol{u})^\mathsf{T})-\frac{2}{3}\mu((\nabla
\cdot\boldsymbol{u})\boldsymbol{I}).
\]</span> 其中 <span class="math inline">\(\boldsymbol{I}\)</span>
是单位矩阵，<span
class="math inline">\(\nabla\boldsymbol{u}+(\nabla\boldsymbol{u})^\mathsf{T}=\boldsymbol{D}\)</span>
也被称为变形率张量。如果流体不可压，则有 <span
class="math inline">\(\text{d}\rho/\text{d}t=0\)</span>，代入方程 <span
class="math inline">\(\eqref{eq_mass}\)</span> 得 <span
class="math inline">\(\nabla\cdot\boldsymbol{u}=0\)</span>。假设流体粘度恒定，此时粘性力项可简化为
<span class="math display">\[
\begin{align*}
\nabla\cdot \boldsymbol{K} & = \nabla\cdot \left[ \mu(\nabla
\boldsymbol{u}+(\nabla\boldsymbol{u})^\mathsf{T})-\frac{2}{3}\mu((\nabla
\cdot\boldsymbol{u})\boldsymbol{I})\right] \\
& = \nabla \cdot \left[ \mu(\nabla
\boldsymbol{u}+(\nabla\boldsymbol{u})^\mathsf{T})\right] \\
& = \mu \nabla \cdot \nabla \boldsymbol{u} = \mu
\nabla^2\boldsymbol{u}.
\end{align*}
\]</span> 则不可压、粘性、层流 NS 方程可写作 <span
class="math display">\[
\begin{equation}\label{eq_ns}
\boxed{\begin{aligned}
\rho \left( \frac{\partial \boldsymbol{u}}{\partial t} +
(\boldsymbol{u}\cdot\nabla) \boldsymbol{u} \right) & = -\nabla p +
\mu \nabla^2\boldsymbol{u} + \boldsymbol{F}, \\
\nabla\cdot\boldsymbol{u} & = 0.
\end{aligned}}
\end{equation}
\]</span> 流体的可压缩性可以通过无量纲数——马赫数 <span
class="math display">\[
Ma = U/c
\]</span> 来判定，其中 <span class="math inline">\(U\)</span>
是流体特征速度， <span class="math inline">\(c\)</span>
是流体声速。按马赫数大小可分为</p>
<ul>
<li>不可压流动（<span class="math inline">\(Ma=0\)</span>）</li>
<li>弱可压缩流动（<span class="math inline">\(Ma<0.3\)</span>）</li>
<li>亚声速流动（<span class="math inline">\(Ma=0.3～0.8\)</span>
左右）</li>
<li>跨声速流动（<span class="math inline">\(Ma=0.8～1.2\)</span>
左右）</li>
<li>超声速流动（<span class="math inline">\(Ma=1.2～5.0\)</span>
左右）</li>
<li>高超声速流动（<span class="math inline">\(Ma>5.0\)</span>）</li>
</ul>
<p>不可压流动可通过方程 <span
class="math inline">\(\eqref{eq_ns}\)</span>
求解，弱可压缩流动可通过方程 <span
class="math inline">\(\eqref{eq_momentum}\)</span> 和 <span
class="math inline">\(\eqref{eq_mass}\)</span> 求解。当 <span
class="math inline">\(Ma>0.3\)</span>
需要考虑湍流效应和热效应，需要额外方程来描述流体流动特征。</p>
<p>对于不可压流动，描述湍流效应还有一重要的无量纲数——雷诺数 <span
class="math display">\[
Re=\frac{\rho UL}{\mu}=\frac{\text{惯性力}}{\text{粘性力}},
\]</span> 其中 <span class="math inline">\(L\)</span>
是流动结构特征长度。尽管不同的流动结构产生湍流效应的临界雷诺数不同，一般可认为当
<span class="math inline">\(Re>2000\)</span> 时产生湍流效应，当 <span
class="math inline">\(Re<2000\)</span> 流体流动可通过方程 <span
class="math inline">\(\eqref{eq_ns}\)</span>
求解。如果惯性力相对于粘性力可以忽略（一般 <span
class="math inline">\(Re<<1\)</span>），流体流动及其缓慢，该种流动也被称为为<strong>蠕动流</strong>。在求解时，方程
<span class="math inline">\(\eqref{eq_ns}\)</span> 中惯性项中对流项
<span class="math inline">\((\boldsymbol{u}\cdot\nabla)
\boldsymbol{u}\)</span> 可以忽略 <span class="math display">\[
\begin{equation}
\boxed{\begin{aligned}
\rho \frac{\partial \boldsymbol{u}}{\partial t} & = -\nabla p + \mu
\nabla^2\boldsymbol{u} + \boldsymbol{F}, \\
\nabla\cdot\boldsymbol{u} & = 0.
\end{aligned}}
\end{equation}
\]</span></p>
<h2 id="湍流">湍流</h2>
<h2 id="热效应">热效应</h2>
<h2 id="两相流">两相流</h2>
<p>在工程上，有时需要追踪流体的界面，此时，表面张力在界面变形中起到重要作用。如最常见的水龙头接水时，液柱界面形状变化。考虑最普通的气液两相不可压流动，如下图所示</p>
<p><img src="https://raw.githubusercontent.com/NChilton/Picture/main/img/202112061710021.png" style="zoom:50%;" /></p>
<p>若空间内气液界面 <span
class="math inline">\(\boldsymbol{s}=\boldsymbol{s}(\boldsymbol{\xi};t)\)</span>
用两个参数 <span
class="math inline">\(\boldsymbol{\xi}=(\xi_1,\xi_2)\)</span>
的参数方程表示，则界面的运动方程满足 <span class="math display">\[
\left( \frac{\partial \boldsymbol{s}(\boldsymbol{\xi};t)}{\partial
t}-\boldsymbol{u}(\boldsymbol{\xi};t) \right)\cdot\boldsymbol{n} = 0,
\]</span> 其中 <span
class="math inline">\(\boldsymbol{u}(\boldsymbol{\xi};t)\)</span>
表示位置 <span
class="math inline">\(\boldsymbol{\xi}=(\xi_1,\xi_2)\)</span>
的速度，该方程通常也简写为动力学边界条件 <span
class="math inline">\(\boldsymbol{n}\cdot(\boldsymbol{u}-\boldsymbol{u}_S)=0\)</span>，<span
class="math inline">\(\boldsymbol{u}_S\)</span>
是气液界面运动速度。如果用等值面 <span
class="math inline">\(S(\boldsymbol{s}(\boldsymbol{\xi};t);t) =
S_0\)</span> 来表示气液界面，上式可写作 <span class="math display">\[
\begin{align}\label{eq_S}
\frac{\partial S}{\partial t} + \boldsymbol{u}\cdot \nabla S = 0.
\end{align}
\]</span> 在气相和液相 <span
class="math inline">\(\Omega_1\cup\Omega_2\)</span>
中，流体流动均满足方程 <span
class="math inline">\(\eqref{eq_ns}\)</span> 。在气液界面 <span
class="math inline">\(S\)</span>
上，两侧的应力差驱动界面流动，满足界面应力平衡方程 <span
class="math display">\[
\begin{align}\label{bc_stress}
\boxed{\boldsymbol{n}\cdot[\boldsymbol{T}_i]_2^1=\gamma\kappa\boldsymbol{n},}
\end{align}
\]</span> 其中 <span class="math inline">\([x_i]_2^1=x_1-x_2\)</span>
表示物理量跨越界面时的差值，<span
class="math inline">\(\boldsymbol{T}_i=-p_i\boldsymbol{I}
\boldsymbol+\mu_i\boldsymbol{D}_i\)</span>
称为水动力应力张量（hydrodynamic stress tensor），<span
class="math inline">\(\kappa=-\nabla\cdot \boldsymbol{n}\)</span>
是界面平均曲率。如果气液两相可当作无粘流体或者均处于静止状态，则两侧压力差为
<span class="math display">\[
\Delta p=p_1-p_2=\gamma\kappa,
\]</span> 该压力差也被称为拉普拉斯压力。注意NS
方程是<strong>体方程</strong>，而方程 <span
class="math inline">\(\eqref{bc_stress}\)</span>
是<strong>表面方程</strong>。追踪界面方法有流体体积（Volume of fluid，
VOF）法、水平集（Level set，LS）法和任意拉格朗日（Arbitrary Lagrangian
Eulerian，ALE）法等。其中 VOF 和 LS 法将界面处理成一定厚度的过渡层。</p>
<h3 id="vof-方法">VOF 方法</h3>
<p>VOF 法引入相体积分数 <span class="math inline">\(c\)</span>
来实现对计算域内界面进行追踪，同时将双流体方程（气液相各一个）转化为单流体方程。流体体积分数一般设置为
<span class="math display">\[
\left\{
\begin{align*}
& c=0 & \text{气相} \\
& 0<c<1 & \text{界面} \\
& c = 1 & \text{液相}
\end{align*}.
\right.
\]</span></p>
<p>通过求出整个计算域内各网格单元的相分数，从而可以构建出界面</p>
<p><img
src="https://raw.githubusercontent.com/NChilton/Picture/main/img/202112071047804.png" /></p>
<p>在 VOF
法中，所有物理量经过界面都是渐变的，如转化为单流体方程中密度和粘度可表示为
<span class="math display">\[
\rho(c)=(1-c)\rho_1+c\rho_2,\ \mu(c)=(1-c)\mu_1+c\mu_2.
\]</span> 此时求解的单流体动量方程为 <span class="math display">\[
\begin{align}
\rho(c) \left( \frac{\partial \boldsymbol{u}}{\partial t} +
(\boldsymbol{u}\cdot\nabla) \boldsymbol{u} \right) = \nabla \cdot
\boldsymbol{T} + \boldsymbol{F_S},\label{eq_mo}
\end{align}
\]</span> 此处我们用 <span class="math inline">\(\boldsymbol{T} =
-p\boldsymbol{I} \boldsymbol+\mu(c)\boldsymbol{D}\)</span> 代替了 <span
class="math inline">\(\eqref{eq_ns}\)</span> 中的压力项和粘性项，<span
class="math inline">\(\boldsymbol{F_S}\)</span>
为表面张力导出的体积力。单流体仍然当作是不可压流体，有 <span
class="math display">\[
\frac{\text{d}\rho}{\text{d}t} = 0 \rightarrow \frac{\text{d}
\left((1-c)\rho_1+c\rho_2 \right)}{\text{d}t}=
(\rho_2-\rho_1)\frac{\text{d}c}{\text{d}t} = 0,
\]</span> 即 <span class="math display">\[
\boxed{\frac{\partial c}{\partial t}+\boldsymbol{u}\cdot\nabla c=0.}
\]</span> 该方程是不可压缩 VOF
模型中的相方程，也是最常见的对流方程，也与方程 <span
class="math inline">\(\eqref{eq_S}\)</span> 一致。VOF
模型中，界面法向向量和曲率可以写为 <span class="math display">\[
\boldsymbol{n}=\frac{\nabla c}{\left |\nabla c \right|},\ \kappa =
-\nabla \cdot \left( \frac{\nabla c}{\left |\nabla c \right|} \right).
\]</span> 为了将表面方程 <span
class="math inline">\(\eqref{bc_stress}\)</span> 考虑到 VOF
方法中，需要用到连续力（Continuum surface
force，CSF）模型，该模型将间断的压力处理成连续过渡的体积力，即（不考虑马兰戈尼效应）
<span class="math display">\[
\boldsymbol{n}\cdot(\boldsymbol{T}_1-\boldsymbol{T}_2)=\gamma\kappa\boldsymbol{n}\rightarrow
\nabla \cdot \boldsymbol{T} = \nabla (\gamma\kappa)=\gamma\kappa\nabla
c,
\]</span> 结合方程 <span class="math inline">\(\eqref{eq_mo}\)</span>
不可压缩 VOF 模型中动量方程 <span class="math display">\[
\boxed{\rho(c) \left( \frac{\partial \boldsymbol{u}}{\partial t} +
(\boldsymbol{u}\cdot\nabla) \boldsymbol{u} \right) = -\nabla p +
\nabla\cdot \left( \mu(c)\boldsymbol{D} \right) - \gamma \nabla \cdot
\left( \frac{\nabla c}{\left |\nabla c \right|} \right) \nabla c.}
\]</span></p>
<h3 id="level-set-方法">Level-set 方法</h3>
<h3 id="ale-方法">ALE 方法</h3>
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