势流理论笔记:01 势流理论基础

前言:
势流理论复习笔记,没想到自己又重新学了一遍势流理论。。。
所以记个笔记

笔记内容基本摘抄自朱仁传老师的《船舶在波浪上的运动理论》,写得好哇!!!

基础理论

均匀、不可压缩理想流体的流场中,连续性方程与欧拉方程可以描述为:

\[\begin{align} \begin{cases} \nabla \boldsymbol v=0 \\ \left( \frac{\partial}{\partial t}+\boldsymbol v\cdot \nabla \right)\boldsymbol v=-\nabla\left(\frac{p}{\rho}+gz \right) \end{cases} \end{align} \]

\({\boldsymbol v} (x,y,z)\text与 p(x,y,z)\)分别为速度矢量与压力场,存在向量关系

\[\begin{align} \nabla\left ( \frac{\boldsymbol v^2}{2}\right)=\nabla\left ( \frac{\boldsymbol v\cdot \boldsymbol v}{2}\right)=(\boldsymbol v\cdot \nabla)\boldsymbol v+\boldsymbol v\times(\nabla \times \boldsymbol v) \end{align} \]

欧拉方程可以改写为以下形式,称为兰姆方程\((Lamb's Equation)\)

\[\begin{align} \frac{\partial \boldsymbol v}{\partial t}+\nabla\left ( \frac{\boldsymbol v^2}{2}\right)-\boldsymbol v\times(\nabla \times \boldsymbol v)=-\nabla\left(\frac{p}{\rho}+gz \right) \end{align} \]

以上共四个方程,四个未知数,方程封闭。如果流体流动无旋有势,方程可以进一步简化

\[\begin{align} \boldsymbol v=\nabla \phi(x,y,z,t) \end{align} \]

\[\begin{equation} \begin{aligned} \nabla^2 \phi(x,y,z,t)=\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}=0\\ \frac{\boldsymbol p}{\rho}+gz+\frac{\boldsymbol v^2}{2}+\frac{\partial \phi}{\partial t}=C(t) \end{aligned} \end{equation} \]

格林函数法

船舶在波浪中的运动问题关键在于求解流畅中的速度势,即求在确定边界条件下的拉普拉斯方程。格林函数法\((Green's \ \ function\ \ method)\)是一类成熟常用的求解方法。格林函数法的基础势格林公式(散度定理)推导得到,对三维空间中有界区域\(\tau\),有以下关系式

\[\begin{align} \iiint _\tau \nabla\cdot \boldsymbol A \text d \tau=\iint_S \boldsymbol n\cdot \boldsymbol A \text d S \end{align} \]

其中,\(S\)为空间域\(\tau\)充分光滑的边界面;\(\boldsymbol n\)为曲面\(S\)的单位外法向矢量(从流体域内指向外部),矢量\(\boldsymbol A\)在封闭区域\(\tau + S\)上连续。

现令\(\boldsymbol A=\phi \nabla \psi\),于是有

\[\begin{align} \nabla \boldsymbol A=\nabla(\phi\nabla\psi)=\nabla\phi \cdot \nabla\psi+\phi\nabla^2\psi \end{align} \]

\[\begin{align} \boldsymbol n\cdot \boldsymbol A=\boldsymbol n\cdot (\phi\nabla \psi)=\phi\frac{\partial \psi}{\partial n} \end{align} \]

\(\boldsymbol A=\phi \nabla \psi\)代入格林公式得到

\[\begin{align} \iint _S \phi \frac{\partial \psi}{\partial n}\text d S= \iiint_\tau \nabla\phi\cdot \nabla \psi\text d \tau+ \iiint_\tau \phi\cdot \nabla^2 \psi\text d \tau \end{align} \]

\(\boldsymbol A=\psi \nabla \phi\)代入格林公式得到

\[\begin{align} \iint _S \psi \frac{\partial \phi}{\partial n}\text d S= \iiint_\tau \nabla\phi\cdot \nabla \psi\text d \tau+ \iiint_\tau \psi\cdot \nabla^2 \phi\text d \tau \end{align} \]

两式作差得到

\[\begin{align} \iint _S \left ( \psi \frac{\partial \phi}{\partial n}- \phi \frac{\partial \psi}{\partial n}\right )\text d S= \iiint_\tau \left ( \phi\cdot \nabla^2 \psi-\psi\cdot \nabla^2 \phi\right ) \text d \tau \end{align} \]

\(\phi,\psi\)\(\tau\)内处处调和,即: \(\nabla^2 \phi=0,\nabla^2\psi=0\),则有

\[\begin{equation} \begin{aligned} \iint _S \left ( \psi \frac{\partial \phi}{\partial n}- \phi \frac{\partial \psi}{\partial n}\right )\text d S=0\\ \iint _S \psi \frac{\partial \phi}{\partial n}\text d S = \iint _S \phi \frac{\partial \psi}{\partial n}\text d S \end{aligned} \end{equation} \]

称作格林第二公式。

引入

现在设\(\phi(x,y,z)\)作为待求速度势,为调和函数;如果能恰当选取调和函数\(\psi\),使得公式11右边端变为

\[\begin{align} \iiint_\tau \phi(Q)\cdot \nabla^2 \psi(P,Q) \text d \tau=\phi(P) \end{align} \]

\(P(x,y,z)\)为流场内任意一点,\(Q(x,y,z)\)为流场内变点,那么势函数\(\phi\)在域内任意一点的值可以有边界上的函数值与法向导数值确定。

满足这种性质的\(\psi\)函数必然满足,\(\delta\)为狄拉克雷函数

\[\begin{align} \nabla^2{\psi(P,Q)}=\delta(P-Q) \end{align} \]

对于三维问题,该方程的一个特解是

\[\begin{align} {\psi(P,Q)}=-\frac{1}{4\pi r(P,Q)} \end{align} \]

式中,\(r(P,Q)\)为P与Q点之间的距离,\(r=\sqrt{(x-\xi)^2+(y-\eta)^2+(z-\zeta)^2}\)\(\psi(P,Q)\)除了在\(P=Q\)点除外,处处满足\(\nabla^2 \psi =0\).

流域内部问题

场点\(P(x,y,z)\)在封闭区域\(\tau+S\)内,由于存在奇异性,围绕场点作一半径为\(r_\varepsilon\)的小球,小球表面积为\(S_\varepsilon\),于是在\(S+S_\varepsilon\)所围成的区域内,\(\psi\)函数处处调和。

由格林第二公式得到:

\[\begin{align} \iint _{S+S_\varepsilon} \psi \frac{\partial \phi}{\partial n}\text d S = \iint _{S+S_\varepsilon} \phi \frac{\partial \psi}{\partial n}\text d S \end{align} \]

\(S_{\varepsilon}\)上,\(\frac{\partial \psi}{\partial n}=-\frac{1}{4\pi}\frac{\partial}{\partial r}(\frac{1}{r})_{r=r_\varepsilon}=-\frac{1}{4\pi}\frac{1}{r_\varepsilon^2}\)利用中值定理,得到

\[\begin{align} \lim\limits_{r_\varepsilon \to 0}\iint _{S_\varepsilon} \phi \frac{\partial \psi}{\partial n}\text d S= \lim\limits_{r_\varepsilon \to 0}\left [\phi\cdot \left( - \frac{1}{4\pi r_\varepsilon^2} \right)\cdot 4\pi r_\varepsilon^2\right] =-\phi(x,y,z) \end{align} \]

\[\begin{align} \lim\limits_{r_\varepsilon \to 0}\iint _{S_\varepsilon} \psi \frac{\partial \phi}{\partial n}\text d S= \lim\limits_{r_\varepsilon \to 0}\left [\frac{\partial \phi}{\partial n}\cdot \left( - \frac{1}{4\pi r_\varepsilon} \right)\cdot 4\pi r_\varepsilon^2\right] =0 \end{align} \]

格林第二公式可以化简为

\[\begin{equation} \begin{aligned} \iint _{S} \psi \frac{\partial \phi}{\partial n}\text d S+ \iint _{S_\varepsilon} \psi \frac{\partial \phi}{\partial n}\text d S= \iint _{S} \phi \frac{\partial \psi}{\partial n}\text d S \iint _{S_\varepsilon} \phi \frac{\partial \psi}{\partial n}\text d S\\ \iint _{S} \psi \frac{\partial \phi}{\partial n}\text d S+ (0)= (-\phi)+ \iint _{S_\varepsilon} \phi \frac{\partial \psi}{\partial n}\text d S \\ \phi(x,y,z)=-\frac{1}{4\pi} \iint _{S}\left[\phi\frac{\partial}{\partial n}\left(\frac{1}{r}\right)-\frac{1}{r}\frac{\partial \phi}{\partial n} \right] \text d S \end{aligned} \end{equation} \]

对于场点落在流场边界,作一半球

\[\phi(x,y,z)=-\frac{1}{2\pi} \iint _{S}\left[\phi\frac{\partial}{\partial n}\left(\frac{1}{r}\right)-\frac{1}{r}\frac{\partial \phi}{\partial n} \right]\text d S \]

对于场点落在流场外部,

\[0=\iint _{S}\left[\phi\frac{\partial}{\partial n}\left(\frac{1}{r}\right)-\frac{1}{r}\frac{\partial \phi}{\partial n} \right] \text d S \]

归纳起来

\[\begin{align} \iint _{S}\left[\phi\frac{\partial}{\partial n}\left(\frac{1}{r}\right)-\frac{1}{r}\frac{\partial \phi}{\partial n} \right]\text d S= \begin{cases} -4\pi \phi(x,y,z),&P \in \tau \\ -2\pi \phi(x,y,z),&P \in S \\ \qquad0,&P\notin \tau+S \end{cases} \end{align} \]

流域外部问题

如果研究的是封闭曲面\(S\)以外区域的流场,则认为流域的边界面可以认为是\(S+S_\infin\)\(S_\infin\)为外部假想球面,半径\(R\rightarrow\infin\),在\(S_\infin\)上有:

\[\begin{equation} \begin{aligned} \iint _{S_\infin}\left[\phi\frac{\partial}{\partial n}\left(\frac{1}{r}\right)-\frac{1}{r}\frac{\partial \phi}{\partial n} \right]\text d S=& -\iint _{S_\infin}\left[\phi\frac{1}{r^2}-\frac{1}{r}\frac{\partial \phi}{\partial r} \right]\text d S\\ =&-\iint _{S_\infin}\left[\phi\frac{1}{R^2}-\frac{1}{R}\frac{\partial \phi}{\partial r} \right]\text d S\\ \end{aligned} \end{equation} \]

当半径\(R\rightarrow\infin\)时,\(\phi\)满足

  1. \(\phi \rightarrow O\left(\frac{1}{R^{\alpha}}\right),\alpha >0\)
  2. \(\frac{\partial \phi}{\partial r} \rightarrow O\left(\frac{1}{R}\right)\)

\[\begin{align} \lim \limits_{R \rightarrow \infin}\iint _{S_\infin}\left[\phi\frac{1}{R^2}-\frac{1}{R}\frac{\partial \phi}{\partial r} \right]\text d S=0 \end{align} \]

此时的法向与之前一节定义的相反,由流场内指向流场外部。

格林函数的选取

对于之前提及的\(\psi=-\frac{1}{4\pi r(P,Q)}\)只是泊松方程的一个特解,但是\(\psi\)的形式并不唯一,如果存在域\(\tau\)内处处调和的函数\(G^*(P,Q)\),则

\[\begin{align} \psi(P,Q)=-\frac{1}{4\pi}\left [ \frac{1}{r(P,Q)}+G^*(P,Q)\right]=-\frac{1}{4\pi}G(P,Q) \end{align} \]

因此,格林第三公式可以写作:

\[\begin{align} \iint _{S}\left[\phi\frac{\partial G}{\partial n}-G\frac{\partial \phi}{\partial n} \right]\text d S= \begin{cases} -4\pi \phi(x,y,z),&P \in \tau \\ -2\pi \phi(x,y,z),&P \in S \\ \qquad0,&P\notin \tau+S \end{cases} \end{align} \]

式中,\(P,Q\)\(S\)边界上的点。若在\(S\)上给定\(\frac{\partial \phi (Q)}{\partial n}\)

posted @ 2021-09-25 14:58  陈橙橙  阅读(2716)  评论(0编辑  收藏  举报