比拟理论与无量纲化导出特征数

\begin{cases} \frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0\\ U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{1}{Re}\frac{\partial^2 U}{\partial Y^2}\\ U\frac{\partial \Theta}{\partial X}+V\frac{\partial \Theta}{\partial Y}=\frac{1}{Re}\frac{\partial ^2 \Theta}{\partial Y^2}\\ \end{cases}\tag{Pr=1}
\frac{1}{X}Nu_X=\frac{\partial \Theta}{\partial Y}|_{y=0}
\frac{\partial U}{\partial Y}|_{y=0}=\frac{C_f}{2}Re_x

忽略体积力，压力的微分方程组如下：

\begin{cases} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\\ u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})\\ u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=\nu(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2})\\ u\frac{\partial t}{\partial x}+v\frac{\partial t}{\partial y}=a(\frac{\partial ^2t}{\partial x^2}+\frac{\partial ^2t}{\partial y^2})\\ h_x=-\frac{\lambda}{t_w-t_f}(\frac{\partial t}{\partial y})_{y=0} \end{cases}\tag{微分方程组}

1.无量纲化

X=\frac{x}{l};Y=\frac{y}{l};P=\frac{p}{\rho u^2_\infty};U=\frac{u}{u_\infty};V=\frac{v}{u_\infty};\Theta=\frac{t-t_w}{t_f-t_w}\tag{无量纲化}

①连续性方程无量纲化

\frac{\partial U}{\partial x}= \frac{\partial \frac{u}{u_\infty}}{\partial x}=\frac{1}{u_\infty}\frac{\partial u}{\partial x}\\ 连续性方程两边除以u_\infty\\ \frac{\partial U}{\partial x}+\frac{\partial V}{\partial x}=0\\ \because X=x/l; l\cdot dX=dx;\\ \therefore \frac{\partial U}{\partial x}=\frac{\partial U}{\partial X\cdot l};\\ l\neq 0等式两边除以\\ \frac{1}{l}\frac{\partial U}{\partial x}+\frac{1}{l}\frac{\partial V}{\partial x}=0\\
\color{#F00}{\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0\tag{1}}

②x方向上动量微分方程无量纲化

动量微分方程两边除以u_\infty^2\\ \frac{u}{u_\infty}\frac{\frac{\partial u}{u_\infty}}{\partial x}+\frac{v}{u_\infty}\frac{\frac{\partial u}{u_\infty}}{\partial y}=\frac{\nu}{u_\infty}(\frac{\partial ^2(\frac{u}{u_\infty})}{\partial x^2}+\frac{\partial ^2(\frac{u}{u_\infty})}{\partial x^2})\\ 变化为：U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial y}=\frac{\nu}{u_\infty}(\frac{\partial^2 U}{\partial x^2}+\frac{\partial^2 U}{\partial y^2})\\ 注意一下，偏微分中\frac{\partial^2 u}{\partial x^2}=\frac{\partial }{\partial x}(\frac{\partial u}{\partial x})，因此对u的偏导项除以一个u_\infty就可以\\ 而对于\partial x^2，相当于平方项需要除以l^2\\ \because X=x/l; x=X\cdot l;x^2=X^2\cdot l^2\\ 微分方程式变为：\\ U\frac{\partial U}{\partial X}\cdot \frac{1}{l}+V\frac{\partial U}{\partial Y}\cdot \frac{1}{l}=\frac{\nu}{u_\infty\cdot l^2}(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2})\\ 上式两边乘以l\\
\color{#F00}{U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{\nu}{u_\infty\cdot l}(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2})=\frac{1}{Re}(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2})\tag{2}}

在(2)式中，\frac{\nu}{u_\infty\cdot l}=\frac{1}{Re}

雷诺数通过对x方向上的动量方程无量纲化导出

（该方法是对已经学过的微分方程处理得到一个已知的雷诺数的范例，其他微分方程也是同样的套路）

③y方向上的动量方程同样可以得到

\color{#F00}{U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=\frac{\nu}{u_\infty\cdot l}(\frac{\partial^2 V}{\partial X^2}+\frac{\partial^2 V}{\partial Y^2})=\frac{1}{Re}(\frac{\partial^2 V}{\partial X^2}+\frac{\partial^2 V}{\partial Y^2})\tag{3}}

④能量微分方程的无量纲化

同样的手段，经过三步分别对u,x,t无量纲化
U\frac{\partial \Theta}{\partial X}+V\frac{\partial \Theta}{\partial Y}=\frac{a}{u_\infty l}(\frac{\partial \Theta^2}{\partial X^2}+\frac{\partial \Theta^2}{\partial Y^2})

\frac{a}{u_\infty l}是什么无量纲数？

已知\frac{\nu}{u_\infty l}=\frac{1}{Re}\\ \frac{a}{u_\infty l}=\frac{a \cdot \nu}{u_\infty l\cdot \nu}\\ \because Pr=\frac{\nu}{a}\\ \therefore \frac{a}{u_\infty l}=\frac{\nu \cdot a}{u_\infty l\cdot \nu}=\frac{1}{Re\cdot Pr}
\color{#F00}{U\frac{\partial \Theta}{\partial X}+V\frac{\partial \Theta}{\partial Y}=\frac{1}{Re\cdot Pr}(\frac{\partial \Theta^2}{\partial X^2}+\frac{\partial \Theta^2}{\partial Y^2})\tag{4}}

二、数据处理

①由(1)式\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0的实验值进行数据处理：

f_1(X,Y,U,V)=0\tag{5}

②由(2)式U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{\nu}{u_\infty\cdot l}(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2})=\frac{1}{Re}(\frac{\partial^2 U}{\partial X^2}+\frac{\partial^2 U}{\partial Y^2})

U是这四个实验值的函数：

U=f_2(X,Y,V,Re)\tag{6}

③由(3)式U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=\frac{\nu}{u_\infty\cdot l}(\frac{\partial^2 V}{\partial X^2}+\frac{\partial^2 V}{\partial Y^2})=\frac{1}{Re}(\frac{\partial^2 V}{\partial X^2}+\frac{\partial^2 V}{\partial Y^2})

V是这四个实验值的函数：

V=f_3(X,Y,U,Re)\tag{7}

④由(4)式U\frac{\partial \Theta}{\partial X}+V\frac{\partial \Theta}{\partial Y}=\frac{1}{Re\cdot Pr}(\frac{\partial \Theta^2}{\partial X^2}+\frac{\partial \Theta^2}{\partial Y^2})

\Theta是六个实验值的函数
\Theta=f_4(X,Y,U,V,Re,Pr)\tag{8}

⑤一般V比U小，把V的表达式(6)带入(7)然后得到

V=f_5(X,Y,Re)\tag{9}

⑥(9)式带回(6)，消去一个因变量

U=f_6(X,Y,Re)\tag{10}

可以看到，无量纲的速度U和V就是坐标X，Y与雷诺数的函数

⑦(9)+(10)带回(8)

\Theta=f_7(X,Y,Re,Pr)\tag{11}

无量纲的温度分布最终需要得到对流换热系数，其是坐标X，Y与Re、Pr数的函数

⑧带回对流换热微分方程:h_x=-\frac{\lambda}{t_w-t_f}(\frac{\partial t}{\partial y})_{y=0}

\Theta=\frac{t-t_w}{t_f-t_w}\\ \therefore h_x=-\frac{\lambda}{-(t_f-t_w)}(\frac{\partial(t-t_w)}{\partial(Y\cdot l)})_{y=0}=\frac{\lambda}{l}(\frac{\partial{(\frac{t-t_w}{t_f-t_w}})}{\partial Y})_{Y=0}=\frac{\lambda}{l}(\frac{\partial \Theta}{\partial Y})_{Y=0}
又Nu_x=h_x\cdot x/\lambda

Nu_x=\frac{x}{l}(\frac{\partial \Theta}{\partial Y})_{Y=0}=X(\frac{\partial \Theta}{\partial Y})_{Y=0}\tag{12}

⑨局部努塞尔数

Nu_x=f_8(X,Y,\Theta)|_{Y=0}
因为Y=0确定，所以局部努塞尔数是X,Re,Pr的函数
Nu_x=f_9(X,Re,Pr)

⑩积分平均掉X，努塞尔数

Nu=f_{10}(Re,Pr)

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