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\title{工程流体力学期末复习}

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作者: Airocéan\footnote{airocean@mail.ustc.edu.cn, a@airocean.cn, airocean@foxmail.com}\footnote{http://airocean.cn}
\begin{align}
		\rho=\cfrac{\mathrm dm}{\mathrm dV}=\cfrac mV \qquad F=\mu Av/h\notag
\end{align}
$\mu$流体动力粘度，$\mu$为Const时该流体为牛顿流体，$\nu$运动粘度，单位$m^2/s$，其微分形式，牛顿切向应力公式：
\begin{align}
& \tau=\mu\cfrac{\mathrm dv_x}{\mathrm dy}\quad\nu=\cfrac{\mu}{\rho}\notag\\
& f_l-\cfrac 1\rho\cfrac{\partial p}{\partial l}=0\notag\\
& \mathrm dp=\rho(f_x\mathrm dx+f_y\mathrm dy+f_z\mathrm dz)\notag\\
& (0=)\mathrm dp=\vec{f}\cdot\mathrm d\vec{r}\notag\\
& z+\cfrac{p}{\rho g}=C\notag\\
& \Delta p=\rho g(\sin \alpha +A_1/A_2)l=kl \notag\\
&p= \rho g\left(\dfrac{\omega^2r^2}{2g}-z\right)+C \notag\\
& \dfrac{\omega^2r^2}{2}-zg=C \notag\\
&F_p=\rho g \sin \alpha x_cA=\rho g h_cA\notag\\
& x_D=x_c+I_{cy}/(x_cA)  \notag\\
& F_{px}=\rho g h_cA_x \quad F_{pz}=\rho g V_p\notag
		\end{align}
		作用点的位置也很好想到直接是线性无关的分而治之；$R_h$水力半径，$\chi$湿周，半径为$r$圆管内充满流体便有$R_h=r/2$
\begin{align}
		& q_V=\iint_A \vec v \cdot \mathrm d \vec A\quad q_m=\iint_A \vec \rho v \cdot \mathrm d \vec A\notag\\
& v_a=q_V/A\quad R_h=\dfrac A\chi\notag\\
& R_h=\dfrac{\pi r^2}{2 \pi r}=\dfrac r 2\notag\\
& \dfrac{\mathrm d N}{\mathrm d t}=\dfrac{\partial}{\partial t}\mathop{\iiint}\limits_{CV}\eta \rho \mathrm d V+\mathop{\iint}\limits_{CS}\eta \rho\vec{v}\cdot \mathrm d \vec{A} \notag
\end{align}
\begin{equation}
\begin{array}{ll}
\dot{\gamma}_{z}=\dfrac{\left(\dfrac{\partial v_{y}}{\partial x}+\dfrac{\partial v_{x}}{\partial y}\right)}{2} &\omega_{z}=\dfrac{\left(\dfrac{\partial v_{y}}{\partial x}-\dfrac{\partial v_{x}}{\partial y}\right)}{2} \\
\dot{\gamma}_{x}=\dfrac{\left(\dfrac{\partial v_{z}}{\partial y}+\dfrac{\partial v_{y}}{\partial z}\right)}{2} &\omega_{x}=\dfrac{\left(\dfrac{\partial v_{z}}{\partial y}-\dfrac{\partial v_{y}}{\partial z}\right)}{2} \\
\dot{\gamma}_{y}=\dfrac{\left(\dfrac{\partial v_{x}}{\partial z}+\dfrac{\partial v_{z}}{\partial x}\right)}{2} &\omega_{y}=\dfrac{\left(\dfrac{\partial v_{x}}{\partial z}-\dfrac{\partial v_{z}}{\partial x}\right)}{2}
\end{array}\notag
\end{equation}
角变速度$2\gamma$，旋转速度$\omega$，无旋则是$\omega_x+\omega_y+\omega_z=0$；
可压缩液体定常流动，若是非压缩流体那么$\rho=Const$
$$\dfrac{\partial}{\partial x}(\rho v_x)+\dfrac{\partial}{\partial y}(\rho v_y)=0$$
$$\vec a=\cfrac{\mathrm d \vec u}{\mathrm d t}=\cfrac{\partial \vec u}{\partial t}+(\vec u \cdot \nabla) \vec u$$\begin{equation}
\left\{\begin{array}{c}
a_{x}=\cfrac{\partial u_{x}}{\partial t}+u_{x} \cfrac{\partial u_{x}}{\partial x}+u_{y} \cfrac{\partial u_{x}}{\partial y}+u_{z} \cfrac{\partial u_{x}}{\partial z} \\
a_{y}=\cfrac{\partial u_{y}}{\partial t}+u_{x} \cfrac{\partial u_{y}}{\partial x}+u_{y} \cfrac{\partial u_{y}}{\partial y}+u_{z} \cfrac{\partial u_{y}}{\partial z} \\
a_{z}=\cfrac{\partial u_{z}}{\partial t}+u_{x} \cfrac{\partial u_{z}}{\partial x}+u_{y} \cfrac{\partial u_{z}}{\partial y}+u_{z} \cfrac{\partial u_{z}}{\partial z}
\end{array}\right. \notag
\end{equation}
\begin{equation}
\left\{\begin{array}{l}
\rho q_{V}\left(\beta_{2} v_{2 x}-\beta_{1} v_{1 x}\right)=F_{f x}+F_{p_{n} x} \\
\rho q_{V}\left(\beta_{2} v_{2 y}-\beta_{1} v_{1 y}\right)=F_{f y}+F_{p_{n} y} \\
\rho q_{V}\left(\beta_{2} v_{2 z}-\beta_{1} v_{1 z}\right)=F_{fz}+F_{p_{nz}}
\end{array}\right. \notag
\end{equation}
其中$\beta$为动量修正系数，一般为1
\begin{equation*}
\beta=\dfrac 1A \mathop{\iint}\limits_{A}\left(\dfrac{v}{v_a}\right)^2\mathrm d A \notag
\end{equation*}
重力作用下的绝热能流方程：
\begin{align}
&\frac{\partial}{\partial t} \mathop{\iiint}\limits_{C V} \rho\left(u+\frac{v^{2}}{2}+
g z\right)\mathrm d V+\notag \\ 
&\mathop{\iint}\limits_{C S} \rho v_{n}\left(u+\frac{v^{2}}{2}+g z\right) \mathrm d A=\mathop{\iint}\limits_{C S} \vec{p}_{n} \cdot \vec{v}\mathrm d A \notag
\end{align}
且有
\begin{equation}
\mathop{\iint}\limits_{C S} \vec{p}_{n} \cdot \vec{v}\mathrm d A=\mathop{\iint}\limits_{C S} \vec{p}_{nn} \cdot \vec{v}\mathrm d A+\mathop{\iint}\limits_{C S} \vec{\tau} \cdot \vec{v}\mathrm d A \notag
\end{equation}
理想流体绝能流动有$u=Const$
\begin{align}
\frac{v_{1}^{2}}{2}+g z_{1}+\frac{p_{1}}{\rho_{1}}=+\frac{v_{2}^{2}}{2}+g z_{2}+\frac{p_{2}}{\rho_{2}} \notag
\end{align}
皮托管和文丘里管
\begin{align}
		& v_B=\sqrt{\frac{2(p_a-p_b)}{\rho}}=\sqrt{2gh} \notag\\
& v_2=\left.\sqrt{\frac{2(p_1-p_2)}{\rho}}\right/\sqrt{1-\dfrac{A_2^2}{A_1^2}}\notag \\
& \dfrac{\partial p}{\partial r}=0 \notag
\end{align}
表明不计重力影响的直线运动，沿着流线法向的压强梯度为0，没有压强差，表明在直管壁内测得的静压强是该截面上任意一点的静压强；弯管中速度内高外低，压强相反；不可压缩粘性流体总流的伯努利方程为：
\begin{align}
&\frac{\alpha_1 v_{1a}^{2}}{2g}+z_{1}+\frac{p_{1}}{\rho g}=\frac{\alpha_2 v_{2a}^{2}}{2g}+z_{2}+\frac{p_{2}}{\rho g} +h_{\mathrm w} \notag \\
&\alpha = \dfrac 1A \mathop{\iint}\limits_{A}\left(\dfrac{v}{v_a}\right)^3\mathrm dA \notag
\end{align}
$\alpha$为动能修正参数一般为1，$v_a$这里都是average velocity\\
重力相似$Fr=\dfrac{v}{\sqrt{gl}}$\\
粘滞力相似$Re=\dfrac{\rho vl}\mu =\dfrac{vl}\nu$\\
压力相似$Eu=\dfrac p{\rho v^2}$\begin{equation}
h_{\mathrm f}=\lambda\dfrac ld \dfrac{v^2}{2g} \notag
\end{equation}一般使用下临界{\it Re}，取2000到2320之间；圆管层流流量Hagen-Poiseuille：
\begin{equation}
q_V=\pi r_0^2 v=\dfrac{\pi d^4 }{128\mu}\dfrac{\rm d}{\mathrm d l}(p+\rho gh) \notag
\end{equation}
因为$v_{\rm a}=-\dfrac{r_0^2}{8\mu}\dfrac{\rm d}{\mathrm d l}(p+\rho gh)$所以流速不变时有$q_V=\dfrac{\pi d^4 }{128\mu}\dfrac{\Delta (p+\rho gh)}{\Delta l}$
因此有层流条件下$\lambda=\dfrac{64}{Re}$和壁面切应力公式$\tau_{\rm w}=\dfrac \lambda 8 \rho v^2$
附加粘性系数$\mu_{t}=\rho l^{2}\left|\dfrac{\mathrm{d} v_{x}}{\mathrm{d} y}\right|$
%\begin{equation}
%\delta=\frac{32.4 d}{\mathrm{Re}^{0.875}}(\mathrm{mm}) ~ \mathrm{or} ~ \delta=\frac{32.8 d}{\operatorname{Re} \lambda^{1 / 2}}(\mathrm{mm}) \notag
%\end{equation}
%层流区、过渡区、紊流光滑管区、紊流粗糙管过渡区、紊流粗糙管平方阻力区
%紊流光滑管区$4000<Re<10^5$有
%$$\lambda=\dfrac{0.3164}{Re^{0.25}}$$
%紊流光滑管区$10^5<Re<3\times10^6$有
%$$\lambda=0.0032+0.221Re^{-0.237}$$
%紊流粗糙管过渡区$26.98(d/\varepsilon)^{8/7}<Re<2308(d/\varepsilon)^{0.85}$有
%$$\dfrac 1{\lambda^{1/2}}=1.42\left[\lg\left(1.273\cdot\dfrac{q_V}{\nu \varepsilon}\right)\right]$$
%紊流粗糙管平方阻力区$2308(d/\varepsilon)^{0.85}<Re$有
%$$\dfrac 1{\lambda^{1/2}}=2\lg\dfrac d{2\varepsilon}+1.74$$
莫迪图推荐的完全紊流粗糙管区与过渡区分界线雷诺数为$Re_{\rm b}=3500(d/\varepsilon)$\\
在非圆轨道中应该使用当量直径$D$描述D-W公式中的$d$\\
管道截面突扩，相当于损失速度水头
$$h_{\rm j}=\dfrac1{2g}(v_1-v_2)^2=\dfrac{v_1^2}{2g}\left(1-\dfrac{A_1}{A_2}\right)^2 \cdots$$
有$\zeta_1=(1-A_1/A_2)^2\cdots$\\
管道截面突缩，有$v_c$
$$h_{\rm j}=\zeta\dfrac{v_2^2}{2g}=\zeta_c\dfrac{v_c^2}{2g}+\dfrac{(v_c-v_2)^2}{2g}$$
$C_c=A_c/A_2$，$\zeta=\dfrac{\zeta_c}{C_c^2}+\left(\dfrac 1{C_c}-1\right)^2$\\
P118，弯管，查图表代入$l_e=\zeta d/  \lambda$
注意：进出的$\zeta$不一样，如二者极限情况，扩是1，缩是0.5；水击现象，可以看看P141
\begin{equation}
\begin{array}{l}
v_{x}=\dfrac{1}{2 \mu}\left[\dfrac{\rm d}{\mathrm d x}(p+\rho g h)\right] y^{2}+C_{1} y+C_{2} \\
\rightarrow v_{x}=-\dfrac{(b-y) y}{2 \mu} \dfrac{\rm d}{\mathrm d x}(p+\rho g h)
\end{array}\notag
\end{equation}
\begin{equation}
-\frac{1}{\rho} \frac{\mathrm d p}{\mathrm d x}=v_{b} \frac{\mathrm d v_{b}}{\mathrm d x} \notag
\end{equation}
\begin{equation}
\begin{array}{l}
\delta=5.84\left(\frac{v x}{v_{\infty}}\right)^{1 / 2}=5.84 x \mathrm{Re}_{x}^{-1 / 2} \\
C_{D f}=\dfrac{F_{D f}}{b l \rho v_{\infty}^{2} / 2}=1.372 \mathrm{Re}_{l}^{-1 / 2}
\end{array} \notag
\end{equation}












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