# 航空动力装置控制规律与特性/第一章

## 1.1 气体动力方程

—般情况下，发动矶各部件中进行的是粘性可压缩气体的空间(三元)流动。空间中每一个点上的气流参数，是由速度向量c， 压力p，密度ρ，和温度T表示。这些参数受空间的坐标x、y、z和时间T制约。发动机气体动力学的题解，以能求出下列关系式描述的整个流动场为前提：

c=c(x,y,z,t);p=p(x,y,z,t);ρ=ρ(x,y,z,t);T=T(x,y,z,t);    (1-1)


${\displaystyle {\frac {\partial \rho }{\partial t}}+div(\rho c)=0}$     (1-2)


${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial (\rho c_{x})}{\partial x}}+{\frac {\partial (\rho c_{y})}{\partial y}}+{\frac {\partial (\rho c_{z})}{\partial z}}=0}$     (1-3)


${\displaystyle \rho {\frac {dc}{dt}}=\mathbf {R} -grad\,\mathbf {p} +\mu \Delta c+{\frac {1}{3}}\mu grad(div\,c)}$    (1-4)


${\displaystyle \rho {\frac {dc}{dt}}=\mathbf {R} -grad\,\mathbf {p} +\mu \Delta c}$    (1-5)


${\displaystyle \rho {\frac {\partial c_{x}}{\partial t}}+\rho (c_{x}{\frac {\partial c_{x}}{\partial x}}+c_{y}{\frac {\partial c_{x}}{\partial y}}+c_{z}{\frac {\partial c_{x}}{\partial z}})=X+{\frac {\partial p}{\partial x}}+\mu ({\frac {\partial ^{2}c_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}c_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}c_{x}}{\partial z^{2}}})}$
${\displaystyle \rho {\frac {\partial c_{y}}{\partial t}}+\rho (c_{x}{\frac {\partial c_{y}}{\partial x}}+c_{y}{\frac {\partial c_{y}}{\partial y}}+c_{z}{\frac {\partial c_{y}}{\partial z}})=Y+{\frac {\partial p}{\partial y}}+\mu ({\frac {\partial ^{2}c_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}c_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}c_{y}}{\partial z^{2}}})}$     (1-6)
${\displaystyle \rho {\frac {\partial c_{z}}{\partial t}}+\rho (c_{x}{\frac {\partial c_{z}}{\partial x}}+c_{y}{\frac {\partial c_{z}}{\partial y}}+c_{z}{\frac {\partial c_{z}}{\partial z}})=Z+{\frac {\partial p}{\partial z}}+\mu ({\frac {\partial ^{2}c_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}c_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}c_{z}}{\partial z^{2}}})}$


${\displaystyle \rho c_{p}{\frac {dT}{dt}}=\lambda \Delta T+\mu \Phi }$    (1-7)


${\displaystyle \Phi \approx 0}$，并引入称之为导温系数的${\displaystyle a={\frac {\lambda }{c_{p}\rho }}}$后，则得

${\displaystyle {\frac {dT}{dt}}=a\Delta T}$    (1-8)


${\displaystyle {\frac {\partial T}{\partial t}}+c_{x}{\frac {\partial T}{\partial x}}+c_{y}{\frac {\partial T}{\partial y}}+c_{z}{\frac {\partial T}{\partial z}}=a({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}})}$    (1-9)


1. 表征进行过程的物体或系统形状和尺寸的几何条件
2. 确定初始瞬间气流状态的时间(初始的)条件
3. 确定系统边界上气流状态的边界条件
4. 确定液体的种类，其物理参数和这些参数与温度关系的物理条件

${\displaystyle G=\rho _{1}c_{1}F_{1}=\rho _{2}c_{2}F_{2}}$    (1-10)


${\displaystyle i_{1}+{\frac {c_{1}^{2}}{2}}+L}$внеш${\displaystyle +Q}$внеш${\displaystyle =i_{2}+{\frac {c_{2}^{2}}{2}}}$    (1.11)