气体动力学讲义（冯喜平编）（修改版） - 图文

气体动力学―――多维流动基础冯喜平编

西北工业大学 二〇一二年三月

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气体动力学―――多维流动基础.......................................................... I 第一章 基础知识（场论概要） ....................................................... 1 1.1矢量代数 ........................................................................................ 1

一、 二、 三、 四、 五、 六、 七、

矢量概念··································································································· 1 矢量的加法、减法··················································································· 2 矢量与数量的乘法··················································································· 3 矢量的数量级（矢量的点积a?b）························································ 4 矢量的矢量积（矢量的叉积?a?b?）···················································· 5 三个矢量的混合积··················································································· 7 三个矢量的混合积··················································································· 8

1.2 矢量分析 .................................................................................... 11

一、矢性函数······································································································· 11 二、

矢性函数的导数和微分········································································· 13

三、矢性函数的积分··························································································· 18

1.3 场论 ............................................................................................ 19

一、 场的概念····································································································· 19 二、数量场的方向导数和梯度··········································································· 22 三、矢量场的通量和散度··················································································· 25 四、矢量场的环量和旋度··················································································· 29

1.4 哈密顿算子的运算规则 ............................................................. 33 1.5 曲线坐标系概念 ........................................................................ 40

1. 曲线坐标的概念 ····························································································· 40 2. 坐标曲线的弧微分 ························································································· 43 3. 正交曲线坐标系中的梯度、散度、旋度及调和量的表达式 ····················· 44 4. 柱坐标系中的梯度、散度、旋度及调和量的表达式 ································· 49 5. 球坐标系中的梯度、散度、旋度及调和量的表达式 ································· 50 6.基矢量e1,e2,e3的导数公式 ············································································· 51

第二章 可压缩流体流动的控制方程 .............................................. 52

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2.1第二章的运算符号 ....................................................................... 52 2.2 引言 ............................................................................................. 53 2.3 连续介质的数学描述 .................................................................. 54

一、系统，控制体和控制面··············································································· 54 二、外延和内涵参数··························································································· 56 三、参数场··········································································································· 56 四、系统或拉格朗日法······················································································· 57 五、控制体或欧拉法··························································································· 58 六、实质导数······································································································· 59

2.4加速度表达式 .............................................................................. 61

一、加速度在直角坐标系中的表达式······························································· 61 二、流体加速度在圆柱坐标系中的表达式······················································· 64

2.5 流体微团运动的分解定理 .......................................................... 69 2.6 系统和控制体之间的关系 .......................................................... 76 2.7 连续方程 ...................................................................................... 81 2.8 动量方程 ...................................................................................... 86 2.9 动量方程 ..................................................................................... 92 2.10 熵方程 ........................................................................................ 98 2.11 小结 ............................................................................................ 99 习 题 ................................................................................................ 100 第三章 无粘性可压缩流体定常多维绝热流动的一般特征 ............. 102 3.1 第三章的运算符号 ................................................................... 102 3.2 引言 .......................................................................................... 102 3.3 关于可压缩流体定常多维绝热无粘性流动的控制微分方程 .. 103

一、笛卡尔坐标系····························································································· 104

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二、圆柱坐标系································································································· 105

3.4 流线、轨迹、流体线和流管 ................................................... 115 3.5 环量、旋转和旋度................................................................... 120

一、环量············································································································· 120 二、旋转············································································································· 121 三、旋度和涡管································································································· 124

3.6 对于可压缩流体定常绝热无粘性流动的欧拉动量方程 ........ 128

一、沿流线的定常运动····················································································· 129 二、定常无旋流动····························································································· 130

3.7 KELVIN定理 ................................................................................ 130 3.8 CROCCO定理 .............................................................................. 132 3.9 HELMHOLTZ定理 ......................................................................... 136 3.10 可压缩流体定常无粘性流动的热力学.................................. 139

一、状态方程····································································································· 140 二、音速方程····································································································· 141

3.11 速度势函数 ............................................................................ 142

一、速度势函数的定义····················································································· 142 二、利用?的运动方程 ····················································································· 143

3.12 流函数 .................................................................................... 148

一、流函数的定义····························································································· 148 二、流函数的物理解释····················································································· 150 三、速度势函数与流函数的关系····································································· 151

3.13 几种简单的势流及其叠加 ..................................................... 154

一、匀直流········································································································· 154 二、点源············································································································· 155 三、偶极子········································································································· 156 四、点涡············································································································· 158 五、匀直流加点源····························································································· 159 六、匀直流加偶极子························································································· 161

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七、匀直流加偶极子加点涡············································································· 163

3.14 小结 ........................................................................................ 166 习 题 ............................................................................................... 171 第四章 固体火箭发动机的流动方程 ............................................... 174 4.1 引言 .......................................................................................... 174 4.2 喷管纯气相流动方程 ............................................................... 174 4.3 燃烧室纯气相流动方程 ........................................................... 178 4.4 二相流的概念 .......................................................................... 182 4.5 燃烧室和喷管中的二相流动方程 ........................................... 185

一、燃烧室中的一维非定常二相流动方程····················································· 185 二、喷管中轴对称二相流动方程····································································· 187

4.6 方程的守恒型式 ...................................................................... 190 4.7 方程的张量表示式 ................................................................... 193

一、应力张量与应变率张量的关系································································· 193 二、关于张量的粗浅概念················································································· 197 三、方程的张量表示形式················································································· 199

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