《空气动力学》（双语版）CHAPTER 9 OBLIQUE SHOCK AND EXPANSION WAVES 斜激波和膨胀波（2/4）

斜波产生的根源 斜激波关系式 普朗特一梅耶膨 胀波 流过尖楔与圆锥 的超音速流 激波干扰与反射 脱体激波 激波-膨胀波理论及其在 超音速翼型中的应用 图9.5第九章路线图

斜波产生的根源 普朗特—梅耶膨 胀波 斜激波关系式 流过尖楔与圆锥 的超音速流 激波干扰与反射 脱体激波 激波-膨胀波理论及其在 超音速翼型中的应用 图9.5 第九章路线图

Wave angle:β激波角,激波与激波上游来流的夹角 Deflection angle: 6.通过斜激波的气流偏转角 M, M1>1 T2>71 M2> < T (a) Concave sort (b)Convex corner FIGURE 9.1 Supersonic flow over a comer

Wave angle: β激波角, 激波与激波上游来流的夹角。 Deflection angle:θ,通过斜激波的气流偏转角 β θ

What is the physical mechanism that creates waves in a supersonic fow?超音速流中产生波的物理机理是什么? The information is propagated upstream at approximately the Disturbance due to body is propa mated upstream local speed of sound m via molecular collisions at approximately the Flo ow noving ed 物体存在的信息以近似等于 sewer than the peed of sound 当地音速的速度传播到上游 去。 (a) If the upstream flow is subsonic as shown in Fig9.2a, the disturbances have no problem working their way upstream, thus giving the incoming flow plenty of time to move out of the way of the body 如图9,2a所示,如果上游是亚音速的,扰动可以毫不困难地传播 到远前方上游,因此,给了来流足够的时间以绕过物体

• What is the physical mechanism that creates waves in a supersonic flow? 超音速流中产生波的物理机理是什么？ If the upstream flow is subsonic , as shown in Fig.9.2a, the disturbances have no problem working their way upstream, thus giving the incoming flow plenty of time to move out of the way of the body. 如图9.2a所示，如果上游是亚音速的， 扰动可以毫不困难地传播 到远前方上游，因此，给了来流足够的时间以绕过物体。 The information is propagated upstream at approximately the local speed of sound. 物体存在的信息以近似等于 当地音速的速度传播到上游 去

Disturbances cannot work thelr way upstream. Instea. they coalesce, forming a stand Disturbance due to body is propagated upstream viu molecular collisions Flow moving at approximately the fasier than the l speed af sound speed of sound FGURE 9.2 Propagation of disturbances. ( a)Subsonic Aow. (b)Supersonic Aow. On the other hand if the upstream flow is supersonic, as shown in Fig 9.2b, the disturbances cannot work their way upstream; rather, at some finite distances from the body, the disturbance waves pile up and coalesce, forming a standing wave in front of the body 在另一方面如图92b所示,如果上游是超音速的扰动不能一直向上 游传播,而是在离开物体某一距离处聚集并接合,形成一静止波

On the other hand, if the upstream flow is supersonic, as shown in Fig.9.2b, the disturbances cannot work their way upstream; rather, at some finite distances from the body, the disturbance waves pile up and coalesce, forming a standing wave in front of the body. 在另一方面,如图9.2b所示，如果上游是超音速的,扰动不能一直向上 游传播，而是在离开物体某一距离处聚集并接合，形成一静止波

Hence, the physical generation of waves in a supersonic flow-both shock and expansion waves--is due to the propagation of information via molecular collisions and due to the fact that such propagation cannot work its way into certain regions of the supersonic flow 因此,超音速流中溦波和膨胀波产生的物理原因是:通 过分子碰撞引起的信息传播和这种传播不能到达超音 速流中某些区域

Hence, the physical generation of waves in a supersonic flow—both shock and expansion waves—is due to the propagation of information via molecular collisions and due to the fact that such propagation cannot work its way into certain regions of the supersonic flow. 因此,超音速流中激波和膨胀波产生的物理原因是: 通 过分子碰撞引起的信息传播和这种传播不能到达超音 速流中某些区域

Why are most waves oblique rather than normal to the upstream fow?为什么大部分激波与来流成斜角而不是垂直的呢? 马赫波 Supersonic Subsonic v>a < 马赫角 at a 1 FIGURE 9.3 Another way of visualizing the propagation of disturbances in(a) subsonic and(b)supersonic dow. SIn M (9.1)

• Why are most waves oblique rather than normal to the upstream flow? 为什么大部分激波与来流成斜角而不是垂直的呢? V M a Vt at 1 sin  = = = M 1 sin −1  = (9.1) 马赫波 马赫角

M>t FIGURE 9A Relation between the oblique shock-wave angle and the Mach angle. If the disturbances are stronger than a simple sound wave then the wave front becomes stronger than a mach wave, creating an oblique shock wave at an angle to the freestream, where B>u. This comparison is shown in Fig. 9.4. However, the physical mechanism creating an oblique shock isis essentially the same as that described above for the mach wave 如果扰动比一个简单声波强,其引起的波前就会比马赫波强,产生 个与来流夹角为β的斜激波,且β>μ。这一比较在图94中给出。然而, 斜激波产生的物理机理与上面描述的马赫波的产生完全相同

If the disturbances are stronger than a simple sound wave, then the wave front becomes stronger than a Mach wave, creating an oblique shock wave at an angle to the freestream, where β>μ. This comparison is shown in Fig. 9.4 . However, the physical mechanism creating an oblique shock is is essentially the same as that described above for the Mach wave. 如果扰动比一个简单声波强,其引起的波前就会比马赫波强,产生一 个与来流夹角为 β的斜激波,且β>μ。这一比较在图9.4中给出。然而， 斜激波产生的物理机理与上面描述的马赫波的产生完全相同

92 OBLIQUE SHOCK RELATIONS(斜激波关系式) o∥⊙ M1 M FIGURE 9 Obilque shock geometry

９.2 OBLIQUE SHOCK RELATIONS (斜激波关系式）

以上图虚线包围区域为控制体,应用连续方程: f11=2l2 (9.2) (9.5)通过斜激波流动的切向 速度分量保持不变 +n11=P2+P2l2(97) 2 (9.12 通过斜激波的流动特性变化只由垂直 于斜激波的速度分量决定

1 u1 = 2 u2 以上图虚线包围区域为控制体，应用连续方程： (9.2) w1 = w2 (9.５) 通过斜激波流动的切向 速度分量保持不变． 2 2 2 2 2 p1 + 1 u1 = p +  u (9.7) 2 2 2 2 2 2 1 1 u h u h + = + (9.12) 通过斜激波的流动特性变化只由垂直 于斜激波的速度分量决定．

方程(9.2)、(9.7、(9.12)与正激波控制方程(8,2)、(86)、(8.10)完全 相同,我们只要将正激波关系式中所有的M用Mn代替,就可以得 到通过斜激波的流动特性变化量: MnI=MISin B (9.13) M21-(y-1)/2(914注意!Ma2是斜激波 1+y 后的法向马赫数 y+1)M (9.15) 2+(y-1)M21 y (Mn-1)(9.16) 方程(914-(9,17)表明对于量热完全气体斜激波的特性只依赖于 上游马赫数的垂直分量Mn1,但是,由(913)知,Mn1即依赖于M又 依赖于β

M n,1 = M1 sin  ( )  ( 1)/ 2 1 1 / 2 2 ,1 2 2 ,1 ,2 − − + − =    n n n M M M 2 ,1 2 ,1 1 2 2 ( 1) ( 1) n n M M + − + =     ( 1) 1 2 1 2 ,1 1 2 − + = + Mn p p   方程(9.2)、(9.7)、(9.12)与正激波控制方程(8.2)、(8.6)、(8.10)完全 相同，我们只要将正激波关系式中所有的M1用Mn,1代替，就可以得 到通过斜激波的流动特性变化量： (9.14) (9.13) (9.15) (9.16) 注意！Mn,2是斜激波 后的法向马赫数． 2 1 1 2 1 2   p p T T = (9.17) 方程(9.14)-(9.17)表明对于量热完全气体,斜激波的特性只依赖于 上游马赫数的垂直分量Mn,1 ,但是,由(9.13)知，Mn,1即依赖于M1又 依赖于 β