空气动力学chapterPPT课件

22222211VhVh22222211uhuh8.4 能量方程的各种特殊表达形式在7.5节中我们得到了定常、绝热、无粘流动的能量方程：其中V1、V2是一条三维流线上的任意两点的速度。 对于我们现在研究的一维流动，能量方程为：（8.28)（8.29)However, keep in mind that all the subsequent results in this section hold in general along a streamline and are by no means limited to just one dimensional flows. 然而，应当记住的是：这一节中所有的结论对于一般的沿流线的问题都适用，并不只是局限于一维流动。 第1页/共28页22222211uTcuTcpp2121222211uRTuRT212122222121uaua(8.30)(8.31)RTa(8.32)以温度表示：以音速表示：第2页/共28页1212022auaconstauaua121212022222121Definition of stagnation speed of sound：滞止声速的定义(8.33)(8.34)对于沿流线的任意两点，我们可将能量方程写成如下形式：第3页/共28页Definition of a*: a*的定义7.5节最后一段引入 T*的定义：Consider a point in a subsonic flow where the local static temperature is T. At this point, imagine that the fluid element is speeded up to sonic velocity, adiabatically. The Temperature it would have at such sonic conditions is denoted as T*. Similarly, consider a point in a supersonic flow, where the local static temperature is T. At this point, imagine that the fluid element is slowed down to sonic velocity, adiabatically. Again, the Temperature it would have at such sonic conditions is denoted as T*. 用*号表示的变量被称为临界参数. 称为临界声速.*RTa第4页/共28页) 1(2*) 1(21222aua2*1*212222aauaIn Equation (8.35), a and u are the speed of sound and velocity, respectively, at any point of flow, and a* is a characteristic value associated with that same point.(8.35)临界音速的计算公式:第5页/共28页.) 1(2*) 1(2121222222121constauaua对于沿一条流线上的任意两点,有:(8.36)constaa1*) 1(21202(8.37)Clearly, these defined quantities, a0 and a* , are both constants along a given in a steady, adiabatic, inviscid flow. If all the streamlines emanate from the same uniform freestream conditions, then a0 and a* are constants throughout the entire flow field. 很明显, a0 和 a*为定义的量, 沿定常、绝热、无粘流动的给定流线为常数。如果所有流线都来自于均匀自由来流，则a0 和 a*在整个流场为常数。第6页/共28页022TcuTcppconstTcuTcuTcppp022221122(8.38) 总温的计算公式回忆7.5节中总温T0的定义，由方程(8.30)可得：(8.39)Equation (8.38) provides a formula from which the defined total temperature T0 can be calculated from the given actual conditions of T and u at any given points in a general flow field. 方程(8.38) 给出了由流场中给定点处的实际温度T和速度u计算总温T0的计算公式。第7页/共28页2220)(211) 1/(2121auRTuTcuTTp20211MTT(8.40)Equation (8.40) is very important; it states that only M (and ,of course, the value of ) dictates the ratio of total temperature to static temperature. 方程(8.40)非常重要；表明只有马赫数（及 的值）决定总温与静温的比。 For a calorically perfect gas, the ratio of total temperature to static temperature, is a function of Mach number only, as follows: （对于量热完全气体，总温和静温的比对于量热完全气体，总温和静温的比 是是马赫数的唯一函数，马赫数的唯一函数，证明如下：）TT0TT0第8页/共28页)1(120)1(20)211 ()211 (MMpp)1(000TTpp 总压、总密度的计算公式：回忆7.5节总压和总密度的定义， 在定义中包含了将气流速度等熵地压缩为零速度。由（7.32）式， 我们有：)1(121212TTpp（8.41)（8.42)（8.43)方程（8.42)和（8.43）表明：总压静压比 、总密度静密度比 只由M 和 决定。因此，对于给定气体，即给定 ， 、 只依赖于马赫数。pp00pp00第9页/共28页 Equation (8.40),(8.42)and (8.43) are very important; they should be branded on your mind. They provided formulas from which the defined , can be calculated from the actual conditions of M ,T ,p and at a given point in general flow field (assuming calorically perfect gas). They are so important that values of and obtained from Eqs. (8.40),(8.42), and (8.43), respectively , are tabulated as functions of M in App.A for (which corresponds to air at standard conditions).000 and ,pTppTT00,4 . 10)1(120)1(20)211 ()211 (MMpp20211MTT（8.42)（8.43)(8.40)第10页/共28页方程(8.40),(8.42)和(8.43) 非常重要；应牢记于心。他们给出了对于量热完全气体的任意流场，由某一给定点实际的M ,T ,p 和 的值来计算定义的量 和 的公式。正因为其重要性，附录A列表给出了 随马赫数M变化的函数关系。（对应 的标准大气条件） 0, 0pT4 . 1000,ppTT)1(120)1(20)211 ()211 (MMpp20211MTT 0第11页/共28页对于 :)1(10)1(00)12(*)12(*12*ppTT634.0* 528.0* 833.0*000ppTT 临界参数的定义与计算公式临界参数的定义：Consider a point in a general flow where the velocity is exactly sonic, i.e. where M=1. Denote the static temperature , pressure, and density at this sonic condition as T*,p*, and *,respectively.考虑流场中速度恰好为音速的这一点，即M=1 的点。我们称这一点（音速条件）的静温、静压、静密度为临界参数，用T*、p*和*表示。(8.44)(8.45)(8.46)4 . 1第12页/共28页*auM 特征马赫数（速度系数）M*的定义及计算公式In the theory of supersonic flow, it is sometimes convenient to introduce a “characteristic” Mach number, M*, defined as: 在超音速流理论中， 有时引入”特征”马赫数(也被称为速度系数）, 其定义如下:Where a* is the value of the speed of sound at sonic conditions, not the actual local value. a* 是音速条件（流动速度u=a*时)的音速值。第13页/共28页) 1(2)/*)(1(211)/(22uaua21)*1() 1(2) 1(1)/1 (22MM) 1(*/ ) 1(222MM222) 1(2) 1(*MMM下面利用能量方程(8.35)得到M与M*的关系：) 1(2*) 1(21222aua（8.35）（8.47）（8.48）第14页/共28页MMMMMMMM if if if if 11*11*11*11*There, M* acts qualitatively in the same fashion as M except M* approaches a finite value when the actual Mach number approaches infinity.可以证明,除了当 时, M*与M定性一致。M第15页/共28页小结：In summary, a number of equations have been derived in this section, all of which stem in one fashion or another from the basic energy equation for steady, inviscid, adiabatic flow.第16页/共28页Example 8.4 用本节推导出的公式解Example 7.3 。（Example 7.3 气流中一点处的压强、温度和速度分别为1atm, 320K,1000m/s。计算这一点的总温和总压。）解：例8.2中解得当地马赫数为2.79；由公式（8.40）得：7 .26)557. 2()211 (4 . 04 . 1)1(20Mpp557. 2)79. 2(24 . 01211220MTT)(818320557. 2557. 20KTT)(7 .2617 .267 .260atmpp第17页/共28页Example 8.5 Consider a point in an airflow where local Mach number, static pressure, static temperature are 3.5, 0.3atm, and 180K, respectively. Calculate the local values of p0, T0, T*, a*, and M* at this point.第18页/共28页解：可以查表A, 也可以直接用公式计算。atmpMp9 .22)211 (120KTMT621)211 (20KTT5 .51712*0smRTa/456*smRTa/9 .268smMaV/94106. 2*aVM26. 4)5 . 3(4 . 02)5 . 3(4 . 2) 1(2) 1(*22222MMM06. 226. 4*M也可以用公式 （8.48）计算M*：第19页/共28页Example 8.6 如图8.5所示翼型流动，假设流动为等熵流动，计算点1处的当地马赫数。atmpppp276. 11276. 1, 0, 0691. 17545. 0276. 1 11 , 0, 01 , 0pppp查表A：得 M=0.9第20页/共28页Example 8.7 如图8.5所示翼型流动，假设流动为等熵流动，当自由来流的温度T=59oF时，计算点1处的速度。KKRRRToo33.288)8 . 1/519(519594600111)( TTpp第21页/共28页KppTT11.2657545. 033.288)( 4 . 114 . 1111smRTa/4 .326)11.265)(287(4 . 111smaMV/76.2934 .3269 . 0111第22页/共28页8.5 WHEN IS A FLOW COMPRESSIBLE? 什么条件下流动是可压缩的？We have stated several times in the preceding chapters the rule of thumb that a flow can be reasonably assumed to be incompressible when M0.3. Why?)1(120)211 (M)1(120)211 (M即第23页/共28页%5 ,32. 00M结论：第24页/共28页VdVdpVdVVppdp2VdVVppdp200)(00)/(/pdppdp(3.12)Hence, the degree by which deviates from unity as shown in Fig.8.5 is related to the same degree by which the fractional pressure change for a given dV/V.0对于一个给定的速度变化， 的变化对压强的影响：0第25页/共28页举例1：储气室中 的气体在管道出口处等熵加速到106.7m/s, 出口处压力用不可压假设和可压流假设的计算结果分别为：不可压：可压缩：KTmNp288,/100132. 10250 )/(104341. 9)7 .106)(226. 1 (5 . 0100132. 121242520mNVppKcVTTp3 .2825 .100427 .1062882220)/(104473. 9)100132. 1 ()2883 .282()(2455 . 3010mNpTTp相对误差：%13. 0104473. 9104341. 9104473. 9444317. 03 .2822874 . 17 .106M此时的马赫数：第26页/共28页举例2：储气室中 的气体在管道出口处等熵加速到274.3m/s, 出口处压力用不可压假设和可压流假设的计算结果分别为：不可压：可压缩：KTmNp288,/100132. 10250 )/(105198. 5)3 .274)(226. 1 (5 . 0100132. 121242520mNVppKcVTTp55.2505 .100423 .2742882220)/(102223. 6)100132. 1 ()28855.250()(2455 . 3010mNpTTp相对误差：%3 .11102223. 6105198. 5102223. 6444864. 055.2502874 . 13 .274M此时的马赫数：第27页/共28页感谢您的观看！第28页/共28页