查看“︁普朗特-迈耶膨胀扇”︁的源代码

{{noteTA
|G1=Aero
|2=zh-cn:膨胀;zh-tw:擴張
}}

[[Image:Expansion fan.png|thumb|300px|普朗特－迈耶膨胀扇]]
'''普朗特－迈耶膨胀扇'''（{{lang-en|Prandtl–Meyer expansion fan}}）是指[[超音速]]流绕外凸转角时所形成的膨胀扇。膨胀扇由无穷多条从尖锐转角发散、角度逐渐偏转的[[马赫波]]组成。而当转角为光滑圆角时，这一系列马赫波则反向交汇于一点。物理上，绕凸角的超音速流不可能仅穿过一条“[[激波]]”，因为那样会违背[[热力学第二定律]]。当流动穿过膨胀扇时，流速与[[马赫数]]增加，[[静压]]、[[温度]]与[[密度]]则减小。由于该过程为[[等熵过程]]，[[滞点|滞止]]性质则保持不变。

== 流动性质 ==
膨胀扇由无穷多条马赫线组成，其中第一条马赫线与入流方向的夹角为<math>\mu_1 = \arcsin \left( \frac{1}{M_1} \right)</math>，最后一条马赫线与出流方向的夹角则为<math>\mu_2 = \arcsin \left( \frac{1}{M_2} \right)</math>。由于马赫线之间的夹角很小、流动转过的角度也很小，故整个过程为等熵过程。由此可以简化流动性质的计算。对等熵过程而言，[[滞止压强]]（<math>p_0</math>）、滞止温度（<math>T_0</math>）、滞止密度（<math>\rho_0</math>）等[[滞点|滞止]]性质为常数，因而出流性质为出流马赫数（<math>M_2</math>）与入流性质的函数：

:<math>\begin{array}{lcl}
\cfrac{T_2}{T_1} &= & \bigg( \cfrac{1+\frac{\gamma -1}{2}M_1^2}{1+\frac{\gamma -1}{2}M_2^2} \bigg) \\
\\
\cfrac{p_2}{p_1} &= & \bigg( \cfrac{1+\frac{\gamma -1}{2}M_1^2}{1+\frac{\gamma -1}{2}M_2^2} \bigg)^{\gamma/(\gamma-1)} \\
\\
\cfrac{\rho_2}{\rho_1} &= &\bigg( \cfrac{1+\frac{\gamma -1}{2}M_1^2}{1+\frac{\gamma -1}{2}M_2^2} \bigg)^{1/(\gamma-1)}.
\end{array}</math>
出流马赫数（<math>M_2</math>）与入流马赫数（<math>M_1</math>）及转角（<math>\theta</math>）之间的关系则为
:<math> \theta = \nu(M_2) - \nu(M_1) \,</math>
其中<math>\nu(M) \,</math>为[[普朗特－迈耶函数]]。

== 参考文献 ==
* {{cite book
  | last = Liepmann | first = Hans W.  |author2=Roshko, A.
  | title = Elements of Gasdynamics    | origyear = 1957
  | publisher = [[Dover Publications]] | year = 2001
  | isbn = 0-486-41963-0 }}
* {{cite book
  | last = Anderson | first = John D. Jr.
  | title = Fundamentals of Aerodynamics | origyear = 1984 | edition = 3rd
  | publisher = [[McGraw-Hill|McGraw-Hill Science/Engineering/Math]] |date=January 2001
  | isbn = 0-07-237335-0 }}
* {{cite book
  | last = Shapiro | first = Ascher H.
  | title = The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1 | url = https://archive.org/details/dynamicsthermody0000shap | year = 1953
  | publisher = [[Ronald Press]]
  | isbn = 978-0-471-06691-0 }}

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[[Category:流体动力学]]