Hsue-Shen Tsien, Yung-Huai Kuo
The problem of flow of a compressible fluid past a body with subsonic flow at infinity is formulated by the hodograph method. The solution in the hodograph plane is first constructed about the origin by superposition of the particular integrals of the transformed equations of motion with a set of constants which would determine, in the limiting case, a known incompressible flow. This solution is then extended outside the circle of convergence by analytic continuation.
The previous difficulty of the Chaplygin method of slow convergence of the series has been overcome by using the asymptotic properties of the hypergeometric functions so that numerical solutions can be obtained without difficulty. It is emphasized that, for a solution covering the whole domain of the field of flow, both fundamental solutions of the hypergeometrical differential equation are required.
Explicit formulas for numerical calculations are given for the flow about a body, such as an elliptic cylinder, and for the periodic flow such as would exist over a wavy surface.
Numerical examples based on the incompressible flow solution of an elliptic cylinder of thickness ratio of 0.6 are computed for free-stream Mach numbers of 0.6 and 0.7.
The results of this investigation indicate an appreciable distortion in the shape of the bodies in compressible flow from that of incompressible flow, which necessitates a series of computations with various values of the geometric parameter in order that the desired body shapes can be selected for a given Mach number. It also is shown that the breakdown of irrotational flow depends solely upon the occurrence of limiting lines, which, in turn, are dependent on the boundary conditions.
The numerical calculations show that at a free-stream Mach number of 0.6, irrotational supersonic flow exists up to a local Mach number of 1.25; whereas breakdown occurs at 1.22 for a Mach number of 0.7.
可压缩流体二维无旋亚声速和超声速混合型流动及上临界马赫数
钱学森 郭永怀
本文采用速度图方法表述了无限远处来流为亚音速的可压缩流体绕物体的流动问题。首先，在原点附近把变换的运动方程的特解线性叠加起来，构成速度图平面上的解。这些常数可以用来确定在极限情况下不可压缩流动的已知解，然后，利用解析延拓把这个解延拓到收敛圆的外部。
由于利用超几何函数的渐近性质克服了Chaplygin方法所得到的级数收敛缓慢的困难，因此比较容易得到数值结果。应着重指出，为了给出全流场的解，需要用超几何微分方程的两个基本解。
对于绕椭圆柱一类物体的流动以及像绕波形壁面可能出现的周期流动，本文给出了它们显式的数值计算公式。
以绕厚度比为0.6椭圆柱不可压缩流动的解为基础，本文给出了来流马赫数分别为0.6和 0.7的两个算例。
本文结果表明，在可压缩的流动中，物体的形状与相应于不可压缩流动物体的形状有明显的畸变，这使得在给定来流马赫数下， 要使物体的形状可以选择，必须对各种不同的几何参数进行一系列的计算。这也表明，无旋流动的破坏只依赖于极限线的出现，于是就依赖于边界条件了。
数值结果计算表明，在来流马赫数为0.6时，直到当地马赫数为1.25时，还存在有无旋的超声速流动，而当来流马赫数为0.7时，当地马赫数仅为1.22时，无旋流动就破坏了。
原文发表于NACA TN No.995(1946)，见《郭永怀文集》北京：科学出版社出版，pp. 22-158, 2009.
