L. J. Pan, Y. H. Kuo

When a thin sharp-nosed body flies at hypersonic speeds, the flow around the body is characterized by its high temperature and low density near the surface, due largely to the loss of kinetic energy of the fluid by the action of viscosity. The consequence of expansion of the gas in the boundary layer is an enormous increase of the viscous deflection of the outer flow, which in turn causes a strong shock at the leading edge. Previous investigations (Refs. 1, 2, 3) have laid special emphasis on the increase of the effective thickness of the body due to viscous deflection of the stream-lines but have neglected the entropy or vorticity generated by the shock.

In view of the fact that at hypersonic Mach numbers the change across the shock, even in the case of thin bodies, can be large, an increase of entropy could be accompanied by an appreciable increase of temperature of the flow. This heating may lead to further expansion of the gas, resulting in additional changes of pressure and skin friction. If, however, these effects were taken into account, the flow behind the shock would involve the interaction of both families of Mach waves and, therefore, would present formidable difficulties for theoretical solution. On the other hand, if the basis for the simple-wave theory can be established, the solution of the problem can be greatly simplified as has been demonstrated in the case of the flat plate. It is, for this reason, important to clarify to what extent entropy will be of significance.

With a motivation to answer this question, it is proposed to consider a special case of an infinite wedge with large wedge angle, for which a strong shock exists even at infinity. Inasmuch as, at high Mach numbers the characteristics of the flow over thin sharp-nosed bodies depend very little on the shape of the body, the results obtained for one case may well cast light on the other. Merely for the purpose of estimating these effects, the investigations can be restricted to the weak interaction zone for which an asymptotic solution can be easily found. It is shown that at hypersonic Mach numbers the contributions due to entropy rise are small even for large wedge angles. It can therefore be concluded that, as in the case of inviscid rotational flows, the waves reflected by the shock play only a minor role in influencing the flow and can be neglected at hypersonic Mach numbers.

楔的高超声速可压缩粘性绕流

潘良儒, 郭永怀

当一个尖头薄体以高超声速飞行时，其绕流的特点是：物面附近温度提高而密度降低，这主要由流体在粘性作用下的动能损失引起的。气体在边界层中发生膨胀，结果使外流因粘性而引起的偏转增大许多，这又使得前缘处产生强激波。以前的工作都侧重研究因流线粘性偏转所引起的物体有效厚度的增加，却都没考虑激波所产生的熵增或涡量。

因为在高超声速的马赫数下，即使对于薄物体而言，激波前后的变化也会很大，所以在发生熵增时气流温度也会显著升高。这种加热导致气体进一步膨胀。它又使压力和表面摩擦发生改变。可是，假若要把这些效应都加以考虑，那么激波后的流动就会涉及两族马赫波的相互作用，要想得到理论解就十分困难了。另一方面，如果能够说明简单波理论是有根据的，那么正如平板情况一样，求解这一问题就可大大简化。因此，重要的是阐明熵增的影响有多大。

为了解答这一问题，本文拟考虑具有大楔角的无穷长楔这样一个特例，对于这样的楔甚至在无穷远处仍有强激波存在。既然在高马赫数下尖头薄体绕流的特性几乎同物形无关，所以对于一种情况所得的结果也完全可以用来说明另一种情况。我们的目的只是估计这些效应，因而可以只研究容易找到渐近解的弱干扰区。本文证明了：在高超声速马赫数下，甚至对于大楔角情形熵增的作用也很小。所以可以得出结论：如同无粘有旋流情形一样，在高超声速马赫数下，从激波反射回来的波对于流动的影响只起次要的作用，可以忽略不计。

原文发表于：Journal of Mathematics and Physics,1956,35（2）：179-193. 见《郭永怀文集》北京：科学出版社出版，pp. 577-591, 2009.