Yung-Huai Kuo
The first part of this paper (Part I) is devoted to the problem of flow past a finite flat plate. From the character of the external potential flow an expansion parameter is deduced, which turns out to be proportional to the maximum thickness ratio of the boundary layer. If this expansion parameter is adopted, then it can be shown that to the second approximation, the problem of solving a viscous flow remains of the boundary-layer type. Namely, from the first-order pressure calculated from the maximum displacement of the stream-lines by viscosity, along the plate and in the wake, the second-order viscous flow in the boundary layer can be determined.
The first-order potential flow outside the boundary layer is shown to be very similar, especially near the leading edge, to that given by the linearized theory of flow over a thin airfoil with round leading edge in an inviscid fluid. At the trailing edge where the flow suffers a sudden change of direction, it produces a strong expansion by the appearance of a logarithmic singularity.
The complete solution of the second-order viscous flow leads to a law of resistance which coincides with the Blasius law for Reynolds number over 104. In the range of Reynolds numbers from 15 to 104, it agrees perfectly with the measurements of Z. Janour.
In PartⅡof the present work, the main problem is the uniformization of the analytic behavior of the successive approximations, taking the boundary layer solution as the basic approximation. It is found that in the case of a flat plate, if Lighthill’s conditions are imposed to the third approximation (as the second approximation needs no modification), there results a second-order shift of one coordinate x, say, measured along the plate. According to this transformation, the original constant-velocity lines, being a family of parabolas with vertices at the leading edge, become a different family of parabolas whose vertices are now all separated and located ahead of the leading edge. Furthermore, the singular line which was formerly normal to the plate, passing through the leading edge, has now been completely eliminated. The interesting fact that, except at the leading edge, the improved Blasius solution will be nowhere singular, was expected to bring out the effect of the leading edge. This was investigated and, indeed, was the case. The flow in the immediate neighborhood of the leading edge is found to satisfy Strokes equation of slow viscous motion, as it should.
In the light of this discussion, two features of the boundary layer approximations deserve to be mentioned. First the dimension of the viscous region around the leading edge in which the boundary layer hypotheses are violated is actually of the order ofv/U∞wherevis the kinematic viscosity andU∞the forward speed and hence is extremely small, compared with the length of the plate when the Reynolds number is large. If the detailed information, such as velocity distribution, is not required, the significance of that region, for all practical purposes, could be ignored. Secondly, as the effect straining the coordinates vanishes on the plate, the very existence of a finite viscous region about the plate does not affect the law of resistance deduced by boundary-layer theory, even to the second order approximation as calculated in Part I. This explains why, with the effect of leading edge poorly represented, the said improved boundary layer solution still yields a correct law of resistance for Reynolds number greater than 15.
关于中等雷诺数下不可压缩粘性流体绕平板的流动
郭永怀
本文第一部分研究有限平板的扰流问题，由外部位势流的特征得到了一个展开参数，发现它正比于边界层的最大厚度比。若取这个展开参数，可以证明，直到二阶近似求解粘性流动仍是边界层类型的问题，即由粘性引起的流线最大位移可计算出沿平板及尾流的一阶近似压力分布，根据这压力分布可确定边界层内的二阶近似粘性流动。证明了边界层外的一阶位势流动与线性化理论描述的绕钝头薄翼的理想流体流动非常类似，在前缘附近更是如此。在流动方向有突变的后缘处，有很强的膨胀，出现了对数奇点。
二阶近似年性流动的完整解答给出的阻力规律在雷诺数大于104的范围内与Blasius规律相一致。在雷诺数有15到104的范围内，它与Z.Janour的测量值非常符合。
在本文第二部分，主要问题是以边界层解作为基本近似，研究逐阶近似解解析性质的一致有效性。在平板情形，若将Lighthill条件应用于三阶近似（因为二阶近似不需要任何改变），可得到沿平板坐标x的二阶平移。按此变换，原来的等速线（它们是顶点在平板前缘的一族抛物线）就变为另一族抛物线，其顶点都在前缘的前方互不重合。而且，原来通过前缘点垂直于平板的奇线现在完全消失了，改进了的Blasius解除了前缘点之外，处处无奇性。期望这个有趣的事实能清楚地显示前缘的影响。对此进行了研究后发现情况果然如此。在前缘附近，正如应该的那样，流动满足缓慢粘性流动的Stokes方程。
根据这个讨论，应该提及边界层近似的两个性质：首先，在前缘附近边界层假设失效的黏性区域的尺度为v/U∞数量级，其中v是流体运动黏性系数，U∞是平板向前运动的速度；因此当雷诺数很大时，这个尺度与平板长度相比是非常小的。若不需要求速度分布等的详细信息，从实用观点来看，可以忽略这个区域。其次，由于在平板表面坐标的变形为零，在平板附近存在有限的黏性区域的事实并不影响由边界层理论导出的阻力规律，即使对于第一部分中算出的二阶近似阻力规律也没有影响。这就解释了为什么即使没有认真考虑前缘影响，所述的改进边界层解仍能给出在雷诺数大于15范围内正确的阻力规律。
原文发表于J. Math. Phys.1953,32:83-101.见《郭永怀文集》北京：科学出版社出版，pp. 265-312, 2009.
