Discontinuous solutions of the boundary-layer equations
Discontinuous solutions of the boundary-layer equations
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DOI：
10.1017/s0022112008003303
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发表时间：
2008-10
影响因子：
3.7
通讯作者：
A. Ruban;K. Vonatsos
中科院分区：
文献类型：
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作者：
A. Ruban;K. Vonatsos
自1904年普朗特尔提出边界层方程以来，由于边界层的粘性性质，人们一直认为普朗特尔方程的解应该在连续函数类中寻求。然而，不连续解存在的数学原因是明确的。而且，在一定条件下，它们是边界层方程的唯一可能解。本文以1933年Schlichting首次研究的层流射流问题的非定常模拟为例。在施里希廷的公式中，射流从一个平坦屏障的窄缝中出现，并穿透到一个充满流体的半无限区域，如果狭缝关闭，流体将保持静止。假设流动稳定，Schlichting能够证明普朗特方程的相应解可以写成显式解析形式。这里我们关注的是非定常流，当狭缝打开时，射流开始渗透到停滞的流体中。为了研究这一过程，我们从非定常边界层方程的数值解开始。由于不连续是预料之中的，所以在有限差分之前，方程被写成保守形式。解表明，射流有一个完善的锋面，表示速度场的不连续，类似于超音速气流中形成的激波。然后，为了揭示激波的“内部结构”，我们转向对不连续面周围小区域的流动进行分析。用Re表示雷诺数，在纵向和横向上估计内部区域的大小为Re−1/2阶。我们发现该区域的流体运动主要是非粘性的，如果在与射流锋面一起运动的坐标系中考虑，可以将其视为准稳定的。通过这些简化，可以推导出一个简单的前速度公式，该公式与Turner （J. Fluid Mech）的实验观察结果非常吻合。第13卷（1962年），第356页。
Since 1904, when Prandtl formulated the boundary-layer equations, it has been presumed that due to the viscous nature of the boundary layers the solution of the Prandtl equations should be sought in the class of continuous functions. However, there are clear mathematical reasons for discontinuous solutions to exist. Moreover, under certain conditions they represent the only possible solutions of the boundary-layer equations. In this paper we consider, as an example, an unsteady analogue of the laminar jet problem first studied by Schlichting in 1933. In Schlichting's formulation the jet emerges from a narrow slit in a flat barrier and penetrates into a semi-infinite region filled with fluid which would remain at rest if the slit were closed. Assuming the flow steady, Schlichting was able to demonstrate that the corresponding solution to the Prandtl equations may be written in an explicit analytic form. Here our concern will be with unsteady flow that is initiated when the slit is opened and the jet starts penetrating into the stagnant fluid. To study this process we begin with the numerical solution of the unsteady boundary-layer equations. Since discontinuities were expected, the equations were written in conservative form before finite differencing. The solution shows that the jet has a well-established front representing a discontinuity in the velocity field, similar to the shock waves that form in supersonic gas flows. Then, in order to reveal the ‘internal structure’ of the shock we turn to the analysis of the flow in a small region surrounding the discontinuity. With Re denoting the Reynolds number, the size of the inner region is estimated as an order Re−1/2 quantity in both longitudinal and lateral directions. We found that the fluid motion in this region is predominantly inviscid and may be treated as quasi-steady if considered in the coordinate frame moving with the jet front. These simplifications allow a simple formula for the front speed to be deduced, which proved to be in close agreement with experimental observation of Turner (J. Fluid Mech. vol. 13 (1962), p. 356).