Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory
Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory
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普朗特边界层理论中速度剖面的渐近行为
DOI：
10.1098/rspa.1967.0151
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
J. Serrin
中科院分区：
文献类型：
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作者：
J. Serrin
考虑不可压缩粘性流体通过刚性壁的定常二维层流的普朗特边界层方程。基于壁面某一初始位置的任意速度分布，分析表明，对于幂律流动速度U（x）<$C（x+d）m（m <$0），下游发展的速度分布由著名的Falkner-Skan相似解渐近给出。此外，对于满足（6）式的流动速度，在下游形成的速度分布是渐近唯一的，尽管所得到的分布的具体形式当然取决于外部流的精确性质。这种渐近行为的收敛速度估计，以及相应的速度收敛的表面摩擦系数。这一结果证实了许多作者的默认假设，即下游速度剖面基本上与初始剖面无关，也为边界层理论中类似解的作用提供了理论依据。我们还证明了凹速度分布的存在时，压力梯度是有利的。由此可见，对于对应于有利压力梯度的流动速度，凹速度剖面与相似剖面对于幂律流动速度所起的作用大致相同。
Consider the Prandtl boundary layer equation for the steady two-dimensional laminar flow of an incompressible viscous fluid past a rigid wall. On the basis of an arbitrary velocity profile at some initial position on the wall, the analysis presented shows that, for a power law streaming speed U(x) ═ C(x+d)m (m ≽ 0), the velocity profile which develops downstream is asymptotically given by the well known Falkner-Skan similarity solution. Moreover, for a streaming speed satisfying (6), the velocity profile which develops downstream is asymptotically unique, though of course the particular form of the resulting profile depends on the precise nature of the exterior stream. The rate of convergence for this asymptotic behaviour is estimated, as well as corresponding rates for the convergence of the skin friction coefficient. This result verifies the tacit assumption of a number of writers that the downstream velocity profile is essentially independent of the initial profile, and also supplies a theoretical justification for the role of similar solutions in boundary layer theory. We also prove the existence of concave velocity profiles whenever the pressure gradient is favourable. It follows that, for streaming speeds which correspond to a favourable pressure gradient, concave velocity profiles play somewhat the same role as similarity profiles do for a power law streaming speed.