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首页> 外文期刊>Journal of Fluid Mechanics >Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations
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Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations

机译：边界层流中的准模式。第1部分。无粘性二维空间谐波摄动

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The work, being the first in a series concerned with the evolution of small perturbations in shear flows, studies the linear initial-value problem for inviscid spatially harmonic perturbations of two-dimensional shear flows of boundary-layer type without inflection points. Of main interest are the perturbations of wavelengths 2 pi /k long compared to the boundary-layer thickness H, kH = epsilon much less than 1. By means of an asymptotic expansion, based on the smallness of e, we show that for a generic initial perturbation there is a long time interval of duration similar to epsilon (-3) ln(l/epsilon), where the perturbation representing an aggregate of continuous spectrum modes of the Rayleigh equation behaves as if it were a single discrete spectrum mode having no singularity to the leading order. Following Briggs et al. (1970), who introduced the concept of decaying wave-like perturbations due to the presence of the 'Landau pole' into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contribution to the entire perturbation field evolves for different field characteristics. We find that over O(epsilon (-3) ln(l/e)) time interval, the quasi-mode dominates the velocity field. In particular, over this interval the share, of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O(epsilon (4)) (O(epsilon (6)) for the Blasius flow). The mode is weakly decaying, as exp(-epsilon (3)t). At larger times the quasi-mode ceases to dominate in the perturbation field and the perturbation decay law switches to the classical t(-2). By definition, the quasi-modes are singular in a critical layer; however, we show that in our context their singularity does not appear in the leading order. From the physical viewpoint, the presence of a small jump in the higher orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Since we have established that the decay rate of the quasi-modes sharply increases with the increase of the wavenumber, one of the major conjectures of the analysis is that the long-wave components prevail in the large-time asymptotics of a wide class of initial perturbations, not necessarily the predominantly long-wave perturbations. Thus, the explicit expressions derived in the long-wave approximation describe the asymptotics of a much wider class of initial conditions than might have been anticipated. The concept of quasi-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics. [References: 39]
机译：这项工作是有关剪切流中小扰动演化的系列文章中的第一篇，它研究了无拐点的边界层类型二维剪切流的无粘性空间谐波扰动的线性初值问题。主要感兴趣的是与边界层厚度H相比长2 pi / k的波长的扰动，kH =ε远小于1。通过渐近展开，基于e的小，我们证明了对于一般初始扰动的持续时间间隔很长，类似于epsilon（-3）ln（l / epsilon），其中表示Rayleigh方程的连续频谱模式集合的扰动的行为就像是一个离散的频谱模式，没有领导秩序的奇异之处。继布里格斯等。 （1970），他将由于“ Landau极点”的存在而引起的衰减波状摄动的概念引入了流体力学，我们将此对象称为准模式。我们分析地分析了在不同的场特征下，对整个扰动场的准模贡献如何演变。我们发现，在O（ε（-3）ln（l / e））时间间隔内，准模式主导了速度场。特别是，在此间隔内，准模式中包含的微分能量的份额非常接近1，一般情况下的差异为O（ε（4））（O（ε（6）））。 Blasius流）。该模式衰减较弱，为exp（-epsilon（3）t）。在更大的时间，准模在扰动场中不再占主导地位，并且扰动衰减定律转换为经典的t（-2）。根据定义，准模式在关键层中是奇异的。但是，我们表明，在我们的上下文中，它们的奇异性没有出现在前导顺序中。从物理角度看，高阶小跃变的存在对流体扰动可以参与线性和非线性共振相互作用的方式意义不大。由于我们已经确定准模的衰减率随波数的增加而急剧增加，因此分析的主要推测之一是，长波分量在较宽的初始类的大渐近渐近现象中占主导地位。扰动，不一定是主要的长波扰动。因此，在长波逼近中得出的显式表示的渐进性比预期的要宽得多。准模态的概念还使我们能够在流体力学中广泛使用的分段线性逼近方法的基础上崭露头角。 [参考：39]

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