On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow
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DOI：
10.1017/s002211206000116x
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发表时间：
1960-11
影响因子：
3.7
通讯作者：
J. T. Stuart
中科院分区：
文献类型：
--
作者：
J. T. Stuart
本文考虑了在给定的波数和雷诺数下，当扰动的放大率足够小时，Navier-Stokes方程的非线性二维解的性质。出现两类问题：（i）跟踪不稳定的无穷小扰动的增长（超临界问题），可能达到稳定平衡状态;（ii）对于不存在不稳定的无穷小扰动的波数和雷诺数的值，跟踪有限扰动从可能的不稳定平衡状态下降到零振幅的衰减（亚临界问题）。在情形（ii）中，不稳定平衡态的存在意味着不稳定扰动的存在。数值计算，这是尚未完成的，需要确定哪两种可能的行为出现在平面Poiffille流，在给定范围内的波数和雷诺数。本文的方法（以及J.沃森在第二部分中所述的推广方法）对中性稳定性曲线内外的雷诺数和波数的范围都是有效的。
This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number. It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.