Instability of stationary unbounded stratified fluid
Instability of stationary unbounded stratified fluid
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静止无界分层流体的不稳定性
DOI：
10.1017/s0022112091000150
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发表时间：
1991
影响因子：
3.7
通讯作者：
J. Nitsche
中科院分区：
文献类型：
--
作者：
G. Batchelor;J. Nitsche
假设静止无界粘性流体的密度是垂直位置坐标 z 的正弦函数。这个流体体的重力是否对小扰动不稳定？如果是，在什么条件下以及对什么类型的扰动？本文考虑了这些问题，答案是，对于任何非零的正弦波振幅值，流体对于大水平波长的扰动确实是不稳定的。这些扰动到处都有近似垂直的速度，并使较重和较轻流体的交替层倾斜，导致前者的流体向下滑动，后者的流体向上滑动，导致垂直平均密度的正弦变化，从而加强垂直运动。这种新颖且有效的全局不稳定机制的识别促使人们考虑其他分层无界流体情况的稳定性。其他两种类型的未受扰动密度分布，第一种是较重或较轻流体的孤立中心层，其密度以高斯函数的形式变化，第二种是密度以高斯函数的导数变化的孤立流体层，发现在密度变化大小的所有值下，对于具有相同全局特征的扰动来说是不稳定的。对于这两种密度分布中的第一种，长水平波长扰动的行为仅取决于中心层单位面积的净过剩质量，而对于第二种，它仅取决于中心层中密度的一阶矩。对于第二种类型，出现了另一种全局不稳定机制，其中轻流体从层的一侧剥离，重流体从另一侧剥离，没有任何倾斜。在所有情况下，中性扰动的特性都是通过数值确定的，并且增长率是作为扰动的瑞利数、普朗特数和水平波数的函数而找到的。当所有扰动增长时，能量论证很容易给出无粘性非扩散极限的结果，并以其最简单的形式揭示了大水平波长扰动的不稳定性的倾斜-滑动机制。
Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.