Absolute and convective instabilities of spatially periodic flows
Absolute and convective instabilities of spatially periodic flows
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空间周期流的绝对和对流不稳定性
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
T. Bridges
中科院分区：
文献类型：
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作者：
L. Brevdo;T. Bridges
单色波的稳定性和波包的演化在无阻流系统向湍流的转变中具有根本的重要性。虽然单色波稳定性的分析和空间均匀或平行流中线性波包的演化一般都很好理解，空间不均匀流的这种分析还没有很好理解。本文研究空间周期介质中线性波包的演化问题。空间均匀流中的绝对和对流不稳定性的数学形式主义推广到空间周期性的情况。的拉普拉斯变换是用来减少初值问题的一个系统的常微分方程的周期系数，然后完全分析使用Floquet理论和傅立叶变换。我们通过Δ（μ，ω）= 0定义广义色散关系，其中μ π mathbb{C} {0}是空间Floquet乘数，ω π mathbb{C}是频率（和拉普拉斯变换参数）。我们发现空间周期流是绝对不稳定的当且仅当Δ（μ，ω）有二重根（或更一般地，是满足碰撞准则的μ at（μ0，ω0）的重根，其中Im ω0 > 0：也就是说，在扰动下二重根分裂，使得在极限情况下，Im（ω - ω0）→ + ∞，Re（ω - ω0）= 0，其中一个根在单位圆的内部，另一个在单位圆的外部。| µ |在复µ平面中，进一步的结果得到的长时间渐近的绝对不稳定和绝对稳定的流动和周期性介质的信号问题。一般来说，不稳定周期波的渐近状态在空间和/或时间上是准周期的。该理论被应用到有限振幅周期行波态的真实的和复杂的Ginzburg-Landau方程。我们发现Eckhaus不稳定总是绝对不稳定的，而在复杂的情况下，有一个有趣的分解区域的不稳定的有限振幅行波到一个绝对不稳定的区域和对流不稳定，但绝对稳定的区域。关于空间周期状态线性化的Navier-Stokes方程的不稳定波包问题，也制定。
Stability of monochromatic waves and wave packet evolution are of fundamental importance in the transition to turbulence in open-flow systems. Although the analysis of monochromatic wave stability and the evolution of linear wave packets in spatially homogeneous or parallel flows is generally well understood, such analysis for spatially inhomogeneous flows is not so well understood. In this paper we consider the problem of linear wave packet evolution in a spatially periodic medium. The mathematical formalism of absolute and convective instabilities in spatially homogeneous flows is generalized to the spatially periodic case. The Laplace transform is used to reduce the initial-value problem to a system of ordinary differential equations with periodic coefficients which is then completely analysed using the Floquet theory and the Fourier transform. We define a generalized dispersion relation by Δ(µ,ω) = 0, where µ Є mathbb{C} {0} is a spatial Floquet multiplier and ω Є mathbb{C} is a frequency (and a Laplace transform parameter). We find that a spatially periodic flow is absolutely unstable if and only if Δ(µ,ω) has a double root (or more generally a multiple root) in µ at (µ0,ω0) with Im ω0 > 0 that satisfies the collision criterion: i.e. the double root splits under perturbation in such a way that in the limit, as Im(ω — ω0) → + ∞, with Re(ω — ω0) = 0, one of the roots goes interior and the other exterior to the unit circle { | µ | = 1} in the complex µ-plane. Further results are obtained on the long-time asymptotics of absolutely unstable and absolutely stable flows and the signalling problem for periodic media is introduced. In general the asymptotic states of unstable periodic waves are quasi-periodic in space and/or time. The theory is applied to the finite-amplitude periodic travelling wave states of the real and complex Ginzburg-Landau equation. We find that the Eckhaus instability is always absolutely unstable whereas in the complex case there is an interesting decomposition of the region of unstable finite-amplitude travelling waves into an absolutely unstable region and a convectively unstable but absolutely stable region. The problem of unstable wave packets for the Navier-Stokes equations, linearized about a spatially periodic state, is also formulated.