A dynamical system approach to the absolute instability of spatially developing localized open flows and media

作者：Brevdo L^*

来源：Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2002, 458(2022): 1375-1397.

DOI：10.1098/rspa.2001.0912

• open spatially developing localized flows and media
• absolute instability
• dynamical system approach
• global dispersion relation
• global discrete and continuous normal modes
• frequency selection mechanism

Absolute instability of spatially developing localized open flows is analysed by applying the Laplace transform in time to the corresponding initial-value linear stability problem, and treating the resulting boundary-value problem on R for the vector equation Z(x) (x, omega) = [A(omega) + R(x)] Z(x, omega) + g(x, omega) as a dynamical system. Here Z(x, omega) is the perturbation, x epsilon R is the spatial coordinate, omega epsilon C is a frequency (and a Laplace transform parameter) I and g (x, w) is the source function. The analysis assumes that the tail matrix R(x) decays faster than any exponential, when x --> +/-infinity. No restriction on the rate of variability of R(x) in the finite domain is imposed. The boundary conditions of decay for Z(x, a;), when x --> +/-infinity, are formulated in terms of two projectors on the subspaces spanned by the eigenvectors and generalized eigenvectors of A(omega) having the eigenvalues with positive and, correspondingly, negative real parts. The boundary-value problem is solved formally, and the dispersion relation function, D-1(omega), for the global modes is expressed in terms of the projectors. It is shown that a spatially developing localized flow or medium is absolutely unstable if and only if either the associated uniform state, i.e. the one with R(x) being zero, is absolutely unstable, or D-1(omega) = 0 has roots in the upper omega-half-plane, or both. When the associated uniform state is absolutely stable then D-1(W) is analytic in (omega epsilon C \ Imomega > 0), the roots of D-1(omega) with Imomega > 0 are the contributors to the absolute instability, and, in the analysis of absolute instability of the spatially localized flow, neither a consideration of the complexified physical coordinate nor the concept of saddle point in the complexified physical space is involved. This is in contrast to the currently widely used WKBJ approach to the absolute instability of spatially non-localized flows. We present a practically implementable procedure for analysing spatially developing localized flows on absolute instability, and suggest a frequency selection criterion for such flows with self-sustained oscillations.

• 出版日期2002-6-8