An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

Náraigh, Lennon Ó; Spelt, Peter D. M.

doi:10.1098/rspa.2013.0171

Cited by 8 publications

(6 citation statements)

References 30 publications

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“…For both these tasks, we warmly suggest the reader to study the exhaustive reviews by Huerre & Monkewitz [10], and by Huerre [11], as well as the textbooks by Criminale et al [12] (see Section 4.4), by Drazin & Reid [13] (see Section 24), and by Charru [14] (see Chapter 3). Fresh new insights into this topic have been also proposed in the recent paper by Ó Náraigh & Spelt [15].…”

Section: Introductionmentioning

confidence: 88%

Transition to Absolute Instability for (not so) Dummies

Barletta,

Alves

2014

Preprint

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These notes are intended as an elementary introduction to the concept of absolute instability. The transition from convective instability to absolute instability is an important issue when the stability of stationary flow solutions is investigated. The arguments here described were first developed in the framework of plasma physics and later applied to the hydrodynamics of mixing layers and shear flows. Far from being a comprehensive analysis of this complicated subject, the aim of these notes is just to sketch a rudimentary and quite elementary ground for physicists or engineers which have a familiarity with the basic features of linear stability in fluid dynamics, but are new to the concept of absolute instability.

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Section: Introductionmentioning

confidence: 88%

Transition to Absolute Instability for (not so) Dummies

Barletta,

Alves

2014

Preprint

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“…Besides standard Orr-Sommerfeld analysis, we also adopted the analytic connection between temporal and spatio-temporal growth rates in the linear regime as presented by Ó Náraigh and Spelt. 38 This approach, which is based on analytical continuation, circumvents possible difficulties in identifying the absolute growth rate that are associated with the multivalued nature of the eigenvalue problem and specifics of the problem at hand. Compared with OS analysis and DNS, this method shows good agreement and is therefore appropriate to accurately estimate the absolute growth rate of the instability.…”

Section: Discussionmentioning

confidence: 99%

“…First, the dispersion relation contains two saddle points, of which both may be dynamically relevant. The confinement of the flow by the channel walls has, furthermore, created a discrete pole (not shown) on the imaginary axis, 38 (α r , α i ) = (0, 3.34), which has implications on the character of the saddle point closer to that particular singularity. Lastly, the multivalued nature of the dispersion relation becomes apparent by the branch cut just below the real axis.…”

Section: Phys Fluids 28 042102 (2016)mentioning

confidence: 99%

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Linear and nonlinear instability in vertical counter-current laminar gas-liquid flows

et al. 2016

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We consider the genesis and dynamics of interfacial instability in gas-liquid flows, using as a model the two-dimensional channel flow of a thin falling film sheared by counter-current gas. The methodology is linear stability theory (Orr-Sommerfeld analysis) together with direct numerical simulation of the two-phase flow in the case of nonlinear disturbances. We investigate the influence of three main flow parameters (density contrast between liquid and gas, film thickness, pressure drop applied to drive the gas stream) on the interfacial dynamics. Energy budget analyses based on the Orr-Sommerfeld theory reveal various coexisting unstable modes (interfacial, shear, internal) in the case of high density contrasts, which results in mode coalescence and mode competition, but only one dynamically relevant unstable internal mode for low density contrast. The same linear stability approach provides a quantitative prediction for the onset of (partial) liquid flow reversal in terms of the gas and liquid flow rates. A study of absolute and convective instability for low density contrast shows that the system is absolutely unstable for all but two narrow regions of the investigated parameter space. Direct numerical simulations of the same system (low density contrast) show that linear theory holds up remarkably well upon the onset of large-amplitude waves as well as the existence of weakly nonlinear waves. In comparison, for high density contrasts corresponding more closely to an air-watertype system, although the linear stability theory is successful at determining the most-dominant features in the interfacial wave dynamics at early-to-intermediate times, the short waves selected by the linear theory undergo secondary instability and the wave train is no longer regular but rather exhibits chaotic dynamics and eventually, wave overturning.

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“…Thus, for this wide class of fluid-mechanical dispersion relations, the condition for absolute instability is simply that the imaginary part of the frequency at the saddle point be positive [12]. Of course, there is a variety of fluid-mechanical problems where naive application of the saddle-point criterion (3b) fails [13]. However, the saddle-point criterion (3b) supplemented with the requirement that the spatial branches emanating from the saddle point should extend into opposite half-planes in the complex α-plane, is sufficient to guarantee absolute instability [3,14].…”

Section: Introductionmentioning

confidence: 99%

Global modes in nonlinear non-normal evolutionary models: Exact solutions, perturbation theory, direct numerical simulation, and chaos

Náraigh

2015

Physica D: Nonlinear Phenomena

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This paper is concerned with the theory of generic non-normal nonlinear evolutionary equations, with potential applications in Fluid Dynamics and Optics. Two theoretical models are presented. The first is a model two-level non-normal nonlinear system that not only highlights the phenomena of linear transient growth, subcritical transition and global modes, but is also of potential interest in its own right in the field of nonlinear optics. The second is the fairly familiar inhomogeneous nonlinear complex Ginzburg-Landau (CGL) equation. The two-level model is exactly solvable for the nonlinear global mode and its stability, while for the spatially-extended CGL equation, perturbative solutions for the global mode and its stability are presented, valid for inhomogeneities with arbitrary scales of spatial variation and global modes of small amplitude, corresponding to a scenario near criticality. For other scenarios, a numerical iterative nonlinear eigenvalue technique is preferred. Two global modes of different amplitudes are revealed in the numerical approach. For both the two-level system and the nonlinear CGL equation, the analytical calculations are supplemented with direct numerical simulation, thus showing the fate of unstable global modes. For the two-level model this results in unbounded growth of the full nonlinear equations. For the spatially-extended CGL model in the subcritical regime, the global mode of larger amplitude exhibits a 'one-sided' instability leading to a chaotic dynamics, while the global mode of smaller amplitude is always unstable (theory confirms this). However, advection can stabilize the mode of larger amplitude.

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An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

Cited by 8 publications

References 30 publications

Transition to Absolute Instability for (not so) Dummies

Transition to Absolute Instability for (not so) Dummies

Linear and nonlinear instability in vertical counter-current laminar gas-liquid flows

Global modes in nonlinear non-normal evolutionary models: Exact solutions, perturbation theory, direct numerical simulation, and chaos

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