Unstable Normal Modes Research Articles

Numerical simulation of laminar-turbulent transition in a supersonic boundary layer under the action of acoustic disturbances

In low disturbance environment, laminar-turbulent transition (LTT) is associated with excitation of unstable normal modes of small initial amplitudes (receptivity problem). These modes grow exponentially to a critical amplitude in accord with the linear stability theory (LST) and trigger the nonlinear breakdown. In the current study numerical simulation of laminar-turbulent transition (LTT) in the boundary layer on the upper surface of a flat plate in a supersonic free stream of Mach number M∞=3 and Reynolds number Re∞=2 × 107 is carried out. The flow is perturbed by fast or slow acoustic waves of low intensity, which excite unstable waves of the first mode. The latter have frequencies corresponding to the integral amplification N ≈ 9.16 typical for flight conditions. Two cases are considered: the angle of attack AoA=0° at which acoustic waves pass through a weak shock induced by viscous-inviscid interaction; AoA=5° at which acoustic waves pass through the expansion fan emanating from the plate leading edge. A holistic modeling of all stages of the transition from receptivity to the birth of turbulent spots is performed using direct numerical simulations. Linear stability theory is used to interpret the results. Feasibility of practical implementation of the amplitude method for predictions of the LTT onset in the considered and similar cases is discussed. It has been shown that to realize the amplitude method in the considered and similar cases, it is sufficient: to calculate the initial amplitudes of instability in a small vicinity of the leading edge and the propagation of instability from the leading edge to the station where the instability reaches the critical amplitude u′max≈4%. Analysis of known numerical and experimental data showed that this criterion weakly depends on the local Mach number in the range 0<Me<7, and it is acceptable for practical predictions of the transition onset points.

Charge instability of JMaRT geometries

We perform a detailed study of linear perturbations of the JMaRT family of non-BPS smooth horizonless solutions of type IIB supergravity beyond the near-decoupling limit. In addition to the unstable quasi normal modes (QNMs) responsible for the ergo-region instability, already studied in the literature, we find a new class of ‘charged’ unstable modes with positive imaginary part, that can be interpreted in terms of the emission of charged (scalar) quanta with non zero KK momentum. We use both matched asymptotic expansions and numerical integration methods. Moreover, we exploit the recently discovered correspondence between JMaRT perturbation theory, governed by a Reduced Confluent Heun Equation, and the quantum Seiberg-Witten (SW) curve of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory with gauge group SU(2) and Nf = (0, 2) flavours.

Flow-induced surface instabilities in a flexible-walled channel with a heavy wall

We consider the stability of laminar high-Reynolds-number flow through a planar channel formed by a rigid wall and a heavy compliant wall under longitudinal tension with motion resisted by structural damping. Numerical simulations indicate that the baseline state (with Poiseuille flow and a flat wall) exhibits two unstable normal modes: the Tollmien–Schlichting (TS) mode and a surface-based mode which manifests as one of two flow-induced surface instabilities (FISI), known as travelling wave flutter (TWF) and static divergence (SD), respectively. In the absence of wall damping the system is unstable to TWF, where the neutrally stable wavelength becomes shorter as the wall mass increases. With wall damping, TWF is restricted to long wavelengths through interaction with the most unstable centre mode, while for wall damping greater than a critical value the system exhibits an SD mode with a two branch neutral stability curve; the critical conditions along the upper and lower branches are constructed in the limit of large wall damping. We compute the Reynolds–Orr and activation energy descriptions of these neutrally stable FISI by continuing the linear stability analysis to the following order in perturbation amplitude. We find that both FISI are primarily driven by the working of normal stress on the flexible wall, lower-branch SD has negative activation energy, while upper-branch SD approaches zero activation energy in the limit of large wall damping. Finally, we elucidate interaction between TS and TWF modes for large wall mass, resulting in stable islands within unstable regions of parameter space.

Eigenvalue Bounds in an Azimuthal Instability Problem of Inviscid Swirling Flows

We consider the eigenvalue problem of azimuthal instability of inviscid swirling ows between coaxial cylinders. It is shown that the complex eigenvalues corresponding to unstable azimuthal normal modes lie inside a semi-ellipse type region whose major axis coincides with the range of the angular velocity of the basic ow while its minor axis depends on the minimum Richardson number, the azimuthal wave number, and the width of the annular region between the coaxial cylinders.

Minimal nonlinear dynamical system for the interaction between vorticity waves and shear flows.

This study is a direct follow-up of the paper by Heifetz and Guha [Phys. Rev. E 100, 043105 (2019)2470-004510.1103/PhysRevE.100.043105] on a minimal nonlinear dynamical system, describing a prototype of linearized two-dimensional shear instability. In that paper, the authors describe the instability in terms of an action at a distance between two vorticity waves, each of which propagates counter to its local mean flow as well as counter to the other. Here we add to the model the effect of mutual interaction between the waves and the mean flow, where growth of the waves reduces the mean shear and vice versa. This addition yields oscillatory Hamiltonian dynamics, including states of phase slipping and libration with finite-size wave amplitude oscillations. We find that wave-mean-flow dynamics emerging from unstable normal modes in the linearized stage are doomed to librate around the antiphased neutral configuration in which the waves hinder each other's counterpropagation rate. We discuss as well how the given dynamics relates to familiar models of phase oscillators.

Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow

Layered flows that are commonly observed in stratified turbulence are susceptible to the Taylor–Caulfield Instability. While the modal stability properties of layered shear flows have been examined, the non-modal growth of perturbations has not been investigated. In this work, the tools of Generalized Stability Theory are utilized to study linear transient growth within a finite time interval of two-dimensional perturbations in an inviscid, three-layer constant shear flow under the Boussinesq approximation. It is found that, for low optimization times, small-scale perturbations utilize the Orr mechanism and achieve growth equal to that in the case of an unstratified flow. For larger optimization times, transient growth is much larger compared to growth for an unstratified flow as the Kelvin–Orr waves comprising the continuous spectrum of the dynamical operator and the gravity edge-waves comprising the discrete spectrum interact synergistically. Maximum growth is obtained for perturbations with scales within the region of instability, but significant growth is maintained for modally stable perturbations as well. For perturbations with scales within the unstable region, the unstable normal modes are excited at high amplitude by their bi-orthogonals. For perturbations with modally stable scales, the Orr mechanism is utilized to excite at high amplitude neutral propagating waves resembling the neutral Taylor–Caulfield modes.

High–low frequency interaction in alternating FPU [formula omitted]-chains

One of the problems of periodic FPU-chains with alternating large masses is whether significant interactions exist between the so-called (high frequency) optical and (low frequency) acoustic groups. We show that for α-chains with 2n particles we have significant interactions caused by external forcing of the acoustic modes by a stable or unstable optical normal mode. In the proofs an embedding theorem plays a part; the analysis is straightforward in the case that n is even, a different approach using invariant manifolds with symmetry is needed if n is odd. For β-chains the interactions are characterised by parametric excitation, there are indications that interaction is negligible.

Spatial structure of unstable normal modes in a glass-forming liquid

The phenomenology of glass-forming liquids is often described in terms of their underlying, high-dimensional potential energy surface. In particular, the statistics of stationary points sampled as a function of temperature provides useful insight into the thermodynamics and dynamics of the system. To make contact with the real space physics, however, analysis of the spatial structure of the normal modes is required. In this work, we numerically study the potential energy surface of a glass-forming ternary mixture. Starting from liquid configurations equilibrated over a broad range of temperatures using a swap Monte Carlo method, we locate the nearby stationary points and investigate the spatial architecture and the energetics of the associated unstable modes. Through this spatially-resolved analysis, originally developed to study local minima, we corroborate recent evidence that the nature of the unstable modes changes from delocalized to localized around the mode-coupling temperature. We find that the displacement amplitudes of the delocalized modes have a slowly decaying far field, whereas the localized modes consist of a core with large displacements and a rapidly decaying far field. The fractal dimension of unstable modes around the mobility edge is equal to 1, consistent with the scaling of the participation ratio. Finally, we find that around and below the mode-coupling temperature the unstable modes are localized around structural defects, characterized by a disordered local structure markedly different from the liquid's locally favored structure. These defects are similar to those associated to quasi-localized vibrations in local minima and are good candidates to predict the emergence of localized excitations at low temperature.

Multiple eigenmodes of the Rayleigh-Taylor instability observed for a fluid interface with smoothly varying density. III. Excitation and nonlinear evolution

A compressible Rayleigh-Taylor (RT) unstable flow with a diffuse interface where the density varies smoothly supports a multiplicity of unstable eigenmodes that grow exponentially in time. This paper studies two relevant problems in a two-dimensional (2-D) RT flow by use of multimode decomposition. The first one is the excitation of these modes by arbitrarily introduced infinitesimal perturbations. The initial amplitude of each mode is calculated by projecting the introduced perturbation into the eigenvector space, and the obtained initial amplitude can be used to predict the linear evolution of the perturbation accurately, as is confirmed by direct numerical simulations. If the initial perturbation includes pressure or velocity components, then in addition to the growth of the RT mode, a strong acoustic wave package would be radiated to the heavy fluid, whereas a mild acoustic wave package would be radiated to the light fluid. As the second problem, in order to reveal the interaction feature of the unstable normal modes in the nonlinear phase, we perform the multimode decomposition based on the instantaneous perturbation field under the Fourier transformation in the direction tangential to the interface. For the current case studies, the emergence of the bubble-spike structure is mainly attributed to the nonlinear amplification of the odd modes decomposed from the fundamental disturbances. As time advances to the nonlinear phase, the projection to the low-order normal modes possesses less portion of the total energy due to the complexity of the multipeak perturbation profiles.

Normal form of synchronization and resonance between vorticity waves in shear flow instability.

A minimal model of linearized two-dimensional shear instabilities can be formulated in terms of an action-at-a-distance, phase-locking resonance between two vorticity waves, which propagate counter to their local mean flow as well as counter to each other. Here we analyze the prototype of this interaction as an autonomous, nonlinear dynamical system. The wave interaction equations can be written in a generalized Hamiltonian action-angle form. The pseudoenergy serves as the Hamiltonian of the system, the action coordinates are the contribution of the vorticity waves to the wave action, and the angles are the phases of the vorticity waves. The term "generalized action angle" emphasizes that the action of each wave is generally time dependent, which allows instability. The synchronization mechanism between the wave phases depends on the cosine of their relative phase, rather than the sine as in the Kuramoto model. The unstable normal modes of the linearized dynamics correspond to the stable fixed points of the dynamical system and vice versa. Furthermore, the normal form of the wave interaction dynamics reveals a new type of inhomogeneous bifurcation: annihilation of a stable and an unstable star node yields the emergence of two neutral center fixed points of opposite circulation.

Non-normality increases variance of gravity waves trapped in a tilted box

ABSTRACTWe study the prototype problem of internal gravity waves in a square domain tilted with respect to the gravity vector by an angle θ. Only for and a countable number of other angles regular normal modes exist, for all other angles wave attractors and singularities dominate the flow. We show that the linear operator of the governing PDE becomes non-normal for giving rise to non-modal transient growth. This growth depends on the underlying norm: for the variance norm significant growth rates can be found whereas for the energy norm, no growth is possible since, in contrast to shear flows for which the mean flow feeds the perturbations, there is no energy source. We continue by showing that the non-normality of the system matrix is increasing with θ and reaches a maximum for . Moreover, with increasing tilt angle the growth rate is increasing as can be expected from the increasing non-normality of the matrix. Our results imply that at least the most simple wave attractors can be seen as those initial flow fields that gain most of the variance during a given time period. We conclude that in a closed confinement with walls tilted with respect to gravity, internal wave variance can be maintained by transient growth of perturbations without a direct contribution of unstable normal modes.

Optimal Perturbations of an Oceanic Vortex Lens

The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed.

Forced oscillations of the string under conditions of 'sonic vacuum'.

We present the results of analytical study of the significant regularities which are inherent to forced nonlinear oscillations of a string with uniformly distributed discrete masses, without its preliminary stretching. It was found recently that a corresponding autonomous system admits a series of nonlinear normal modes with a lot of possible intermodal resonances and that similar synchronized solutions can exist in the presence of a periodic external field also. The paper is devoted to theoretical explanation of numerical data relating to one of possible scenarios of intermodal interaction which was numerically revealed earlier. This is unidirectional energy flow from unstable nonlinear normal mode to nonlinear normal modes with higher wavenumbers under the conditions of sonic vacuum. The mechanism of such a scenario has not yet been clarified contrary to alternative mechanisms consisting in almost simultaneous energy flow to all nonlinear normal modes with breaking the above-mentioned conditions of sonic vacuum. We begin with a description of single-mode manifolds and then show that consideration of arbitrary double mode manifolds is sufficient for solution of the problem. Because of this, the two-modal equations of motion can be reduced to a linear equation which describes a perturbation of initially excited nonlinear normal mode of the forced system in the conditions of sonic vacuum. We have found analytical representation (in the parametric space) of the thresholds for all possible energy transfers corresponding to unidirectional energy flow from unstable nonlinear normal modes. The analytical results are in a good agreement with previous numerical calculations.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Determining the stiffness of the equation of state using low T/W dynamical instabilities in differentially rotating stars

We investigate the nature of low T/W dynamical instabilities in various ranges of the stiffness of the equation of state in differentially rotating stars. Here T is the rotational kinetic energy, while W the gravitational binding energy. We analyze these instabilities in both a linear perturbation analysis and a three-dimensional hydrodynamical simulation. An unstable normal mode of a differentially rotating star is detected by solving an eigenvalue problem along the equatorial plane of the star. The physical mechanism of low T/W dynamical instabilities is also qualitatively confirmed by a scattering of sound waves between corotation and the surface caused by the corotation barrier. Therefore, we can draw a picture of existing pulsation modes unstabilized due to an amplified reflection of sound waves from the corotation barrier. The feature in the eigenfrequency and eigenfunction of the unstable mode in the linear analysis roughly agrees with that in the three-dimensional hydrodynamical simulation in Newtonian gravity. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface for an unstable star requires reinterpretation of pulsation modes in differentially rotating stars. Finally, we propose a manner by which to constrain the stiffness of the equation of state by the direct detection of mode decomposed gravitational waveforms.

Connections in the development of tropical cyclone Mitch (1998): A tribute to the human loss in this calamity

Tropical cyclone Mitch, which occurred in October 1998 on the Caribbean Sea, is studied. Mitch affected countries in Central America, where it caused great economic and human losses, and the authors believe that the scientific community has not paid enough attention to this phenomenon. Two NE-SE oriented dipole patterns associated with their convergence-divergence zones contributed significantly to its formation and intensification, which are analyzed from the observational synoptic point of view. The most remarkable characteristics of this phenomenon in the mean field occurred between October 18 and November 4, 1998 when another NW-SE dipolar pattern was formed, consisting of a high-level anticyclone from North America (HLA) and a high-level anticyclone from South America (HLC). To discern these patterns, two data sources were used: the NCAR-ds627.0 (ERA-Interim) reanalysis and the NCEP/NCAR reanalysis. The simplest idealization of the NW-SE dipolar vortex is a tropical modon, which is an exact solution of the barotropic vorticity equation (BVE) on a rotating sphere. The role of the HLA and the HLC in the excitation of tropical waves is explored using a numerical spectral method for sutdying the normal mode stability of exact solutions of the BVE. The initiation of the tropical wave at low levels could be associated with disturbances at high levels of the most unstable normal mode.

Instabilities in flexible channel flow with large external pressure

We examine the stability of laminar high-Reynolds-number flow through an asymmetric flexible-walled channel driven by a fixed upstream flux and subject to a (large) uniform external pressure. We construct a long-wavelength, spatially one-dimensional model using a flow profile assumption, modelling the flexible wall as a thin tensioned membrane subject to a large axial pre-stress. We numerically construct the non-uniform static shape of the flexible wall and consider its stability using both a global eigensolver and numerical simulation of the nonlinear governing equations. The system admits multiple static solutions, including a highly collapsed steady state where the membrane has a single constriction which increases with increasing external pressure. We demonstrate that the non-uniform static state is unstable to two distinct (infinite) families of normal modes which we characterise in the limit of large external pressure. In particular, there is a family of low-frequency oscillatory modes which each persist to low membrane tensions, where the most unstable mode has an oscillating membrane profile which is outwardly bulged at the centre of the domain with a narrow constriction at the downstream end. In addition, there is a family of high-frequency oscillatory modes which are each unstable beyond a critical value of the tension within a two-branch neutral curve. Unstable modes along the lower branch of the neutral curve are sustained by a leading-order balance between unsteady inertia and the restoring force of membrane tension along the channel. In addition, we elucidate the mechanism of energy transfer to sustain the self-excited oscillations: oscillations decrease the mean maximal constriction of the channel over a period, which reduces the overall dissipation of the mean flow and releases energy to sustain the instability. Fully nonlinear simulations indicate that as the Reynolds number increases these unstable normal modes can grow supercritically into sustained large-amplitude ‘slamming’ oscillations, where the membrane is periodically drawn very close to the opposite rigid wall before recovering.

Unstable buoyant flow in a vertical porous layer with convective boundary conditions

The instability of natural convection in a vertical porous layer is analysed. The plane parallel boundaries of the vertical layer are modelled as open and subject to Robin-type temperature conditions. The latter conditions describe heat transfer to the external environment, with a finite conductance. The basic state is given by a stationary fully-developed flow with linear velocity and temperature profiles. Instability arises when the Darcy-Rayleigh number exceeds its critical value. This value depends on the Biot number associated with the temperature boundary conditions. The most unstable normal modes turn out to be transverse. By solving numerically the stability eigenvalue problem, it is shown that the critical Darcy-Rayleigh number is a decreasing function of the Biot number when the Biot number is sufficiently small. For larger Biot numbers, a minimum is attained, and then the critical Darcy-Rayleigh number becomes an increasing function of the Biot number.

Unstable normal modes of lowT/Wdynamical instabilities in differentially rotating stars

We investigate the nature of low T/W dynamical instabilities in differentially rotating stars by means of linear perturbation. Here, T and W represent rotational kinetic energy and the gravitational binding energy of the star. This is the first attempt to investigate low T/W dynamical instabilities as a complete set of the eigenvalue problem. Our equilibrium configuration has "constant" specific angular momentum distribution, which potentially contains a singular solution in the perturbed enthalpy at corotation radius in linear perturbation. We find the unstable normal modes of differentially rotating stars by solving the eigenvalue problem along the equatorial plane of the star, imposing the regularity condition on the center and the vanished enthalpy at the oscillating equatorial surface. We find that the existing pulsation modes become unstable due to the existence of the corotation radius inside the star. The feature of the unstable mode eigenfrequency and its eigenfunction in the linear analysis roughly agrees with that in three-dimensional hydrodynamical simulations in Newtonian gravity. Therefore, our normal mode analysis in the equatorial motion proves valid to find the unstable equilibrium stars efficiently. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface radius for unstable stars requires reinterpretation of the pulsation modes in differentially rotating stars.

Kramers' theory for diffusion on a periodic potential.

Kramers' turnover theory, based on the dynamics of the collective unstable normal mode (also known as PGH theory), is extended to the motion of a particle on a periodic potential interacting bilinearly with a dissipative harmonic bath. This is achieved by considering the small parameter of the problem to be the deviation of the collective bath mode from its value along the reaction coordinate, defined by the unstable normal mode. With this change, the effective potential along the unstable normal mode remains periodic, albeit with a renormalized mass, or equivalently a renormalized lattice length. Using second order classical perturbation theory, this not only enables the derivation of the hopping rates and the diffusion coefficient, but also the derivation of finite barrier corrections to the theory. The analytical results are tested against numerical simulation data for a simple cosine potential, ohmic friction, and different reduced barrier heights.

Mode parameterization of structures with very low symmetry: PZT and magnetite

The parameterization of distorted structures in terms of symmetry modes is an effective and efficient method for both their description and refinement [1]. A basis of symmetry-adapted modes transforming according to irreducible representations not only provides a hierarchical division of the degrees of freedom consistent with the mechanism at the origin of the distorted phase, but it allows the avoidance of false refinement minima, typical of highly pseudo-symmetric phases. A reduction of the number of free parameters by setting to zero negligible marginal modes is also possible. The mode description is nowadays easily applicable through freely available programs [2,3], while direct single crystal and powder diffraction refinements under this parameterization are possible combining these programs with some of the most popular refinement codes. The mode description is especially effective when dealing with distorted structures of very low symmetry compared with that of the parent phase. In these cases, the hierarchy between strong primary modes and weak marginal ones is specially pronounced, minimizing the role of many secondary modes. The physical origin of each primary distortion is usually a set of unstable degenerate normal modes. This introduces correlations among the different phases in the phase diagram that become patent in a mode description and can be used both to characterize the evolution of the relevant order parameters and as a stringent test of proposed structural models. Furthermore, the fact that each of the primary mode distortions is basically associated with the activity (instability) of a single normal mode can yield a "single mode" signature in the mode decomposition, which represents a set of subtle additional structural constrains beyond conventional crystallography. We will illustrate these considerations using the examples of the monoclinic phases of ferroelectric PbZr1-xTixO3 (PZT) and the Verwey phase of magnetite.