Unstable Modes Research Articles

Three-dimensional elastic Rayleigh–Taylor instability at the cylindrical interface

This paper presents a linear analysis of elastic Rayleigh–Taylor instability at both cylindrical column and cylindrical shell interfaces. By considering the rotational part of the disturbance flow field, an exact solution is derived, revealing that the most unstable mode is two-dimensional in the cross section. As the column radius decreases, the maximum growth rate increases, while the corresponding azimuthal wave number decreases incrementally until it reaches 1. Thinning the cylindrical shell is found to be a destabilizing effect, leading to an increase in both the cutoff wave number and the most unstable azimuthal wave number. The maximum growth rate usually increases as the shell becomes thinner, except in cases with small radii where feedthrough effects occur. For thin shells with small radii, the cutoff axial wave number is determined by the radius rather than the shell thickness. Comparisons between the growth rates derived from the potential flow theory and the exact solution show significant discrepancies in cylindrical shells, mainly due to substantial deviations in the cutoff wave number.

Design theory for section model tests considering multimode coupling and imperfect modeshape similarity of full bridges

Section model tests are the most fundamental wind-tunnel technique widely employed in the design of flexible cable-supported bridges. To guarantee the accuracy of aeroelastic stability assessment, the dynamic behavior of a section model test should reflect the aeroelastic characteristics of the full bridge. Conventional method only employs the fundamental vertical and torsional modes for the design of dynamic parameters, typically ignoring multimode coupling effect and assuming perfect modeshape similarity. This paper introduces a novel theoretical framework that integrates multimode coupling and imperfect modeshape similarity into the design of a section model test. The framework establishes a direct equivalence between the complex modal parameters of the unstable mode in the multimode coupling system and the two degrees-of-freedom coupling system of the section model. It quantifies each mode's contribution to coupled flutter through a single parameter denoted as mode participation factor. Validation of this improved design theory is performed using numerical models of four representative cable-supported bridges, highlighting its effectiveness. The study also sheds light on the complexities of multimode aeroelastic coupling and the limitations of section model tests.

Three-dimensional eigensolutions of the unsteady compressible boundary-region equations

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

Suppression of the Effect of Parametric Instability in the LIGO Voyager Gravitational Wave Detector

The number of unstable combinations of elastic and Stokes optical modes of the LIGO Voyager gravitational wave detector, frequencies and spatial distributions of displacement vectors of elastic modes of mirrors were calculated. The values of overlap factors for unstable modes up to ninth-order optical modes are calculated, taking into account the azimuthal condition of parametric instability. An analysis of the influence of the temperature dependence of the Young’s modulus of the mirror material on the number of unstable modes in the Fabry-Perot resonator was performed.

On Trade-Off Relationship between Static and Dynamic Lateral Stabilities of Articulated Heavy Vehicles

Articulated heavy vehicles exhibit poor lateral stability, which may lead to unstable motion modes, e.g., trailer-sway and jackknifing, causing severe accidents. Varying relevant vehicle parameters improves the static stability but degrades the dynamic stability. The past studies focused either on the static or dynamic stability alone. However, little attention has been paid to exploring the trade-off between the static and dynamic stabilities. To gain design insights for active safety systems for AHVs, this article studies this trade-off systematically. To this end, a systematic method is proposed to conduct the linear stability and trade-off analysis. To implement and demonstrate the proposed method, a linear three-degrees-of-freedom yaw-plane model is generated to represent a tractor/semi-trailer. A trade-off analysis is conducted considering two tractor rear-axle configurations and three trailer payload arrangements. In each case, simulation is performed in both steady-state and transient testing maneuvers. To validate the linear stability analysis based on the linear yaw-plane model, two nonlinear TruckSim models are introduced, and the corresponding simulation is conducted. Insightful understanding of the trade-off is gained through analyzing the simulation results, and the linear stability analysis will provide valuable guidelines for the design and development of active safety systems for AHVs.

Detecting neoclassical tearing modes in high-temperature ITER plasma scenarios with the ITER prototype electron cyclotron emission diagnostic

Abstract Electron cyclotron emission (ECE) diagnostics for ITER serve two key purposes. The diagnostics will measure plasma electron temperature with high spatial and temporal resolution. Additionally, they will be used to detect neoclassical tearing modes (NTMs), a deleterious and nonlinearly unstable mode causing the growth of magnetic ‘seed’ islands. Interpreting ECE requires anticipation of physical limits including frequency cut-offs and harmonic overlap. In high temperature plasmas, the relativistic shift and broadening of the emission must also be considered to accurately reconstruct the electron temperature spatial profile. Accounting for these effects allows ECE diagnostics to be used for accurate measurement of the equilibrium electron temperature profile, as well as fluctuations about this equilibrium. One such fluctuation is caused by the fast radial transport of heat across rotating magnetic islands. ECE diagnostics can detect this change as an oscillation at the plasma rotation frequency to determine the existence and location of NTMs. This paper presents work on a synthetic diagnostic for ECE. The synthetic diagnostic tests simulated ECE signals, which are inferred from ITER scenarios perturbed by magnetic islands after accounting for all ECE physics. The synthetic diagnostic tests conventional ECE detection algorithms for NTMs in real-time on ITER-recommended hardware. Combined, these two areas of focus help determine design of the ECE system.

Wave patterns of the coupled nonlinear Schrödinger equations in photonic crystal fibers with four-wave mixing

Abstract In this paper, we examine the behavior of modulation instability within photonic crystals. The model employed is the coherent coupled nonlinear Schrödinger equation, incorporating weak birefringence and four-wave mixing, which arises at the edge of the optical mode. The linear analysis is used to derive the modulation instability spectrum. Throughout the modulation instability spectrum, we identify both stable and unstable modes, thereby confirming the breakdown of the plane wave. For certain four-wave mixing parameters, the amplitude of the modulation instability spectrum and its bandwidths expand, creating an opening for localized structures to emerge. Another aspect of this study has been demonstrated in normal and anomalous dispersion regimes where an increasing initial amplitude of the plane wave is fulfilled. Specifically, numerical simulations highlight the occurrence of Benjamin-Feir instability, where wave patterns emerge under the influence of four-wave mixing. Additionally, solitonic waves are generated, demonstrating the presence of Akhmediev breathers and other modulated structures, confirming that photonic crystals with four-wave mixing are conducive to these formations. The findings from this study could inform future research in the development of nonlinear photonic waveguides.

Analysis on stall mechanism of axial flow compressor with rotating inlet distortion by dynamic mode decomposition

To study the impact of rotating inlet distortion on stall mechanism of an axial compressor, the full-annulus unsteady simulation results are analysed using dynamic mode decomposition (DMD). Firstly, the frequency and energy ratio of each mode at near stall condition for the compressor with uniform inflow are obtained. A dominating frequency band 0.3–0.6 BPF regarded as the characteristic frequency of unsteady tip leakage flow is observed. The distribution of eigenvalues further confirms that there are a few critical unstable modes in the flow field. With further throttling, four large-scale disturbance structures are captured by the second-order mode, whose frequency is twice as the rotor frequency. The circumferential propagation velocity of the stall cell is thus deduced to be about 50% of the rotor speed. Further, when the compressor is subjected to rotating inlet distortion, the change of stalling process is related to the relative position of distortion plate and stall cell according to the DMD mode frequency and shape. The addition of rotating distortion does not change the stall type of the compressor, but the unstable modes indicate that the number and shape of the stall cells have been varied.

Baroclinic instability in the Eady model for two coupled flows

ABSTRACT The linear instability of baroclinic flows in two superimposed and coupled, immiscible fluids is studied theoretically. These flows are westerlies, and they are thermally or mechanically coupled. The fluids are stably stratified, internally and mutually (the upper fluid is lighter than the lower fluid). In each fluid, two-level surface quasi-geostrophy governs the evolution of the perturbed westerly flow. The perturbations are horizontal normal modes. Firstly, the models are not coupled and the flow instability in each fluid is validated separately against the results of the classical Eady model of baroclinic instability. Secondly, the two fluids are thermally and/or mechanically coupled. With thermal coupling, and for meridionally uniform perturbations, a new mode of instability appears for long waves. This pair of unstable modes converges towards the modes of the uncoupled fluids at medium wavelengths. For perturbations with a non trivial meridional structure, the thermal coupling essentially damps the instability. For an upper flow with a larger deformation radius than in the lower flow, the growth rates of the perturbation are therefore more strongly altered in the former than in the latter. With mechanical coupling, the instability is essentially damped at large to medium scales, while the short-wave cut-off is extended towards smaller waves. When the fluids are both thermally and mechanically coupled, these effects add up. This very idealised study is a first step towards studying more realistic cases.

Limit cycle oscillations in the zonal-flow-catalyzed interactions of ion-temperature-gradient turbulence

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized.

Propagation and stabilization properties of rotating detonation waves in a hollow combustion chamber with heated air inflow

The application of rotating detonation to conventional engine poses challenges such as high-temperature inflow conditions. In this study, experiments were conducted to discuss the propagation and stabilization properties of rotating detonation waves within a hollow rotating detonation combustor using pure heated air/ethylene as propellant. The temperature and equivalence ratio are varied to explore the boundary of detonative limits. The results show that the pressure of detonation waves is influenced by temperature fluctuations, whereas changes in air temperature have minimal impact on velocity losses. In addition to detonation, the deflagration zone is observed within the combustor. When the air temperature is increased, the deflagration zone expands gradually. The static temperature of air increases from 304 to 618 K, resulting in an increase in the area of the glowing zone from 17.7% to 60.0%. The expansion of the zone indicates a rise in deflagration percentage, and a stable single-wave mode is achieved mainly under fuel-rich conditions. The detonative equivalence ratio range is narrowed with increasing heated temperature at 484–618 K. Unstable propagation modes, characterized by sawtooth waves, are frequently observed at detonative lower limit, where the equivalence ratio is around 1.

Tayler Instability Revisited

Tayler instability of toroidal magnetic fields B ϕ is broadly invoked as a trigger for turbulence and angular momentum transport in stars. This paper presents a systematic revision of the linear stability analysis for a rotating, magnetized, and stably stratified star. For plausible configurations of B ϕ , instability requires diffusive processes: viscosity, magnetic diffusivity, or thermal/compositional diffusion. Our results reveal a new physical picture, demonstrating how different diffusive effects independently trigger instability of two types of waves in the rotating star: magnetostrophic waves and inertial waves. It develops via overstability of the waves, whose growth rate sharply peaks at some characteristic wavenumbers. We determine instability conditions for each wave branch and find the characteristic wavenumbers. The results are qualitatively different for stars with magnetic Prandtl number Pm ≪ 1 (e.g., the Sun) and Pm ≫ 1 (e.g., protoneutron stars). The parameter dependence of unstable modes suggests a nonuniversal scaling of the possible Tayler–Spruit dynamo.

Investigation of streamwise streak characteristics over a compression ramp at Mach 4

Experiments of shock wave/boundary layer interactions over a nominally two-dimensional compression ramp are conducted in a Mach 4 Ludwieg tube tunnel. Measurements of Schlieren, Rayleigh scattering, and surface pressure are performed to present the relevant flow features. The effects of two parameters, namely the Reynolds number based on the length of the flat plate and the ramp angle, on the flow stabilities are focused on. Four ramp angles of 6°, 8°, 10°, and 12° are tested under a Reynolds number of 7.22 × 105, while two other Reynolds numbers (3.66 × 105 and 9.19 × 105) are investigated with a ramp angle of 10°. Streamwise streaks are observed downstream of the reattachment point. The spanwise wavelength of the streaks remains unchanged with different ramp angles, whereas it slightly decreases as the Reynolds number increases. Power spectral density results show that the flow is transitional in the streak region and becomes turbulent where streaks break down. When increasing the ramp angle or the Reynolds number, the streamwise length of streaks shrinks. Two different patterns are distinguished at the breakdown, resembling the two unstable modes observed in the breakdown of Görtler vortices. To clarify the underlying physics of the formation of streaks, global stability analysis and resolvent analysis are carried out. Two regions of maximum optimal gain are identified, which are associated with Mack's first mode and streaks. The former can serve as an initial seed of Görtler instability via nonlinear interaction, while the latter can be associated with transient growth due to the lift-up mechanism and Görtler instability.

Can boundary slip destabilize rotating microchannel flows?

Deviation from the traditional no-slip boundary condition due to factors like surface roughness and wettability is of paramount importance in microfluidics and nanofluidics, as it is attributable to its significance in drag reduction, flow control and enhancement and improved mixing. Augmentation in mixing, in turn, is known to strongly correlate with potential instabilities in the flow structure. Reported research studies indicate that slip is an inherent flow stabilizer in microfluidics, to the extent that with sufficient slip, the flow becomes linearly stable against all wavelike disturbances for all wavelengths and Reynolds numbers [“The linear stability of slip channel flows,” Phys. Fluids 34,074103(2022)]. Contrary to such intuitive proposition, here we show that slip effects can destabilize microchannel flows under spanwise rotation, delving on the interplay of rotational forces and slippery hydrodynamics. Our results reveal that increasing the slip length decreases the critical rotation speed, indicating lower rotational effort required to destabilize the flow, whereas the critical Reynolds number for the flow remains effectively unaltered for different slip lengths in a spanwise rotating system. As the slip length increases progressively, the critical rotation number (dimensionless rotational speed) for the onset of instability decreases further, then remains constant up to a certain limit, and subsequently declines with additional enhancement in the slip length. This indicates the potential for deploying customized hydrophobic (slippery) substrates to facilitate transitions from stable to unstable modes by simple tuning of the rotational speed—a paradigm that offers great promise in various applications ranging from materials synthesis to biomedical technology.

Effects of combustor wall temperature on the outer recirculation zone combustion modes transition in lean premixed DME/air swirling flames

In this study, the effects of wall thermal boundary conditions on the outer recirculation zone (ORZ) combustion modes transition in the DME/air swirling flames were investigated using the Large Eddy Simulation (LES) and Flamelet Generated Manifold (FGM) method. The results show that the ORZ chemistry and temperature are highly sensitive to the wall thermal boundary condition. Based on the correlation between the temperature and species observed in the time-averaged numerical results, the relation between the combustion mode and temperature in the ORZ was discovered. Through temporal analysis of species and heat release rate in the ORZ, the stable Mode II and Mode IV as well as the unstable Mode III were identified. The chemical reaction pathway analysis for typical probe in the ORZ reveals that different combustion modes undergo distinct (low-temperature or high-temperature) chemical reaction pathways. Afterwards, the definition of ORZ combustion modes used in experiments was refined by utilizing the chemical reaction data from the simulations. And it was applied to identify the combustion mode across the entire computational domain. The results confirm the consistency between the combustion modes determined in experiments based on OH and CH2O and those determined in simulations using chemical reaction rates. Finally, this explains the mechanism of the flickering flame in Mode III is the competition between the low-temperature and high-temperature reaction in the ORZ. Whereas the unstable Mode III can be stabilized by changing the wall temperature to reach the stable Mode II or Mode IV.

The effect of hydrogen enrichment on the conical premixed methane–air flame response and thermoacoustic modes coupling

This paper investigates the thermoacoustic dynamic responses of hydrogen-enriched laminar premixed conical methane–air flames under dual-mode coupling. Utilizing the level set method and the G-equation model, one meticulously derives the flame describing function (FDF) in relation to variations in hydrogen enrichment levels (ηH). This precisely derived FDF is then integrated into the low-order network model of the Rijke tube to analyze the thermoacoustic unstable modes of the system. Finally, dual-frequencies incoming flow velocity perturbations are reintroduced as inputs to obtain the flame response under thermoacoustic modes coupling. Results show that while keeping the unstretched steady flame aspect ratio constant, an increase in ηH not only raises the FDF’s cut-off frequency but also enhances the FDF gain within the nonlinear frequency region, leading to more unstable modes, especially high frequency modes within the Rijke tube system. Furthermore, the flame response is further altered under the excitation of dual unstable modes. When both modes are within the linear frequency region of the FDF, the flame response is co-controlled by the two modes, predominantly by the mode with a larger velocity perturbation amplitude, with weaker modes coupling leading to the additional frequency perturbation having a suppressive effect on the flame response at the original frequency. Conversely, when one or two modes are within the nonlinear frequency region of the FDF, the flame response is dominated by the lower-frequency mode, with higher nonlinear modes coupling allowing the additional frequency perturbation to both promote and suppress the response at the original frequency and also couple to produce a significant difference frequency response. Novelty and significance The novelty of this paper lies in the determination of the flame describing function (FDF) with varying hydrogen enrichment levels and the stability of thermoacoustic systems, and more significantly, conducting an in-depth and comprehensive investigation into the nonlinear dynamic responses of hydrogen-enriched flames to different types of dual-mode coupling perturbations. The dual-mode perturbations result in either attenuation or amplification of the flame response, depending on whether the corresponding frequencies lie within the linear or nonlinear frequency regions of the FDF, which innovatively incorporates the influence of the FDF phase and examines the responses at the difference frequencies generated by the coupling. This lays a foundation for further exploration into the mechanisms of modes coupling in hydrogen-enriched and other thermoacoustic systems.

Accuracy of kinetic equilibrium reconstruction of NSTX and NSTX-U plasmas and its impact on the transport and stability analysis

Abstract An accurate magnetohydrodynamic (MHD) equilibrium reconstruction is an essential starting point for stability and transport plasma analysis. This work describes an approach for obtaining kinetic equilibrium reconstructions using the OMFIT framework, which has been applied for the first time to spherical tokamak data from NSTX and NSTX-U. The EFIT equilibrium solver is integrated with experimental data analysis procedures and subsequent TRANSP transport simulations to enhance the accuracy of the reconstruction, in particular at the edge region, by adding constraints on the total pressure and current density profiles, based on the transport code solution. The accuracy of the equilibrium depends on the uncertainty and number of constraints, as well as the choice of basis functions to represent the pressure and current density profiles. An improved fidelity of the equilibrium reconstruction is demonstrated by reducing the variability of the magnetic axis and boundary locations from several centimeters for reconstructions based on magnetic and experimental pressure constraints to only several millimeters for kinetic reconstructions based on transport code constraints, when different representations for basis functions were tested. Variability of the safety factor on-axis was reduced ten times in the same sensitivity study. The accuracy of the equilibrium reconstruction and subsequent mapping of experimental kinetic profile data have a significant impact on TGLF and linear CGYRO turbulence simulations, which predict different spectra of unstable modes and turbulent fluxes for cases with different numbers of constraints in the equilibrium reconstruction. Conversely, the stability analysis performed with the GATO code shows plasmas stable to n = 1 MHD modes in both equilibria using magnetic and experimental pressure constraints as well as the transport code constrained equilibrium. However, a scan of parameters away from these conditions shows considerable deviation in the threshold of unstable modes between these reconstructions. Therefore, for reliable plasma analysis and use in turbulence and stability calculations, a high fidelity equilibrium reconstruction with accurate kinetic constraints based on transport code solutions is necessary.

Vertical Shear Instability with Partially Reflecting Boundary Conditions

Abstract The vertical shear instability (VSI) is widely believed to be effective in driving turbulence in protoplanetary disks. Prior studies on VSI exclusively exploit the reflecting boundary conditions (BCs) at the disk surfaces. VSI depends critically on the boundary behaviors of waves at the disk surfaces. We extend earlier studies by performing a comprehensive numerical analysis of VSI with partially reflecting BCs for both the axisymmetric and non-axisymmetric unstable VSI modes. We find that the growth rates of the unstable modes diminish when the outgoing component of the flow is greater than the incoming one for high-order body modes. When the outgoing wave component dominates, the growth of VSI is notably suppressed. We find that the non-axisymmetric modes are unstable and they grow at a rate that decreases with the azimuthal wavenumber. The different BCs at the lower and upper disk surfaces naturally lead to non-symmetric modes relative to the disk midplane. The potential implications of our studies for further understanding planetary formation and evolution in protoplanetary disks (PPDs) are also briefly discussed.

A microscopic approach to crystallization: Challenging the classical/non-classical dichotomy.

We present a fundamental framework for the study of crystallization based on a combination of classical density functional theory and fluctuating hydrodynamics that is free of any assumptions regarding order parameters and that requires no input other than molecular interaction potentials. We use it to study the nucleation of both droplets and crystalline solids from a low-concentration solution of colloidal particles using two different interaction potentials. We find that the nucleation pathways of both droplets and crystals are remarkably similar at the early stages of nucleation until they diverge due to a rapid ordering along the solid pathways in line with the paradigm of "non-classical" crystallization. We compute the unstable modes at the critical clusters and find that despite the non-classical nature of solid nucleation, the size of the nucleating clusters remains the principle order parameter in all cases, supporting a "classical" description of the dynamics of crystallization. We show that nucleation rates can be extracted from our formalism in a systematic way. Our results suggest that in some cases, despite the non-classical nature of the nucleation pathways, classical nucleation theory can give reasonable results for solids but that there are circumstances where it may fail. This contributes a nuanced perspective to recent experimental and simulation work, suggesting that important aspects of crystal nucleation can be described within a classical framework.

Plasmon scattering at a junction in double-layer two-dimensional electron gas plasmonic waveguide

Abstract The scattering of plasmons at a junction within a double-layer two-dimensional electron gas plasmonic waveguide is studied via a full electromagnetic method. The dispersion relation is derived by utilizing the transfer matrix method and can be extended to the situation of an arbitrary number of layers. By numerically solving the dispersion equations, both the acoustic and optical plasmon modes are identified in this double layer system, and the unstable plasmon modes arising from plasmon coupling in different layers are discussed elaborately. Subsequently, the total fields are expanded with eigenmodes and matched at the interface to analyze the scattering characteristics at the junction. The results indicate that the total power of the plasmon mode is amplified when the electron fluid flows from a high concentration region to a low concentration region, and the amplification is more evident at a higher drift velocity. Additionally, we address the scattering of unstable plasmons caused by the two-stream instability and find that the transmitted plasmons are excited intensively at the incidence of the growing plasmon, leading to the plasmon amplification. The detailed examination of plasmon scattering at junction is the prerequisite for studying more complex structures of terahertz plasmonic devices and comprehending the corresponding amplification mechanism.