Linear Stability Analysis Research Articles

Acoustic tones generated by impinging jets: Differences between laminar and highly-disturbed nozzle-exit boundary layers

The differences between the acoustic tones generated by impinging jets with laminar and highly-disturbed nozzle-exit boundary layers are investigated. For that, jets at Mach numbers between 0.6 and 1.3 impinging on a flat plate at a distance of 8 nozzle radii from the nozzle exit are computed using large-eddy simulations. The amplitudes of the tones generated by the jets through feedback loops establishing between the nozzle and the plate are found to be significantly affected by the exit turbulent disturbances. In the present study, overall, they are lower for the initially laminar jets than for the initially disturbed ones. The level decrease varies from a few dB up to 15 dB, depending on the tones, which can change the frequencies of the dominant tones and the numbers and azimuthal structures of their associated feedback modes. For Mach numbers 0.75 and 0.8, for instance, the dominant tone frequencies are approximately two times lower for the initially laminar jets than for the other ones, yielding a better agreement with experiments of the literature in the former case. For a Mach number of 1.1, as a second example, the dominant tone is associated with the axisymmetric third feedback mode in the laminar case but with the helical fifth feedback mode in the disturbed case. The differences in the tone amplitude are finally discussed by estimating the power gains of the shear-layer instability waves between the nozzle and the plate using linear stability analysis for the axisymmetric mode. In most cases, at the frequency of a specific tone, the higher the gain, the stronger the acoustic tone.

Modulation instability in nonidentical metamaterial waveguide arrays by graph Laplacian approach

We theoretically investigate modulation instability (MI) in a nonidentical waveguide array, which is made up of positive and negative index metamaterial waveguides. The unit cell of the optical waveguide array consists of three waveguides arranged in a triangular manner. Waveguides 1 and 3 are made up of positive index material (PIM) channels and waveguide 2 is by negative index material (NIM) channels, as a result, they show different light propagation characteristics. We model this array of waveguides using a generalized nonlinear Schro¨dinger equation, replacing the Laplacian operator with the graph Laplacian. Following linear stability analysis, we will discuss MI for different values of transverse wave number, as it determines the order of the Brillouin zone. We also discuss the effect of input power on periodic MI in normal and anomalous dispersion regimes. Thus we report a comprehensive study on the MI and hence the better ways to generate and manipulate the solitons or ultra-short pulses in NIM PIM waveguide arrays.

When discrete fronts and pulses form a single family: FPU chain with hardening-softening springs

We consider a version of the classical Hamiltonian FPU (Fermi–Pasta–Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink–antikink bundle. Direct numerical simulations and Floquet analysis of linear stability suggest that all of the obtained solutions are effectively stable. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem both qualitatively and quantitatively.

Snap-induced flow in a closed channel

Snap-through is a buckling instability that allows slender objects, including those in plant and biological systems, to generate rapid motion that would be impossible if they were to use their internal forces exclusively. In microfluidic devices, such as micromechanical switches and pumps, this phenomenon has practical applications for manipulating fluids at small scales. The onset of this elastic instability often drives the surrounding fluid into motion – a process known as snap-induced flow. To analyse the complex dynamics resulting from the interaction between a sheet and a fluid, we develop a prototypical model of a thin sheet that is compressed between the two sides of a closed channel filled with an inviscid fluid. At first, the sheet bends towards the upstream direction and the system is at rest. However, once the pressure difference in the channel exceeds a critical value, the sheet snaps to the opposite side and drives the fluid dynamics. We formulate an analytical model that combines the elasticity of thin sheets with the hydrodynamics of inviscid fluids to explore how external pressure differences, material properties and geometric factors influence the system's behaviour. To analyse the early stages of the evolution, we perform a linear stability analysis and obtain the growth rate and the critical pressure difference for the onset of the instability. A weakly nonlinear analysis suggests that the system can exhibit a pressure spike in the vicinity of the inverted configuration.

Flow instabilities driven by Prandtl number effect and rotation-depth coupling effect in the cylinder with a top disk

We employed linear stability analysis to investigate the Prandtl number (Pr) effect and rotation-depth coupling effect on the complex flow instability in a cylinder with a top disk. The dependence of the critical Rayleigh number on the Pr number, as well as the relationship between the critical disk rotation rate and the aspect ratio at Pr = 6.7, were obtained. Results reveal that the flow stability increases with increasing Pr number, and the convection instability stems from the inertial mechanism for 0.011 ≤ Pr ≤ 0.0258 and thermal buoyancy mechanism for 1.4 ≤ Pr ≤ 28.01. Regarding the rotation-depth coupling effect for the melt with Pr = 6.7, a decreasing melt level leads to a general increase in the critical disk rotation rate. Furthermore, within the interval of aspect ratio (ratio of depth to cylinder radius) 0.7 ≤ Γ ≤ 1.62, multiple transitions of the flow state from stable to unstable and then back to stable were observed with increasing disk rotation rate before finally becoming unstable. Energy analysis reveals that multiple transitions in the flow state are attributed to the competition between thermal buoyancy and inertial mechanisms.

Onset of rotating convection of a nanofluid with helical force: Stability analysis

Employing linear stability analysis of nanofluids holds promising advantages across diverse domains. The impact of rotation and helical force on the onset of convection in the nanofluid is investigated using the linear stability analysis. The method of normal modes is used to solve the governing nondimensional equations, which results in an eigenvalue problem for the linear stability analysis. The bvp4c in MATLAB R2021a is employed to solve the eigenvalue problem. Three different lower-upper boundary conditions are considered: free-free, rigid-free, and rigid-rigid. The Rayleigh and wave number are determined for various required values of the other physical parameters and graphically displayed. Specifically, as the Taylor number rises, there is a corresponding grow in the critical Rayleigh number. This indicates that higher values of the Taylor number stabilize the system, making it less prone to convective instabilities. Conversely, the Lewis number and Helical force parameter destabilize the critical Rayleigh number, making the system more prone to convective instabilities across all boundaries. This information provides valuable insights into the system’s dynamics under study and can be crucial for predicting and understanding its behavior under different conditions.

Flow-induced vibration of two tandem square cylinders at low Reynolds number: transitions among vortex-induced vibration, biased oscillation and galloping

The flow-induced vibrations (FIVs) of two identical tandem square cylinders with mass ratio m* = 3.5 at Reynolds number Re = 150 are investigated through two-dimensional direct numerical simulation (DNS) and linear stability analysis over a parameter range of spacing ratio 1.5 ≤ L* ≤ 5 and reduced velocity 3 ≤ Ur ≤ 34. Three kinds of FIV responses, namely vortex-induced vibration (VIV), biased oscillation (BO) and galloping (GA), are identified. The FIVs are then further classified into the branches of initial VIV (IV), resonant VIV (RV and RV′), flutter-induced VIV (FV), desynchronized VIV (DV), VIV developing from GA (GV), transitional state between VIV and GA (TR), BO and GA based on the characteristics of the vibration responses. The transitions among different FIV branches are examined by combining the DNS with linear stability analysis, where the transition boundaries among the VIV, BO and GA branches over the concerned parameters are identified on the branch maps. The transition from IV to RV or RV′ is found to be related to the unstable wake mode, while the FV, transiting from RV or RV′, is induced by the unstable structural factor in the wake-structure mode. The structural instability is considered as the physical origin of GA, whereas the mode competition between unstable wake and structure leads to DV, GV and TR, and thus delays the appearance of GA. The transition from DV to BO with biased equilibrium position, accompanied by the even-order harmonic frequencies, is essentially induced by the symmetry breaking bifurcation.

A toxicant-taxis reaction–diffusion model for the interaction between two species and a toxicant in an aquatic ecosystem

In this paper, we develop a toxicant-taxis reaction–diffusion model that describes the interaction dynamics between two species and a toxicant in a polluted aquatic ecosystem. We analyze the existence and stability of the equilibria of the corresponding space-absent ordinary differential equation system. We then perform linear stability analysis for our reaction–diffusion system to obtain sufficient conditions on model parameters under which the stable spatially homogeneous steady states become unstable. Numerical simulations are performed to illustrate that spatially inhomogeneous patterns and spatiotemporal patterns between the two species and the toxicant will typically emerge. We also numerically analyze the effects of spatially varying toxicant input and the intensity of toxicant-taxis on the population persistence and species diversity. It turns out that heterogeneous toxicant inputs and substantial toxicant taxis can boost the coexistence of two species, thereby promoting species diversity in ecosystems. By comparing the impact of a toxicant on a single species versus its impact on two species, we find that the collective sharing of the toxic burden among multiple species proves advantageous for biodiversity.

Prediction of pellet mass thresholds for ELM triggering in low-collisionality, ITER-like discharges

In ITER, pellets are calculated to require more than 8 times the mass than currently planned to reliably trigger edge-localized modes (ELMs). Unmitigated heat flux impulses from ELMs are intolerable in ITER at full power and current. Therefore, ITER operation relies on multiple approaches to control ELM heat fluxes. One method is pellet ELM pacing to instigate small rapid ELMs with low heat flux. Predicting the performance of pellet pacing is critical for ITER, which is expected to operate in a regime with a low-collisionality, peeling-limited pedestal. However, to trigger ELMs the local pressure increase in the expanding pellet cloud pushes the equilibrium over the ballooning stability limit. In this work, linear and nonlinear M3D-C1 simulations are used to predict pellet mass thresholds in DIII-D discharges and ITER scenarios with peeling-limited pedestals. It is found that the distance of the equilibrium’s operational point from the ballooning branch of the pedestal stability boundary strongly changes thresholds. Linear M3D-C1 simulations find a strong dependence of the pellet mass threshold on the poloidal injection location for ITER’s 15 MA, Q = 10 scenario. The required pellet mass at the planned injection locations is 8 to 17 times larger than currently considered. However, such linear simulations do not include pellet ablation physics or time evolution of density and temperature. A new scheme of 2D nonlinear simulations, coupled with linear stability analysis at various steps throughout the nonlinear time evolution, was developed to include such physics and improve on the linear results. These new nonlinear-to-linear simulations confirm previous findings. This result suggests that pellet ELM triggering in ITER could require pellets much larger than those currently planned, which makes ELM-pacing operationally challenging. On the other hand, fueling pellets injected from the high-field side will likely not unintentionally trigger ELMs in an otherwise ELM-stable plasma.

Orthogonal multi-peak solitons from the coupled fractional nonlinear Schrödinger equation

In this paper, we investigate the dynamics and stability of multi-peak solitons from the coupled nonlinear Schrödinger equation with the fractional dimension based on Lévy random flights. By implementing linear stability analysis and direct simulations, we demonstrate regions where the single and multi-peak modes are stable. Analysis of perturbed coupled solitons confirms the stability of higher-order modes compared to lower-order modes under the same configurations. The stability diagrams show that the force coupling, Lévy index, power, and the nonlinear intensity significantly influence the stability of high-order modes. Our findings indicate that stability is favored in self-defocusing systems with high Lévy indices under weak coupling conditions, with higher-order states exhibiting smaller stability regions.

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási–Albert (BA), Erdös–Rényi (ER), and Watts–Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

Instability mechanisms initiating laminar–turbulent transition past bioprosthetic aortic valves

Bioprosthetic heart valves create turbulent flow during early systole which might be detrimental to their durability and performance. Complex mechanisms in the unsteady and heterogeneous flow field complicate the isolation of specific instability mechanisms. We use linear stability analysis and numerical simulations of the flow in a simplified model to study mechanisms initiating the laminar–turbulent transition. The analysis of a modified Orr–Sommerfeld equation, which includes a model for fluid–structure interaction (FSI), indicates Kelvin–Helmholtz and FSI instabilities for a physiological Reynolds number regime. Two-dimensional parametrized FSI simulations confirm the growth rates and phase speeds of these instabilities. The eigenmodes associated with the observed leaflet kinematics allow for decoupled leaflet oscillations. A detailed analysis of the temporal evolution of the flow field shows that the starting vortex interacts with the aortic wall leading to a secondary vortex which moves towards the shear layer in the wake of the leaflets. This appears to be connected to the onset of the shear layer instabilities that are followed by the onset of leaflet motion leading to large-scale vortex shedding and eventually to a nonlinear breakdown of the flow. Numerical results further indicate that a narrower aorta leads to an earlier onset of the shear layer instabilities. They also suggest that the growing perturbations of the shear layer instability propagate upstream and may initiate the FSI instabilities on the valve leaflets.

Morphological stability of electrostrictive thin films

A large electric field is typically present in anodic or passive oxide films. Stresses induced by such a large electric field are critical in understanding the breakdown mechanism of thin oxide films and improving their corrosion resistance. In this work, we consider electromechanical coupling through the electrostrictive effect. A continuum model incorporating lattice misfit and electric field-induced stresses is developed. We perform a linear stability analysis of the full coupled model and show that, for typical oxides, neglecting electrostriction underestimates the film’s instability, especially in systems with a large electric field. Moreover, a region where electrostriction can potentially provide a stabilizing effect is identified, allowing electrostriction to enhance corrosion resistance. We identified an equilibrium electric field intrinsic to the system and the corresponding equilibrium film thickness. The film’s stability is very sensitive to the electric field: a 40 percent deviation from the equilibrium electric field can change the maximum growth rate by nearly an order of magnitude. Moreover, our model reduces to classical morphological instability models in the limit of misfit-only, electrostatic-only, and no-electrostriction cases. Finally, the effect of various parameters on the film’s stability is studied.

Thermohydrodynamics of Bödewadt hybrid-nanofluid flow in a horizontal magnetic field

Since disks, rotating or stationary, are integral in many thermal systems, such as rotating heat exchangers, thermoelectric coolers, solar energy systems, and geothermal plants, this study scrutinizes the thermohydrodynamics of rotating hybrid nanofluid flow over a stretchable disk at rest. Further, the shape of the nanoparticles and the choice of base fluid substantially impact the enhancement of the heat transfer rate in various such thermal systems. Therefore, the present investigation considers engineered colloids made of four distinct (sphere, cylinder, column, lamina) titania and copper nanoparticle shapes in two different base fluids: water and engine-oil. The problem is formulated mathematically in an externally applied horizontal magnetic field and incorporating a non-Fourier heat flux model in the presence of solar radiation. Analogous to the case of the conventional vertical magnetic field, a similarity solution is also possible when magnetic forces act horizontally, both towards and opposite to the direction of rotation. Thus, the governing partial differential equations (PDEs) are first reduced to a set of highly non-linear and coupled ordinary differential equations (ODEs) via similarity transformations. These resulting set of ODEs are then solved numerically using a rapid and efficient spectral quasilinearization method (SQLM). Obtained results show that lamina-shaped TiO2−Cu/ engine-oil hybrid-nanofluid is a better choice than the other shapes of nanoparticle suspension in water. The horizontally applied magnetic field exhibits a stronger influence on the flow and heat transfer characteristics when compared to a vertical magnetic field. Additionally increasing the thermal relaxation parameter αt from 0.1 to 0.35, the Nusselt number boost by 24.44%, 23.41%, 22.54%, and 17.93% for sphere, cylinder, column, and lamina nanoparticles of TiO2−Cu/water hybrid nanofluid. In the case of TiO2−Cu/engine-oil hybrid-nanofluid augmenting αt from 0.1 to 0.35, the Nusselt number rise by 23.95%, 22.83%, 21.82%, and 15.89% for sphere, cylinder, column, and lamina nanoparticles. Based on their increasing NS and Be values, the various shapes follow the sequence: lamina ¡ column ¡ cylinder ¡ sphere. Furthermore, entropy generation can be optimized through the augmentation of solar radiation factors QSR, δ, along with the reduction of the magnetic interaction parameter M, and by a proper selection of nanoparticle’s shape. Interestingly, dual solutions are observed for the case of a shrinking disk, i.e., for S<0, and a linear temporal stability analysis reveals that only one of these two branches, namely the first solution branch, is stable and the second branch unstable.

Two-dimensional modulation instability and soliton clusters in nonlocal media with competing cubic–quintic nonlinearities

Using linear-stability analysis, two-dimensional modulation instability (MI) of plane wave is studied in nonlocal media with competing cubic–quintic (CQ) nonlinearities, which shows that MI can be effectively eliminated by strong nonlocality and competing quintic nonlinearity. Furthermore, propagation properties of higher-order soliton clusters, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) solitons are also investigated. Bifurcated solutions of these solitons are obtained analytically with variational approach. We also demonstrate in detail the propagation dynamics of the HG and LG solitons with split-step Fourier transform.

Revisiting the mechanism responsible for the stratified-slug transition in two-phase flows

Linear stability analysis is extensively used for predicting the transition between stratified and slug flow. In the present work, a one-dimensional two-fluid flow is linearly perturbed to evaluate the behavior of the dispersion curves through parameters such as the maximum wave growth rate (ωI,max), the fastest-growing wave (kmax) and the wave that makes the problem permanently stable (kc) as a function of the gas and liquid superficial velocities. The novelty of this article relies upon coupling the analysis of the behavior of transition-related parameters to the physical effects that are responsible for stabilizing and destabilizing the flow interface. The coupling of the transition analysis with the physical parameters showed potential as a reliable way of explaining the obtained transition behavior. By doing this, the stabilizing effects of gravity and surface tension are found to be invariable to the superficial velocities of the phases. On the other hand, the destabilizing effect of inertia increased with phase superficial velocities, while the shear stress increases with the liquid superficial velocity and shows a non-monotonic behavior with the gas superficial velocity. Although the overall trend of ωI,max, kmax and kc was to increase with the superficial velocities of the phases, they were directly affected by the shear stress behavior, also showing a non-monotonic trend.

Modulation instability in nonlinear media with sine-oscillatory nonlocal response function and pure quartic diffraction.

Modulation instability of one-dimensional plane wave is demonstrated in nonlinear Kerr media with sine-oscillatory nonlocal response function and pure quartic diffraction. The growth rate of modulation instability, which depends on the degree of nonlocality, coefficient of quartic diffraction, type of the nonlinearity and the power of plane wave, is analytically obtained with linear-stability analysis. Different from other nonlocal response functions, the maximum of the growth rate in media with sine-oscillatory nonlocal response function occurs always at a particular wave number. Theoretical results of modulation instability are confirmed numerically with split-step Fourier transform. Modulation instability can be controlled flexibly by adjusting the degree of nonlocality and quartic diffraction.

Linear Stability Analysis of Relativistic Magnetized Jets: The Minimalist Approach

Linear Stability Analysis of Relativistic Magnetized Jets: The Minimalist Approach

Emergence of a hexagonal pattern in shear-thickening suspensions under orbital oscillations

A dense particle suspension under shear may lose its uniform state to large local density and stress fluctuations, which challenge the mean-field description of the system. Here, we explore the novel dynamics of a non-Brownian suspension under orbital oscillations, where localized density waves along the flow direction appear beyond an excitation frequency threshold and self-organize into a hexagonal pattern across the system. The spontaneous occurrence of the inhomogeneity pattern arises from a coupling between particle advection and the shear-thickening nature of the suspension. Through linear stability analysis, we show that they overcome the stabilizing effects of particle pressure at sufficient particle volume fraction and oscillation frequency. In addition, the long-standing density waves degenerate into random fluctuations when replacing the free surface with rigid confinement. It indicates that the shear-thickened state is intrinsically heterogeneous, and the boundary conditions are crucial for developing local disturbance.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.