Numerical Linear Stability Analysis Research Articles

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.

Transition to fully developed turbulence in liquid-metal convection facilitated by spatial confinement

Using thermal convection in liquid metal, we show that strong spatial confinement not only delays the onset Rayleigh number $Ra_c$ of Rayleigh–Bénard instability but also postpones the various flow-state transitions. The $Ra_c$ and the transition to fully developed turbulence Rayleigh number $Ra_f$ depend on the aspect ratio $\varGamma$ with $Ra_c\sim \varGamma ^{-4.05}$ and $Ra_f\sim \varGamma ^{-3.01}$ , implying that the stabilization effects caused by the strong spatial confinement are weaker on the transition to fully developed turbulence when compared with that on the onset. When the flow state is characterized by the supercritical Rayleigh number $Ra/Ra_{c}$ ( $Ra$ is the Rayleigh number), our study shows that the transition to fully developed turbulence in strongly confined geometries is advanced. For example, while the flow becomes fully developed turbulence at $Ra\approx 200Ra_c$ in a $\varGamma =1$ cell, the same transition in a $\varGamma =1/20$ cell only requires $Ra\approx 3Ra_c$ . Direct numerical simulation and linear stability analysis show that in the strongly confined regime, multiple vertically stacked roll structures appear just above the onset of convection. With an increase of the driving strength, the flow switches between different-roll states stochastically, resulting in no well-defined large-scale coherent flow. Owing to this new mechanism that only exists in systems with $\varGamma <1$ , the flow becomes turbulent in a much earlier stage. These findings shed new light on how turbulence is generated in strongly confined geometries.

Spontaneous oscillation of an active filament under viscosity gradients.

We investigate the effects of uniform viscosity gradients on the spontaneous oscillations of an elastic, active filament in viscous fluids. Combining numerical simulations and linear stability analysis, we demonstrate that a viscosity gradient increasing from the filament's base to tip destabilises the system, facilitating its self-oscillation. This effect is elucidated through a reduced-order model, highlighting the delicate balance between destabilising active forces and stabilising viscous forces. Additionally, we reveal that while a perpendicular viscosity gradient to the filament's orientation minimally affects instability, it induces asymmetric ciliary beating, thus generating a net flow along the gradient. Our findings offer new insights into the complex behaviours of biological and artificial filaments in complex fluid environments, contributing to the broader understanding of filament dynamics in heterogeneous viscous media.

Numerical Analysis on Shimmying Wheels with Dry Friction Damper

The dynamics of the 1.5-degree-of-freedom model of towed wheel is investigated. Dry friction at the king pin is considered, leading to a non-smooth dynamical system. Beyond analytical and numerical linear stability analysis, the nonlinear vibrations are investigated by numerical bifurcation analysis with smoothing and by numerical simulations with event handling. The effect of dry friction at the king pin on the birth of separated periodic branches is presented on bifurcation diagrams. The presence of bistable parameter domains is also shown. The effect of smoothing is investigated by comparing bifurcation diagrams of the smoothed and the original non-smooth systems.

Bifurcations to transient and oscillatory excitations in inhomogeneous excitable media: Insights into arrhythmogenesis in long QT syndrome.

Ventricular arrhythmias are the leading cause of sudden cardiac death. Understanding the mechanisms of arrhythmia initiation is important for developing effective therapeutics for prevention. Arrhythmias can be induced via premature external stimuli or occur spontaneously via dynamical instabilities. Computer simulations have shown that a large repolarization gradient due to regional prolongation of the action potential duration can result in instabilities leading to premature excitations and arrhythmias, but the bifurcation remains to be elucidated. In this study we carry out numerical simulations and linear stability analyses using a one-dimensional heterogeneous cable consisting of the FitzHugh-Nagumo model. We show that a Hopf bifurcation leads to local oscillations, which, once their amplitudes are large enough, lead to spontaneous propagating excitations. Depending on the degree of heterogeneities, these excitations can range from one to many and to be sustained oscillations, manifesting as premature ventricular contractions (PVCs) and sustained arrhythmias. The dynamics depends on the repolarization gradient and the length of the cable. Complex dynamics is also induced by the repolarization gradient. The mechanistic insights from the simple model may help in the understanding of the genesis of PVCs and arrhythmias in long QT syndrome.

High-Order-Accurate Incompressible Flow Solver for High-Fidelity Numerical Simulations and Linear Stability Analysis

The development of high-fidelity versatile incompressible Navier–Stokes solver that is applicable for both direct numerical simulations (DNSs) and linear global stability investigations is presented and discussed. The solver is based on a vorticity–velocity formulation of the Navier–Stokes equations for orthogonal curvilinear grids. A rigorous and efficient approach is proposed that guarantees the divergence-free condition for the velocity and vorticity fields. The code incorporates advanced numerical algorithms including compact finite differences in combination with a pseudo-spectral method, together with an efficient and high-order-accurate Poisson solver. Linear stability modules based on the linearized Navier–Stokes equations (LNSEs) tailored for primary and secondary instability analyses are developed and incorporated into the code, such that the solver can switch from DNS to LNSE in a consistent manner by turning off the nonlinear terms and vice versa. The linear framework includes all the nonparallel effects with respect to the steady/unsteady baseflow and the form of disturbances, and because it is formulated as an initial value problem, both convective and absolute/global instability mechanisms can be investigated. An additional significant advantage of the linear module is that, for the secondary stability analysis, time-periodic assumption is not required and/or the unsteady baseflow does not have to be computed a priori. In the present paper, results obtained with the new Navier–Stokes solver are compared with benchmark solutions for the flow past a static and oscillating circular cylinder. The new code then was employed for a three-dimensional DNS of the flow for a wing section with a modified NACA 6 series airfoil at a chord Reynolds number of . Furthermore, the capability of the linear stability modules was demonstrated by investigating the primary and secondary (convective and absolute) instability mechanisms for boundary layers and the flow past a circular cylinder.

Asymmetric localized states at a nonlinear interface of fractional systems with optical lattices

We investigate the existence and stability of localized gap states at a non-linear interface of non-linear fractional systems in a one-dimensional photonic lattice. By using the direct numerical simulations and linear stability analysis, we obtain the stability of the asymmetric localized gap states in the first and second finite gaps. Our theoretical results show that the power of the localized gap states decrease gradually as the increase of propagation constant and the non-linear landscape (non-linear coefficient ratio between the left and right interface), providing insights into soliton physics in non-linear periodic systems with fractional-order diffraction.

Vorticity phase separation and defect lattices in the isotropic phase of active liquid crystals.

We use numerical simulations and linear stability analysis to study the dynamics of an active liquid crystal film on a substrate in the regime where the passive system would be isotropic. Extensile activity builds up local orientational order and destabilizes the quiescent isotropic state above a critical activity, eventually resulting in spatiotemporal chaotic dynamics akin to the one observed ubiquitously in the nematic state. Here we show that tuning substrate friction yields a variety of emergent structures at intermediate activity, including lattices of flow vortices with associated regular arrangements of topological defects and a new state where flow vortices trap pairs of +1/2 defect that chase each other's tail. These chiral units spontaneously pick the sense of rotation and organize in a hexagonal lattice, surrounded by a diffuse flow of opposite rotation to maintain zero net vorticity. The length scale of these emergent structures is set by the screening length of the flow, controlled by the shear viscosity η and the substrate friction Γ, and can be captured by simple mode selection of the vortical flows. We demonstrate that the emergence of coherent structures can be interpreted as a phase separation of vorticity, where friction plays a role akin to that of birth/death processes in breaking conservation of the phase separating species and selecting a characteristic scale for the patterns. Our work shows that friction provides an experimentally accessible tuning parameter for designing controlled active flows.

Nonlinear exceptional-point lasing with ab initio Maxwell–Bloch theory

We present a general analysis for finding and characterizing nonlinear exceptional point (EP) lasers above threshold using steady-state ab initio Maxwell–Bloch equations. For a system of coupled slabs, we show that a nonlinear EP is obtained for a given ratio between the external pumps in each resonator and that it is associated with a kink in the output power and lasing frequency, confirming coupled-mode theory predictions. Through numerical linear stability analysis, we confirm that the EP laser can be stable for a large enough inversion relaxation rate. We further show that the EP laser can be characterized by scattering a weak signal off the lasing cavity so that the scattering frequency spectrum exhibits a quartic divergence. Our approach can be applied to arbitrary scatterers with multi-level gain media.

Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

Influence of Centrifugal Buoyancy in Thermal Convection within a Rotating Spherical Shell

The dynamo action, which is of importance in the study of the geomagnetism mechanism, is considered to be caused by the convection structure formed inside a rotating spherical shell. This convection structure elongated in the rotation axis is generated by the action of both heat and rotation on the fluid inside a spherical shell. In this study, we analyzed thermal convection in such a rotating spherical shell and attempted to understand the phenomenon of this convective structure. It is known that each value of the Prandtl number, the Ekman number and the Rayleigh number and their balance are important for the generation of such convective structure. We fixed these three parameters and considered the effect of centrifugal buoyancy as the Froude number additionally. To investigate how the effects of centrifugal buoyancy affect the convective structure, we carried out both three-dimensional numerical simulations and linear stability analyses. In particular, we focused on the transition from axisymmetric flow to non-axisymmetric flow having wavenumbers in the toroidal direction and investigated both growth rate and phase velocity of the disturbance. It was found that axisymmetric flow tends to be maintained as the effect of centrifugal buoyancy increases.

Investigation on transition correlation of conical boundary layer in hypersonic wind tunnel experiments

The wind tunnel experiment is an important approach to studying boundary layer transition. In this study, the effects of bluntness, unit Reynolds number, and wall temperature ratio on hypersonic conical boundary layer transition were studied using numerical simulations and linear stability analyses based on the wind tunnel experimental data. Through the flow similarity principle, the treatment of the bluntness-varying problem was determined. Combining the experimental transition data and theoretical analysis results of sharp and blunt cones at home and abroad, the relation between the transition factor Ntr and the unit Reynolds number before ‘N factor reversal’, as well as the relation between the transition value Rtr and the unit Reynolds number from ‘N factor reversal’ to ‘transition reversal’ were determined too. Finally, according to the wind tunnel test data of China Aerodynamics Research and Development Center, the effect of wall temperature ratio on the receptivity of blunt-cone boundary layer was stated out, and the correlation of transition Reynolds number under different wall temperature ratios was determined. The influence law of parameters in this correlation was also determined.

T-shaped micromixers aligned in a row: characterization of the engulfment regime

Accurate control of mixing between two fluids is a fundamental aspect in many applications and generally implies the use of small devices operating at low velocities. This is often achieved using micromixers which, due to the combination of small dimensions and low velocities, work in the laminar regime and generally process very limited flow rates. The flow rates can be increased using more mixers at the same time. In this respect it is appealing to use several micromixers connected in a unique device. In this paper we propose and characterize a simple strategy to connect several T-mixers together by aligning them in a row and feeding them by shared inlet channels. Since engulfment is the preferred flow regime for mixing, we investigate the proposed devices specifically focusing on the onset and on the properties of the engulfment regime. This investigation is carried out by a combined use of numerical simulation and linear stability analysis. The results reported here show that the proposed strategy can lead to compact devices in which the single mixers manifest engulfment, although some important differences in comparison with isolated T-mixers may exist depending on the spacing between the inlet/outlet channels in the device.

Active Extensile Stress Promotes 3D Director Orientations and Flows.

We use numerical simulations and linear stability analysis to study an active nematic layer where the director is allowed to point out of the plane. Our results highlight the difference between extensile and contractile systems. Contractile stress suppresses the flows perpendicular to the layer and favors in-plane orientations of the director. By contrast extensile stress promotes instabilities that can turn the director out of the plane, leaving behind a population of distinct, in-plane regions that continually elongate and divide. This supports extensile forces as a mechanism for the initial stages of layer formation in living systems, and we show that a planar drop with extensile (contractile) activity grows into three dimensions (remains in two dimensions). The results also explain the propensity of disclination lines in three dimensional active nematics to be of twist type in extensile or wedge type in contractile materials.

Stability of multi-hump localized solutions in the Holstein model for linear acoustic and soft nonlinear optical interactions

We consider the energy/electron transport problem along a lattice holding linear acoustic and soft cubic non-dispersive interactions in the long wave limit for Holstein’s approach. It is found that the model equations, consisting of the coupling between the linear Schrödinger and a nonlinear Klein–Gordon equations, support one- and two-hump solutions of the first and second kind. We derive these multi-humped solutions by using a compatible reduction of the equations of motion and by means of an adaptation of the direct method, both in the traveling frame. Remarkably, the solution profiles for the mechanical deformation recall the one and two soliton solutions of the Korteweg–de Vries equation. Then, based on the Fourier collocation method, we perform a numerical linear stability analysis of these solutions at the steady state to find the parameters regime for stability. Additionally, we are able to implement the Vakhitov–Kolokolov criterion to get the bifurcation value for the one-hump solution both of the first and second kind. We show that the two-hump solutions are linearly unstable.

Rayleigh–Taylor instability of classical diffusive density profiles for miscible fluids in porous media: a linear stability analysis

A Rayleigh–Taylor instability typically develops when a denser layer overlies a less dense one in the gravity field. In that case, the initial base state density profile is a step function for which linear stability analysis results are well known. We investigate here analytically the linear stability analysis of other classical diffusive density profiles for porous media flows. We find that, for a species A initially distributed in the upper half of the domain with an initial concentration profile of \((-X)^m\) for \(0<m<1\) where X is the vertical coordinate, and absent from the bottom half of the domain, for large times the eigenfunctions grow like \(\exp \left( \omega _0 T^{(m+1)/2} + \omega _1 \ln (T) \right) \) where \(\omega _0\) and \(\omega _1\) are constants and T is time. Thus, the growth rate defined by \((1/A) (\mathrm{{d}}A/\mathrm{{d}}T)\) decays like \(c_1 T^{(m-1)/2} +c_2 T^{(m-2)/3}\) whilst the maximum growing wavenumber scales with \(T^{(m-2)/6}\). These results are compared to the growth rates obtained using numerical linear stability analysis. Our analytical predictions provide a set of generalised results that pave the way to the analysis of Rayleigh–Taylor instabilities of nontrivial density profiles.

Magnetoconvection in a horizontal duct flow at very high Hartmann and Grashof numbers

Direct numerical simulations and linear stability analysis are carried out to study mixed convection in a horizontal duct with constant-rate heating applied at the bottom and an imposed transverse horizontal magnetic field. A two-dimensional approximation corresponding to the asymptotic limit of a very strong magnetic field effect is validated and applied, together with full three-dimensional analysis, to investigate the flow's behaviour in the previously unexplored range of control parameters corresponding to typical conditions of a liquid metal blanket of a nuclear fusion reactor (Hartmann numbers up to $10^4$ and Grashof numbers up to $10^{10}$ ). It is found that the instability to quasi-two-dimensional rolls parallel to the magnetic field discovered at smaller Hartmann and Grashof numbers in earlier studies also occurs in this parameter range. Transport of the rolls by the mean flow leads to magnetoconvective temperature fluctuations of exceptionally high amplitudes. It is also demonstrated that quasi-two-dimensional structure of flows at very high Hartmann numbers does not guarantee accuracy of the classical two-dimensional approximation. The accuracy deteriorates at the highest Grashof numbers considered in the study.

Instability of a planar fluid interface under a tangential electric field in a stagnation point flow

The interface between two immiscible fluids can become unstable under the effect of an imposed tangential electric field along with a stagnation point flow. This canonical situation, which arises in a wide range of electrohydrodynamic systems including at the equator of electrified droplets, can result in unstable interface deflections where the perturbed interface gets drawn along the extensional axis of the flow while experiencing strong charge build-up. Here, we present analytical and numerical analyses of the stability of a planar interface separating two immiscible fluid layers subject to a tangential electric field and a stagnation point flow. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a local linear stability analysis in the vicinity of the stagnation point to study the behaviour of the system in terms of the relevant dimensionless groups of the problem. The local theory is complemented with a numerical normal-mode linear stability analysis based on the full system of equations and boundary conditions using the boundary element method. Our analysis demonstrates the subtle interplay of charge convection and conduction in the dynamics of the system, which oppose one another in the dominant unstable eigenmode. Finally, numerical simulations of the full nonlinear problem demonstrate how the coupling of flow and interfacial charge dynamics can give rise to nonlinear phenomena such as tip formation and the growth of charge density shocks.

Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations

The instability of surface lenticular vortices is investigated using a comprehensive suite of laboratory experiments combined with numerical linear stability analysis as well as nonlinear numerical simulations in a two-layer Rotating Shallow Water model. The development of instabilities is discussed and compared between the different methods. The linear stability analysis allows for a clear description of the origin of the instability observed in both the laboratory experiments and numerical simulations. While global qualitative agreement is found, some discrepancies are observed and discussed. Our study highlights that the sensitivity of the instability outcome is related to the initial condition and the lower-layer flow. The inhibition or even suppression of some unstable modes may be explained in terms of the lower-layer potential vorticity profile.

Linear and Nonlinear Stability Analyses of Double-Diffusive Convection in a Vertical Brinkman Porous Enclosure under Soret and Dufour Effects

Analytical and numerical investigations were performed to study the influence of the Soret and Dufour effects on double-diffusive convection in a vertical porous layer filled with a binary mixture and subject to horizontal thermal and solute gradients. In particular, the study was focused on the effect of Soret and Dufour diffusion on bifurcation types from the rest state toward steady convective state, and then toward oscillatory convective state. The Brinkman-extended Darcy model and the Boussinesq approximation were employed to model the convective flow within the porous layer. Following past laboratory experiments, the investigations dealt with the particular situation where the solutal and thermal buoyancy forces were equal but acting in opposite direction to favor the possible occurrence of the rest state condition. For this situation, the onset of convection could be either supercritical or subcritical and occurred at given thresholds and following various bifurcation routes. The analytical investigation was based on the parallel flow approximation, which was valid only for a tall porous layer. A numerical linear stability analysis of the diffusive and convective states was performed on the basis of the finite element method. The thresholds of supercritical, RTCsup, and overstable, RTCover, convection were computed. In addition, the stability of the established convective flow, predicted by the parallel flow approximation, was studied numerically to predict the onset of Hopf’s bifurcation, RTCHopf, which marked the transition point from steady toward unsteady convective flows; a route towards the chaos. To support the analytical analyses of the convective flows and the numerical stability methodology and results, nonlinear numerical solutions of the full governing equations were obtained using a second-order finite difference method. Overall, the Soret and Dufour effects were seen to affect significantly the thresholds of stationary, overstable and oscillatory convection. The Hopf bifurcation was marked by secondary convective flows consisting of superposed vertical layers of opposite traveling waves. A good agreement was found between the predictions of the parallel flow approximation, the numerical solution and the linear stability results.