Floquet Stability Analysis Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Floquet stability analysis of pulsatile flow in toroidal pipes

The linear temporal stability of the fully developed pulsatile flow in a torus with high curvature is investigated using Floquet theory. The baseflow is computed via a Newton-Raphson iteration in frequency space to obtain basic states at supercritical Reynolds numbers in the steady case for two curvatures, δ=0.1 and 0.3, exhibiting structurally different linear instabilities for the steady flow. The addition of a pulsatile component is found to be overall stabilizing over a wide range of pulsation amplitudes, in particular for high values of curvature. The pulsatile flows are found to be at most transiently stable with large intracyclic growth rate variations even at small pulsation amplitudes. While these growth rates are likely insufficient to trigger an abrupt transition at the parameters in this work, the trends indicate that this is indeed likely for higher pulsation amplitudes, similar to pulsatile flow in straight pipes. At the edge of the considered parameter range, subharmonic eigenvalue orbits in the local spectrum of the time-periodic operator, recently found in pulsating channel flow, have been confirmed also for pulsatile flow in toroidal pipes, underlining the generality of this phenomenon. Published by the American Physical Society 2024

Instability mechanisms of symmetry breaking in the wakes of two anti-phase pitching foils at zero free-stream velocity

In the present study, we investigate into the symmetry-breaking phenomenon in the wakes produced by two anti-phase pitching foils at zero free-stream velocity. Numerical simulations are performed in a gap ratio range of 0.6≤g≤2.0 and a Reynolds number range of 70≤Re≤200. Six regions are identified in the plane (g, Re), based on distinct symmetry-breaking (or symmetry-preserving) behaviors of the wakes. Floquet stability analysis and dynamic mode decomposition are performed on some selected cases. By means of modal analysis, the behaviors of wakes in different regions are rationalized and the instability mechanisms of symmetry breaking in the unstable regions are unveiled. In addition, the influence of symmetry-breaking instability on propulsive performance is also discussed.

Periodic flow features in a planar sudden expansion with pulsatile inflow velocity

Flow through sudden expansion finds its application in several engineering and biological processes. Though the stability of flow through steady sudden expansion has garnered much attention, little to none is given to the pulsatile flow through sudden expansion. Hence, in the present work we study the influence of inflow pulsatility on flow characteristics in a sudden expansion. The inflow velocity is a sinusoidal waveform that is modulated to encompass a wide range of amplitudes, ${{a}}$ , and reduced velocities, ${{U_{r}}}$ . We report four different modes, namely, synchronized growth of the recirculation region (at high ${{U_{r}}}$ ), necking and diffusion of the recirculation region (at moderately high ${{U_{r}}}$ ), splitting and convection of the recirculation region (at moderate ${{U_{r}}}$ ) and inverse growth of the recirculation region (at low ${{U_{r}}}$ ). In each mode, the symmetry-breaking critical Reynolds number is obtained through numerical experiments and compared with those of Floquet stability analysis. We found that diffusion and the convection mode of the recirculation region increases the stability of the flow while the inverse growth mode of the recirculation region decreases the same. The effect of the expansion ratio, ${{ER}}$ , is also explored, and we found that as ${{ER}}$ increases, the absolute stability of flow decreases, but relative stability between the modes remains similar. Finally, we explain the dynamics of the modes by using terms involving the vorticity transport equation.

Floquet stability analysis of a two-layer oscillatory flow near a flexible wall

We investigate the linear Floquet stability of two fluid layers undergoing oscillations in the direction parallel to the flexible wall that separates them. This canonical configuration is inspired by the cerebrospinal fluid flow in the spinal canal of subjects with hydromyelia/syringomyelia. The analysis focuses on the marginal conditions for the onset of instability, and how these depend on the spatial wavelength of the perturbation, and on the values of the control parameters, which are the two channel widths, the Reynolds number and the wall stiffness. Unstable perturbations are found to oscillate synchronous with the base flow. The wavelength of the most unstable perturbation, of the order of the stroke length of the basic oscillatory motion, depends strongly on the wall stiffness, but is only weakly influenced by the channel widths and the Reynolds number. In general, around criticality, it was found that increasing the Reynolds number has a destabilizing effect, and that decreasing the canal widths stabilizes the instability. The wall stiffness on the other hand has a non-monotonic effect, exhibiting an intermediate value for which the instability is maximally amplified. The present analysis is a first step towards a better understanding of the physical mechanisms that govern many (bio)fluid mechanical problems that involve oscillatory flows near compliant walls.

Flow regime identification and flow instability analysis of oscillatory flows over twin circular cylinders

Oscillatory flows past two identical circular cylinders are investigated by two-dimensional direct numerical simulations in the parameter space of gap ratio (0.5 ≤ G ≤ 4.0), angle of flow incidence (0° ≤ α ≤ 90°) and Keulegan–Carpenter number (4 ≤ KC ≤ 12) with a constant Reynolds number Re = 150, where G = L/D, KC = UmT/D and Re = UmD/υ with D being the dimeter of the identical cylinders, L the shortest surface-to-surface distance between the two cylinders, Um and T being the velocity amplitude and period of the sinusoidal oscillatory flow, respectively, and α is defined as the angle between the flow direction to the line connecting the centers of the two cylinders. Comparing with the tandem or side-by-side arrangements of twin circular cylinders in oscillatory flows, the staggered twin cylinders (0° < α < 90°) involve more diverse flow regimes, including the periodic, quasi-periodic and chaotic flow states, due to the inherent asymmetric flow interference around the cylinder pair. In addition to introducing four flow regimes for the tandem and side-by-side arrangements, this study newly identifies 11 flow regimes for the staggered twin cylinders. The newly reported flow regimes in this work are collaboratively identified through the flow visualizations, steady streaming, frequency spectra of fluid forces and Lissajous phase diagrams, as well as the temporal-spatial symmetry features of the wake flows. Connecting with the previous work by Zhao and Cheng [“Two-dimensional numerical study of vortex shedding regimes of oscillatory flow past two circular cylinders in side-by-side and tandem arrangements at low Reynolds numbers,” J. Fluid Mech. 751, 1–37 (2014)], this study presents overall regime maps in the KC-α plane for varied gap ratios. It is found that the flow regimes previously and presently identified for the tandem and side-by-side arrangements may also appear for the staggered twin cylinders. The present numerical results suggest the sensitive dependence of the flow regimes on the parameters of KC, α, and G. It is also found that a specific flow regime with narrow parameter bands may appear within another flow regime, forming the abnormal regime hole in the regime map. To understand the profound influence of the control parameters on the flow regime transition, and the relevant temporal-spatial symmetry breaking, the linear Floquet stability analysis is conducted in this work. It was confirmed that the variation of the KC number may result in the Ky symmetry breaking over several periodic flow regimes, while the change of the angle of flow incidence may account for the H2 symmetry covering various periodic and quasi-periodic flow regimes. The stability analysis also explains the temporal flow transition and the abnormal occurrence of the regime holes within either quasi-periodic or chaotic flow regimes.

Two- and three-dimensional wake transitions of a NACA0012 airfoil

Flow transitions are an important fluid-dynamic phenomena for many reasons, including the direct effect on the aerodynamic forces acting on the body. In the present study, two-dimensional (2-D) and three-dimensional (3-D) wake transitions of a NACA0012 airfoil are studied for angles of attack in the range $0^\circ \leq \alpha \leq 20^\circ$ and Reynolds numbers $500 \leq {\textit {Re}} \leq 5000$ . The study uses water-channel experiments and 2-D and 3-D numerical simulations based on the nodal spectral-element method, level-set function-based immersed-interface method and Floquet stability analysis. The different wake states are categorised based on the time-instantaneous wake structure, non-dimensional frequency and aerodynamic force coefficients. The wake states and transition boundaries are summarised in a wake regime map. The critical angle of attack and Reynolds number for the supercritical Hopf bifurcation (i.e. steady to periodic wake transition) varies as $\alpha _1 {\sim} {\textit {Re}}^{-0.65}$ , while the critical angle of attack for the onset of three dimensionality varies as $\alpha _{3D} {\sim} {\textit {Re}}^{-0.5}$ . Over the entire Reynolds number range, the transition to 3-D flow occurs through a mode C (subharmonic) transition. Beyond this initial transition, further instabilities of the 2-D periodic base flow arise and are investigated. For instance, at $ {\textit {Re}}=2000$ and $\alpha _{3D,2}=11.0^\circ$ , mode C coexists together with modes related to modes A and QP seen in a stationary circular cylinder wake. In contrast, at $ {\textit {Re}}=5000$ and $\alpha _{3D,2}=8.0^\circ$ , the dominant mode C coexists with mode QP. Three-dimensional simulations well beyond critical angles indicate that 2-D vortex-street transitions are approximately maintained in the fully saturated 3-D wakes in a spanwise-averaged sense.

Floquet stability analysis of equilibrium points and numerical exploration of pulsating zero-velocity curves and Newton–Raphson basins of attraction

An important and pivot aspect of a dynamical system is the stability property and its range, which play a crucial role towards the stabilisation of a mission design. Present paper deals with Floquet stability analysis of equilibrium points and estimation of pulsating zero velocity curves and Newton–Raphson basins of attraction associated to these equilibrium points under the radiation pressure effect in the photo-gravitational planar elliptic restricted four body problem. Floquet stability test for the equilibrium points is performed with the help of the transition curves, which are generated under the influence of radiation pressure. It is noticed that stability range of the respective equilibrium points have deviated due to radiation pressure. To observe the possible regions of motion for restricted body, the pulsating zero velocity curves are numerically explored after establishing an invariant relation. It is found that not only the radiation parameter but also eccentricity and true anomaly reflect a considerable impact on the shape and size of forbidden regions. To see the tendency of randomly selected points in the phase space, Newton–Raphson basins of attraction associated to each attractor are estimated and effects of perturbing parameters are analysed. It is observed that effects of radiation and mass parameters are considerable however, that of eccentricity and true anomaly are negligible. Moreover, from probability distribution bar diagram, most probable number of iterations is 10 for the case when two primaries are of equal masses and third one is different, whereas it is 20 for the case when all primaries are of identical mass. These results will be helpful to study the generalised problem along with other kind of perturbations.

Aeroelastic modal dynamics of floating wind turbines in anisotropic conditions based on Floquet analysis

A reduced order model of a floating wind turbine with 22 degrees of freedom has been implemented and validated in comparison to the multibody, hydro-servo-aero-elastic code hGAST. The multi-blade Coleman transformation is applied and Floquet stability analysis is performed. This allows on one hand, to verify the actual Floquet implementation against the standard Coleman-based eigenvalue analysis in cases the transformed dynamic system has constant coefficients and on the other hand to analyze the effects of different anisotropies on stability. Mass imbalance of the rotor and yaw misalignment have been specifically addressed. In the mass imbalance scenario, the homogeneous results are not substantially altered while in the wind yaw scenario, there is slight change of the flapwise natural frequencies and of the tower lateral frequency as well as of the corresponding damping characteristics.

Along-the-path exponential integration for Floquet stability analysis of wind turbines

Traditionally, stability assessment of wind turbines has been performed by eigenanalysis of the azimuthally-averaged linearized system after applying the Multi-Blade Coordinate (MBC) transformation. However, due to internal or external anisotropy, the MBC transform does not produce an exact Linear Time-Invariant (LTI) system, and a Floquet analysis is required to capture the influence of all periodic terms, leading to a more accurate stability analysis. In this paper exponential integration methods that use system linearizations at different blade azimuth positions are used to integrate the perturbed system state and compute the Floquet monodromy matrix. The proposed procedure is assessed for a simple 6 DOF wind turbine model and a more complex aeroelastic model of a 5MW onshore wind turbine. The defined along-the-path or moving-point exponential integrator is found to be the suitable in order to perform a Floquet stability analysis even using a coarse linearization grid.

Floquet solitons in square lattices: Existence, stability, and dynamics.

In the present work, we revisit a recently proposed and experimentally realized topological two-dimensional lattice with periodically time-dependent interactions. We identify the fundamental solitons, previously observed in experiments and direct numerical simulations, as exact, exponentially localized, periodic in time solutions. This is done for a variety of phase-shift angles of the central nodes upon an oscillation period of the coupling strength. Subsequently, we perform a systematic Floquet stability analysis of the relevant structures. We analyze both their point and their continuous spectrum and find that the solutions are generically stable, aside from the possible emergence of complex quartets due to the collision of bands of continuous spectrum. The relevant instabilities become weaker as the lattice size gets larger. Finally, we also consider multisoliton analogs of these Floquet states, inspired by the corresponding discrete nonlinear Schrödinger (DNLS) lattice. When exciting initially multiple sites in phase, we find that the solutions reflect the instability of their DNLS multi-soliton counterparts, while for configurations with multiple excited sites in alternating phases, the Floquet states are spectrally stable, again analogously to their DNLS counterparts.

Sub-terahertz-repetition-rate frequency comb generated by filter-induced instabilities in passive driven fiber resonators

Ultrahigh-repetition-rate frequency comb generation exhibits great potential in applications of optical waveform synthesis, direct comb spectroscopy, and high capacity telecommunications. Here we present the theoretical investigations of a filter-induced instability mechanism in passive driven fiber resonators with a wide range of cavity dispersion regimes. In this novel concept of modulation instability, coherent frequency combs are demonstrated numerically with rates up to sub-terahertz level. Floquet stability analysis based on the Ikeda map is utilized to understand the physical origin of the filter-induced instability. Comparison with the well-known Benjamin–Feir instability and parametric instability is performed, revealing the intrinsic distinction in the family of modulation instabilities. Our investigations might benefit the development of ultrahigh-repetition-rate frequency comb generation, providing an alternative method for the microresonators.

Transition control of cylinder wake via Lorentz force

The flow around a cylinder is a typical flow acting as the oscillator, and the perturbations can grow with time at a fixed position. This phenomenon can induce the laminar-turbulent transition in the wake, thus increasing the vibrations of the cylinder as well as the noise in the flow system. There exist three control strategies, i.e. the streamwise Lorentz force from the electromagnetic actuator set on the windward surface named windward control, the leeward surface named leeward control, and the whole surface of cylinder named global control, which are adopted to modify the two-dimensional base flow, thereby reducing the growth rates of perturbations in the wake and changing the transition mode. According to the Floquet stability analysis, it is found that the growth rates of the transition modes A and B present small changes in the windward control, while the growth rates of the two modes decrease with the increase of the control number in the other two control cases. Comparing the inviscid growth rates induced by the elliptic instability and the hyperbolic instability with each other, it is observed that the high inviscid growth rate in the windward control can be similar to those without control, while the inviscid growth rates can decrease with the increase of control number in the other two control cases. Three-dimensional direct numerical simulations are performed to validate the control effects. The results shows that the three-dimensional shape of the wake is changed from mode B to mode A when the actuator is set on the leeward surface or the whole surface of the cylinder. This is consistent with the result from the Floquet stability analysis. In addition, the drag of the cylinder reduces 15.2% for the leeward control and 14.4% for the the global control.

Dynamics and stability of the wake behind a circular cylinder in the vicinity of a plane moving wall

This paper identifies and characterizes modes of the three-dimensional instability of the flow over a circular cylinder of diameter D near a plane moving wall with a gap height of G=0.1D beyond the onset of vortex shedding through the Floquet stability analysis. The associated effects of the transitions on the hydrodynamic forces of the cylinder are examined by performing direct numerical simulations. Six distinct three-dimensional modes are identified and their characteristics are discussed in detail. It is found that the physical mechanism responsible for the synchronous mode 1 appears to be associated to an elliptic instability. The quasi-periodic mode 4 is observed to transform to a synchronous mode (mode 6) on increasing the Reynolds number, indicating that they are likely to be resulted from the same physical mechanism. For mode 6, its physical nature seems to be a hyperbolic instability. Fully three-dimensional simulations show that the wake undergoes a rapid transition to complex chaotic flow state after the most dominant mode starts to saturate, due to the strongly nonlinear interactions of many modes with different spanwise wavelengths.

Influence of Spectral Filtration on Pulse Dynamics in Ring-Cavity Mamyshev Oscillator

Here we present a numerical study of pulsing build-up dynamics inside the fiber Mamyshev Oscillator (MO). The main scope of the investigation is to describe the influence of the spectral separation between the filters on self-starting MO dynamics and transition from multipulse to single-pulse generation regimes. It was found that Floquet stability analysis provides a straightforward way to determine whether the system will be self-starting or if it has to be excited by external source and predicts the repetition rate of the pulse train. We showed that spectrally overlapped bandpass filters provide reliable multi-pulse generation due to Faraday instability. Adiabatic increase in the spectral separation between the filters decreases the number of pulses down to single-pulse regime, therefore providing a flexible way to generate adjustable number of mode-locked pulses on demand.

A stability perspective of bioinspired unmanned aerial vehicles performing optimal dynamic soaring

The phenomenon of dynamic soaring, as exhibited by soaring birds, has long been a biological inspiration for aerospace and control engineers. If this fascinating phenomenon, which allows soaring birds to perform almost unpowered flight using wind shear, can be mimicked by unmanned aerial vehicles (UAVs), then there is substantial potential for technological and economic enhancement of UAV performance. Although there has been a considerable amount of research covering the modeling, optimization, control and simulation aspects of different UAVs performing dynamic soaring, there is little to no conclusive work analyzing the stability of such UAVs in soaring orbits. In this paper we present a comprehensive framework for determining the stability of soaring UAVs utilizing both linear (Floquet-based) and nonlinear (contraction theory-based) techniques. Floquet stability analysis was inconclusive, which led to the use of a nonlinear contraction formulation to reach a conclusive stability assessment for an actual nonlinear fixed-wing UAV performing dynamic soaring. Furthermore, parametric variation along with numerical simulations were conducted to ascertain the response of the actual nonlinear system when perturbed from the nominal motion studied in this paper. The analysis and simulations revealed that the system possesses instability as the UAV motion diverges from its nominal trajectory and follows an undesirable path. From this result we conclude, for the first time in the literature as far as we are aware, that UAVs performing dynamic soaring in an optimal way to reduce wind shear requirements are inherently unstable. The results of this work suggest that mimicking of dynamic soaring by UAVs will require careful investigation of tracking and regulatory controls that need to be implemented.

Floquet modes and stability analysis of periodic orbit-attitude solutions along Earth\u2013Moon halo orbits

Future space programmes pose some interesting research problems in the field of non-Keplerian dynamics, being the Moon and the cislunar space central in the proposed roadmap for the future space exploration. In these regards, the deployment of a cislunar space station on a non-Keplerian orbit in the lunar vicinity is a fundamental milestone to be achieved. The paper investigates the natural orbit-attitude dynamics and the attitude stabilisation of coupled motions for extended bodies in the Earth–Moon system. The discussion is carried out analysing the phase space of natural dynamics, constituted by both the orbital and the rotational periodic motions of a spacecraft in cislunar orbits. Floquet theory is applied to periodic orbit-attitude solutions in lunar proximity, to characterise their attitude stability properties and their attitude manifolds, which are discussed and analysed focusing on their dynamical features applicable to cislunar environment. Attitude stabilisation methods are proposed and developed, with particular attention to spin-stabilised solutions. Periodic orbit-attitude dynamics are studied to highlight possible favourable conditions that may be exploited to host a cislunar space station with a simplified control action. The focus of the analysis is dedicated to halo orbits and near-rectilinear halo orbit in the circular restricted three-body problem Earth–Moon system.

Effects of abdomen undulation in energy consumption and stability for monarch butterfly

The flight of monarch butterflies is characterized by a relatively large wing, flapping at a relatively low frequency coupled with abdomen undulation. This paper presents the dynamics of a flapping wing flyer that can be applied to the coupled motion of the wing, body, and abdomen at the monarch butterfly scale, which is formulated directly on the configuration manifold. The resulting thorax and abdomen motion as well as the resultant forces are consistent with the flight of a live monarch butterfly. Based on these, beneficial effects of the abdomen undulation in the flight of monarch butterflies are illustrated. For both hover and forward-climbing trajectories, the abdomen undulation results in a reduction of the energy and power consumption. Furthermore, the Floquet stability analysis shows that the periodic orbits associated with both flight modes are stable. In particular, the abdomen undulation improves the stability. Compared to the dynamics of hawkmoth, bumblebee, and fruitfly models, the monarch possesses superior stability properties.

Flow past a transversely oscillating cylinder at lock-on region and three-dimensional Floquet stability analysis of its wake

A numerical investigation is conducted on the flow past a transversely oscillating square cylinder at various amplitudes with a Reynolds number of 200 in the “lock-on” region. Floquet stability analysis was then applied to study the three-dimensional instability of its wake. The results show that mean recirculating region sizes decrease and force coefficients increase as the oscillating amplitude increases. The dynamic mode decomposition analysis reveals that although oscillating amplitude changes the topology of high order harmonic modes, the primary modes share similar features, which result in the classical “2S” mode at various amplitudes. The distinct high-order-harmonic modes lead to the difference of the flow pattern at various amplitudes. For the three-dimensional instability, the oscillating amplitudes can be divided into three stages, I (0 ≤ A/D < 0. 3), II (0.3 ≤ A/D < 0.4), and III (0.4 ≤ A/D < 0.7). In region I, the dominant Floquet mode is Mode-A corresponding to a long wavelength λz/D ≈ 3.49. As the oscillating amplitude increases to region III, the wavelength of the dominant mode abruptly changes to a much shorter wavelength λz/D ≈1.14 (a new mode called Mode-SA). Mode-SA shows similar features to Mode-A, except that this mode dissipates rapidly in the wake. Region II is the transitional region.

Fluid–solid Floquet stability analysis of self-propelled heaving foils

Abstract