Floquet Stability Analysis Research Articles

Nonlinear dynamics of a thin liquid film deposited on a laterally oscillating corrugated surface in the high-frequency limit

The nonlinear dynamics of thin liquid films deposited on various periodically corrugated substrates, both left-right symmetric and asymmetric, subjected to lateral vibration in the high-frequency limit is investigated. The method used to derive the governing evolution equation is based on the long-wave approximation, multiscale time expansion, and averaging over the fast time scale. The resulting evolution equation contains the effects of gravity, capillarity, vibration, and the substrate topology. The initial-boundary-value problem associated with this evolution equation is numerically solved and the system behavior is investigated for a variety of parameter sets. Typical patterns emerging as a result of the film evolution include hump formation within the troughs of the substrate and homogenized coatings whose configuration resembles that of the substrate, as well as the possibility of film rupture. We show that the choice of the vibration parameters and the topological features of the substrate may be used for controlling the shape of the film interface as well as its properties such as the amplitude, continuity, or rupture. Together with the film profiles stationary in terms of the averaged film interface with respect to the slow time scale, time evolution of the total (comprising of the averaged and pulsating components) interfacial profiles and streamline maps is presented to illustrate the film flow. We carry out Floquet stability analysis of periodically replicated steady states for the time-independent problem, linear stability analysis based on a reduced low-order projection approximation for the time-dependent problem, and stability analysis with respect to disturbances of a larger wavelength. We have found also that in the case of two-dimensional corrugated substrates, the vast majority of two-dimensional steady-state flows in terms of the averaged film interface exhibit stability in three dimensions with respect to small perturbations in the transverse spatial direction.

Instabilities of interacting vortex rings generated by an oscillating disk.

We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number β=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of β, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},β) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.

A MIMO periodic ARX identification algorithm for the Floquet stability analysis of wind turbines

The paper presents a new stability analysis approach applicable to wind turbines. At first, a reduced order periodic model is identified from response time histories, and then stability is assessed using Floquet theory. The innovation of the proposed approach is in the ability of the algorithm to simultaneously consider multiple response time histories, for example in the form of measurements recorded both on the rotor and in the stand still system. As each different measurement carries a different informational content on the system, the simultaneous use of all available signals improves the quality and robustness of the analysis.

Liquid film flow along a substrate with an asymmetric topography sustained by the thermocapillary effect

We investigate flow in a thin liquid film over a “thick” asymmetric corrugated surface in a gas-liquid bi-layer system. Using long-wave approximation, we derive a nonlinear evolution equation for the spatiotemporal dynamics of the liquid-gas interface over the corrugated topography. A closed-form expression indicating a non-zero value for a liquid flow rate is derived in a steady state of the system. Through numerical investigations we study the nonlinear dynamics of the liquid-gas interface with respect to topographical variations of the solid surface, different thermal properties of the liquid and the solid, and different values of the Marangoni number. We find the existence of a critical value for the Marangoni number Mc, so that for M > Mc, the liquid film ruptures, whereas for M < Mc, the interface will remain continuous. In a broad variety of parameters, the interface attains a deformed steady state with a nonzero average flow rate through the system, thus the described mechanism may be used as a means of transport in microfluidic devices. We carry out the Floquet stability analysis of periodic steady states with respect to spatial replication and show that in the framework of the time-independent evolution equation, the system is unstable to long wave perturbations. We demonstrate that in a finite periodic setting, the system may evolve within a certain parameter range into a metastable state which may be manipulated by varying the Marangoni number M in time in order to increase, control, and sustain the average flow rate through the system. We also show that in the case of a solid substrate with the thermal conductivity lower than that of the liquid, the flow rate through the system may be significantly increased with respect to the opposite case.

Three-dimensional instabilities in the wake of a flapping wing at low Reynolds number

We present a stability analysis of a plunging and pitching wing of infinite aspect ratio at low Reynolds number. Four cases are considered by varying the mean pitch angle and the phase shift between pitching and plunging. Each case is studied by performing two dimensional DNS and Floquet stability analysis. The four cases considered display different wake structures resulting in different mean aerodynamic forces. Two cases produce thrust and lift, one case only thrust (with symmetric plunging and pitching) and the remaining case mainly lift (with the highest mean pitch angle). In addition, the latter case displays a period doubling phenomenon, and it is found to be linearly unstable for long wavelengths, with an instability mode that resembles that of mode A found in the wake of cylinders. Other cases, although being linearly stable, present a convective instability at smaller wavelengths. Finally, the unstable case has been studied with a fully 3D DNS to evaluate the effect of the three-dimensionality on the forces. The resulting flow structure is consistent with the linear stability analysis in the near wake. Further downstream nonlinearities lead to a fully 3D wake. Despite this, the aerodynamic forces on the 3D wing are very similar to those obtained in the 2D simulation.

Flow instabilities in the wake of a rounded square cylinder

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.

Parametric Resonance in Spherical Immersed Elastic Shells

We perform a stability analysis for a fluid-structure interaction problem in which a spherical elastic shell or membrane is immersed in a 3D viscous, incompressible fluid. The shell is an idealised structure having zero thickness, and has the same fluid lying both inside and outside. The problem is formulated mathematically using the immersed boundary framework in which Dirac delta functions are employed to capture the two-way interaction between fluid and immersed structure. The elastic structure is driven parametrically via a time-periodic modulation of the elastic membrane stiffness. We perform a Floquet stability analysis, considering the case of both a viscous and inviscid fluid, and demonstrate that the forced fluid-membrane system gives rise to parametric resonances in which the solution becomes unbounded even in the presence of viscosity. The analytical results are validated using numerical simulations with a 3D immersed boundary code for a range of wavenumbers and physical parameter values. Finally, potential applications to biological systems are discussed, with a particular focus on the human heart and investigating whether or not fluid-structure interaction-mediated instabilities could play a role in cardiac fluid dynamics.

Dependence on aspect ratio of symmetry breaking for oscillating foils: implications for flapping flight

Using two-dimensional direct numerical simulations, we investigate the flow in a fluid of kinematic viscosity ${\it\nu}$ and density ${\it\rho}$ around elliptical foils of density ${\it\rho}_{s}$ with major axis $c$ and minor axis $b$ for three different aspect ratios: $AR=b/c=1$ (a circle); $AR=0.5$; and $AR=0.1$. The vertical location of these foils $y_{s}(t)=A\sin (2{\rm\pi}f_{0}t)$ oscillates with amplitude $A$ and frequency $f_{0}$ in two distinct ways: ‘pure’ oscillation, where the foils are constrained to remain in place; and ‘flying’ oscillation, where horizontal motion is allowed. We simulate the flow for a range of the two appropriate control parameters, the non-dimensional amplitude, or Keulegan–Carpenter number $KC=2{\rm\pi}A/c$, and the non-dimensional frequency, or Stokes number ${\it\beta}=f_{0}c^{2}/{\it\nu}$. We observe three distinct patterns of asymmetry, labelled ‘S-type’ for synchronous asymmetry, ‘$\text{QP}_{\text{H}}$-type’ and ‘$\text{QP}_{\text{L}}$-type’ for quasi-periodic asymmetry at sufficiently high and sufficiently low (i.e. $AR=0.1$) aspect ratios, respectively. These patterns are separated at the critical locus in $KC$–${\it\beta}$ space by a ‘freezing point’ where the two incommensurate frequencies of the QP-type flows combine, and we show that this freezing point tends to occur at smaller values of $KC$ as $AR$ decreases. We find for the smallest aspect ratio case ($AR=0.1$) that the transition to asymmetry, for all values of $KC$, occurs for a critical value of an ‘amplitude’ Stokes number ${\it\beta}_{A}={\it\beta}(KC)^{2}=4{\rm\pi}^{2}f_{0}A^{2}/{\it\nu}\simeq 3$. The $\text{QP}_{\text{L}}$-type asymmetry for $AR=0.1$ is qualitatively different in physical and mathematical structure from the $\text{QP}_{\text{H}}$-type asymmetry at higher aspect ratio. The flows at the two ends of the ellipse become essentially decoupled from each other for the $\text{QP}_{\text{L}}$-type asymmetry, the two frequencies in the horizontal force signature being close to the primary frequency, rather than twice the primary frequency as in the $\text{QP}_{\text{H}}$-type asymmetry. Furthermore, the associated coefficients arising from a Floquet stability analysis close to the critical thresholds are profoundly different for low aspect ratio foils. Freedom to move slightly suppresses the transition to S-type asymmetry, and for certain parameters, if a purely oscillating foil subject to S-type asymmetry is released to move, flow symmetry is rapidly recovered due to the negative feedback of small horizontal foil motion. Conversely, for the ‘higher’ aspect ratios, the transition to $\text{QP}_{\text{H}}$-type asymmetry is encouraged when the foil is allowed to move, with strong positive feedback occurring between the shed vortices from successive oscillation cycles. For $AR=0.1$, freedom to move significantly encourages the onset of asymmetry, but the newly observed ‘primary’ $\text{QP}_{\text{L}}$-type asymmetry found for pure oscillation no longer occurs when the foil flies, with S-type asymmetry leading ultimately to locomotion at a constant speed occurring all along the transition boundary for all values of $KC$ and ${\it\beta}$.

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

We consider a chain of torsionally-coupled, planar pendula shaken horizontally by an external sinusoidal driver. It has been known that in such a system, theoretically modeled by the discrete sine-Gordon equation, intrinsic localized modes, also known as discrete breathers, can exist. Recently, the existence of multifrequency breathers via subharmonic driving has been theoretically proposed and numerically illustrated by Xu et al. (2014) [21]. In this paper, we verify this prediction experimentally. Comparison of the experimental results to numerical simulations with realistic system parameters (including a Floquet stability analysis), and wherever possible to analytical results (e.g. for the subharmonic response of the single driven–damped pendulum), yields good agreement. Finally, we report the period-1 and multifrequency edge breathers which are localized at the open boundaries of the chain, for which we have again found good agreement between experiments and numerical computations.

Floquet and transient growth stability analysis of a flow through a compressor passage

In this paper we investigate the instabilities arising in a flow through a compressor passage using BiGlobal stability analysis. The adopted geometry comes from the results of previous experimental and numerical investigations on a linear low-pressure (LP) compressor cascade [6,19,20]. Specifically, we address the role of laminar separation of the boundary layers at Re=138,500, where such separation effects are enhanced by the strong adverse pressure gradients that the flow experiences, in contrast to the more commonly studied low-pressure (LP) turbines. The vortical structures downstream the separation bubble on the suction surface were recognised to show a well-defined time periodicity, which could be precisely detected. Floquet stability analysis was then used to investigate the response of the flow to infinitesimal perturbations. To overcome the difficulty of performing a Floquet stability analysis when the periodicity is restricted just to a small region of the domain, a phase-averaged base flow was computed, such that only the organised motions are extracted, neglecting all the background unsteadiness. The same technique allowed us to confirm the presence of strong energy transient growth phenomena, which are directly associated with convective instabilities occurring in the region downstream from the separation bubble.

An Immersed Boundary Model of the Cochlea with Parametric Forcing

The cochlea or inner ear has a remarkable ability to amplify sound signals. This is understood to derive at least in part from some active process that magnifies vibrations of the basilar membrane (BM) and the cochlear partition in which it is embedded, to the extent that it overcomes the effect of viscous damping from the surrounding cochlear fluid. Many authors have associated this amplification ability to some type of mechanical resonance within the cochlea, however there is still no consensus regarding the precise cause of amplification. Our work is inspired by experiments showing that the outer hair cells within the cochlear partition change their lengths when stimulated, which can in turn cause periodic distortions of the BM and other structures in the cochlea. This paper investigates a novel fluid-mechanical resonance mechanism that derives from hydrodynamic interactions between an oscillating BM and the surrounding cochlear fluid. We present a model of the cochlea based on the immersed boundary method, in which a small-amplitude periodic internal forcing due to outer hair cells can induce parametric resonance. A Floquet stability analysis of the linearized equations demonstrates the existence of resonant (unstable) solutions within the range of physical parameters corresponding to the human auditory system. Numerical simulations of the immersed boundary equations support the analytical results and clearly demonstrate the existence of resonant solution modes. These results are then used to illustrate the influence of parametric resonance on wave propagation along the BM and explicit comparisons are drawn with results from another two-dimensional cochlea model.

Delocalized periodic vibrations in nonlinear LC and LCR electrical chains

We consider electrical LC- and LCR-chains consisting of N cells. In the LC-chain each cell contains a linear inductor L and a nonlinear capacitor C, while the cell in the LCR-chain include additionally a resistor R and an voltage source. It is assumed that voltage dependence of capacitors represents an even function. Such capacitors have implemented by some experimental groups studying propagation of electrical signals in the lines constructed on MOS and CMOS substrates. In these chains, we study dynamical regimes representing nonlinear normal modes (NNMs) by Rosenberg. We prove that maximum possible number of symmetry-determined NNMs which can be excited in the considered chains is equal to 5. The stability of these modes for different N is studied with the aid of the group-theoretical method [Physical Review E 73 (2006) 36216] which allows to simplify radically the variational systems appearing in the Floquet stability analysis. For NNMs in LC-chain, the scaling of the voltage stability threshold in the thermodynamic limit (N→∞) is determined. It is shown that the above group theoretical method can be also used for studying stability of NNMs in the LCR-chains.

Gain sideband splitting in dispersion oscillating fibers

We analyze the modulation instability spectrum in a varying dispersion optical fiber as a function of the dispersion oscillation amplitude. For large dispersion oscillations, we predict a novel sideband splitting into different sub-sidebands. The emergence of the new sidebands is observed whenever the classical perturbation analysis for parametric resonances predicts vanishing sideband amplitudes. The numerical results are in good quantitative agreement with Floquet or Bloch stability analysis of four-wave mixing in the periodic dispersion fiber. We have also shown that linear gain or loss may have a dramatic influence in reshaping the new sidebands.

Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate

AbstractWhile the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

A review of rotating cylinder wake transitions

Recent work on the flow past a rotating cylinder is reviewed and further investigated at low Reynolds numbers. The various two- and three-dimensional transitions that occur as the rotation rate is increased are detailed. Two steady states, steady state I and steady state II, are identified based on the physical characteristics of the wake and the drag force on the body. Steady state I occurs at lower rotation rates, while state steady state II occurs at higher rotation rates. Linear stability analysis shows that two three-dimensional modes become unstable on steady state I and steady state II. Floquet stability analysis of the unsteady base flows that occur at very low rotation rates shows the presence of five three-dimensional modes. The curves of marginal stability are presented, followed by a comparison of numerical simulations to their experimentally obtained counterparts. Furthermore, the spatio-temporal characteristics of each mode and the likely underlying physical mechanisms are briefly discussed.

Three-dimensional instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder

Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D) instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder with various aspect ratios (AR). The conventional modes A, B, and QP vanish as AR decreases towards a thin normal flat plate, whereas new modes A2 and QP2 emerge. For “thin” rectangular cylinders, mode QP2 becomes unstable first with increasing Reynolds number (Re). In a certain range of AR, mode A becomes unstable and then stable again as Re increases, resulting in a closed curve on the neutral stability diagram. For validation, 3D simulations were also performed.

Regime Change Thresholds in Recorder-Like Instruments: Influence of the Mouth Pressure Dynamics

Since they correspond to a jump from a given note to another one, the mouth pressure thresholds leading to regime changes are particularly important quantities in flute-like instruments. In this paper, a comparison of such thresholds between an artificial mouth, an experienced flutist and a non player is provided. It highlights the ability of the experienced player to considerabily shift regime change thresholds, and thus to enlarge its control in terms of nuances and spectrum. Based on recent works on other wind instruments and on the theory of dynamic bifurcations, the hypothe- sis is tested experimentally and numerically that the dynamics of the blowing pressure influences regime change thresholds. The results highlight the strong influence of this parameter on thresholds, suggesting its wide use by experienced musicians. Starting from these observations and from an analysis of a physical model of flute-like instruments, involving numerical continuation methods and Floquet stability analysis, a phenomenological modelling of regime change is proposed and validated. It allows to predict the regime change thresholds in the dynamic case, in which time variations of the blowing pressure are taken into account.

Floquet stability analysis of the wake of an inclined flat plate

The route from a time-periodic two-dimensional wake flow to a three-dimensional flow has been investigated by means of linear Floquet stability analysis. The critical Reynolds number for the onset of three-dimensional instabilities in the wake behind a flat plate with an angle of attack α in the range from 20° to 30° with respect to the free stream was determined. For all three angles considered, in the lower wavelength range, the two-dimensional base flow first became unstable with respect to the sub-harmonic mode C. Although the critical Reynolds number decreased with increasing angle of attack, the spanwise wavelength remained close to two times the projected plate width. Qualitatively different transition scenarios were obtained for the three angles of attack with a particularly simple scenario for α = 30°.

Three-dimensional instability in the flow past two side-by-side square cylinders

The onset of the three-dimensional (3D) instability in the flow past two side-by-side square cylinders has been studied via Floquet stability analysis. The gap between the two cylinders is the key parameter that alters flow topology, leading to change of stability characteristics. Depending on the gap, four distinct flow patterns have been identified for base flow. For each pattern of base flow, distinctive Floquet modes of 3D instability have been found. The spatial and temporal characteristics of the Floquet modes are similar to those of mode A or mode B of the single square cylinder case, namely, odd or even Reflection-Translation symmetries. Six distinct Floquet modes have been identified in total, and the spatial and temporal characteristics of each mode are described in detail. The criticality of Floquet modes, including their critical Reynolds number and the associated dominant spanwise wave number, is presented for each gap considered in the current study. For a small gap, the flow past the two cylinders resembles that of a single object, resulting in an approximately doubled characteristic length. Consequently, the critical Reynolds number becomes low with a small gap between the two cylinders. When the two cylinders are moderately apart, the interaction between the two wakes diminishes, and the criticality of 3D instabilities approaches that of the single cylinder case. Coexisting 3D instabilities for a specific base-flow topology are also noticed with a moderate gap. Neutral stability curves are computed and presented for some selected values of the gap.

Flow instability in obstructed channel flow

Flow instability in a channel obstructed by an infinite array of equi-spaced circular cylinders has been numerically studied. An immersed boundary method was employed to facilitate the placement of the cylinders within a Cartesian grid system. This flow configuration is relevant to a heat exchanger in which vortex generators play an important role in enhancing its heat-transfer capacity. The presence of the circular cylinders arranged periodically in the streamwise direction causes a significant topological change of the flow, leading to increasing susceptibility to flow instability. A parametric study has been carried out to investigate the effects of Reynolds number (Re) and the gap (G) between the cylinders and the channel wall not only on the primary instability for Hopf bifurcation, but also on the secondary instability leading to three-dimensional flow. The blockage ratio (d/H) and the distance between two neighboring cylinder centers were fixed as 0.2 and 3.333H, respectively, where d and H represent the cylinder diameter and the channel height, respectively. The Stuart-Landau equation was used to compute the growth rate of the primary instability. The characteristics of the primary instability, including the critical Reynolds numbers and the patterns of the subsequent unsteady flow, were identified. In particular, the crossover of flow topology from flow separation on the channel wall to separated free shear layers from the cylinders turned out to be the key point to explain the flow characteristics of an obstructed channel flow. In addition, the effects of Re and G on the flow-induced forces and the frequency of vortex shedding are also reported. A Floquet stability analysis was employed to study the secondary instability at a higher Reynolds number. It reveals the critical Reynolds number for the three-dimensional instability along with the most unstable spanwise wave number associated with each G considered here. The dependency of the secondary instability on G is also addressed.