High-amplitude Limit Research Articles

Reduced-order modelling of thermoacoustic instabilities in can-annular combustors

Thermoacoustic instabilities in stationary gas turbines may cause high-amplitude limit cycles, leading to damaged components and costly down-time. To better understand the physical origin of such instabilities in a can-annular combustor configuration, we study the properties of the spectrum of a reduced-order can-annular thermoacoustic system. Increased focus is placed on representing the aeroacoustic interaction between the longitudinal eigenmodes of the individual cans with physically relevant models. To represent the acoustic pressure dynamics in the combustor, we combine an analytical, experimentally validated model for the can-to-can impedance with a frequency-dependent model of the flame response in the cans to acoustic perturbations. By using this approach, we perform a parametric study of the linear stability of an atmospheric can-annular thermoacoustic system, and emphasize general features of the structure and properties of the eigenvalues and the eigenvectors of can-annular combustors. Lastly, we emphasize the differences in the can-to-can coupling that arise when considering open-end boundary conditions – representative of atmospheric set-ups – or closed-end boundary conditions – representative of real gas turbine combustors.

Experimental investigation on effects of injection distribution on combustion instability in a model rocket combustor

Self-excited combustion instabilities of transverse modes were experimentally investigated in a rectangular multi-injector model combustor, operating with the bipropellants O2/CH4. The propellants were injected through a linear array of five oxidizer-centered shear coaxial injectors into the combustor. High-amplitude limit cycles obtained in hot-fire tests were analyzed in detail. Different combustion instability modes, including first and second width modes, were observed in cases with three different injection distribution schemes. Hence, the injection distribution strongly determined the combustion dynamics. One insight can be gained that the stable combustion could be achieved by properly designing the propellants' injection distributions.

Aeroelastic Simulation of Stall Flutter Undergoing High ‎and Low Amplitude Limit Cycle Oscillations

The aeroelastic behaviour of an airfoil oscillating in large and small pitch amplitudes due to nonlinearity ‎in aerodynamics is examined. The phenomenon of stall flutter resulted in the limit cycle oscillations of ‎NACA 0012 at low to intermediate Reynolds number is investigated numerically through the unsteady ‎two-dimensional aeroelastic simulation. The simulations employed unsteady Reynolds Average Navier ‎Stokes shear stress transport k-ω turbulent model with the low Reynolds number correction. The ‎simulations of the fluid-structure interaction were performed by coupling the structural equation of ‎motion with a fluid solver through the user-defined function utility. Numerical simulations were executed ‎at three different elastic axis positions; the leading-edge, 18% and 36% of the airfoil chord length. The ‎airfoil chord measures 0.156 m. The simulations were executed at the free stream velocity ranging from ‎‎5.0 m/s to 13 m/s corresponding to the Reynolds number between 51618 and 134207. Two types of ‎oscillation amplitudes were observed at each elastic axis position. At the leading-edge and 18% case, ‎small amplitude oscillations were observed while at 36%, the system underwent high amplitude ‎oscillations. The analysis revealed the cause for small oscillation amplitude is due to the separation of the ‎laminar boundary layer on the suction side of the airfoil starting at the trailing edge. High amplitude ‎oscillations occurred due to the existence of the dynamic stall phenomenon beginning at the leading-edge. ‎Small amplitude LCOs only occurred within a limited range of airspeed before it disappeared due to ‎increasing airspeed‎‎.

Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators.

We apply the framework of optimal nonlinear control to steer the dynamics of a whole-brain network of FitzHugh-Nagumo oscillators. Its nodes correspond to the cortical areas of an atlas-based segmentation of the human cerebral cortex, and the internode coupling strengths are derived from diffusion tensor imaging data of the connectome of the human brain. Nodes are coupled using an additive scheme without delays and are driven by background inputs with fixed mean and additive Gaussian noise. Optimal control inputs to nodes are determined by minimizing a cost functional that penalizes the deviations from a desired network dynamic, the control energy, and spatially nonsparse control inputs. Using the strength of the background input and the overall coupling strength as order parameters, the network's state-space decomposes into regions of low- and high-activity fixed points separated by a high-amplitude limit cycle, all of which qualitatively correspond to the states of an isolated network node. Along the borders, however, additional limit cycles, asynchronous states, and multistability can be observed. Optimal control is applied to several state-switching and network synchronization tasks, and the results are compared to controllability measures from linear control theory for the same connectome. We find that intuitions from the latter about the roles of nodes in steering the network dynamics, which are solely based on connectome features, do not generally carry over to nonlinear systems, as had been previously implied. Instead, the role of nodes under optimal nonlinear control critically depends on the specified task and the system's location in state space. Our results shed new light on the controllability of brain network states and may serve as an inspiration for the design of new paradigms for noninvasive brain stimulation.

Dynamics of the vacuum state in a periodically driven Rydberg chain

We study the dynamics of the periodically driven Rydberg chain starting from the state with zero Rydberg excitations (vacuum state denoted by $|0\rangle$) using a square pulse protocol in the high drive amplitude limit. We show, using exact diagonalization for finite system sizes ($L\le 26$), that the Floquet Hamiltonian of the system, within a range of drive frequencies which we chart out, hosts a set of quantum scars which have large overlap with the $|0\rangle$ state. These scars are distinct from their counterparts having high overlap with the maximal Rydberg excitation state ($|\mathbb{Z}_2\rangle$); they coexist with the latter class of scars and lead to persistent coherent oscillations of the density-density correlator starting from the $|0\rangle$ state. We also identify special drive frequencies at which the system undergoes perfect dynamic freezing and provide an analytic explanation for this phenomenon. Finally, we demonstrate that for a wide range of drive frequencies, the system reaches a steady state with sub-thermal values of the density-density correlator. The presence of such sub-thermal steady states, which are absent for dynamics starting from the $|\mathbb{Z}_2\rangle$ state, imply a weak violation of the eigenstate thermalization hypothesis in finite sized Rydberg chains distinct from that due to the scar-induced persistent oscillations reported earlier. We conjecture that in the thermodynamic limit such states may exist as pre-thermal steady states that show anomalously slow relaxation. We supplement our numerical results by deriving an analytic expression for the Floquet Hamiltonian using a Floquet perturbation theory in the high amplitude limit which provides an analytic, albeit qualitative, understanding of these phenomena at arbitrary drive frequencies. We discuss experiments which can test our theory.

Active dynamic vibration absorber for flutter suppression

A novel flutter suppression technique is proposed using an active dynamic vibration absorber (ADVA). The ADVA is introduced by adding an active element to a classical mass-spring-damper system. This active element drives the mass by a controlling force using feedback signal from the response of the aeroelastic system. An aeroelastic mathematical model of a 2 degrees of freedom (DOFs) airfoil equipped with the proposed ADVA is established based on unsteady aerodynamics theory. The proposed ADVA is experimentally realized by a cantilever beam with a bonded Macro-fiber Composite (MFC) and a lumped mass. MFC nonlinearities are compensated by applying Prandtl-Ishlinskii approach to design a proportional integral (PI)-based hysteresis compensator controller. A test rig is designed, fabricated and the parameters of the physical airfoil model and the connected ADVA are identified. The feasibility of the ADVA for flutter suppression is assessed via a wind tunnel testing. The experimental results show that applying the closed-loop ADVA increases the airfoil flutter speed by 42.9%. Whereas the open-loop ADVA results in 26.6% improvement in the flutter speed. Moreover, if the airfoil experiences a high amplitude limit cycle oscillation (LCO), the ADVA efficiently suppresses up to 93.3% of its amplitude when turned to the closed-loop control.

Nucleation and propagation of excitation fronts in self-excited systems

The paper considers the transition between spatially localized and propagating stick–slip motion in frictional systems. To overcome the difficulties related to the non-smoothness of the friction law we have studied two models: a friction-excited chain of weakly coupled oscillators excited by a frictional moving belt (model I) and a chain of Van der Pol oscillators (model II). The two models show very similar dynamical features. Both exhibit discrete breathers (i.e. spatially localized periodic solutions) for low coupling. For higher coupling, both exhibit propagation of switch fronts from low to high-amplitude limit cycle oscillations. Besides, in both models the transition from discrete breathers to propagating limit cycles happens through a nucleation process and, above a certain critical coupling, the velocity of propagation scales linearly with the elastic coupling coefficient. We suggest that dynamical features similar to what was demonstrated in this work are expected for models that are different quantitatively but preserve the same structure of the state space in every single unit.

Efficient potential well escape for bi-stable Duffing oscillators

The problem of escape from a potential well of bi-stable oscillators has attracted attention given the diversity of physical and engineering systems described by this mathematical model. Most previous studies have considered quasi-static dynamics leading to escape. In devising efficient escape strategies for structures, transient conditions have not yet received adequate consideration. In this study, the intra-well nonlinear resonant dynamics of bi-stable systems are studied and exploited, yielding a time-efficient strategy for triggering minimal amplitude escape by employing transient perturbations. The response characteristics of both, the symmetric and asymmetric double-well Duffing oscillators are explored analytically to identify the stable solution branches for any given forcing configuration. Based on the basins of attraction of the stable attractors, a novel actuation methodology employing controlled perturbations in the phase of the forcing for driving the system into a series of high-amplitude limit cycle oscillations and eventual escape to the desired stable solution is proposed. Additionally, accelerated settling to the desired configuration is achieved by implementing state feedback techniques. The proposed algorithm serves as a potential tool for implementing fast shape adaptation in bi-stable structural systems.

Identification of freeplay and aerodynamic nonlinearities in a 2D aerofoil system with via higher-order spectra

ABSTRACTHigher-Order Spectra (HOS) are used to characterise the nonlinear aeroelastic behaviour of a plunging and pitching 2-degree-of-freedom aerofoil system by diagnosing structural and/or aerodynamic nonlinearities via the nonlinear spectral content of the computed displacement signals. The nonlinear aeroelastic predictions are obtained from high-fidelity viscous fluid-structure interaction simulations. The power spectral, bi-spectral and tri-spectral densities are used to provide insight into the functional form of both freeplay and inviscid/viscous aerodynamic nonlinearities with the system displaying both low- and high-amplitude Limit Cycle Oscillation (LCO). It is shown that in the absence of aerodynamic nonlinearity (low-amplitude LCO) the system is characterised by cubic phase coupling only. Furthermore, when the amplitude of the oscillations becomes large, aerodynamic nonlinearities become prevalent and are characterised by quadratic phase coupling. Physical insights into the nonlinearities are provided in the form of phase-plane diagrams, pressure coefficient distributions and Mach number flowfield contours.

Analysis of the Rudenko-Solodov equation in the theory of highly nonlinear shear vibrations

An analytic solution is found using the discrete model of highly nonlinear shear vibrations in the high-amplitude approximation, and its spectrum is determined. It is shown that, in the high-amplitude limit, the period of vibrations tends to a linear value, while the vibrations remain nonsinusoidal.

Interference mechanisms of acoustic/convective disturbances in a swirl-stabilized lean-premixed combustor

Interference mechanisms of acoustic/convective disturbances were experimentally investigated in a swirl-stabilized lean-premixed gas turbine combustor operated with natural gas fuel and air at atmospheric pressure and elevated temperature. Interference between azimuthal and acoustic velocity disturbances at high-amplitude limit cycle oscillations is characterized in detail as a function of axial swirler location, oscillation frequency, and mean nozzle velocity. We show that both the frequency and the intensity of self-excited instabilities in a model gas turbine combustor are correlated with axial swirler position, which indicates that a vorticity wave generated at the swirl vanes is a primary source of convective disturbances in the absence of equivalence ratio nonuniformities. Flame transfer function measurements confirm that the linear/nonlinear heat release response is a strong function of axial swirler location, even when unforced flame structures remain unchanged. The key parameter controlling this phenomenon is the phase difference between the azimuthal and acoustic velocity perturbations at the combustor dump plane; the phase difference is affected by swirler location, frequency, mean velocity, and the speed of sound. It was found that out-of-phase interference between azimuthal and acoustic velocity disturbances at the combustor inlet yields large flame angle fluctuations in relation to swirl number fluctuations, and therefore the formation of a coherent structure is hindered due to high kinematic viscosity within the vortex formation region. In-phase interference mechanisms, on the other hand, lead to high-amplitude limit cycle oscillations. This interference mechanism is then explored in the presence of temporal equivalence ratio nonuniformities, in which two different sources of convective mechanisms should be considered simultaneously in connection with acoustic velocity perturbations and the vortex dynamics. Results reveal that equivalence ratio oscillation has a significant effect on the strength of combustion-acoustic interactions. Strong self-excited instabilities of partially premixed flames are produced by in-phase interactions between acoustic velocity and equivalence ratio oscillations, which are governed by fuel injection location, frequency, mean nozzle velocity, and fuel injector impedance. At this phase condition, unburned reactants with high equivalence ratio impinge on the flame front with high inlet velocity, potentially causing large fluctuations of heat release rate.

Flutter of a two-dimensional wing with asymmetry at low Mach numbers

A two-dimensional wing aeroelastic model with a modified Leishman–Beddoes model in state-space form is performed to investigate the flutter behavior at low Mach numbers. The two-dimensional wing aeroelastic system employs highly aerodynamic non-linear features. The modifications to the Leishman–Beddoes model are validated by comparing with the experimental results of airfoil airloads at low Mach numbers, and the wing flutter experimental results also verify the model of the two-dimensional wing aeroelastic system. Results demonstrate that a fold bifurcation, which is related to the high-amplitude limit cycle oscillation, stems from the phenomenon of dynamic stall and the emergence of the period-2 limit cycle oscillation is due to the asymmetry of the aeroelastic system.

Direct flutter and limit cycle computations of highly flexible wings for efficient analysis and optimization

The usefulness of flutter as a design metric is diluted for wings with destabilizing (softening) nonlinearities, as a stable high-amplitude limit cycle (subcritical) may exist for flight speeds well below the flutter point. It is thus desired to design aeroelastic structures such that the post-flutter behavior is as benign (i.e., supercritical) as possible, among the other constraints commonly considered in the optimization process. In order to account for these metrics in an accurate and efficient manner, direct tools are utilized to first locate the Hopf-point (flutter speed), and then to obtain a nonlinear perturbation solution via the method of multiple scales. The latter scheme provides a scalar variable whose sign and magnitude dictate the nature of the limit cycle. The accuracy of these methods is demonstrated with a high-aspect-ratio highly flexible wing, modeled with nonlinear beam finite elements and the ONERA dynamic stall tool. Stiffness and inertial design variables are allowed to vary spatially throughout the wing, in order to conduct gradient-based optimization of the limit cycle under flutter and mass constraints. The resulting wing structure demonstrates strongly supercritical behavior, as well as several design conflicts between linear (flutter) and nonlinear (limit cycles) sensitivities, which are not present in the uniform baseline wing.

Measurements of triggering and transient growth in a model lean-premixed gas turbine combustor

Instability triggering and transient growth of thermoacoustic oscillations were experimentally investigated in combination with linear/nonlinear flame transfer function (FTF) methodology in a model lean-premixed gas turbine combustor operated with CH 4 and air at atmospheric pressure. A fully premixed flame with 10 kW thermal power and an equivalence ratio of 0.60 was chosen for detailed characterization of the nonlinear transient behaviors. Flame transfer functions were experimentally determined by simultaneous measurements of inlet velocity fluctuations and heat release rate oscillations using a constant temperature anemometer and OH ∗/CH ∗ chemiluminescence emissions, respectively. The phase-resolved variation of the local flame structure at a limit cycle was measured by planar laser-induced fluorescence of OH. Simultaneous measurements of inlet velocity, OH ∗/CH ∗ emission, and acoustic pressure were performed to investigate the temporal evolution of the system from a stable to a limit cycle operation. This measurement allows us to describe an unsteady instability triggering event in terms of several distinct stages: (i) initiation of a small perturbation, (ii) exponential amplification, (iii) saturation, (iv) nonlinear evolution of the perturbations towards a new unstable periodic state, (v) quasi-steady low-amplitude periodic oscillation, and (vi) fully-developed high-amplitude limit cycle oscillation. Phase-plane portraits of instantaneous inlet velocity and heat release rate clearly show the presence of two different attractors. Depending on its initial position in phase space at infinitesimally small amplitude, the system evolves towards either a high-amplitude oscillatory state or a low-amplitude oscillatory state. This transient phenomenon was analyzed using frequency- and amplitude-dependent damping mechanisms, and compared to subcritical and supercritical bifurcation theories. The results presented in this paper experimentally demonstrate the hypothesis proposed by Preetham et al. based on analytical and computational solutions of the nonlinear G-equation [J. Propul. Power 24 (2008) 1390–1402]. Good quantitative agreement was obtained between measurements and predictions in terms of the conditions for the onset of triggering and the amplitude of triggered combustion instabilities.

Existence and stability of the resonant phenomena in the dc- and ac-driven overdamped Frenkel-Kontorova model with the incommensurate structure

Dynamical mode-locking phenomena in the incommensurate structures of the dc- and ac-driven overdamped Frenkel-Kontorova model are studied by molecular-dynamics simulations. The obtained results have shown that Shapiro steps exhibit significantly different amplitude and frequency dependence from the one observed in the commensurate structures. Due to the incommensurability of the system the special symmetry of the motion of particles is broken, and in the amplitude dependence of Shapiro steps, this will result in the appearance of anomalies and deviation from the well-known Bessel-like behavior. In the frequency or period dependence, oscillations have been observed in the high-amplitude limit; however, they exhibit strong anomalies compared with those in the commensurate structures. The oscillatory behavior and the anomalies have been also be revealed in the (F(ac),F) and the (ν(0),F) phase diagrams where several phases are defined.

Flutter and Stall Flutter of a Rectangular Wing in a Wind Tunnel

The aeroelastic behavior of a rectangular wing with pitch and plunge degrees of freedom was observed experimentally using pressure, acceleration, and particle image velocimetry measurements. The wing was set at different static angles of attack and wind-tunnel airspeeds. The wing’s dynamic behavior was governed by a twoparameter bifurcation from steady to limit cycle oscillations, with the two parameters being the airspeed and the static angle of attack. At the lowest static angle, the wing underwent a classical flutter phenomenon that was transformed into a supercriticalHopf bifurcation at higher angles. The latterwas combinedwith a fold bifurcation at intermediate angles of attack. All limit cycle oscillations observed were either low-amplitude oscillations with timevarying amplitude or high-amplitude oscillations with nearly steady amplitude. They were caused by two different types of dynamic stall phenomena. During low-amplitude limit cycle oscillations the periodically stalled flow covered only the rear part of the wing. During high-amplitude limit cycle oscillations, trailing-edge and leading-edge separation occurred. Trailing-edge separation was characterized by a significant amount of unsteadiness, varying visibly from cycle to cycle. The occurrence of leading-edge separation was muchmore regular and had the tendency to stabilize the amplitude of the limit cycle oscillation motion.

Frequency dependence of the subharmonic Shapiro steps

Frequency dependence of the subharmonic Shapiro steps has been studied in the ac driven overdamped Frenkel-Kontorova model with deformable substrate potential. As potential gets deformed, in addition to the harmonic steps, subharmonic steps appear in the number and size that increase as the frequency of the external force increases. It was found that size of both harmonic and subharmonic steps strongly depend on the frequency where in the high-amplitude limit oscillatory dependence appears. When expressed as a function of period these oscillations of the step size with frequency have the same form as the oscillations of the step size with amplitude. Deformation of the potential has strong influence on these oscillations, and as in the case of amplitude dependence, with the increase of deformation, the same three distinctive types of behavior have been classified.

Tilted phase-space measurements in the eight-port homodyne detection scheme

We show that the phase shift of {\pi}/2 is crucial for the phase space translation covariance of the measured high-amplitude limit observable in eight-port homodyne detection. However, for an arbitrary phase shift {\theta} we construct explicitly a different nonequivalent projective representation of R$^2$ such that the observable is covariant with respect to this representation. As a result we are able to determine the measured observable for an arbitrary parameter field and phase shift. Geometrically the change in the phase shift corresponds to the tilting of one axis in the phase space of the system.

Realistic eight-port homodyne detection and covariant phase space observables

We consider the quantum optical eight-port homodyne detection scheme in the case that each of the associated photon detectors is assigned with a different quantum efficiency. We give a mathematically rigorous and strictly quantum mechanical proof of the fact that the measured observable (positive operator measure) in the high-amplitude limit is a smearing of the covariant phase space observable related to the ideal measurement, that is, the measurement performed with fully efficient detectors. The result is proved for an arbitary parameter field. Furthermore, we investigate some properties of the measured observable. In particular, we show that detector inefficiencies do not affect the observable's ability to distinguish between different states.

A note on the measurement of phase space observables with an eight-port homodyne detector

It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically [see e.g. Leonhardt, U.; Paul, H. Phys. Rev. A 1993, 47, R2460–R2463]. Using recent rigorous results on the quantum theory of homodyne detection observables [Kiukas, J.; Lahti, P. J. Mod. Opt., in press (see arXiv:0706.4436v1 [quant-ph])], we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.