Littérature scientifique sur le sujet « Non-Modal stability theory »

In the context of geologic sequestration of carbon dioxide in saline aquifers, much interest has been focused on the process of density-driven convection resulting from dissolution of CO2 in brine in the underlying medium. Recent investigations have studied the time and length scales characteristic of the onset of convection based on the framework of linear stability theory. It is well known that the non-autonomous nature of the resulting matrix does not allow a normal mode analysis and previous researchers have either used a quasi-static approximation or solved the initial-value problem with arbitrary initial conditions. In this manuscript, we describe and use the recently developed non-modal stability theory to compute maximum amplifications possible, optimized over all possible initial perturbations. Non-modal stability theory also provides us with the structure of the most-amplified (or optimal) perturbations. We also present the details of three-dimensional spectral calculations of the governing equations. The results of the amplifications predicted by non-modal theory compare well to those obtained from the spectral calculations.

Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.

In this paper, we study the linear stability of a plane Couette flow of a power-law fluid. The influence of shear-thinning effect on the stability is investigated using the classical eigenvalue analysis, the energy method and the non-modal stability theory. For the plane Couette flow, there is no stratification of viscosity. Thus, for the stability problem the stress tensor is anisotropic aligned with the strain rate perturbation. The results of the eigenvalue analysis and the energy method show that the shear-thinning effect is destabilizing. We focus on the effect of non-Newtonian viscosity on the transition from laminar flow towards turbulence in the framework of non-modal stability theory. Response to external excitations and initial conditions has been studied by examining the ε-pseudospectrum and the transient energy growth. For both Newtonian and non-Newtonian fluids, it is found that there can be a rather large transient growth even though the linear operator of the Couette flow has no unstable eigenvalue. The results show that shear-thinning significantly increases the amplitude of response to external excitations and initial conditions.

In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$.

Layered flows that are commonly observed in stratified turbulence are susceptible to the Taylor–Caulfield Instability. While the modal stability properties of layered shear flows have been examined, the non-modal growth of perturbations has not been investigated. In this work, the tools of Generalized Stability Theory are utilized to study linear transient growth within a finite time interval of two-dimensional perturbations in an inviscid, three-layer constant shear flow under the Boussinesq approximation. It is found that, for low optimization times, small-scale perturbations utilize the Orr mechanism and achieve growth equal to that in the case of an unstratified flow. For larger optimization times, transient growth is much larger compared to growth for an unstratified flow as the Kelvin–Orr waves comprising the continuous spectrum of the dynamical operator and the gravity edge-waves comprising the discrete spectrum interact synergistically. Maximum growth is obtained for perturbations with scales within the region of instability, but significant growth is maintained for modally stable perturbations as well. For perturbations with scales within the unstable region, the unstable normal modes are excited at high amplitude by their bi-orthogonals. For perturbations with modally stable scales, the Orr mechanism is utilized to excite at high amplitude neutral propagating waves resembling the neutral Taylor–Caulfield modes.

The linear stability properties of viscous circular flows in a rotating environment are studied with respect to symmetric perturbations. Through the use of an effective energy or Lyapunov functional, we derive sufficient conditions for Lyapunov stability with respect to such perturbations. For circular flows with swirl velocity V(r) we find that Lyapunov stability is determined by the properties of the function ℱ(r) = (2V/r + f)/Q (with f the Coriolis parameter, r the radius and Q the absolute vorticity) instead of the customary Rayleigh discriminant Φ(r) = (2V/r + f)Q. The conditions for stability are valid for flows with non-zero Q everywhere. Further, the flows are presumed stationary, incompressible and velocity perturbations are required to vanish at rigid boundaries. For Lyapunov stable flows an upper bound for the increase of the total perturbation energy due to transient non-modal growth is derived which is valid for any Reynolds number. The theory is applied to Couette flow and the Lamb–Oseen vortex.

AbstractHomboe (Geophys. Publ., vol. 24, 1962, pp. 67–112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (broken-line profiles), homogeneous or density-stratified, inviscid shear layers. Here we have generalized Holmboe’s mechanistic picture of linear shear instabilities by (i) not initially specifying the wave type, and (ii) providing the option for non-normal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Doppler-shifted, progressive interfacial waves moving in opposite directions. Moreover, we have found a necessary and sufficient (N&S) condition for the existence of exponentially growing instabilities in idealized shear flows. The two interfacial waves, starting from arbitrary initial conditions, eventually phase-lock and resonate (grow exponentially), provided the N&S condition is satisfied. The theoretical underpinning of our wave interaction model is analogous to that of synchronization between two coupled harmonic oscillators. We have re-framed our model into a nonlinear autonomous dynamical system, the steady-state configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normal-mode theory, the steady-state/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is non-modal, favouring rapid transient growth. Depending on the wavenumber and initial phase-shift, non-modal gain can exceed the corresponding modal gain by many orders of magnitude. Instability is also observed in the parameter space which is deemed stable by the normal-mode theory. Using our model we have derived the discrete spectrum non-modal stability equations for three classical examples of shear instabilities: Rayleigh/Kelvin–Helmholtz, Holmboe and Taylor–Caulfield. We have shown that the N&S condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the normal-mode theory.

Linear stability of the non-parallel Batchelor vortex is studied using global modes. This family of swirling wakes and jets has been extensively studied under the parallel-flow approximation, and in this paper we extend to more realistic non-parallel base flows. Our base flow is obtained as an exact steady solution of the Navier–Stokes equations by direct numerical simulation (with imposed axisymmetry to damp all instabilities). Global stability modes are computed by numerical simulation of the linearized equations, using the implicitly restarted Arnoldi method, and we discuss fully the numerical and convergence issues encountered. Emphasis is placed on exploring the general structure of the global spectrum, and in particular the correspondence between global modes and local absolute modes which is anticipated by weakly non-parallel asymptotic theory. We believe that our computed global modes for a weakly non-parallel vortex are the first to display this correspondence with local absolute modes. Superpositions of global modes are also studied, allowing an investigation of the amplifier dynamics of this unstable flow. For an illustrative case we find global non-modal transient growth via a convective mechanism. Generally amplifier dynamics, via convective growth, are prevalent over short time intervals, and resonator dynamics, via global mode growth, become prevalent at later times.

The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285–303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

L'objectif de cette thèse est d'étudier les mécanismes physiques qui gouvernent l'évolution et le mélange des jets latéraux au sein des jets ronds de mélange binaire à faible densité, au moyen d'une double approche numérique et expérimentale. Les mécanismes physiques à l'origine des jets latéraux restent encore incertains et sont en lien direct avec le développement d'instabilités secondaires responsables de la tridimensionalisation de l'écoulement. Ainsi, une meilleure compréhension de ces mécanismes en jeu constitue un prérequis indispensable à la conception d'une stratégie de contrôle qui vise à promouvoir le mélange entre le jet et l'environnement ambiant.L'objectif de l'étude numérique est d'identifier les mécanismes transitoires qui influencent la croissance des perturbations tridimensionnelles dans le jet rond à faible masse volumique, en particulier dans les conditions physiques pour lesquelles apparaissent les jets latéraux. À cet effet, nous mettons en oeuvre une analyse de stabilité linéaire non-modale de l'évolution non-linéaire et axisymétrique de l'anneau tourbillonnaire de Kelvin-Helmholtz qui se développe dans les jets ronds à faible masse volumique en réponse à l'instabilité primaire de Kelvin-Helmholtz. Cette analyse de stabilité est réalisée à l'aide du code académique dalsa qui a été adapté pour ce régime d'écoulement. L'utilisation d'une méthode d’optimisation directe-adjointe permet d'identifier la structure spatiale et l'évolution temporelle des perturbations tridimensionnelles qui maximisent leur gain d'énergie généralisée, ainsi que les mécanismes physiques sous-jacents et leur lien avec l'apparition des jets latéraux. En particulier, l'objectif est d'apporter de nouveaux éléments permettant de statuer entre les deux hypothèses proposées dans la littérature pour expliquer le mécanisme à l'origine des jets latéraux. La première proposée par Monkewitz & Pfizenmaier (1991) repose sur un mécanisme d'induction de vitesse par les tourbillons longitudinaux contra-rotatifs qui se développent dans les écoulement cisaillés à masse volumique constante. La seconde est basée sur le mécanisme de tri-dimensionalisation associé aux stries de vitesse longitudinale de signe opposé qui se développent de part et d'autre du point de stagnation hyperbolique dans la tresse identifiée par Lopez-Zazuetaet al. (2016) dans la couche de mélange plane à densité variable.L'étude numérique conduite dans cette thèse est conçue en lien étroit avec la campagne expérimentale visant à étudier la structure des jets latéraux dans un jet de mélange binaire hélium-air. Ainsi, les paramètres de l'étude numérique comme les nombres de Reynolds et d'Atwood ainsi que le rapport d'aspect du jet, sont basés sur les conditions expérimentales. Ceci permet une comparaison directe des résultats théoriques aux observations expérimentales. Pour ce faire, nous avons conduit une campagne de mesures d'anémométrie à fil chaud du profil radial de vitesse du jet, et de la fréquence de l'instabilité primaire, afin de le caractériser et de le placer dans le contexte de la littérature scientifique.L'objectif de l'approche expérimentale est d'analyser la structure des jets latéraux et le mélange qu'ils induisent. Nous avons ainsi conçu et assemblé un banc de Background Oriented Schlieren tomographique (3DBOS). Ce banc est conçu afin de pouvoir observer les déviations optiques de l'ordre de 0,5 mrad induites par le champ hétérogène d'indices de réfraction créé par le jet de mélange hélium-air. Les mesures de 3DBOS obtenues permettent de reconstruire les champs de masse volumique du jet à faible densité en présence de jets latéraux. Ces champs de masse volumique sont originaux dans la littérature scientifique sur le sujet, et permettent d'augmenter notre compréhension de la structure des jets latéraux et de leur impact sur le mélange, et ainsi de corroborer les prédictions issues de l'analyse de stabilité
The aim of this PhD thesis is to study the physical mechanisms which govern the evolution and the mixing of side-jets in low-density binary mixing round jets, using a complementary numerical and experimental approach. The physical mechanisms which are responsible for the generation of side-jets, closely related to the three-dimensionalisation of the jet through the development of secondary instabilities, are as of yet not fully understood. As such, a better understanding of the mechanisms at play is a prerequisite for the design of an efficient control strategy to promote the mixing between the jet and ambient fluid.The objective of the numerical study is to identify the transient mechanisms which influence the growth of three-dimensional disturbances in the low-density round jet, specifically under the physical conditions in which side-jets appear. To that aim, a linear non-modal stability analysis was conducted over the non-linear evolution of a two dimensional axisymmetric Kelvin-Helmholtz vortex ring which develops in low-density round jets due to the Kelvin-Helmholtz primary instability.The stability analysis was implemented through further numerical development of the existing dalsa academic code. Through the use of a direct-adjoint optimisation method, we identify the spatial structure and temporal evolution of three-dimensional disturbances which yield the highest growth of generalised energy, as well as the underlying physical mechanisms and their relation to side-jets generation in low-density round jets at low Atwood numbers. In particular, we seek to bring a new perspective in order to settle between the two current hypotheses concerning the physical mechanisms at the origin of side-jets. The first hypothesis suggested by Monkewitz & Pfizenmaier (1991) relies on a velocity induction mechanism induced by the longitudinal counter-rotating vortex dipoles developing in the constant-density case. The second one is based on the three-dimensionalisation mechanism associated with longitudinal velocity streaks of opposite sign developing on either side of the hyperbolic stagnation point in the braid identified by Lopez-Zazueta et al. (2016) in the case of variable-density plane mixing layers.The numerical analysis is conducted in close relation to an experimental investigation of the structure of side-jets in a helium-air binary mixture round jet. The parameters used in the numerical analysis, such as the Reynolds number, the Atwood number and jet aspect ratio, are based on the operating conditions used in the experiment, allowing the theoretical predictions to be compared with the empirical evolution of the helium-air jet. To that aim, we conduct hot-wire anemometry measurements of the jet radial profile and frequency of the primary instability under several operating conditions to characterise the evolution of the governing parameters and relate the experimental conditions to the existing scientific literature.The objective of the experimental investigation is to study the structure of side-jets and their effect on the mixing of the jet and ambient fluids. To do so, we have designed and assembled a tomographic Background Oriented Schlieren (3DBOS) experimental bench. This bench is designed to observe the deviations of light-rays of the order of 0.5 mrad induced by the change in refractive index in the helium-air jet. The 3DBOS technique employed in this study provides novel reconstructions of three-dimensional density maps of the side-jets which develop over the helium-air jet. Through these novel density maps, we can provide new insight into the structure of side-jets and their induced mixing, and relate them to the predictions of the stability analysis

Coupled powerplant and rotorcraft flight dynamics simulations are commonly carried out in the non-linear time-domain framework (e.g. for pilot-in-the-loop handling qualities assessments), although these integrated models are generally not fully accurate from drivetrain dynamics perspective. Nevertheless, there is interest to verify that usual assumption of decoupled torsional stability (including rigid drivetrain analysis) and aircraft rigid body stability is valid, and up to what extent. The process described in the paper entails the automatic assembly of relevant subsystems (bare aircraft flight dynamics, Flight Control System including fly-by-wire actuation, sensors, and Control Laws software, drivetrain dynamics, powerplant dynamics) state space matrices through a Company developed Matlab toolbox. The proposed approach is control system design oriented, i.e. it does not require detailed flexible multibody modelling of the entire aircraft including dynamic systems and it is a natural extension of the process generally carried out by Control Laws and Flight Mechanics discipline for rigid-body stability assessment. The paper specifically addresses, as case study, the coupled torsional dynamics analysis of Next Generation Civil Tiltrotor Technology Demonstrator (NGCTR-TD) in Helicopter configuration. The NGCTR-TD project is developed under the EU Clean Sky 2 program for defining novel technology for future civil Tiltrotor platforms. The proposed coupled torsional stability modelling process is described, and the main outcomes valid for NGCTR-TD trimmed in hover and near-hover conditions presented. It is shown that the proposed methodology provides consistent outputs against the original NGCTR-TD linear stability results (from individual flight dynamics and drive system models). Analysis of Modal Participation Factors is used to quantify the coupling existing between drive system and flight dynamics state vectors, in nominal and failure conditions (Inter Connect Drive Shaft failure).

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

The non-linear behavior and stability under static and dynamic loads of an inverted spatial pendulum with rotational springs in two perpendicular planes, called Augusti’s model, is analyzed in this paper. This 2DOF lumped-parameter system is an archetypal model of modal interaction in stability theory representing a large class of structural problems. When the system displays coincident buckling loads, several post-buckling paths emerge from the bifurcation point (critical load) along the fundamental path. This leads to a complex potential energy surface. Herein, we aim to investigate the influence of nonlinear modal interactions on the dynamic behavior of Augusti’s model. Coupled/uncoupled dynamic responses, bifurcations, escape from the pre-buckling potential well, stability and space–time-varying displacements, attractor-manifold-basin phase portraits are numerically evaluated with the aim of enlightening the system complex response. The investigation of basins evolution due to variation of system parameters leads to the determination of erosion profiles and integrity measures which enlighten the loss of safety of the structure due to penetration of eroding fractal tongues into the safe basin.

The stalling behavior in a single-stage low-speed axial compressor under inlet distortion is investigated. A blade-passage-scale flow mechanism is proposed to explain the stability deterioration caused by inlet distortion for the tested compressor exhibiting spike stall inception. In contrast to the existing understanding of inlet distortion based on system scale dynamics, the main elements of this flow mechanism are the unsteady behavior of tip leakage vortices (TLV) under inlet distortion; its effect on the initiation of spike flow disturbances, and its interaction with distorted sectors. Rotating inlet distortion (RID) is used as a tool because RID makes it possible to directly compare the flows between distorted and clean flow sectors with fixed measurement stations on the casing, and the fact that the stationary inlet distortion is only a special case of RID makes the results generic. The tests demonstrate that the blade loading in the distorted sector is heavier than that in the non-distorted sector, causing the TLV in the distorted sector move closer to the leading edge of the rotor blade and thus be the first to initiate the spike-like disturbance. The unsteady CFD simulation further confirms that such a disturbance corresponds to a vortex spinning out of the leading edge of the blades. However, the initiation of this spike-like disturbance doesn’t necessarily trigger the full stall immediately. The tracking of the disturbances indicates that most of such spike-like disturbances will be smeared by non-distorted sector and the growth of the spike-like disturbances actually relate closely to how and how often the path of the propagating disturbances come across the path of the rotating distorted sector. The proposed blade-passage-scale flow mechanism also offers an alternative explanation to the “resonance” phenomenon in rotating inlet distortion research, which was explained with excitation-and-response theory for compressors that exhibit modal stall inception.

This work analyzes the nonlinear vibrations of a simply supported functionally graded cylindrical shell considering the effects of an internal fluid and static preloading. The cylindrical shell is subjected to a time dependent axial loading. The fluid is considered to be incompressible, non-viscous and irrotational and its effect on the shell wall is obtained using the potential flow theory. The shell is modeled by Donnell nonlinear shallow shell theory. The axial and circumferential displacement fields are described in terms of lateral displacement, thus generating a low-dimensional model, while the lateral displacement field is determined by a perturbation procedure which provides a general expression for the nonlinear vibration modes. These modal expansions satisfy the boundary and symmetry conditions of the problem. The discretized equations of motion are obtained by applying the Galerkin method. Various numerical techniques are employed to obtain the resonance curves and time responses of the cylindrical shell, showing the influence of the geometry, the internal fluid, static preloading and functionally graded material law on the shell dynamics and stability.

This paper analyzes transversal thermoacoustic oscillations in an experimental gas turbine combustor utilizing dynamical system theory. Limit cycle acoustic motions related to the first linearly unstable transversal mode of a given 3D combustor configuration are modeled, and reconstructed by means of a low order dynamical system simulation. The source of nonlinearity is solely allocated to flame dynamics, saturating the growth of acoustic amplitudes, while the oscillation amplitudes are assumed to always remain within the linearity limit. First, a Reduced Order Model (ROM), which reproduces the combustor’s modal distribution and damping of acoustic oscillations is derived. The ROM is a low-order state-space system, which results from a projection of the Linearized Euler Equations (LEE) into their truncated eigenspace. Second, flame dynamics are modeled as a function of acoustic perturbations by means of a nonlinear transfer function. This function has a linear and a nonlinear contribution. The linear part is modeled analytically from first principles, while the nonlinear part is mathematically cast into a cubic saturation functional form. Additionally, the impact of stochastic forcing due to broadband combustion noise is included by additive white noise sources. Then, the acoustic and the flame system is interconnected, where thermoacoustic non-compactness due to the transversal modes’ high frequency is accounted for by a distributed source term framework. The resulting nonlinear thermoacoustic system is solved in frequency and time domain. Linear growth rates predict linear stability, while envelope plots and probability density diagrams of the resulting pressure traces characterize the thermoacoustic performance of the combustor from a dynamical systems theory perspective. Comparisons against experimental data are conducted, which allow the rating of the flame modes in terms of their capability to reproduce the observed combustor dynamics. Ultimately, insight into the physics of high-frequency, transversal thermoacoustic systems is created.

Abstract Flutter is one of the important issues in turbomachinery analysis. There are four common types of flutter, including sub/transonic stall flutter, choke flutter, supersonic stall flutter, and supersonic non-stall flutter. Flutter may occur under many different operating conditions. Therefore, it is important to study the aeroelastic stability of blades when the compressor operates under different conditions. Based on the energy method proposed by Carta [1], this paper studied the aeroelastic stability of the second-stage rotor blade of an axial compressor under different operating conditions. It is found that the aerodynamic damping of the blade under the near-stall operating point of the compressor is negative. Three typical operating points are selected to study the differences in flutter mechanism between different operating conditions. The 90% span section is selected as the reference section to analyze the variation of the aerodynamic work at different operating points. The influence of reduced frequency, modal component, and tip clearance on aerodynamic damping are analyzed under three operating points.

Abstract The geared turbofan engines bring the potential to rotate the fan at lower speed and allow an increase in diameter, which in turn leads to an increase in propulsive efficiency through high by-pass ratio. The low-pressure turbine stages driving the fan can also rotate at high speed resulting in fewer stages when compared to traditional turbofans. However, when operating at high speed, pressure fluctuations due to self-excited vibrations increase and may provoke flutter instabilities. In a geared architecture, to deliver the high power required by the fan and the intermediate-pressure compressor, the low-pressure turbine system operates at higher temperatures compared to its predecessors. This phenomenon requires structural materials with higher heat resistance, which carries the inconvenience of poor welding suitability. That is the reason why alternative non-welded blade shroud joint techniques are so important, techniques as the blade interlock mechanism studied in this work. This manuscript examines the effects of different design parameters of a low-pressure turbine blade shroud interlock on flutter stability, to make future recommendations for geared engines. The shrouded turbine rotor blades feature blade interlocks, which enhances the dynamic stability by providing stiffness to the rotor blade row. To assess the stability of the system, a parametric design of a turbine blade-disk assembly was prepared. In the parametric model the design variables that define the blade interlock are the interlock angle, interlock axial position, interlock contact length and height, knife seal position and pre-twist angle. After parametrization, a finite element model of the turbine blade and disk assembly was prepared with cyclic symmetry boundary condition. The stresses caused by rotation were calculated in a static structural analysis and these were used as pre-stress boundary conditions in modal analysis. The modal results were afterwards exchanged with an aerodynamic model to obtain the aerodynamic damping for different blade interlock design configurations. In the present work, the dynamic response of the first three excitation modes was analyzed. It was found that the third mode was stable for all the design points, whereas first and second modes were unstable at least for the reference design point. Among the considered six different parameters that define the blade interlock geometry, the interlock contact position turned to be the most influential parameter for modal response and for flutter stability. Moving the interlock contact position towards the trailing edge gave the most beneficial results. On the other hand, the interlock angle showed the least influence on both, the modal analysis and flutter behavior. The accomplished Design of Experiments and subsequent optimization process also conclude that there exists an interdependency between the studied parameters.

Of the multitude of available control techniques, modal control is a favourite amongst structural dynamicists because of its representation in modal coordinates. The term modal control is used to describe a wide variety of control techniques which find their origin in a description of the system through the main coordinates, defined by the modes of vibration of the system. This approach stems from the consideration that the response of a mechanical system to a disturbance is the sum of the independent responses of its vibrational modes. This motivates the desire to design a control that does not alter these mode shapes, but allows to change the natural frequency and the damping of each mode. In active vibration control the purpose is to increase the damping of modes interested in the vibratory phenomenon. The paper shows how stability, spillover effects, system controllability and sensors and actuators position are all linked to the analysis of the controlled system damping matrix and to the possibility that the forces introduced by the control is non-dissipative. Theoretical aspects are supported by numerical simulations.

Heavy duty gas turbine front stages compressor blades aero-elastic behavior is deeply analyzed and investigated by means of an uncoupled, non-linear and time-accurate CFD URANS solver. The travelling-wave approach and the energy method have been applied in order to assess the aerodynamic damping (in terms of logarithmic decrement) for each inter blade phase angle (IBPA) and thus to localize the flutter stability region. The work is mainly focused on a sensitivity analysis with respect to blade operating conditions, eigen-mode shapes and frequency in order to improve the understanding of flutter mechanism and to identify the key parameters. Transonic, supercritical and subsonic blades are investigated at different operating conditions with their corresponding eigenmode and eigen-frequency (first and second flexural mode and first torsional). The results show that non-linear effects can be neglected for subsonic blades. Besides, the modal-shape and the shock structure, if any, are identified to play a key role for flutter stability.