Non-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Computations

A time-domain harmonic balance method for the analysis of almost-periodic (multi-harmonics) flows is presented. This method relies on Fourier analysis to derive an efficient alternative to classical time marching schemes for such flows. It has recently received significant attention, especially in the turbomachinery field where the flow spectrum is essentially a combination of the blade passing frequencies. Up to now, harmonic balance methods have used a uniform time sampling of the period of interest, but in the case of several frequencies, non-necessarily multiple of each other, harmonic balance methods can face stability issues due to a bad condition number of the Fourier operator. Two algorithms are derived to find a non-uniform time sampling in order to minimize this condition number. Their behavior is studied on a wide range of frequencies, and a model problem of a 1D flow with pulsating outlet pressure, which enables to prove their efficiency. Finally, the flow in a multi-stage axial compressor is analyzed with different frequency sets. It demonstrates the stability and robustness of the present non-uniform harmonic balance method regardless of the frequency set

Time-domain harmonic balance method for aerodynamic and aeroelastic simulations of turbomachinery flows

A time-domain Harmonic Balance method is applied to simulate the blade row interactions and vibrations of state- of-the-art industrial turbomachinery configurations. The present harmonic balance approach is a time-integration scheme that turns a periodic or almost-periodic flow problem into the coupled resolution of several steady computations at different time samples of the period of interest. The coupling is performed by a spectral time-derivative operator that appears as a source term of all the steady problems. These are converged simultaneously making the method parallel in time. In this paper, a non-uniform time sampling is used to improve the robustness and accuracy regardless of the considered frequency set. Blade row interactions are studied within a 3.5-stage high-pressure axial compressor representative of the high-pressure core of modern turbofan engines. Comparisons with reference time-accurate computations show that four frequencies allow a fair match of the compressor performance, with a reduction of the computational time up to a factor 30. Finally, an aeroelastic study is performed for a counter-rotating fan stage, where the rear blade is submitted to a prescribed harmonic vibration along its first torsion mode. The aerodynamic damping is analysed, showing possible flutter

Multi-frequential harmonic balance approach for the computation of unsteadiness in multi-stage turbomachines

In turbomachines, the relative motion of fixed and rotating blades gives rise to deterministic unsteady interactions at frequencies termed BPFs (Blade Passing Frequencies). In a multi-stage turbomachine, a row sandwiched between two other rows is submitted to (at least) two BPFs, hence the need for multiple frequency methods. Initially developed for single frequency problems, harmonic methods have been extended to account for multiple frequencies. All the variations of the Harmonic Balance (HB) technique proposed in the literature rely on a uniform time sampling of the longest period of interest (though the number of samples can differ). This can compromise the efficiency of the method, as too many time samples are computed. Besides, as demonstrated in the present contribution, uniform time sampling can also raise stability issues. To overcome these computational limitations, a new approach using non-uniform time sampling is proposed in the present contribution. This paper will be organized as follows: first, the multi-frequency HB methods is presented, and the impact of time sampling on numerical stability is discussed. Then, algorithms for an automatic choice of the time samples are presented and compared. The proposed non-uniform sampling is assessed for a model problem (i.e. a pulsating channel). Finally, a section is dedicated to the application to a turbomachinery configuration, with emphasis on the choice of frequencie

Attachement et troubles du comportement

John Bowlby élabore la théorie de l’attachement après la Deuxième Guerre mondiale, à la lumière de ses travaux sur les réactions à la séparation et sur la carence de soins maternels. Par la suite, ses élèves créent des outils d’évaluation remarquables tels que la Strange Situation, l’Adult Attachment Interview qui permettent d’identifier différentes catégories d’attachement chez l’enfant ou d’état d’esprit vis-à-vis de l’attachement chez l’adulte (sécure ou équilibré, insécure ambivalent : ou évitant, insécure résistant ou coléreux, désorganisé). Cette théorie rend compte de la façon dont les relations précoces et leur développement peuvent influencer ce à quoi l’enfant s’attend de la part de l’autre, et l’évolution de ses stratégies adaptatives en fonction de l’évolution de ses relations aux figures d’attachement. Elle offre donc des perspectives prometteuses sur la compréhension de la genèse du comportement social et antisocial. Il existe un lien manifeste entre attachement désorganisé et troubles du comportement, à la fois sur le plan clinique comme sur le plan théorique. Cependant les différents travaux de recherche entrepris pour relier les catégories d’attachement aux troubles du comportement ne révèlent pour le moment que des liens modestes, bien que significatifs. Nous reprenons alors la conceptualisation de Fonagy et de Greenberg qui envisagent l’insécurité de l’attachement comme un facteur de vulnérabilité qui, combiné à d’autres facteurs de risque, serait une voie vers l’apparition de troubles du comportement

Multifrequential Harmonic Balance Computations for a Multistage Compressor

To reduce the cost of unsteady flow simulations, a pseudo-spectral method (the so-called Time Spectral Method – TSM) has been proposed in the literature, which takes into account the flow time periodicity. Thanks to Fourier spectral analysis, the unsteady Navier-Stokes equations can be read as 2N+1 steady equations coupled by a source term. This approach has been implemented in the structured multiblock flow solver elsA .It efficiently computes unsteady flows and shows major improvements regarding the computational time (to a factor up to ten). However this method is only able to compute a flow field composed of one frequency and its harmonics in each blade row. The present work extends this method to a more general case in which the frequencies are not necessarily multiples of a base frequency. A multistage compressor is computed using increasing levels of complexity regarding the unsteady rows interactions. In the first step, two rows (a rotor and a stator) are modeled with an injection boundary condition accounting for the inlet guide vane wakes. Therefore, two fundamental frequencies are seen by the rotor. The influence of the frequency content in the rotor (i.e. number of harmonics of the blade passing frequencies and their combinations) is analyzed. The validation of the results is done against results obtained with classical time-marching URANS computations. Then the method is applied to the three-stage compressor CREATE

Attachement et troubles du comportement

John Bowlby élabore la théorie de l’attachement après la Deuxième Guerre mondiale, à la lumière de ses travaux sur les réactions à la séparation et sur la carence de soins maternels. Par la suite, ses élèves créent des outils d’évaluation remarquables tels que la Strange Situation, l’Adult Attachment Interview qui permettent d’identifier différentes catégories d’attachement chez l’enfant ou d’état d’esprit vis-à-vis de l’attachement chez l’adulte (sécure ou équilibré, insécure ambivalent : ou évitant, insécure résistant ou coléreux, désorganisé). Cette théorie rend compte de la façon dont les relations précoces et leur développement peuvent influencer ce à quoi l’enfant s’attend de la part de l’autre, et l’évolution de ses stratégies adaptatives en fonction de l’évolution de ses relations aux figures d’attachement. Elle offre donc des perspectives prometteuses sur la compréhension de la genèse du comportement social et antisocial. Il existe un lien manifeste entre attachement désorganisé et troubles du comportement, à la fois sur le plan clinique comme sur le plan théorique. Cependant les différents travaux de recherche entrepris pour relier les catégories d’attachement aux troubles du comportement ne révèlent pour le moment que des liens modestes, bien que significatifs. Nous reprenons alors la conceptualisation de Fonagy et de Greenberg qui envisagent l’insécurité de l’attachement comme un facteur de vulnérabilité qui, combiné à d’autres facteurs de risque, serait une voie vers l’apparition de troubles du comportement