Bibliographies thématiques / Phase-lagged method

Littérature scientifique sur le sujet « Phase-lagged method »

Auteur : Grafiati
Publié le 10 mars 2023

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Articles de revues sur le sujet "Phase-lagged method"

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Pascual-Marqui, Roberto D., Dietrich Lehmann, Martha Koukkou, Kieko Kochi, Peter Anderer, Bernd Saletu, Hideaki Tanaka et al. « Assessing interactions in the brain with exact low-resolution electromagnetic tomography ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 369, no 1952 (13 octobre 2011) : 3768–84. http://dx.doi.org/10.1098/rsta.2011.0081.

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Scalp electric potentials (electroencephalogram; EEG) are contingent to the impressed current density unleashed by cortical pyramidal neurons undergoing post-synaptic processes. EEG neuroimaging consists of estimating the cortical current density from scalp recordings. We report a solution to this inverse problem that attains exact localization: exact low-resolution brain electromagnetic tomography (eLORETA). This non-invasive method yields high time-resolution intracranial signals that can be used for assessing functional dynamic connectivity in the brain, quantified by coherence and phase synchronization. However, these measures are non-physiologically high because of volume conduction and low spatial resolution. We present a new method to solve this problem by decomposing them into instantaneous and lagged components, with the lagged part having almost pure physiological origin.
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Newton, Marcus C., Yoshinori Nishino et Ian K. Robinson. « Bonsu : the interactive phase retrieval suite ». Journal of Applied Crystallography 45, no 4 (14 juillet 2012) : 840–43. http://dx.doi.org/10.1107/s0021889812026751.

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Coherent X-ray diffraction imaging has received considerable attention as a nondestructive method for probing material structure at the nanoscale. However, tools for reconstructing and analysing data in both two and three dimensions have lagged somewhat behind.Bonsu, the interactive phase retrieval suite, is the first software package that allows real-time visualization of the reconstruction of phase information in both two and three dimensions. It comes complete with an inventory of algorithms and routines for data manipulation and reconstruction.Bonsuis open source, is designed around the Python language (with C++ bindings) and is largely platform independent.Bonsuis made available under version three of the GNU General Public License and can be found at https://code.google.com/p/bonsu/.
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Chen, Yuan Ying, Xiao Ling Yin, Dong Lin Bai et Li Cheng Li. « Spectral Analysis on Sub-Tidal Variability of Salinity in Modaomen Waterway of Pearl River Estuary, China ». Advanced Materials Research 518-523 (mai 2012) : 1982–85. http://dx.doi.org/10.4028/www.scientific.net/amr.518-523.1982.

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Salinity and tidal range time series observed in Modaomen waterway was analyzed in power spectrum method, and both showed the period of half month (14.22d). Moreover, the salinity and tidal range time series were coherent at that period through cross spectrum analysis. Besides, the phase analysis at the period of 14.22d showed that, within the estuary, the salinity time series upstream lagged that of the downstream, and the response time of salinity time series to the tidal range time series was about 9-10d, increasing upstream. But the phase of salinity time series and response time to the tidal range at the estuary mouth did not correspond with the laws within the estuary. The response time of salinity to the tidal range was about 12d there.
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Hirvonen, Timo P., Heikki Aalto, Ilmari Pyykkö et Martti Juhola. « Comparison of two head autorotation tests ». Journal of Vestibular Research 9, no 2 (1 avril 1999) : 119–25. http://dx.doi.org/10.3233/ves-1999-9206.

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The head autorotation test is a novel method for studying the high-frequency vestibuloocular reflex without heavy machinery to generate whole-body rotation. Despite many studies with the test, the method is far from standardized, and no comparison has been made of different versions of the test. The objective of this study was to compare the vestibuloocular reflex of 100 healthy subjects measured simultaneously with two versions of the head autorotation test. Gain, phase, asymmetry, and the frequency bands reached were determined in the frequency bands of 1, 2, 3, 4, and 5 Hz. The gain measured with both tests was close to unity (range 0.95–1.04) from 1 to 4 Hz and about 0.9 at 5 Hz. In the test developed by Vorteq the phase lagged ( − 7 to − 21 ∘ ) in all the frequency bands, and it differed significantly from the phase lead of 2 to 5 ∘ that was measured by the other test. The asymmetry measured with the Vorteq test increased continuously from 1.5% variation, which was larger in the higher frequency bands. In conclusion, the high-frequency vestibuloocular reflex of healthy subjects can be quantified with active head oscillation. Both tests produced similar gain results, but the phase results differed systematically. Thus, the results of different head autorotation tests may not be directly comparable.
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Ma, Zhongtian, Hok Sum Fok et Linghao Zhou. « GRACE-Derived Time Lag of Mekong Estuarine Freshwater Transport in the Western South China Sea Validated by Isotopic Tracer Age ». Remote Sensing 13, no 6 (20 mars 2021) : 1193. http://dx.doi.org/10.3390/rs13061193.

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Estuarine freshwater transport has a substantial impact on the near-shore ecosystem and coastal ocean environment away from the estuary. This paper introduces two independent methods to track the Mekong freshwater-induced mass transport by calculating the time lag (or equivalently, the phase) between in situ Mekong basin runoff and the equivalent water height (EWH) time series over the western South China Sea from a gravity recovery and climate experiment (GRACE). The first method is the harmonic analysis that determines the phase difference between annual components of the two time series (called the P-method), and the other is the cross-correlation analysis that directly obtains the time lag by shifting the lagged time series forward to attain the highest cross-correlation between the two time series (called the C-method). Using a three-year rolling window, the time lag variations in three versions of GRACE between 2005 and 2012 are computed for demonstrating the consistency of the results. We found that the time lag derived from the P-method is, on average, slightly larger and more variable than that from the C-method. A comparison of our gridded time lag against the age determined via radium isotopes in September, 2007 by Chen et al. (2010) revealed that our gridded time lag results were in good agreement with most isotope-derived ages, with the largest difference less than 6 days. Among the three versions of the GRACE time series, CSR Release 05 performed the best. The lowest standard deviation of time lag was ~1.6 days, calculated by the C-method, whereas the mean difference for all the time lags from the isotope-derived ages is ~1 day by P-method. This study demonstrates the potential of monitoring Mekong estuarine freshwater transport over the western South China Sea by GRACE.
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Lewis, J. P., R. A. Delaney et E. J. Hall. « Numerical Prediction of Turbine Vane-Blade Aerodynamic Interaction ». Journal of Turbomachinery 111, no 4 (1 octobre 1989) : 387–93. http://dx.doi.org/10.1115/1.3262285.

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A time-accurate analysis for turbine vane-blade interaction is presented. The analysis simultaneously solves the unsteady isentropic Euler equations in the vane and blade rows on a blade-to-blade surface of revolution. The equations are integrated on overlapped O-type grids using a rapid and robust explicit hopscotch algorithm. Vane and blade rows with unequal numbers of airfoils in each row are treated using a single passage model with phase-lagged periodic boundary conditions. Boundary conditions between the rows are set by a combination of bilinear interpolation and a reference plane method of characteristics. Nonreflective inflow and outflow boundary point calculation procedures are incorporated to ensure that outward-radiating planar waves pass out of the solution domain without reflection. Presented results for a turbine stage show significant effects of the interaction on the time-mean airfoil surface pressure distributions.
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Zhang, Jing, Sen Yang, Xiyue Wang, Ming Tang, Heng Yin et Ling He. « Automatic hypernasality grade assessment in cleft palate speech based on the spectral envelope method ». Biomedical Engineering / Biomedizinische Technik 65, no 1 (28 janvier 2020) : 73–86. http://dx.doi.org/10.1515/bmt-2018-0181.

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AbstractDue to velopharyngeal incompetence, airflow overflows from the oral cavity to the nasal cavity, which results in hypernasality. Hypernasality greatly reduces speech intelligibility and affects the daily communication of patients with cleft palate. Accurate assessment of hypernasality grades can provide assisted diagnosis for speech-language pathologists (SLPs) in clinical settings. Utilizing a support vector machine (SVM), this paper classifies speech recordings into four grades (normal, mild, moderate and severe hypernasality) based on vocal tract characteristics. Linear prediction (LP) analysis is widely used to model the vocal tract. Glottal source information may be included in the LP-based spectrum. The stabilized weighted linear prediction (SWLP) method, which imposes the temporal weights on the closed-phase interval of the glottal cycle, is a more robust approach for modeling the vocal tract. The extended weighted linear prediction (XLP) method weights each lagged speech signal separately, which achieves a finer time scale on the spectral envelope than the SWLP method. Tested speech recordings were collected from 60 subjects with cleft palate and 20 control subjects, and included a total of 4640 Mandarin syllables. The experimental results showed that the spectral envelope of normal speech decreases faster than that of hypernasal speech in the high-frequency part. The experimental results also indicate that the SWLP- and XLP-based methods have smaller correlation coefficients between normal and hypernasal speech than the LP method. Thus, the SWLP and XLP methods have better ability to distinguish hypernasal from normal speech than the LP method. The classification accuracies of the four hypernasality grades using the SWLP and XLP methods range from 83.86% to 97.47%. The selection of the model order and the size of the weight function are also discussed in this paper.
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Guo, Tongqing, Di Zhou et Zhiliang Lu. « A Double-Passage Shape Correction Method for Predictions of Unsteady Flow and Aeroelasticity in Turbomachinery ». Advances in Applied Mathematics and Mechanics 9, no 4 (18 janvier 2017) : 839–60. http://dx.doi.org/10.4208/aamm.2016.m1478.

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AbstractIn this paper, a double-passage shape correction (DPSC) method is presented for simulation of unsteady flows around vibrating blades and aeroelastic prediction. Based on the idea of phase-lagged boundary conditions, the shape correction method was proposed aimed at efficiently dealing with unsteady flow problems in turbomachinery. However, the original single-passage shape correction (SPSC) may show the disadvantage of slow convergence of unsteady solutions and even produce nonphysical oscillation. The reason is found to be related with the disturbances on the circumferential boundaries that can not be damped by numerical schemes. To overcome these difficulties, the DPSC method is adopted here, in which the Fourier coefficients are computed from flow variables at implicit boundaries instead of circumferential boundaries in the SPSC method. This treatment actually reduces the interaction between the calculation of Fourier coefficients and the update of flow variables. Therefore a faster convergence speed could be achieved and also the solution stability is improved. The present method is developed to be suitable for viscous and turbulent flows. And for real three-dimensional (3D) problems, the rotating effects are also considered. For validation, a 2D oscillating turbine cascade, a 3D oscillating flat plate cascade and a 3D practical transonic fan rotor are investigated. Comparisons with experimental data or other solutions and relevant discussions are presented in detail. Numerical results show that the solution accuracy of DPSC method is favorable and at least comparable to the SPSC method. However, fewer iteration cycles are needed to get a converged and stable unsteady solution, which greatly improves the computational efficiency.
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Katsafados, P., A. Papadopoulos, G. Varlas, E. Papadopoulou et E. Mavromatidis. « Seasonal predictability of the 2010 Russian heat wave ». Natural Hazards and Earth System Sciences 14, no 6 (18 juin 2014) : 1531–42. http://dx.doi.org/10.5194/nhess-14-1531-2014.

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Abstract. The atmospheric blocking over eastern Europe and western Russia that prevailed during July and August of 2010 led to the development of a devastating Russian heat wave. Therefore the question of whether the event was predictable or not is highly important. The principal aim of this study is to examine the predictability of this high-impact atmospheric event on a seasonal timescale. To this end, a set of dynamical seasonal simulations have been carried out using an atmospheric global circulation model (AGCM). The impact of various model initializations on the predictability of this large-scale event and its sensitivity to the initial conditions has been also investigated. The ensemble seasonal simulations are based on a modified version of the lagged-average forecast method using different lead-time initializations of the model. The results indicated that only a few individual members reproduced the main features of the blocking system 3 months ahead. Most members missed the phase space and the propagation of the system, setting limitations in the predictability of the event.
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Chen, Shu-Ya, Ching-Yuang Huang, Ying-Hwa Kuo et Sergey Sokolovskiy. « Observational Error Estimation of FORMOSAT-3/COSMIC GPS Radio Occultation Data ». Monthly Weather Review 139, no 3 (1 mars 2011) : 853–65. http://dx.doi.org/10.1175/2010mwr3260.1.

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Abstract The Global Positioning System (GPS) radio occultation (RO) technique is becoming a robust global observing system. GPS RO refractivity is typically modeled at the ray perigee point by a “local refractivity operator” in a data assimilation system. Such modeling does not take into account the horizontal gradients that affect the GPS RO refractivity. A new observable (linear excess phase), defined as an integral of the refractivity along some fixed ray path within the model domain, has been developed in earlier studies to account for the effect of horizontal gradients. In this study, the error statistics of both observables (refractivity and linear excess phase) are estimated using the GPS RO data from the Formosa Satellite 3–Constellation Observing System for Meteorology, Ionosphere and Climate (FORMOSAT-3/COSMIC) mission. The National Meteorological Center (NMC) method, which is based on lagged forecast differences, is applied for evaluation of the model forecast errors that are used for estimation of the GPS RO observational errors. Also used are Weather Research and Forecasting (WRF) model forecasts in the East Asia region at 45-km resolution for one winter month (mid-January to mid-February) and one summer month (mid-August to mid-September) in 2007. Fractional standard deviations of the observational errors of refractivity and linear excess phase both show an approximately linear decrease with height in the troposphere and a slight increase above the tropopause; their maximum magnitude is about 2.2% (2.5%) for refractivity and 1.1% (1.3%) for linear excess phase in the lowest 2 km for the winter (summer) month. An increase of both fractional observational errors near the surface in the summer month is attributed mainly to a larger amount of water vapor. The results indicate that the fractional observational error of refractivity is about twice as large as that of linear excess phase, regardless of season. The observational errors of both linear excess phase and refractivity are much less latitude dependent for summer than for winter. This difference is attributed to larger latitudinal variations of the specific humidity in winter.
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Thèses sur le sujet "Phase-lagged method"

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GIOVANNINI, MATTEO. « Development of a Phase-Lagged approach for unsteady turbomachinery analysis ». Doctoral thesis, 2012. http://hdl.handle.net/2158/866093.

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This PhD thesis is aimed at developing a computational method which allows to solve blade row interactions by using a reduced computational domain. In this model (named \Phase-lagged") the traditional assumption of circumferential direct-periodicity is replaced by one of phase-shifted periodicity, resulting in an impressive reduction of the computational costs. This model was implemented in the Q3D as well as in the 3D releases of the TRAF code, a CFD solver previously developed at the \S. Stecco" Department of Energy Engineering of the University of Florence by the research group of Prof. Arnone. The Phase-lagged model was developed for single stage problems and then generalized in order to handle multi-stage environment. The currently implemented formulation can solve both single-passage and multi-passage computational domains. The latter have been introduced to assure greater stability to the numerical scheme. The validation of the implemented model was carried out for single and multi-stage test-cases for both Q3D and 3D approach. The validation was mainly performed comparing Phase-lagged results with the time accurate results of the Full Annulus approach. For a single 3D test-case the comparisons with some experimental data will be presented as well. Finally, the results of an unsteady analysis of a whole LPT module are shown: these results conrm the robustness and accuracy of the model and can be seen as a feasible method for the introduction of the unsteady simulations in the aerodynamic design.
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Actes de conférences sur le sujet "Phase-lagged method"

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Burgos, Manuel Antonio, et Roque Corral. « Application of Phase-Lagged Boundary Conditions to Rotor/Stator Interaction ». Dans ASME Turbo Expo 2001 : Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/2001-gt-0586.

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The phase-lagged boundary condition concept is briefly reviewed, both for its classical application to vibrating blades and rotor/stator interaction. A simple conceptual model for rotor/stator interaction, compatible with the phase lagged boundary conditions is proposed. Careful comparisons between single-passage and full annulus inviscid simulations have been made to validate the method and demonstrate that potential differences between both approaches are associated to slight changes in the scheme due to the phase-shifted periodic boundary conditions. Finally the Erdos and Shape Correction methods are compared.
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Yang, Rongfei, Yuanyuan Gu, Yan Xue et Ning Ge. « Unsteady simulation of a 3.5 stage compressor using multi-frequency phase-lagged method ». Dans 2018 15th International Bhurban Conference on Applied Sciences and Technology (IBCAST). IEEE, 2018. http://dx.doi.org/10.1109/ibcast.2018.8312293.

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Srivastava, R., Milind A. Bakhle, Theo G. Keith et G. L. Stefko. « Phase-Lagged Boundary Condition Methods for Aeroelastic Analysis of Turbomachines : A Comparative Study ». Dans ASME 1999 International Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/99-gt-019.

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In the present work a comparative study of phase-lagged boundary condition methods is carried out. The relative merits and advantages of time-shifted and the Fourier decomposition methods are compared. Both methods are implemented in a time marching Euler/Navier-Stokes solver and are applied to a flat plate helical fan with harmonically oscillating blades to perform the study. Results were obtained for subsonic as well as supersonic inflows. Results for subsonic inflow showed good comparisons with published results and between the two methods along with comparable computational costs. For the supersonic inflow, despite the presence of shocks at the periodic boundary results from both the methods compared well, however, Fourier decomposition method was computationally more expensive. For linear flowfield Fourier decomposition method is best suited, especially for work-station environment. The time-shifted method is better suited for CRAY category of computers where fast input-output devices are available.
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Mouret, Gaelle, Nicolas Gourdain et Lionel Castillon. « Adaptation of Phase-Lagged Boundary Conditions to Large-Eddy Simulation in Turbomachinery Configurations ». Dans ASME Turbo Expo 2015 : Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/gt2015-42516.

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With the increase in computing power, large eddy simulation (LES) emerges as a promising technique to improve both knowledge of complex physics and reliability of turbomachinery flow predictions. However these simulations are very expensive for industrial applications, especially when a 360° configuration should be considered. The objective of this paper is thus to improve the well-known phase lagged conditions to the LES approach by replacing the traditional Fourier Series Decomposition (FSD) by a compression method which does not make any assumption on the spectrum of the flow. Several methods are reviewed and the Proper Orthogonal Decomposition (POD) is retained. This new method is first validated on a flow around a circular cylinder with rotating downstream blocks. The results show significant improvements with respect to FSD. It is then applied to URANS simulations of a single stage compressor in 2.5D and 3D as a first validation step toward single-passage LES of turbomachinery configuration.
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Burgos, Manuel A., Jesus Contreras et Roque Corral. « Validation of an Efficient Unstructured Time-Domain Rotor/Stator Interaction Method ». Dans ASME Turbo Expo 2009 : Power for Land, Sea, and Air. ASMEDC, 2009. http://dx.doi.org/10.1115/gt2009-59385.

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A method to address the rotor/stator interaction problem on an edge-based Reynolds Averaged Navier-Stokes solver is presented. The implementation of the phase-lagged boundary conditions and a new original conservative discretization of the sliding plane in unstructured grids is discussed in detail. Results of the present method are compared with the experimental results obtained at NLR for a subsonic fan.
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Li, H. D., et L. He. « Prediction of Flutter and Inlet Distortion Driven Response of a Transonic Fan Rotor Using Phase-Lagged Boundary Conditions ». Dans ASME Turbo Expo 2001 : Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/2001-gt-0272.

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Prediction of blade forced response and flutter is of great importance to turbomachinery designers. However, calculations of unsteady turbomachinery flows using conventional time-domain methods typically would lead to the use of multi-passage/whole-annulus domains due to the required direct periodic condition. This makes numerical computations extremely time-consuming and is one of the major difficulties for nonlinear unsteady calculations to be applied in a blading design environment. A single-passage approach to three-dimensional unsteady Navier-Stokes calculations using the Fourier-series based Shape-Correction method has been developed, and been applied to analyze inlet distortion driven response and flutter of a transonic fan rotor (NASA Rotor-67). The key feature is that the Shape-Correction method enables a single-passage solution to unsteady flows in blade rows under influences of multiple disturbances with arbitrary inter-blade phase angles. The results show that the single-passage solution can capture deterministic unsteadiness as well as time-averaged flows in good agreement with conventional multi-passage solutions, while the corresponding computing time can be reduced dramatically.
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Mokulys, Thomas, Stephen C. Dewhurst et Reza S. Abhari. « Numerical Validation of Characteristic and Linearized Unsteady Boundary Conditions for Non-Integer Blade Ratios in a Non-Linear Navier Stokes Solver ». Dans ASME Turbo Expo 2005 : Power for Land, Sea, and Air. ASMEDC, 2005. http://dx.doi.org/10.1115/gt2005-68670.

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Unsteady single passage methods for turbomachinery applications are becoming a more and more mature method in predicting unsteady blade-to-blade aerodynamics in a more efficient and accurate way compared to multipassage computations with scaled airfoil geometries. The outlined work presents results of a fourier-based method for phase-lagged boundary conditions using a usual conservative approach at the interfaceplane and compares it to a new characteristic treatment as well as a linearized single row approach. The new method provides results of similar accuracy with improved stability of the algorithm and faster convergence times.
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Laumert, Bjo¨rn, Hans Ma˚rtensson et Torsten H. Fransson. « Simulation of Rotor/Stator Interaction With a 4D Finite Volume Method ». Dans ASME Turbo Expo 2002 : Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30601.

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A finite volume method for the computation of rotor/stator interaction for stages with arbitrary rotor/stator pitch ratios is presented and partly validated in this paper. The method which solves the unsteady three-dimensional Euler equations is formulated in the four-dimensional time-space domain. The method of time inclination is utilized to account for unequal pitchwise periodicity by distributing time co-ordinates at the grid nodes such that phase lagged boundary conditions can be employed. Calculated results show excellent agreement with the results of a reference solver for the validation test case. Furthermore the method was applied to the simulation of the unsteady flow field in a transonic test turbine stage with a stator/rotor pitch ratio of 1.875. The results were compared with measurements of the unsteady rotor blade pressure and a reference solver calculation where an approximate pitch ratio of 2.0 with a 6.7% scaled rotor geometry was employed. Both computational cases show satisfactory agreement with the experiments for both time averaged pressure distributions and pressure perturbation amplitudes.
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Mata, Sanjay, Luca Di Mare, Imregun Mehmet et Jeffrey Green. « Generalised Single-Passage Multi-Bladerow Method for Turbomachinery Forced Response ». Dans ASME Turbo Expo 2009 : Power for Land, Sea, and Air. ASMEDC, 2009. http://dx.doi.org/10.1115/gt2009-59624.

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This paper describes the formulation and validation of a novel computational method for a quick and accurate assessment of multi-bladerow turbomachinery forced response using 3D, viscous, time-domain, unsteady aerodynamics. Phase-lagged boundary conditions are applied at the periodic boundaries and information is exchanged using a 0-thickness whole-annulus representation of the bladerow boundary. The proposed method is general in terms of rotor/stator blade numbers and preserves the accuracy of an equivalent multi-bladerow whole-annulus time-domain computation. The method is first validated on simple wave propagation cases for cylinders with known exact solutions. A turbine configuration is studied next, and results from the single-passage multi-bladerow method are checked against those from equivalent whole-annulus computations. The two sets of results are found to be in excellent agreement. The computational requirements of the two methods are compared and the advantages of the generalized single-passage method over its whole-annulus counterpart are highlighted in terms of CPU effort and memory storage. Implementation details, such as convergence, solution acceleration and the effect of non-reflecting boundary condition treatments at the inter-bladerow boundaries, are also discussed.
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Srivastava, R., M. A. Bakhle, T. G. Keith et G. L. Stefko. « Flutter Analysis of a Transonic Fan ». Dans ASME Turbo Expo 2002 : Power for Land, Sea, and Air. ASMEDC, 2002. http://dx.doi.org/10.1115/gt2002-30319.

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This paper describes the calculation of flutter stability characteristics for a transonic forward swept fan configuration using a viscous aeroelastic analysis program. Unsteady Navier-Stokes equations are solved on a dynamically deforming, body fitted, grid to obtain the aeroelastic characteristics using the energy exchange method. The non-zero inter-blade phase angle is modeled using phase-lagged boundary conditions. Results obtained show good correlation with measurements. It is found that the location of shock and variation of shock strength strongly influenced stability. Also, outboard stations primarily contributed to stability characteristics. Results demonstrate that changes in blade shape impact the calculated aerodynamic damping, indicating importance of using accurate blade operating shape under centrifugal and steady aerodynamic loading for flutter prediction. It was found that the calculated aerodynamic damping was relatively insensitive to variation in natural frequency.
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