Proper Orthogonal Decomposition of Turbine Rotor-Stator Interaction

Cizmas, Paul G. A.; Palacios, Antonio Lozano

doi:10.2514/2.6108

Cited by 51 publications

(24 citation statements)

References 16 publications

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“…The method has been employed for industrial applications such as supersonic jet modeling [4], turbine flows [5], thermal processing of foods [2], and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.…”

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confidence: 99%

Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

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confidence: 99%

Error Estimation for Reduced‐Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2007

SIAM Rev.

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“…The relative energy captured by the kth mode is defined as Cizmas and Palacios, 2003). As the relative energy of the first modes increases, fewer POD modes are needed in the reconstruction…”

Section: Acceleration Methods For the Rommentioning

confidence: 99%

“…A set of time coefficients were calculated directly by projecting the snapshots onto the basis functions (Cizmas and Palacios, 2003). This gave the best case solution for the flow at reference conditions.…”

Section: Augmented Proper Orthogonal Decompositionmentioning

confidence: 99%

A Reduced-Order Model of Transport Phenomena for Power Plant Simulation

Cizmas¹,

Richardson²,

Brenner³

et al. 2009

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This report summarizes the objectives and tasks of this project, and describes how they were achieved. The report presents the progress made towards the program milestones and deliverables and summarizes the additional work done in the pursuit of these goals.A reduced-order model based on proper orthogonal decomposition (POD) has been developed to simulate transient two-and three-dimensional isothermal and non-isothermal flows in a fluidized bed. Reduced-order models of void fraction, gas and solids temperatures, granular energy, and z-direction gas and solids velocity have been added to the previous version of the code. These algorithms are presented and their implementation is discussed. Verification studies are presented for each algorithm.A number of methods to accelerate the computations performed by the reduced-order model are presented. The errors associated with each acceleration method are computed and discussed. Using a combination of acceleration methods, a two-dimensional isothermal simulation using the reduced-order model is shown to be 114 times faster than using the full-order model.In the pursue of achieving the objectives of the project and completing the tasks planned for this program, several unplanned and unforeseen results, methods and studies have been generated. These additional accomplishments are also presented and they include: (1) a study of the effect of snapshot sampling time on the computation of the POD basis functions, (2) an investigation of different strategies for generating the autocorrelation matrix used to find the POD basis functions, (3) the development and implementation of a bubble detection and tracking algorithm based on mathematical morphology, (4) a method for augmenting the proper orthogonal decomposition to better capture flows with discontinuities, such as bubbles, and (5) a mixed reduced-order/full-order model, called point-mode proper orthogonal decomposition, designed to avoid unphysical due to approximation errors.The limitations of the proper orthogonal decomposition method in simulating transient flows with moving discontinuities, such as bubbling flows, are discussed and several methods are proposed to adapt the method for future use.

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“…Annaswamy et al (2002) examined Ôedge-tonesÕ of an aircraft nozzle by analyzing the POD modes of azimuthal pressure distribution of a circular jet. Cizmas and Palacios (2003) gained insight on the turbine rotor-stator interaction with a lower-order POD investigation. They effectively utilized the time history and phase-plane plots of the POD coefficients to unravel the key dynamics in the flow behavior.…”

Section: Introductionmentioning

confidence: 99%