Nonlinear Modeling of Mechanical Gas Face Seal Systems Using Proper Orthogonal Decomposition

2006 ◽  
Vol 128 (4) ◽  
pp. 817-827 ◽  
Author(s):  
Haojiong Zhang ◽  
Brad A. Miller ◽  
Robert G. Landers
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Face Seal ◽  
Reduced Order ◽  
Wide Range ◽  
Gas Face Seal ◽  
Proper Orthogonal

An approach based on proper orthogonal decomposition and Galerkin projection is presented for developing low-order nonlinear models of the gas film pressure within mechanical gas face seals. A technique is developed for determining an optimal set of global basis functions for the pressure field using data measured experimentally or obtained numerically from simulations of the seal motion. The reduced-order gas film models are shown to be computationally efficient compared to full-order models developed using the conventional semidiscretization methods. An example of a coned mechanical gas face seal in a flexibly mounted stator configuration is presented. Axial and tilt modes of stator motion are modeled, and simulation studies are conducted using different initial conditions and force inputs. The reduced-order models are shown to be applicable to seals operating within a wide range of compressibility numbers, and results are provided that demonstrate the global reduced-order model is capable of predicting the nonlinear gas film forces even with large deviations from the equilibrium clearance.

Nonlinear Modeling of Mechanical Gas Face Seal Systems Using Proper Orthogonal Decomposition

2005 ◽  
Author(s):  
Haojiong Zhang ◽  
Brad A. Miller ◽  
Robert G. Landers
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Orthogonal Decomposition ◽  
Simulation Studies ◽  
Face Seal ◽  
Model Order ◽  
Reduced Order ◽  
Gas Face Seal ◽  
Proper Orthogonal

A nonlinear reduced-order modeling approach based on Proper Orthogonal Decomposition (POD) is utilized to develop an efficient low order model, based on ordinary differential equations, for mechanical gas face seal systems. An example of a coned mechanical gas face seal in a flexibly mounted stator configuration is presented. The axial mode is modeled, and simulation studies are conducted using different initial conditions and forcing inputs. The results agree well with a fully meshed finite difference model, while the resulting model order is significantly decreased.

Error Estimates for Reduced Order POD Models of Navier-Stokes Equations

2008 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Error Estimates ◽  
Incompressible Flows ◽  
Orthogonal Decomposition ◽  
Galerkin Projection ◽  
Dimensional Model ◽  
Reduced Order ◽  
Infinite Dimensional ◽  
Low Dimensional ◽  
Proper Orthogonal

Reduced order modeling for the purpose of constructing a low dimensional model from high dimensional or infinite dimensional model has important applications in science and engineering such as fast model evaluations and optimization/control. A popular method for constructing reduced-order model is based on finding a suitable low dimensional basis by proper orthogonal decomposition (POD) and forming a model by Galerkin projection of the infinite dimensional model onto the basis. In this paper, we will discuss error estimates for Galerkin proper orthogonal decomposition method for an unsteady nonlinear coupled partial differential equations arising in viscous incompressible flows. A specific finite element in space and finite difference in time discretization scheme will be discussed.

A Dual-Weighted Approach to Order Reduction in 4DVAR Data Assimilation

2008 ◽  
Vol 136 (3) ◽  
pp. 1026-1041 ◽  
Author(s):  
D. N. Daescu ◽  
I. M. Navon
Keyword(s):  
Data Assimilation ◽  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Order Reduction ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Water Model ◽  
Reduced Order ◽  
Model Dynamics ◽  
Proper Orthogonal

Abstract Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the order-reduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin–Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

Real-Time Force and Moment Estimation for Mechanical Gas Face Seal Systems Using Reduced-Order Kalman Filters

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Haojiong Zhang ◽  
Robert G. Landers ◽  
Brad A. Miller
Keyword(s):  
Real Time ◽  
Axial Force ◽  
Kalman Filters ◽  
Face Seal ◽  
Reduced Order ◽  
Moment Estimation ◽  
Wide Range ◽  
Motion Measurements ◽  
Gas Face Seal ◽  
Proper Orthogonal

A method using reduced-order Kalman filters is developed to estimate the thin gas film axial force and moments in real time for mechanical gas face seal systems in a flexibly mounted stator configuration. First-order Gauss–Markov stochastic models, combined with the stator motion equations, form the basis for the reduced-order Kalman filter estimators. Two schemes are presented to estimate axial force and moments based on stator motion measurements. In one scheme, the force and moments are directly estimated and, in another scheme, a set of proper orthogonal decomposition weighting functions is estimated, from which the gas film force and moments are computed. Both estimators are shown to approximate the gas film axial force and moments successfully for different forcing functions over a wide range of compressibility numbers.

Modeling the Turbulent Wake Behind a Wall-Mounted Square Cylinder

2020 ◽  
Author(s):  
Christian Amor ◽  
José M Pérez ◽  
Philipp Schlatter ◽  
Ricardo Vinuesa ◽  
Soledad Le Clainche
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Turbulent Wake ◽  
Orthogonal Decomposition ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Square Cylinder ◽  
Complex Flow ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Proper Orthogonal

Abstract This article introduces some soft computing methods generally used for data analysis and flow pattern detection in fluid dynamics. These techniques decompose the original flow field as an expansion of modes, which can be either orthogonal in time (variants of dynamic mode decomposition), or in space (variants of proper orthogonal decomposition) or in time and space (spectral proper orthogonal decomposition), or they can simply be selected using some sophisticated statistical techniques (empirical mode decomposition). The performance of these methods is tested in the turbulent wake of a wall-mounted square cylinder. This highly complex flow is suitable to show the ability of the aforementioned methods to reduce the degrees of freedom of the original data by only retaining the large scales in the flow. The main result is a reduced-order model of the original flow case, based on a low number of modes. A deep discussion is carried out about how to choose the most computationally efficient method to obtain suitable reduced-order models of the flow. The techniques introduced in this article are data-driven methods that could be applied to model any type of non-linear dynamical system, including numerical and experimental databases.

Reduced-Order Model of a Bladed Rotor With Geometric Mistuning

2007 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Orthogonal Decomposition ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Coordinate Measurement ◽  
Coordinate Measurement Machine ◽  
Bladed Disk ◽  
Bladed Rotor ◽  
Proper Orthogonal

This paper deals with the development of an accurate reduced-order model of a bladed disk with geometric mistuning. The method is based on vibratory modes of various tuned systems and proper orthogonal decomposition of coordinate measurement machine (CMM) data on blade geometries. Results for an academic rotor are presented to establish the validity of the technique.

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

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