Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis

Morzyński, Marek; Afanasiev, Konstantin; Thiele, Frank

doi:10.1016/s0045-7825(98)00183-2

Cited by 93 publications

(84 citation statements)

References 15 publications

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“…Details relating to the methods and parallel implementation of the linear stability analysis algorithms can be found in [24,12,13,25]. The analysis begins at a given steady state solution point.…”

Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

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Summary:Stability analysis algorithms coupled with a robust Newton-Krylov steady state iterative solver are used to understand the behavior of the 2D model problem of thermal convection in a 8:1 differentially heated cavity. Parameter continuation methods along with bifurcation and linear stability analysis are used to study transition from steady to transient flow as a function of Rayleigh number. To carry out this study the steady state form of the governing PDEs is discretized using a Galerkin/Least Squares Finite Element formulation, and solved on parallel computers using a fully coupled Newton method and preconditioned Krylov iterative linear solvers. Linear stability analysis employing a large scale eigenvalue capability is used to determine the stability of the steady solutions. The boundary between steady and time dependent flows is determined by a Hopf bifurcation tracking capability that is used to directly track the instability with respect to the aspect ratio of the system and with respect to mesh resolution. The effect of upwinding stabilization terms in the finite element formulation on the computed value of critical Rayleigh number is investigated. The Hopf bifurcation signaling the onset of flow is determined to occur at a critical Rayleigh number of .

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Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

View full text Add to dashboard Cite

show abstract

“…Details relating to the methods and parallel implementation of the linear stability analysis algorithms can be found in [11][12][13]. The analysis begins at a given steady state solution point.…”

Section: Linear Stability Analysis Algorithmsmentioning

confidence: 99%

Understanding the 8: 1 cavity problem via scalable stability analysis algorithms

Salinger¹,

Lehoucq²,

Pawlowski³

et al. 2001

Computational Fluid and Solid Mechanics

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Stability analysis algorithms coupled with a robust steady state solver are used to understand the behavior of the 2D model problem of thermal convection in a 8 : 1 differentially heated cavity. The system is discretized using a Galerkin=Least Squares Finite Element formulation, and solved to steady state on parallel computers using a fully coupled Newton method and iterative linear solvers. An eigenvalue capability is used to probe the stability of the solutions, and the neutral stability curves are tracked directly using a Hopf tracking algorithm.

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“…Noack & Eckelmann 1994a, b), physical (e.g. Bangia et al 1997;Morzyński, Afanasiev & Thiele 1999) and empirical approaches. Successes, difficulties and limitations of each approach have been reported in the literature (see for example Noack et al 2003;Bergmann et al 2005).…”

mentioning

confidence: 99%

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

2009

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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. IntroductionA number of practical engineering problems requires the repeated simulation of unsteady fluid flows for a large number of parameter values. These problems include the control, optimization and uncertainty quantification of fluid systems. To make many of these problems tractable, reduced-order modelling has been used to minimize the simulation requirements. The use of reduced-order modelling in control and optimization has led to practical solutions for extremely challenging problems (Ito & Ravindran 1996) Copyright by the Cambridge University Press. Hay, A.; Borggaard, J. T.; Pelletier, D., "Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition,"

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Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis

Cited by 93 publications

References 15 publications

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Understanding the 8: 1 cavity problem via scalable stability analysis algorithms

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

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