Elimination of Numerical Damping in the Stability Analysis of Noncompact Thermoacoustic Systems With Linearized Euler Equations

2021 ◽  
Vol 143 (3) ◽  
Author(s):  
Thomas Hofmeister ◽  
Tobias Hummel ◽  
Frederik Berger ◽  
Noah Klarmann ◽  
Thomas Sattelmayer
Keyword(s):  
Euler Equations ◽  
State Of The Art ◽  
Linearized Euler Equations ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
University Of Munich ◽  
Numerical Damping ◽  
Artificial Diffusion ◽  
The Stability ◽  
Stability Results

Abstract The hybrid computational fluid dynamics/computational aeroacoustics (CFD/CAA) approach represents an effective method to assess the stability of noncompact thermoacoustic systems. This paper summarizes the state-of-the-art of this method, which is currently applied for the stability prediction of a lab-scale configuration of a perfectly premixed, swirl-stabilized gas turbine combustion chamber at the Thermodynamics institute of the Technical University of Munich. Specifically, 80 operational points, for which experimentally observed stability information is readily available, are numerically investigated concerning their susceptibility to develop thermoacoustically unstable oscillations at the first transversal eigenmode of the combustor. Three contributions are considered in this work: (1) flame driving due the deformation and displacement of the flame, (2) visco-thermal losses in the acoustic boundary layer and (3) damping due to acoustically induced vortex shedding. The analysis is based on eigenfrequency computations of the Linearized Euler Equations with the stabilized finite element method (sFEM). One main advancement presented in this study is the elimination of the nonphysical impact of artificial diffusion schemes, which is necessary to produce numerically stable solutions, but falsifies the computed stability results.

Elimination of Numerical Damping in the Stability Analysis of Non-Compact Thermoacoustic Systems With Linearized Euler Equations

2020 ◽  
Author(s):  
Thomas Hofmeister ◽  
Tobias Hummel ◽  
Frederik Berger ◽  
Noah Klarmann ◽  
Thomas Sattelmayer
Keyword(s):  
Euler Equations ◽  
Linearized Euler Equations ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
University Of Munich ◽  
Numerical Damping ◽  
Artificial Diffusion ◽  
Physical Impact ◽  
The Stability ◽  
Stability Results

Abstract The hybrid Computational Fluid Dynamics/Computational AeroAcoustics (CFD/CAA) approach represents an effective method to assess the stability of non-compact thermoacoustic systems. This paper summarizes the state-of-the-art of this method, which is currently applied for the stability prediction of a lab-scale configuration of a perfectly-premixed, swirl-stabilized gas turbine combustion chamber at the Thermodynamics institute of the Technical University of Munich. Specifically, 80 operational points, for which experimentally observed stability information is readily available, are numerically investigated concerning their susceptibility to develop thermoacoustically unstable oscillations at the first transversal eigenmode of the combustor. Three contributions are considered in this work: (1) flame driving due the deformation and displacement of the flame, (2) visco-thermal losses in the acoustic boundary layer and (3) damping due to acoustically induced vortex shedding. The analysis is based on eigenfrequency computations of the Linearized Euler Equations with the stabilized Finite Element Method (sFEM). One main advancement presented in this study is the elimination of the non-physical impact of artificial diffusion schemes, which is necessary to produce numerically stable solutions, but falsifies the computed stability results.

Elimination of Numerical Damping in the Stability Analysis of Non-Compact Thermoacoustic Systems With Linearized Euler Equations

2021 ◽  
Author(s):  
Thomas Hofmeister ◽  
Tobias Hummel ◽  
Frederik Magnus Berger ◽  
Noah Klarmann ◽  
Thomas Sattelmayer
Keyword(s):  
Stability Analysis ◽  
Euler Equations ◽  
Linearized Euler Equations ◽  
Numerical Damping ◽  
The Stability

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

Characteristic stabilized finite element method for non-stationary conduction-convection problems

2019 ◽  
Vol 30 (2) ◽  
pp. 625-658
Author(s):  
Yongshuai Wang ◽  
Md. Abdullah Al Mahbub ◽  
Haibiao Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Time Derivative ◽  
Content Type ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Advection Term ◽  
The Stability ◽  
Element Method ◽  
Low Order

Purpose This paper aims to propose a characteristic stabilized finite element method for non-stationary conduction-convection problems. Design/methodology/approach To avoid difficulty caused by the trilinear term, the authors use the characteristic method to deal with the time derivative term and the advection term. The space discretization adopts the low-order triples (i.e. P1-P1-P1 and P1-P0-P1 triples). As low-order triples do not satisfy inf-sup condition, the authors use the stability technique to overcome this flaw. Findings The stability and the convergence analysis shows that the method is stable and has optimal-order error estimates. Originality/value Numerical experiments confirm the theoretical analysis and illustrate that the authors’ method is highly effective and reliable, and consumes less CPU time.

A Stabilized Finite Element Method for Non-Stationary Conduction-Convection Problems

2011 ◽  
Vol 3 (2) ◽  
pp. 239-258 ◽  
Author(s):  
Ke Zhao ◽  
Yinnian He ◽  
Tong Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimal Error Estimates ◽  
Two Dimensional ◽  
Element Level ◽  
Optimal Error ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
The Stability ◽  
Element Method

AbstractThis paper is concerned with a stabilized finite element method based on two local Gauss integrations for the two-dimensional non-stationary conduction-convection equations by using the lowest equal-order pairs of finite elements. This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf-sup condition. The stability of the discrete scheme is derived under some regularity assumptions. Optimal error estimates are obtained by applying the standard Galerkin techniques. Finally, the numerical illustrations agree completely with the theoretical expectations.

scholarly journals Two-Level Brezzi-Pitkäranta Stabilized Finite Element Methods for the Incompressible Flows

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Rong An ◽  
Xian Wang
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Incompressible Flows ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Linearization Methods ◽  
Correction Scheme ◽  
Iteration Methods ◽  
Level Method ◽  
The Stability

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.

Impact of the Stabilized Finite Element Method on Acoustic and Vortical Perturbations in Thermoacoustic Systems

2020 ◽  
Author(s):  
Thomas Hofmeister ◽  
Tobias Hummel ◽  
Thomas Sattelmayer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mode Shapes ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Damping Rate ◽  
Artificial Diffusion ◽  
Eigen Mode ◽  
The Impact ◽  
Element Method

Abstract This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized Finite Element Method (sFEM). Specifically, this work analyzes and quantifies the impact of the Streamline-Upwind-Petrov-Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic Linearized Euler Equations (LEE) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This non-physical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigen-frequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.

Impact of the Stabilized Finite Element Method On Acoustic and Vortical Perturbations in Thermoacoustic Systems

2020 ◽  
Author(s):  
Thomas Hofmeister ◽  
Tobias Hummel ◽  
Thomas Sattelmayer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mode Shapes ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Damping Rate ◽  
Artificial Diffusion ◽  
Eigen Mode ◽  
The Impact ◽  
Element Method

Abstract This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized Finite Element Method (sFEM). Specifically, this work analyzes and quantifies the impact of the Streamline-Upwind-Petrov-Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic Linearized Euler Equations (LEE) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This non-physical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigenfrequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.

Epitaxial Growth in Dislocation-Free Strained Alloy Films

2002 ◽  
Vol 715 ◽  
Author(s):  
Zhi-Feng Huang ◽  
Rashmi C. Desai
Keyword(s):  
Deposition Rate ◽  
Elastic Moduli ◽  
Composition Dependence ◽  
Critical Thickness ◽  
Lattice Misfit ◽  
Misfit Strain ◽  
Joint Stability ◽  
Alloy Films ◽  
The Stability ◽  
Stability Results

AbstractThe morphological and compositional instabilities in the heteroepitaxial strained alloy films have attracted intense interest from both experimentalists and theorists. To understand the mechanisms and properties for the generation of instabilities, we have developed a nonequilibrium, continuum model for the dislocation-free and coherent film systems. The early evolution processes of surface pro.les for both growing and postdeposition (non-growing) thin alloy films are studied through a linear stability analysis. We consider the coupling between top surface of the film and the underlying bulk, as well as the combination and interplay of different elastic effects. These e.ects are caused by filmsubstrate lattice misfit, composition dependence of film lattice constant (compositional stress), and composition dependence of both Young's and shear elastic moduli. The interplay of these factors as well as the growth temperature and deposition rate leads to rich and complicated stability results. For both the growing.lm and non-growing alloy free surface, we determine the stability conditions and diagrams for the system. These show the joint stability or instability for film morphology and compositional pro.les, as well as the asymmetry between tensile and compressive layers. The kinetic critical thickness for the onset of instability during.lm growth is also calculated, and its scaling behavior with respect to misfit strain and deposition rate determined. Our results have implications for real alloy growth systems such as SiGe and InGaAs, which agree with qualitative trends seen in recent experimental observations.

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