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.

Stability and Unstability of the Standing Wave to Euler Equations

2017 ◽  
Vol 9 (4) ◽  
pp. 818-838
Author(s):  
Xiuli Tang ◽  
Xiuqing Wang ◽  
Ganshan Yang
Keyword(s):  
Saddle Point ◽  
Euler Equation ◽  
Euler Equations ◽  
Standing Wave ◽  
Well Posedness ◽  
Compressible Euler Equation ◽  
Compressible Euler ◽  
Local Solutions ◽  
The Stability

AbstractIn this paper, we first discuss the well-posedness of linearizing equations, and then study the stability and unstability of the 3-D compressible Euler Equation, by analysing the existence of saddle point. In addition, we give the existence of local solutions of the compressible Euler equation.

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 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

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 BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

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