Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation
Abstract
:1. Introduction
- Its function should not take a negative value for all parameter fields.
- It should be equal to zero in uniform fields.
- If there is a steady solution to the studied system, then all possible disturbances should have a greater value for this function. It is supposed that there are no external disturbances to the system because, in this work, self-sustained oscillations are under consideration.
- This function should not depend on ambient conditions because sonic boundary conditions in the inlet and outlet of the combustion chamber and low heat transfer from the combustion chamber casing prevent the transfer of information about ambient conditions.
- This function should give a limit from above for pressure oscillations that could emerge in the combustion chamber.
2. Simple Model
- -
- It is not negative;
- -
- The internal exergy of the whole system is not less than the internal exergy of its parts;
- -
- Internal exergy in a closed system may only decrease;
- -
- Internal exergy does not depend on ambient parameters;
- -
- The mean temperature could be found through internal exergy and internal entropy production to reach equilibrium.
3. The Simple Model’s Extension to the Moving System
4. Variational Principles for Studying Combustion Instability
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
acoustic power | ||
adiabatic index | - | |
absolute pressure | ||
pulsating pressure | ||
heat source | ||
pulsating heat | ||
frequency | ||
mean by volume pressure | ||
velocity of flow | ||
pulsating velocity | ||
heat capacity | ||
entropy | ||
pulsating entropy | ||
averaging by volume | ||
energy | ||
mass | ||
exergy | ||
density | ||
entropy of system | ||
small parameter | - | |
volume of system under consideration | ||
applied to system work | ||
components of velocity vector | ||
external velocity | ||
external pressure | ||
field of temperature | ||
reference temperature | ||
components of viscous stress tensor |
Appendix A
Appendix B
- (1)
- It introduced two time scales: fast and slow. Pulsating pressure and velocity are represented as a series of time-dependent linear equation eigenvectors in which the amplitude of such eigenvectors changes on a slow time scale.
- (2)
- The author used Mayers’ equations to find a nonlinear form of acoustic energy see Equation (13).
- (3)
- The author using Mayer’s equations, found source terms for this form of acoustic energy and found its function
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Skiba, D.V.; Zubrilin, I.A.; Yakushkin, D.V. Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation. Appl. Sci. 2024, 14, 1453. https://doi.org/10.3390/app14041453
Skiba DV, Zubrilin IA, Yakushkin DV. Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation. Applied Sciences. 2024; 14(4):1453. https://doi.org/10.3390/app14041453
Chicago/Turabian StyleSkiba, Dmitriy Vladimirovich, Ivan Alexandrovich Zubrilin, and Denis Vladimirovich Yakushkin. 2024. "Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation" Applied Sciences 14, no. 4: 1453. https://doi.org/10.3390/app14041453
APA StyleSkiba, D. V., Zubrilin, I. A., & Yakushkin, D. V. (2024). Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation. Applied Sciences, 14(4), 1453. https://doi.org/10.3390/app14041453