Exergy as Lyapunov Function for Studying the Dynamic Stability of a Flow, Reacting to Self-Oscillation Excitation

Abstract

This article introduces a new, physically-based Lyapunov function definition that can be used as acoustic energy for finite-amplitude pressure oscillations in reacting systems, especially in combustion chambers. Reacting flow is seen as an open, non-equilibrium (parameters are distributed unevenly locally) thermodynamic system. This Lyapunov function is defined as the maximum mechanical work that could be extracted by a heat engine from the studied system if this system were disconnected from the inlet and outlet and from any other surrounding environment and the engine could transfer to the surrounding environment only mechanical work.

1. Introduction

Combustion instability is one of the most significant problems for low-emission combustion chambers and high-energy-density rocket engines currently under development. Combustion instability has a significant meaning in land-based gas turbine installations due to the widespread method of burning lean, preliminarily blended mixtures. This approach allows for significantly reducing the most harmful nitrous oxides (NOx) while sacrificing combustion stability [1].

Under these operating conditions, burners are severely damaged by self-exciting oscillations caused by combustion. This effect has been investigated with different types of combustion chambers in the following works [2,3,4]. Particularly, the instability problem emerges in the combustion of such a perspective type of fuel as hydrogen for both aviation engines and land-based gas turbine installations. Combustion instability, a coupling between resonant combustor acoustics and heat release rate fluctuations, is one of the leading challenges in developing and operating both aircraft and power-generation gas turbines [5]. Combustion stability and velocity of heat dissipation, depending also on acoustic oscillations, impact both the engine’s characteristics and its reliability. It is also known that the method of vortex burning, which is commonly employed because of its numerous advantages, also has a tendency to instability onset, which is documented in the following sources [6,7,8,9,10]. Therefore, as for producing a complete mathematical model of self-oscillating processes in the combustion chamber, it is necessary to consider the total energy, including the acoustic component.

In some devices, such as thermoacoustic engines and pulsejets, this instability may be used to produce work. In systems where combustion instability is inhibited, such phenomena produce work but are structurally destructive. Pressure oscillations, which occur in the combustion instability process, have high availability to produce work. Moreover, a system that uses pressure oscillations can be added to any internal combustion engine to create additional work.

The first study for singing flames was produced by Rayleigh [11]. In his work, the criteria for maintaining the oscillations were elaborated: “…it follows that at the phase of greatest condensation heat is received by the air, and at the phase of greatest rarefaction heat is given up from it, and thus there is a tendency to maintain the vibrations”. Analyzing combustion stability has been carried out in two ways. The first way was started by work [12] using Kovaznay ideas. The purpose of this work was a physical-based explanation of Rayleigh criteria by dividing the volume of the studied system into small parts and studying the interaction between them. It is assumed that these parts work as the cylinder and piston of a reciprocating engine and move with the main flow. These parts interact with acoustic waves; if parts produce work, then this work is used to increase the energy of acoustic waves; if these parts consume work, then it decreases the energy of acoustic waves.

The second way was started from the Putnam interpretation [13] of Rayleigh criteria; in this work, mean flow was omitted from the study, and from the linear acoustic analysis for pressure wave equation, it was derived that the mean power added to the acoustic wave for a unit of volume is equal to ${\mathrm{N}}_{\mathrm{A}}$:

$${\mathrm{N}}_{\mathrm{A}}=\underset{\mathsf{\tau }\to \infty }{{\displaystyle \lim }}\frac{1}{\mathsf{\tau }}{\displaystyle \underset{0}{\overset{\mathsf{\tau }}{\int }}\frac{\left(\mathrm{k}-1\right)}{\mathrm{k}{\mathrm{P}}_0}{\mathrm{P}}^{\prime }{\mathrm{Q}}^{\prime }\mathrm{d}\mathrm{t}.}$$

(1)

Polifke, in his work [14], divided acoustic emission from the combustor into combustion noise and self-excited fluctuation. The combustion noise does not have feedback from the sound emitted behind the combustion region, which does not influence the combustion process or species concentration. The self-excited fluctuations are produced by the right-phase correlation between combustion and reflected sound waves. In his explanation of the thermodynamic cycle, which increases the internal energy of sound waves, he also neglected the mean flow effect, so he gets, as a result, the Putnam interpretation of Rayleigh’s criterion Equation (1). In the following chapters of his work, the author studied moving systems in which the produced work can be compared with Equation (1):

$${\mathrm{N}}_{\mathrm{A}}=\underset{\mathsf{\tau }\to \infty }{{\displaystyle \lim }}\frac{1}{\mathsf{\tau }}{\displaystyle \underset{0}{\overset{\mathsf{\tau }}{\int }}\frac{\left(\mathrm{k}-1\right)}{\mathrm{k}{\mathrm{P}}_0}{\mathrm{P}}^{\prime }{\mathrm{Q}}^{\prime }\left(1-\frac{{\mathrm{P}}^{\prime }\mathrm{Q}}{{\mathrm{P}}_0{\mathrm{Q}}^{\prime }}\right)\mathrm{d}\mathrm{t}.}$$

(2)

If Rayleigh’s criterion (in Putnam’s interpretation [13]) is analyzed from a thermodynamic point of view, it can be shown that it is another form of the Carnot formula:

$$\begin{array}{l}{\mathrm{N}}_{\mathrm{A}}=\underset{\mathsf{\tau }\to \infty }{{\displaystyle \lim }}\frac{1}{\mathsf{\tau }}{\displaystyle \underset{0}{\overset{\mathsf{\tau }}{\int }}\frac{\left(\mathrm{k}-1\right)}{\mathrm{k}{\mathrm{P}}_0}{\mathrm{P}}^{\prime }{\mathrm{Q}}^{\prime }\mathrm{dt}}\approx \\ \approx \mathrm{f}{\displaystyle \oint \frac{{\mathrm{T}}^{\prime }}{{\mathrm{T}}_0}}\mathrm{d}{\mathrm{Q}}^{\prime }\approx \mathrm{f}{\displaystyle \oint \left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}}\right)}\mathrm{d}{\mathrm{Q}}^{\prime }.\end{array}$$

(3)

In other words, Rayleigh’s criteria only means that the system is unstable if more heat has been added when it has a higher temperature and less heat has been added when it has a lower temperature.

This analogy between Rayleigh criteria and the Carnot formula clearly shows that, in the same way as the efficiency of real internal combustion engines is less than the efficiency of the Carnot cycle, the real combustion instability efficiency in transforming heat into acoustic energy is less than in the model used to formulate Rayleigh’s criteria. And Equation (2) from work [14] can be seen as a correction to Brayton cycle efficiency in comparison with the Carnot cycle.

The definitions of stability criteria are usually produced from some adaptation of the Lyapunov function; in the first work, it was acoustic energy; later, it was extended to a function that takes into account entropy pulsations. However, the Lyapunov function was defined using only algebraic manipulations, without a physical-based explanation.

This method to derive an equation for acoustical energy was described in Mayers’ work [15]. In works [16,17,18,19,20,21], this method to derive acoustic energy was extended for non-isentropic flow and to analyze combustion systems. In all these works, acoustic energy is studied as a second-order polynomial with disturbances. In their work [20], the authors studied which type of acoustic energy quadratic form does not support spurious solution growth. In their work [21], authors extend Mayers’ method to derive the equation for a third-order polynomial by disturbances, which is used to study nonlinear interaction between acoustic waves having different frequencies, but the physical meaning of this quantity was not explained. In this study, the origin of acoustical energy is analyzed at the base of the thermodynamics of open systems, and the definition of acoustic energy is the system’s availability to produce work (or, in other words, the exergy of a system). Previously, this method was used in works [22,23], but the definition of exergy used in those works was only a first-order polynomial by disturbances, and the proposed approximation of acoustic energy has a zero-mean value. Due to this, it could not be used to study the growth and attenuation of combustion instability.

The task of this work is to purposefully extend the acoustic energy conception for entropically reacting flows. This function should comply with the following criteria:

• 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

Thermodynamic formulation of the Lyapunov function for reacting flow is performed according to Bystrai [24]. The studied system is seen as having internal and external parameters. If the system is steady, it means it reaches equilibrium between external thermodynamic forces and dissipation; internal parameters are fully defined; and additional work extraction from this system is impossible. Thus, the simplest case is when there are no external thermodynamic forces at all; we just have a non-equilibrium closed system.

To study the influence of pressure, velocity, and entropy fluctuations on the availability of work, we start our analysis with a closed volume that contains ideal gases with uneven pressure, entropy, and velocity fields. Suppose these fluctuations are small enough compared to the mean internal energy in the volume, where $\mathsf{\epsilon }$ is a small parameter:

$$\mathrm{p}={\mathrm{p}}_0+\mathsf{\epsilon }{\mathrm{p}}^{\prime }(x); \mathrm{s}=\mathsf{\epsilon }{\mathrm{s}}^{\prime }(x); \overrightarrow{v}=\mathsf{\epsilon }\overrightarrow{v^{\prime }} (x)$$

(4)

Mean parameters were taken in such a manner that these conditions were met, where $\langle \hspace{0.17em}\hspace{0.17em}\rangle$ is an operator to find the mean value in volume:

$$\langle \overrightarrow{{\mathrm{v}}^{\prime }}\rangle =\frac{1}{\mathrm{V}}{\displaystyle \underset{\mathrm{V}}{\int }\overrightarrow{\mathrm{v}}}\mathrm{d}\mathrm{V}=\overrightarrow{0};$$

(5)

$$\langle {\mathrm{p}}^{\prime }\rangle =\frac{1}{\mathrm{V}}{\displaystyle \underset{\mathrm{V}}{\int }}{\mathrm{p}}^{\prime }\mathrm{d}\mathrm{V} = 0;$$

(6)

$$\langle {\mathrm{s}}^{\prime }\rangle =\frac{1}{\mathrm{V}}{\displaystyle \underset{\mathrm{V}}{\int }{\mathrm{s}}^{\prime }}\mathrm{d}\mathrm{V} = 0.$$

(7)

For ideal gas, we will use the equation of state represented by variables $\mathrm{p}$ and $\mathrm{s}$ at mean pressure, and the mean entropy density is equal to ${\mathsf{\rho }}_0$:

$$\mathsf{\rho }={\mathsf{\rho }}_0{\left(\frac{\mathrm{p}}{{\mathrm{p}}_0}\right)}^{1/\mathrm{k}}\exp \left(-\frac{\mathrm{s}}{{\mathrm{C}}_{\mathrm{p}}}\right).$$

(8)

It is supposed that no dissipative processes are present in this volume; it is also supposed that we have some engine that could produce work from fluctuations in this volume but without working gas’ mass loss. Moreover, this engine has no connection with the surrounding environment except for mechanical work’s transfer. The initial distribution of parameters in this system will be named “state 1”. The final distribution of parameters will be named “state 2”. When this final distribution is achieved, this engine will have extracted all available work. In “state 2”, the distribution of parameters should be uniform.

To find the maximum work that can be produced from such a system, we will use the statement from Landau and Lifshits [25]: «A system does maximum work when its entropy remains constant, i.e., when the process of reaching equilibrium is reversible». The same method to derive the availability of work is used in [26] for the optics field. This maximal work will be named internal exergy in this text.

According to this statement, the maximum work $\mathrm{Ex}$ will be:

$$\mathrm{Ex}={\mathrm{E}}_1-{\mathrm{E}}_2,$$

(9)

where ${\mathrm{E}}_1$ is this system’s energy in “state 1”, ${\mathrm{E}}_2$ is this system’s energy in “state 2”. The systems in “state 1” and in “state 2” should have the same mass (M), the same entropy (S), and the same volume (V), but the system in “state 2” should have uniform parameters. According to this, we have:

$$\begin{array}{l}\mathrm{M}={\mathsf{\rho }}_2\mathrm{V}={\displaystyle \underset{\mathrm{V}}{\int }{\mathsf{\rho }}_1\mathrm{d}\mathrm{V}=}\\ ={\displaystyle \underset{\mathrm{V}}{\int }{\mathsf{\rho }}_0{\left(\frac{{\mathrm{p}}_0+\mathsf{\epsilon }{\mathrm{p}}^{\prime }\left(x\right)}{{\mathrm{p}}_0}\right)}^{1/\mathrm{k}}\exp \left(-\frac{\mathsf{\epsilon }{\mathrm{s}}^{\prime }\left(x\right)}{{\mathrm{C}}_{\mathrm{p}}}\right)}\mathrm{d}\mathrm{V}\end{array}$$

(10)

$$\begin{array}{l}\mathrm{S}={\mathsf{\rho }}_2{\mathrm{s}}_2\mathrm{V}={\displaystyle \underset{\mathrm{V}}{\int }{\mathsf{\rho }}_1{\mathrm{s}}_1\mathrm{d}\mathrm{V}=}\\ ={\displaystyle \underset{\mathrm{V}}{\int }{\mathsf{\rho }}_0{\left(\frac{{\mathrm{p}}_0+\mathsf{\epsilon }{\mathrm{p}}^{\prime }\left(x\right)}{{\mathrm{p}}_0}\right)}^{1/\mathrm{k}}\exp \left(-\frac{\mathsf{\epsilon }{\mathrm{s}}^{\prime }\left(x\right)}{{\mathrm{C}}_{\mathrm{p}}}\right)}\mathsf{\epsilon }{\mathrm{s}}^{\prime }\left(x\right)\mathrm{d}\mathrm{V}.\end{array}$$

(11)

The full derivation of energies in “state 1” and “state 2” is presented in Appendix A. The final result is:

$$\begin{array}{l}\frac{\mathrm{Ex}}{\mathrm{V}{\mathsf{\epsilon }}^2}=\frac{{\mathrm{E}}_1 - {\mathrm{E}}_2}{\mathrm{V}{\mathsf{\epsilon }}^2}=\frac{{\mathsf{\rho }}_0\langle {\left|\overrightarrow{{\mathrm{v}}^{\prime }}\right|}^2\rangle }{2}+\frac{\langle {\left({\mathrm{p}}^{\prime }\right)}^2\rangle }{{2\mathrm{kp}}_0}+\frac{{\mathrm{p}}_0\langle {\left({\mathrm{s}}^{\prime }\right)}^2\rangle }{{2\mathrm{kRC}}_{\mathrm{P}}}+\\ -\frac{\mathsf{\epsilon }\left(2\mathrm{k} - 1\right)\langle {\left({\mathrm{p}}^{\prime }\right)}^3\rangle }{{6\mathrm{k}}^2{\left({\mathrm{p}}_0\right)}^2}+\frac{\langle {\mathrm{p}}^{\prime }{\left({\mathrm{s}}^{\prime }\right)}^2\rangle }{{2\mathrm{kRC}}_{\mathrm{P}}}-\frac{\mathsf{\epsilon }{\mathrm{p}}_0\langle {\left({\mathrm{s}}^{\prime }\right)}^3\rangle }{3\mathrm{R}{\left({\mathrm{C}}_{\mathrm{P}}\right)}^2}+\\ +\frac{\mathsf{\epsilon }{\mathsf{\rho }}_0\langle {\mathrm{p}}^{\prime }{\left|\overrightarrow{{\mathrm{v}}^{\prime }}\right|}^2\rangle }{{2\mathrm{kp}}_0}-\frac{\mathsf{\epsilon }{\mathsf{\rho }}_0\langle {\mathrm{s}}^{\prime }{\left|\overrightarrow{{\mathrm{v}}^{\prime }}\right|}^2\rangle }{{2\mathrm{C}}_{\mathrm{P}}}+O\left({\mathsf{\epsilon }}^2\right).\end{array}$$

(12)

If we take only the quadratic part of this expression, it is the same as in works [17,18,21]. From Strumpe and Furletov’s work [17], it is:

$${\mathrm{E}}_{\mathrm{III}}=\frac{\mathsf{\rho }{\left({\mathrm{v}}^{\prime }\right)}^2}{2}+\frac{{\left({\mathrm{p}}^{\prime }\right)}^2}{{2\mathsf{\rho }\mathrm{c}}^2}+\frac{\mathsf{\rho }\mathrm{T}{\left({\mathrm{s}}^{\prime }\right)}^2}{{2\mathrm{C}}_{\mathrm{P}}}.$$

(13)

From Thierry Poinsot and Denis Veynante’s work [8], it is:

$$e_{tot}{=\mathsf{\rho }}_0\frac{{\left(\overrightarrow{{\mathrm{v}}^{\prime }}\right)}^2}{2}+\frac{1}{{\mathsf{\rho }\mathrm{c}}^2}\frac{{\left({\mathrm{p}}^{\prime }\right)}^2}{2}+\frac{{\mathrm{p}}_0{\left({\mathrm{s}}^{\prime }\right)}^2}{{2\mathrm{RC}}_{\mathrm{P}}}.$$

(14)

The third power term part of Expression (12) can be found only in Jacob’s work [27]:

$$\begin{array}{l}{\mathrm{E}}_3=\frac{1}{2}{\mathsf{\rho }}_0{\left(\overrightarrow{{\mathrm{v}}^{\prime }}\right)}^2+\frac{1}{{\mathsf{\rho }}_0{\mathrm{c}}^2}\frac{{\left({\mathrm{p}}^{\prime }\right)}^2}{2}+\\ +\frac{{\mathsf{\rho }}_0{\mathrm{T}}_0{\left({\mathrm{s}}^{\prime }\right)}^2}{2{\left({\mathrm{C}}_{\mathrm{P}}\right)}^2}+\\ +\frac{\left(1 - 2\mathrm{k}\right){\left({\mathrm{p}}^{\prime }\right)}^3}{6{\left({\mathsf{\rho }}_0\right)}^2{\mathrm{c}}^4}+\frac{{\mathrm{p}}^{\prime }{\left({\mathrm{s}}^{\prime }\right)}^2}{{2\mathrm{kRC}}_{\mathrm{P}}}-\\ -\frac{{\mathsf{\rho }}_0{\mathrm{T}}_0{\left({\mathrm{s}}^{\prime }\right)}^3}{3{\left({\mathrm{C}}_{\mathrm{P}}\right)}^3},\end{array}$$

(15)

but it does not have all the terms in Equation (12) that can be found in this work, mainly because the author of that work think they are negligible.

It is valuable to note that this equation for internal exergy does not contain the mean temperature of volume. This peculiarity can be explained as follows: no heat is added or removed from the system, and the entropy is kept the same during this thermodynamic process, so the thermodynamic state is fully described by pressure and entropy.

We can also study such situations as when fluctuations of parameters diminish due to the dissipation process while mass in volume and internal energy are kept constant. This will lead to an increase in system’s entropy. This state will be named as “state 3”. Using asymptotic expansion with a small parameter $\mathsf{\epsilon }$ we will get the following result:

$$\frac{{\mathrm{p}}_0}{\mathrm{R}}\Delta {\mathrm{s}}_{1\to 3}=\mathrm{Ex}.$$

(16)

The mean thermodynamic temperature may be determined according to the Gouy-Stodola theorem [28]:

$${\mathrm{T}}_0=\frac{\mathrm{Ex}}{\langle \mathsf{\rho }\rangle \Delta {\mathrm{s}}_{1\to 3}}=\frac{{\mathrm{p}}_0}{\mathrm{R}\langle \mathsf{\rho }\rangle },$$

(17)

or the mean temperature in this case is calculated from the mean pressure and density.

This method to find the internal exergy of a closed volume may be extended to a reacting mixture according to the internal exergy definition. In this case, we can study a closed volume with a reacting mixture and the mass fraction of its components, which initially has an uneven distribution around the volume. Engines can produce work using such unevenness until concentrations of components become uniform and all chemical reactions are fully completed to equilibrium. The work will be maximal if entropy is kept constant by such an engine, and in “state 2”, it will achieve minimal internal energy in this volume compared to all states if the constraints are fulfilled.

The defined internal exergy has such valuable properties:

-

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

The case studied in the previous section may be extended to the case of a closed thermodynamic system interacting with its surroundings through work and heat. This interaction with the surroundings will lead to changes in mean entropy and pressure. In this article, we study such systems as having internal structure, which can be analyzed by internal exergy. To define such a system’s internal exergy, we will derive an evolutionary differential equation using the assumption that irreversible entropy generation is fully defined by dissipative function and by heat exergy definition based on non-isothermal heat transfer. The initial conditions for this equation will be defined according to the results of the previous section.

In this section, we will derive an equation for a second-order internal exergy. In this definition, it is assumed that the volume of averaging is fixed. The defining equation for higher-order exergy can be derived using the same method.

We will use the following identity:

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\langle \frac{{\left(\mathsf{\phi }-\langle \mathsf{\phi }\rangle \right)}^2}{2}\rangle =\langle \left(\mathsf{\phi }-\langle \mathsf{\phi }\rangle \right)\frac{\mathrm{d}\mathsf{\phi }}{\mathrm{d}\mathrm{t}}\rangle ,$$

(18)

with the following second-order expression of internal exergy:

$${\mathrm{Ex}}_2=\frac{\langle \mathsf{\rho }\rangle \langle {\left|\overrightarrow{{\mathrm{v}}^{\prime }}\right|}^2\rangle }{2}+\frac{\langle {\left({\mathrm{p}}^{\prime }\right)}^2\rangle }{2\mathrm{k}\langle \mathrm{p}\rangle }+\frac{\langle \mathrm{p}\rangle \langle {\left({\mathrm{s}}^{\prime }\right)}^2\rangle }{{2\mathrm{kRC}}_{\mathrm{P}}},$$

(19)

and equations of gas dynamics for ideal gas in form:

$$\mathsf{\rho }\frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{t}}+\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=\nabla \left({\mathsf{\sigma }}_x\right),$$

(20)

$$\mathsf{\rho }\frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{t}}+\frac{\partial \mathrm{p}}{\partial \mathrm{y}}=\nabla \left({\mathsf{\sigma }}_y\right),$$

(21)

$$\mathsf{\rho }\frac{\mathrm{d}\mathrm{w}}{\mathrm{d}\mathrm{t}}+\frac{\partial \mathrm{p}}{\partial \mathrm{z}}=\nabla \left({\mathsf{\sigma }}_z\right),$$

(22)

$$\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{t}}+\mathrm{k}\mathrm{p}\left(\nabla ·\overrightarrow{\mathrm{v}}\right)=\left(\mathrm{k}-1\right)\left(\mathrm{Q}+\Phi \right),$$

(23)

$$\frac{\mathrm{p}}{\mathrm{R}}\frac{\mathrm{d}\mathrm{s}}{\mathrm{d}\mathrm{t}}=\left(\mathrm{Q}+\Phi \right).$$

(24)

After representing all quadratic parts in Equation (19) by Equation (18) and inserting Equations (20)–(24) into Equation (19), we will get the following result:

$$\begin{array}{l}\frac{{\mathrm{dEx}}_2}{\mathrm{d}\mathrm{t}}\mathrm{V}=-{\displaystyle \underset{\Gamma }{\int }{\mathrm{p}}^{\prime }\overrightarrow{{\mathrm{v}}^{\prime }}·\mathrm{d}\Gamma }+\\ +{\displaystyle \underset{\mathrm{V}}{\int }\left(\left(\left(1 - \frac{{\mathrm{T}}_0}{{\mathrm{T}}_0+{\mathrm{T}}^{\prime }}\right)-1\right)\left({\mathrm{Q}}^{\prime }+{\Phi }^{\prime }\right)-{\Phi }^{\prime }\right)\mathrm{d}\mathrm{V}}+\\ +{\displaystyle \underset{\Gamma }{\int }\left(\frac{\mathsf{\rho }{\left|\overrightarrow{{\mathrm{v}}^{\prime }}\right|}^2}{2}+\frac{{\left({\mathrm{p}}^{\prime }\right)}^2}{2\mathrm{k}\langle \mathrm{p}\rangle }+\frac{{\left({\mathrm{s}}^{\prime }\right)}^2\langle \mathrm{p}\rangle }{{2\mathrm{kRC}}_{\mathrm{P}}}+\frac{\left(\mathrm{k}-1\right)\langle {\left({\mathrm{p}}^{\prime }\right)}^2\rangle }{2\mathrm{k}\langle \mathrm{p}\rangle }\right)\overrightarrow{\mathrm{v}}}·\mathrm{d}\Gamma +\\ +\left(\frac{\langle \mathrm{p}\rangle \langle {\left({\mathrm{s}}^{\prime }\right)}^2\rangle }{{2\mathrm{kRC}}_{\mathrm{P}}}-\frac{\langle {\left({\mathrm{p}}^{\prime }\right)}^2\rangle }{2\mathrm{k}\langle \mathrm{p}\rangle }\right)\frac{\left(\mathrm{k}-1\right)}{\langle \mathrm{p}\rangle }\langle \mathrm{Q}\rangle \mathrm{V},\end{array}$$

(25)

where the first term is the power at which work is transferred to the studied system, the second term is the exergy of heat transferred to the system and loss of exergy by dissipation, the third term is additional exergy transfer, which is usually neglected due to the small velocity in the studied system (and this is the effect of flowing), and the last term is Brayton cycle correction.

4. Variational Principles for Studying Combustion Instability

The analysis in Section 2 clearly shows that for computational fluid dynamic methods based on finite volume methods, the exergy loss will occur at each time step because of averaging values of pressure, density, temperature, and species concentration in control volumes. Calculation by first-order methods fully follows the procedure described in Section 1. In second-order methods, the loss of exergy will be less, but even this method could not prevent exergy loss because values reconstruction in control surfaces between volumes achieved on the basis of averaged values. In special cases of isentropic flow and in the case of turbulence flow, we have examples in which there are introduced equations for parameters that could not be resolved during calculations with energy conservation. Equation (19) derived in Section 3 could be used in the same way.

The nonlinear isentropic flow has a Hamiltonian structure and may be formulated as a minimal action principle in which the Hamiltonian has a direct relation to exergy [29]. The variational principle for exergy was formulated for incompressible flow in work [30]. The variational principle for exergy may be derived from the principle of least dissipation of energy described in [31]: the difference between the production of irreversible entropy in the system under study, represented by thermodynamic coordinates and forces, and dissipative function in real processes is extremum.

$$\mathsf{\delta }{\displaystyle \underset{\mathrm{V}}{\int }\left(\mathsf{\rho }\mathrm{T}\frac{\mathrm{d}{\mathrm{s}}^i}{\mathrm{dt}}-\Phi \right)}\mathrm{d}\mathrm{V}=0.$$

(26)

Production of irreversible entropy could be found from exergy in such a way:

$$\mathsf{\rho }\mathrm{T}\frac{\mathrm{d}{\mathrm{s}}^i}{\mathrm{d}\mathrm{t}}=\mathrm{A}+\mathrm{Q}\left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}}\right)+\Phi \left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}}\right)-\frac{\mathrm{d}\mathrm{Ex}}{\mathrm{d}\mathrm{t}},$$

(27)

and the variational principle will have such a form

$$\mathsf{\delta }{\displaystyle \underset{\mathrm{V}}{\int }\left(\mathrm{A}+\mathrm{Q}\left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}}\right)+\Phi \left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}}\right)-\frac{\mathrm{d}\mathrm{Ex}}{\mathrm{dt}}-\Phi \right)}\mathrm{d}\mathrm{V}=0.$$

(28)

This principle is formulated for moving closed volume in an ambient field, where the $\mathrm{A}$ (power at which work is transferred to the studied system) should be formulated as having two parts representing the influence of ambient pressure and velocity on the studied system.

$$\mathrm{A}=-\overrightarrow{\mathrm{v}}·\nabla {\mathrm{p}}^{\mathrm{e}}-\mathrm{p}\left(\nabla ·{\overrightarrow{\mathrm{v}}}^{\mathrm{e}}\right),$$

(29)

where ${\mathrm{p}}^{\mathrm{e}}$ and ${\overrightarrow{\mathrm{v}}}^{\mathrm{e}}$ represent external influences by pressure and velocity on the studied volume.

After the variational procedure, these external influences should be set equal to pressure and velocity in the system, and the derived equations can be seen as a model of self-consistent field theory.

If we use second-order exergy, then this variational procedure allows us to recover equations of compressible flow, which are used in work [32], with additions representing addition heat by viscous dissipation:

$$\begin{array}{l}\mathsf{\delta }{\displaystyle \underset{\mathrm{V}}{\int }\left(\mathrm{A}+\left(\mathrm{Q}+\Phi \right)\left(1-\frac{{\mathrm{T}}_0}{\mathrm{T}\left(\mathrm{p},\mathrm{s}\right)}\right)-\Phi -\frac{\mathrm{d}\mathrm{Ex}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}=\\ ={\displaystyle \underset{\mathrm{V}}{\int }\mathsf{\delta }\mathrm{u}\left(-\frac{\partial {\mathrm{p}}^{\mathrm{e}}}{\partial x}+\nabla \left({\mathsf{\sigma }}_x\right)-\mathsf{\rho }\frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}+\\ +{\displaystyle \underset{\mathrm{V}}{\int }\mathsf{\delta }\mathrm{v}\left(-\frac{\partial {\mathrm{p}}^{\mathrm{e}}}{\partial y}+\nabla \left({\mathsf{\sigma }}_y\right)-\mathsf{\rho }\frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}+\\ +{\displaystyle \underset{\mathrm{V}}{\int }\mathsf{\delta }\mathrm{w}\left(-\frac{\partial {\mathrm{p}}^{\mathrm{e}}}{\partial z}+\nabla \left({\mathsf{\sigma }}_z\right)-\mathsf{\rho }\frac{\mathrm{d}\mathrm{w}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}+\\ {\displaystyle \underset{\mathrm{V}}{\int }\mathsf{\delta }\mathrm{p}\left(-\nabla ·{\overrightarrow{\mathrm{v}}}^{\mathrm{e}}+\left(\mathrm{Q}+\Phi \right)\frac{\mathrm{k}-1}{{\mathrm{kp}}_0}-\frac{1}{{\mathrm{kp}}_0}\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}+\\ +{\displaystyle \underset{\mathrm{V}}{\int }\mathsf{\delta }\mathrm{s}\left(\left(\mathrm{Q}+\Phi \right)\frac{{\mathrm{p}}_0}{{\mathrm{pC}}_{\mathrm{P}}}-\frac{{\mathrm{p}}_0}{{\mathrm{RC}}_{\mathrm{P}}}\frac{\mathrm{d}\mathrm{s}}{\mathrm{d}\mathrm{t}}\right)}\mathrm{d}\mathrm{V}.\end{array}$$

(30)

Equation (24) may be seen as a Galerkin formulation for flow calculation, which, if $\mathsf{\delta }\mathrm{p}=\mathrm{p},$ $\mathsf{\delta }\mathrm{s}=\mathrm{s},$ $\mathsf{\delta }\mathrm{u}=\mathrm{u},$ $\mathsf{\delta }\mathrm{w}=\mathrm{w}$, represents the internal exergy of flow. This equation could be used in numerical computation to study dynamical flame instability and find the upper limit of self-sustained oscillations.

5. Conclusions

The novelty of the presented results resides in the physical-based interpretation of the Lyapunov function, usually used in analyzing stability in reacting flows.

To find the difference between oscillating flow and steady flow, it is usually assumed that steady flow exists and that the mean values calculated in oscillating flow are equal to steady ones. However, this is true only up to and inclusive of the first order of pressure oscillation magnitude. In this work, the ground state is the uniform state from which mechanical work cannot be extracted. The amount of mechanical work that could be extracted may be attributed to pressure, velocity, and entropy waves, as well as turbulent pulsations of flow. In addition, this amount of work could be attributed to work that could be applied to destroy the structure, and this is the upper limit of this mechanical work.

The results of this work show that the results of the algebraic manipulations described in works [17,18,21] have clear physical meaning: The value for which the transport equation was derived is maximal mechanical work, which could be extracted by a heat engine from the studied system if this system were disconnected from the inlet and outlet and from any other surroundings and the engine could transfer to the surroundings only mechanical work. This value in this article is named internal exergy.

The integral of the transport equation for internal exergy by volume may be seen as a variational functional for compressible Navie-Stockes equations, which allows to “physically correct” model wave dynamics in numerical calculations because this prevents too high numerical dissipations and at the same time avoids incorrect internal entropy negative production. In Appendix B, a brief overview of the application of internal exergy formulation to the construction of the master equation is presented. A detailed discussion of this problem should be the topic of a separate article.

The increase in effectiveness in determining the characteristics of acoustic oscillations in numerical models, with the help of the aforementioned “physical correction”, could allow more efficient evaluation of both peak and integral heat dissipation [33]. Additionally, the use of the present model allows for more efficiently solving during the design phase one of the main problems in the development and operation of both aviation and land-based gas turbine installations, with the additional effect of minimizing the destabilization of the combustion process through acoustic oscillations.