Dynamical response of a perfectly premixed flame and limit behavior for high power density systems
Introduction
Combustion instabilities are an important technical concern when designing industrial systems which incorporate combustion chambers. They often result from the coupling between the unsteady heat released by the flame and the acoustic modes of the combustor. Understanding the phenomena driving the dynamics and response of a flame submitted to acoustic perturbations is of theoretical and practical interest.
Chen et al. [1] provided recently an interesting description of the dynamical behavior of lean premixed flames. They formulated the discontinuity between upstream and downstream of the flame using the Rankine–Hugoniot relationships, introducing the flame velocity in the laboratory frame of reference . This mathematical description provides a way to study two apparent problems that appear when the flame is considered immobile. The first one is that the mass flow rate is not conserved through an acoustic discontinuity. The second one is that velocity fluctuations upstream of the flame translates in significant entropy fluctuations downstream even for a perfectly premixed flame, which is not observed in experiments. Considering the fact that the flame position changes with time to adapt to upstream conditions allows to solve elegantly these conceptual issues. The impact of this displacement on the stability analysis of an actual system is not investigated in their study.
In parallel, an analysis of the impact of the motion of a flame on the acoustic source term is presented in [2]. An energetic approach is derived to estimate analytically the acoustic source term related exclusively to the flame displacement. One strong assumption of this study is that the flame follows exactly the acoustic velocity, which is expressed by the balance equation: where is the local heat release rate and is the velocity of the flame in the laboratory frame of reference and chosen equal to the acoustic velocity. It is not true in general. This assumption was brought forth to isolate the contribution of the motion on the acoustic source term. In that case, it was proven that the contribution of the flame displacement to the acoustic balance equation was strictly positive and could become significant for systems with high power density. One reaches significant growth rate by rising the global heat release rate or by decreasing the dimensions of the system. This generally corresponds to an increase in the resonant frequencies of the system.
The first idea of the present work is to use the dynamical representation of the premixed flame provided in [1], apply it on a concrete case and estimate the different contributions, including the one due to the flame displacement discussed in [2]. As mentioned above, the displacement contribution becomes significant when the power density of the system increases. A particular focus is made to estimate the system behavior in the asymptotic condition corresponding to very high power density.
An energetic approach is chosen to account for the different contributions independently. There are two major contributors to the system stability: the coupling between flame and acoustics and the acoustic fluxes at the boundary conditions. In this study, this problem has been considered with much scrutiny because it appeared that some interesting features arise when the Mach number starts to grow. In addition, the formulation of the jump conditions by Chen et al. introduces an entropy term that is convected in the burnt gases. Therefore, an energetic formulation for the boundary contribution that includes entropy effect has been used [3].
The complex way premixed flames respond geometrically to acoustic perturbations is out of the scope of this paper. The G-Equation was used successfully in the past to model such phenomena [4], [5], [6], [7]. Instead, the flame response is captured here using an model. The parameters n and τ are varied to study the influence on linear stability. Luzzato and Morgans [8] recently considered the effect of the flame motion on stability. The mathematical solution was obtained using the characteristic method in the temporal domain. This methodology provides interesting results, in particular because it gives access to non-linear features of the instability. Nevertheless the physical mechanisms which drive the impact on stability are not clearly identified and discussed. The study does not address the limit cycle for high-frequency modes and the role of entropy convection, which is investigated here. Using an energetic approach allows to provide a deeper physical understanding of why and in which context the implication of the motion is important.
The objectives of this article are the following:
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Use the dynamical description of the flame proposed by Chen et al. [1] to model the displacement contribution in an actual simple system.
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Introduce an energetic approach representing independently the different contributions from the flame and boundaries.
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Propose an asymptotic modeling of the flame contributions when the resonant frequencies and power density of the system increase.
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Present a parametric study to demonstrate the validity of the models and illustrate the influence of some key parameters: power density, Mach number, flame response, size of the cavity among others.
In Section 2, the formulation of the flame dynamics introduced in [1] is reviewed. Section 3 provides the various energetic contributions that will be compared in the parametric study. Section 3.1 introduces αRay, which accounts for the complete flame contribution, motion and flame response. Section 3.2 reminds the E-FAME model of [2]. In Section 3.3, the flame response is decomposed in two terms, a flame velocity response linked to velocity perturbations upstream of the flame and a flame pressure response. In Section 3.4, an asymptotic model is proposed for high frequency/high power density conditions. The boundary conditions contributions are estimated in Section 3.5. Section 4 describes the classical method to obtain the wave structure in the system. It also provides a reference growth rate which is used to compare to the energetic approach. Finally, Section 5 presents the results of the parametric study.
Section snippets
Formulation of the perfectly premixed flame problem
The idea of a moving discontinuity to represent a heat source has been introduced by Chu [9], [10]. Subsequently, this idea was used in various studies ([11], [12], [13] among others). The particular feature of Chen et al. approach [1] is to link the entropy creation at the flame location and its velocity in the laboratory frame of reference. Using this formalism allows to provide meaningful explanations for apparent issues discussed previously in the literature, as already mentioned in the
Energetic approach
The energetic approach was used quite early in the development of methodologies to study the acoustic stability of systems [17]. For example, Cantrell and Hart [18] provided a rigorous formalism to estimate the impact of the boundary fluxes on the acoustic stability of the system (see Section 3.5 for more details on this subject). A useful introduction to this method, as well as the basis relationships between acoustic fluxes, combustion/acoustic coupling terms and growth of the acoustic energy
Derivation of the plane wave solution
The geometry is represented in Fig. 2. Two cavities are considered, with a flame located at the interface.
In this section, one assumes that the acoustic and entropic variables behave like 1D plane waves, which is a common assumption when studying simple geometries. The principle is to use the boundary conditions and jump conditions at the flame location to express linear relationships between the waves amplitudes. This provides two important results. First, the complex frequency, which
Results of the parametric study
The purpose of this section is to implement the different growth rates expressed so far and to calculate them in a representative case which can be related to practical applications. Various parameters are modified to illustrate the impact on the stability of the system.
A few growth rate expressions have been introduced along this document. Their purpose is to identify accurately the contributions and their physical origin. The scheme presented in Fig. 3 aims at showing the link between the
Conclusion
This article focuses on the modeling of the response of a perfectly premixed flame in a 1D cavity. The modeling is based on the set of equations provided by Chen et al. [1] who described the response and dynamics of a lean premixed infinitely thin flame in a quasi-1D flow configuration. The flame described by these equations is included in a simple acoustic element and the different acoustic contributors are carefully evaluated.
The flame contribution to the acoustic behavior of the system is
Acknowledgments
We thank the reviewers of this article for the fruitful discussions throughout the reviewing process, which have significantly contributed to increasing the quality of the paper.
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