تجزیه و تحلیل حساسیت از توابع انتقال شعله های ورقه های نازک

The sensitivity of laminar premixed methane/air flames responses to acoustic forcing is investigated using direct numerical simulation to determine which parameters control their flame transfer function. Five parameters are varied: (1) the flame speed sL, (2) the expansion angle of the burnt gases α, (3) the inlet air temperature Ta, (4) the inlet duct temperature Td and (5) the combustor wall temperature Tw. The delay of the flame transfer function is computed for the axisymetric flames of Boudy et al. [1] and the slot flames of Kornilov et al. [2]. Stationary flames are first computed and compared to experimental data in terms of flame shape and velocity fields. The flames are then forced at different frequencies. Direct numerical simulations reproduce the flame transfer functions correctly. The sensitivity analysis of the flame transfer function is done by changing parameters one by one and measuring their effect on the delay. This analysis reveals that the flame speed sL and the inlet duct temperature Td are the two parameters controlling the flame delay and that any precise computation of the flame transfer function delay must first have proper models for these two quantities.

The prediction of acoustically coupled instabilities has become a major issue in combustion [3] and [4]. Numerous authors have proposed approaches to predict the resonant modes between acoustics and combustion [5], [6], [7], [8], [9] and [10]. In all theories, a crucial ingredient is the flame transfer function (FTF) first introduced by Crocco [11] and [12] and Tsien [13]. In its simplest form, the FTF F (ω ) measures the response of the global unsteady reaction rate in the flame View the MathML source(q′/q¯) to an inlet velocity perturbation View the MathML source(u′/u¯) measured at a fixed reference point: equation(1) View the MathML sourceF(ω)=q′/q¯u′/u¯ Turn MathJax on Although many of these studies were performed for complex geometry turbulent burners [14], [10], [15] and [16], they are usually limited and difficult to extrapolate to other regimes or other geometries because turbulent systems combine the difficulties of acoustic/flame coupling and turbulent flows. To isolate the mechanisms controlling FTF results, some groups have started investigating simpler laminar flames where the validity of acoustic/combustion theories can be tested in the absence of complex turbulent effects [17], [18], [19] and [20]. Studies dedicated to the FTF of laminar flames in multiple configurations [21], [2], [22], [23] and [1] are now available, providing both experimental and numerical methods to obtain FTFs. In these cases, only acoustic perturbations imposed on perfectly premixed flames are investigated. Equivalence ratio fluctuations are out of the scope of the present study. In all these configurations, the values obtained for the FTFs parameters are a gain n and a phase ϕ (or delay τ = ϕ/ω), which depend on the forcing frequency ω and in certain cases on the forcing amplitude (see for example the recent developments on the flame describing function [21]). These parameters are critical to predict stability in acoustic solvers [11], [24], [25] and [26]. Small errors on the phase ϕ can lead to drastic changes in stability so that the question of uncertainties in measurement and simulation of FTF becomes an interesting issue. When computing the FTF of a flame, being able to evaluate the sensitivity of the results to modeling parameters is a critical question. For example, Kaess et al. [27] computed the FTF of a laminar flame and concluded that an accurate computation was impossible without the knowledge of the temperature of the stabilizing plate. More generally, many other input parameters of a FTF simulation may affect results and it is important to identify their relative importance. Experimentally, the same question arises: if FTF measurements depend critically on parameters which are not measured with accuracy, results will be useless. For example, the temperature of the plate on which flame are stabilized is rarely measured with precision but it could have a strong effect on the FTF. A good solution to guess which parameters can modify FTFs is to start from theoretical models for the delay τ [17], [19] and [14]. The global heat release rate q (t ) of a flame is written as [7] and [17]: equation(2) View the MathML sourceq(t)=∫sρusLΔqdA Turn MathJax on where the integral is performed over the flame surface, ρ u is the unburnt gas density, s L the flame speed and Δq is the heat release per unit mass of mixture. From Eq. (2), fluctuations in the density ρ u, the flame speed s L, the heat of reaction Δq and in the flame surface A contribute to heat release oscillations View the MathML sourceq′/q¯. Considering a perfectly premixed flow with a constant density and neglecting the effect of the stretch due to flame wrinkling on flame speed [28], the FTF can be expressed in terms of two dimensionless parameters ω ∗ and View the MathML sourcesL∗ [7], [17], [19] and [14]: equation(3) View the MathML sourceF(ω)=q′/q¯u′/u¯=F(ω∗,sL∗)=FωHfVe,sLVe Turn MathJax on where Ve is the convective velocity at the burner inlet and Hf is the flame height. It is generally complex to express directly the fluctuation of the heat release as a function of the fluctuating velocity. Nevertheless, since it is observed that the phase ϕ increases regularly with ω∗, it is possible to describe ϕ as a time lag τ = ϕ/ω. The simplest way to evaluate τ is to express it as the mean time necessary for a velocity perturbation to be convected from the exit plane to the effective position of concentrated heat release [17] and [14]: equation(4) View the MathML sourceτ=HfβVe Turn MathJax on where β is a coefficient depending on the configuration. Values of β ranging from 1 to 3 are typically measured. Since the flame height depends on the flame speed sL and on the convective velocity Ve, Eq. (4) suggests that τ changes only with sL and Ve, hence that kinetic parameters (controlling sL) but also temperatures of gas and walls (controlling Ve) must be important input data for τ. In this paper, FTFs of laminar premixed flames were computed using direct numerical simulation (DNS) to evaluate the influence of five critical input parameters (Fig. 1): (1) the flame speed sL, (2) the shape of the domain characterized by its expansion angle α, (3) the inlet air temperature Ta, (4) the inlet duct temperature Td and (5) the combustor wall temperature Tw. Full-size image (17 K) Fig. 1. Parameters controlling the FTF of a laminar premixed flame. Figure options All these parameters have a direct effect on the FTF delay τ (or phase ϕ ). The flame speed s L obviously controls the flame length and therefore the delay of the flame to react to velocity changes. The shape of the domain determines the expansion of the burnt gases and the flow velocity, thereby also changing the FTF delay: here it is supposed to have a conical shape of angle α . Many experiments (and computations) are designed to perfectly match periodic arrays of flame [21] and [2] where α should be zero. Note that the confinement of the flames comes from the proximity of neighboring flames and not from a closed burner. In practice however, these flames are only partially confined: the gases produced by each individual flame can expand both in the axial and transverse directions. This can be accounted for in the DNS by using an expanding computation and values of α up to ten degrees are commonly observed experimentally. The inlet air temperature T a affects both the gas velocity and the flame speed whereas the inlet duct temperature T d changes the temperature and velocity profiles at the burner inlet. The combustor walls temperature T w determines the lift-off of the flame and can also control the FTF delay. Obviously, other uncertainties and phenomena can affect the FTF as radiation heat losses, geometric imperfections, inlet velocity profiles (steady and forcing parts), flame to flame interactions, three-dimensional effects or position of the reference point for the velocity View the MathML sourceu′/u¯ measurement. Nevertheless, the study is restricted to these five parameters which are difficult to determine precisely, have an important impact on FTF, and are easily manageable with a CFD solver. The objective of this work is to determine the sensitivity of the FTF to these five parameters. This identification will be done using simple differentiation methods (i.e. changing only one parameter and measuring its effect on the FTF delay). The exercise will be performed on two recent laminar flame experiments (Fig. 2) for which extensive sets of experimental results are available: the experiment of Boudy et al. [1] corresponds to 49 conical flames stabilized on a perforated plate while the configuration of Kornilov et al. [2] corresponds to an array of 12 slot flames. Full-size image (21 K) Fig. 2. The two laminar flame experiments computed in this work. Left: the experiment of Boudy et al. [1]. Right: the experiment of Kornilov et al. [2]. Figure options The paper is organized as follows. First the Boudy et al. and Kornilov et al. experimental facilities are presented. The numerical methodology used to predict the FTFs is then described. Uncertainty sources in FTF phase determination are identified and the methodology for the sensitivity analysis is exposed. Finally, results on steady and forced flames are analyzed.

Flame transfer functions (FTFs) measure the response of flames submitted to acoustic forcing. Their determination is critical to predict the stability of combustors. The present work has focused on the determination of the sensitivity of FTF to five important sources of uncertainty on two laminar premixed flames: (1) the flame speed sL, (2) the shape of the domain characterized by its expansion angle α, (3) the inlet air temperature Ta, (4) the inlet duct temperature Td and (5) the combustor wall temperature Tw. Results show that these five modeling parameters directly impact velocity profiles and laminar flame speeds of steady configurations and thus the FTF phases. Nevertheless, due to associated typical error margins, two parameters play a dominant role in the FTF phase error calculation: • The flame speed sL has a direct effect on the delay: increasing sL leads to decrease the FTF phase. Unfortunately, this is typically a quantity which is not well known and difficult to specify with precision even when using the most advanced flame solvers. In this field, research on flame dynamics is conditioned by progress in chemical kinetics. • The duct wall temperature Td induces significant errors on the FTF phase: an increase in the duct wall temperature induces an acceleration of the fresh mixture at the burner inlet as well as an increase of the local flame speed leading to a decrease of the FTF delay. Knowing the wall temperature of the inlet duct is needed to predict the phase correctly. A direct implication of this result is that coupled computations of flame and heat transfer through the stabilization plate are needed to obtain Td and be able to predict FTFs in such configurations.