Elsevier

Combustion and Flame

Volume 156, Issue 11, November 2009, Pages 2201-2212
Combustion and Flame

Linear and non-linear forced response of a conical, ducted, laminar premixed flame

https://doi.org/10.1016/j.combustflame.2009.06.027Get rights and content

Abstract

This paper presents an experimental study on the dynamics of a ducted, conical, laminar premixed flame subjected to acoustic excitation of varying amplitudes. The flame transfer function is measured over a range of forcing frequencies and equivalence ratios. In keeping with previous works, the measured flame transfer function is in good agreement with that predicted by linear kinematic theory at low amplitudes of acoustic velocity excitation. However, a systematic departure from linear behaviour is observed as the amplitude of the velocity forcing upstream of the flame increases. This non-linearity is mostly in the phase of the transfer function and manifests itself as a roughly constant phase at high forcing amplitude. Nonetheless, as predicted by non-linear kinematic arguments, the response always remains close to linear at low forcing frequencies, regardless of the forcing amplitude. The origin of this phase behaviour is then sought through optical data post-processing.

Introduction

The dynamics of premixed laminar flames under acoustic excitation have been the subject of many investigations for a long time. Over the last half century a wealth of research has been devoted to this topic, and the field remains active (e.g. [1], [2], [3], [4], [5]). Studies of laminar flame dynamics form part of ongoing research into the broader problems of combustion generated sound and thermo-acoustic combustion stability. In these problems, flames both respond to and generate sound as well as other disturbances which, in turn, may generate sound through interaction with other system elements [3], [4], [6], [7], [8], [9].

Theoretical analyses of premixed laminar flame dynamics usually involve a kinematic description of the flame front as a hydrodynamic discontinuity, through use of the so called ‘G equation [2], [3], [4], [5]. For example, detailed analysis of the shape of a conical flame under low amplitude acoustic forcing and incident vortices was conducted by Boyer and Quinard [10]. They argued that the evolution of the flame front under weak acoustic forcing can be predicted reasonably well by considering only the kinematics of the problem. This was confirmed and advanced theoretically and experimentally by Baillot et al. [11]. Considering the same problem configuration as Boyer and Quinard, Baillot et al. showed that the flame response includes a wave originating at the flame base which then propagates along the flame with a convective velocity close to the mean flow velocity. A kinematic approach was again shown to predict the geometric characteristics of the excited flame satisfactorily.

Fleifil et al. [12] then used a kinematic model to calculate the (linear) transfer function between the heat release and upstream velocity modulation of a flame stabilised in a simple Poiseuille flow. Such arguments were extended to general axi-symmetric conical flames by Ducruix et al. [13], who assumed a uniform, one-dimensional velocity excitation along the flame and the calculated flame transfer function was observed to be in good agreement with experiment at lower frequencies. An extension to this linear theory was made by Schuller et al. [14], [15] by substitution of the uniform velocity excitation with excitation by a convective wave, with the following flame transfer functionq/|q|u/|u|=2St211-cos2α1-exp(jSt)+exp(jStcos2α)-1cos2α,whereSt=(ω.R)/(SL.cosα).Here, the terms q and q¯ are the fluctuating and mean flame heat release (W), u and u¯ are the fluctuating and mean velocity just upstream of the flame (m/s), St is a Strouhal number and ω, R,SL and α are, respectively, the forcing frequency (rad/s), flame holder radius (m), the laminar flame speed (m/s) and the flame half apex angle.

Schuller et al. [14] observed that Eq. (1) agreed with experiment better than the model of Ducruix et al. [13]. However, some disparities persisted at higher frequencies. Birbaud et al. [16] later showed that assuming convection of flame disturbances at cold flow convective velocity is physically justifiable. Other works by Lieuwen and co-workers [5], [17] have extended the Schuller et al. transfer function, and suggested that the phase speed of the velocity disturbance should be considered as an independent variable. Thus, the problem is controlled by a Strouhal number based on the cold flow velocity and the steady flame length, the ratio of the flame length to the flame holder radius and the ratio of the speed of the velocity disturbance to the mean flow velocity. Following [18], the two-dimensional influence of the near field acoustics upon the flame upstream velocity and pressure field in a confined configuration was investigated [19]. These two-dimensional effects were then coupled with the kinematic model to calculate the flame transfer function of a ducted conical flame [20]. The coupling resulted in significant changes in both the amplitude and phase of the transfer function compared to those predicted by the linear theory of Schuller et al. [14].

Most experimental works on non-linear flame dynamics focus more on the evolution of the flame front and the flow configuration rather than measure the flame transfer function [21], [22], [23]. The effects of frequency and amplitude of velocity forcing on the stability and appearance of an open, conical, laminar premixed flame has been studied experimentally by Bourehla and Baillot [21]. By varying the forcing frequency up to 1000 Hz and excitation amplitude ϵ=|u/u¯| up to 75%, with a cold flow mean velocity of about 1 m/s on a burner diameter of 22 mm, Bourehla and Baillot reported three main flame regimes. The first regime, occurring at frequencies and excitation amplitude below 120 Hz and 20%, respectively, was called ‘wrinkling’. This included anchored flames wrinkling axi-symmetrically around the burner centre line. For frequencies between 120 and 240 Hz and excitation amplitudes of 25–36% the flame front vibrates at sub-harmonics of the forcing frequency, this regime was therefore termed ‘sub-harmonic’. Higher excitation amplitudes in both the wrinkling and sub-harmonic regimes were limited by flame flash back and blow-off. This led to the third regime, ‘chaotic flames’, in which the flame is lifted and disordered and covered a frequency range from 120 Hz to 600 Hz. This study showed that for excitation amplitudes below 20% the spatially averaged laminar flame speed of the excited flame and the flame’s mean length were close to their steady values. No heat release or flame transfer function measurements were reported in this study.

Durox et al. [22] reported a study at higher frequencies and forcing amplitudes. At frequencies of order a 1000 Hz and above and velocity excitation amplitudes about 3 times the laminar flame velocity of a stoichiometric methane–air, open conical flame with mean cold flow velocity of about 0.9 m/s, Durox et al. [22] demonstrated that the oscillating flow dominated the mean flow and the conical flame flattened into a roughly hemispherical shape. Further, no further flame surface area change occurred once the flame was flattened, and therefore it was concluded that the reaction rate remained unaffected by the acoustic perturbation. Prior to the transition from a conical to a hemispherical configuration, the interaction of acoustic waves with intrinsic flame instabilities led to a parametric instability [23]. Similar to Bourehla and Baillot [21], these studies [22], [23] were also not concerned with flame transfer function or heat release measurement.

Recently, the transfer function of a number of different laminar flame configurations under higher amplitude forcing was studied experimentally by Durox et al. [24]. These authors observed varying degrees of non-linearity in each configuration, with a saturation in the phase of an open conical laminar flame at excitation amplitudes of about 20%.

Theoretical treatment of the non-linear laminar flame dynamics also mainly includes use of the G equation. To capture the cusp formed on an acoustically excited flame, Baillot et al. [25] presented an analytical solution for the non-linear equation describing the front motion. The calculated front for an anchored conical flame closely resembled experiment. Assuming one-dimensional uniform, unsteady velocity field and constant flame speed, Lieuwen [26] solved the G equation numerically for the two cases of conical and bluff-body stabilised flames. Formation of cusps under varying forcing frequency was analysed. Lieuwen argued [26] that the transfer function of the conical flame remains closely linear for all amplitudes of velocity excitation, as long as flame is anchored at its base. Conversely, the dynamics of a bluff-body stabilised flame was observed to be highly dependent upon the velocity excitation amplitude, and demonstrated non-linearity at high excitation amplitude. Similar conclusions on the dynamics of conical and bluff-body stabilised flames were made by Schuller et al. [14].

The dynamics of conical laminar flames have therefore received close theoretical attention. This has led to a number of models for the linear and non-linear flame transfer functions under confined and unconfined configurations (e.g. [5], [14], [17], [20], [26]). These models treat the oscillating flow and the flame in significantly different ways, resulting in qualitative and quantitative differences in their modelled flame dynamics. These differences are consistent with experimental evidence indicating that the flame dynamics is sensitive to factors such as the flame anchoring and the flow field [5], [27]. The inclusion of such effects into low-order flame models is not straight-forward.

These different theoretical approaches suggest that further experimental study is required. In particular, measurements of the flame transfer function at higher forcing amplitudes are rather limited. Studies of the transfer function of conical flames focus on low to moderate amplitude forcing, no greater than ϵ=22% [2], [4], [13], [24], [27]. Attempts to measure the transfer function of a conical laminar flames when the amplitude of the velocity forcing has the same order as the mean velocity are rare. However, it is now commonly accepted that the dominant non-linearity in thermo-acoustic systems even during limit-cycle is within the flame and can feature ϵ1 [2], [6], [9], [5]. Thus, experimental investigation of any non-linear mechanisms in the flame dynamics at high excitation amplitude is important.

This paper, therefore, reports a series of experiments on a ducted, conical, laminar premixed flame which is acoustically excited from upstream. The use of a confined flame allows flame imaging and measurement of the flame transfer function at forcing amplitudes up to ϵ=|u/u¯|=0.9. The study includes investigations of varying equivalence ratio, upstream flow velocity and excitation amplitude on the flame transfer function. Results are compared to existing linear theories. Linear and non-linear flame responses are then distinguished. Camera images are further post-processed to explain the observed non-linear behaviour.

Section snippets

Overall experimental set-up

The rig used in this study resembles those used in similar investigations [11], [13]. Fig. 1 shows the experimental set up and Fig. 2 shows a cross-sectional view of the burner itself. The burner consists of a plenum which contains several layers of mesh and honeycomb to provide a laminar flow, followed by a contraction and a flame holder. The flame is stabilised on the rim of an annular flame holder with an internal diameter of 25 mm, and is within a quartz tube of 50 mm internal diameter and 120

PMT results for low amplitude forcing

The flame was first excited by forcing in the range ϵ=|u/u¯|0.1 to 0.15. A number of upstream mean flow velocities and equivalence ratios were examined over a range of forcing frequency from 10 to 200 Hz. Frequencies below 10 Hz were not tried as a sinusoidal speaker response at these very low frequencies could not be generated. Table 2 summarises these experimental conditions. The flame transfer function used throughout this study is (q/q¯)/(u/u¯), as used in Eq. (1), with the Strouhal

Conclusions

The dynamic response of a ducted, conical, laminar premixed flame to acoustic excitation was studied experimentally. The flame transfer function between the acoustic velocity forcing and the inferred flame heat release was measured for varying forcing amplitudes and frequencies. Flame images were then used to further investigate the physical mechanisms involved in the flame dynamics. For low amplitude excitation (|u/u¯|0.15), the measured flame transfer function was in good agreement with

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