Nonlinear flame response dependencies of a V-flame subjected to harmonic forcing and turbulence

Abstract

We study the dependence of the flame sheet response of an open V-flame when subjected to simultaneous but independently controlled – (1) narrowband, coherent disturbances, (2) mean flow velocity, and (3) turbulent broadband disturbances. We re-examine the data presented in Humphrey et al. (2018). Flame edge and flow field were obtained from Mie scattering images and PIV vector fields, respectively. We determine the flame sheet response, which allows us to understand the effects of kinematic restoration on the local and global flame dynamics. We find some highly nonlinear behaviors in the flame sheet response, which have not been reported previously. In particular, we observe (1) oscillations in the rise and peak region of the flame response, and (2) early onset of the nonlinear coupling between narrowband and broadband disturbances, resulting in oscillatory decay immediately downstream of the flame holder region. The spatial oscillations seen in the flame response arise only when unalike disturbances such as vortical and flame wrinkle disturbances have a comparable wavelength and, thus, interfere. We then find that higher nominal velocity and turbulent intensities increase asymmetry in the flame response, and subsequently affect the local and global heat release response (HRR). We analytically calculate HRR from flame sheet response. We find that there is a monotonic increase in the response with increasing flame length for lower nominal velocities, possibly due to higher flame symmetry. Moreover, the HRR calculated from flame sheet response captures the effect of kinematic restoration very well. Notably, there is a decrease in HRR with increasing turbulence intensity due to enhanced smoothing from kinematic restoration at higher turbulence levels. We further find that the flame response dependence on harmonic forcing manifest in the low-pass filter characteristics of the global HRR response.

Introduction

Lean combustion systems are prone to the coupling between unsteady heat release rate and unsteady pressure oscillations, and give rise to large amplitude pressure oscillations, typically known as combustion or thermoacoustic instability [1], [2]. Thermoacoustic instability is a result of the positive feedback between unsteady pressure and heat release rate oscillations such that the energy from combustion adds to the acoustic energy of the combustor, leading to amplification of the duct acoustic modes [1]. These large amplitude pressure oscillations induce cyclic loads on the combustor, seriously undermining their operational life, and have cost billions of dollars in revenue to propulsion and power ventures. Thus, understanding the underlying mechanisms behind thermoacoustic instability is crucial for developing targeted control strategies.

Practical combustion systems are in general turbulent. During thermoacoustic instability, the reacting flow-field is acted upon by coherent narrowband flow fluctuations induced by the acoustic field in the combustor and by broadband flow fluctuations due to turbulence. These complex interactions have been known to lead to highly nonlinear phenomena such as intermittency, quasiperiodicity, chaos, and n-period oscillations [3], [4], [5], [6]. Thus, understanding the contribution of both the narrowband and broadband fluctuations on the flame response assumes particular significance.

The interaction between turbulence and combustion is quite complicated, even in the absence of any self-excited narrowband excitations. The overall dynamics of the reacting flow field is controlled by the balance between, but not limited to, the following processes: the susceptibility of the background flow to flow instability [7], [8], [9], alteration in the flow stability due to combustion [10], [11], [12], [13], [14], [15], flame propagation normal to itself [16], [17], [18], [19], intrinsic flame instability such as Darrieus–Landau or Rayleigh–Taylor instability [20], [21], [22]. As such, in the absence of flow instabilities, the balance between the characteristic scales (time and length) of the flow and the flame leads to various types of flame structures as indicated by the flame regimes in a Borghi diagram [23], [24].

In turbulent combustion systems, flow instabilities play a crucial role in flame stabilization. So, flow instabilities are induced in practical combustors through flame holding mechanisms such as bluff body or dump plane, such that an unstable shear layer is created. The shear layer breaks down through Kelvin–Helmholtz instability and leads to periodic vortex shedding or Bénard-von Kárman (BVK) instability. The periodic vortices create recirculation zones where combustion takes place. This process is thus responsible for fluctuations in the heat release rate response in such systems [8], [9].

Many studies have shown that measuring the flame sheet response can shed light on the effect that different flame-flow processes have on the overall flame response [18], [25], [26], [27], [28], [29]. While some of the effects are destabilizing, flame propagating normal to itself (or Huygens propagation [30]) has a net stabilizing effect through local smoothing of any modulations on the flame surface. This flame stabilizing process is known as kinematic restoration [17], [30]. Accordingly, any induced impulse response normal to the flame will decay in the absence of any other destabilizing mechanism. In general, any given material line (such as streamlines, vorticity contours, the level-set defining the flame surface) is exponentially stretched under turbulence induced strain, so any disturbance normal to the material line diminishes at an exponential rate [31]. As a result, in the absence of kinematic restoration (i.e., non-propagating flames), laminar flames lose stability with monotonic and unbounded growth of the flame area [32]. Thus kinematic restoration is essential in maintaining the stability of premixed flames.

Next, we consider the effect of turbulence on the flame response. For bluff-body stabilized conical flames in the limit of corrugated flamelet and thin reaction regime, increase in turbulence intensity have shown to (1) increase the thickness of the preheat zone, (2) broaden the strain rate and curvature probability density functions, (3) increase the flame area ratio, and (4) increase and saturate the flame brush thickness [33], [34]. Further, the average corrugated flame burning speed depends crucially on the turbulence intensity, with quadratic, 4/3 power law and linear dependence at low, intermediate and high turbulence levels, respectively [35], [36], [37]. The effect of turbulence on kinematic restoration is of particular interest in this study. Kinematic restoration is strengthened with increasing turbulence, as is evidenced by the decrease in the ensemble averaged flame fluctuations in numerical [25], [26], [27] and experimental [29], [38] studies. Figure 1(a) shows the increase in flame smoothing for a turbulent simulation (solid red line) and Fig. 1(b) shows the decrease in flame wrinkle amplitude, $|〈\widehat{L}(y,f_0)〉|$. The effect of kinematic restoration is indeed significant as it can alter the gain and phase of the flame transfer function considerably through leading order corrections in the asymptotic analysis of the heat release rate response [19].

The imposition of harmonic excitation, self-excited or external, on the flame–turbulence interaction brings in a characteristic acoustic length and time scale in addition to the spectrum of length and time scales present due to the underlying flow turbulence. Consequently, the flame response to acoustic [39], [40], [41], [42] and vortical [43], [44], [45], [46] disturbances have been found to be highly nonlinear functions of the frequency and amplitude of the imposed disturbances.

The flame sheet response dynamics depends critically on the frequency (or wavelength) of the imposed acoustic disturbances [19]. Kinematic restoration will cause the random fluctuation in the flame position to tend to the baseline imposed by the harmonic acoustic disturbances, as indicated by the dotted blue line in Fig. 1(c). Further, the baseline curvature can locally increase/decrease the turbulent flame consumption speed [29]. Thus, the flame sheet response at the frequency of the harmonic acoustic disturbances, $\widehat{L}\left(f_0\right),$ is of particular interest. For bluff-body stabilized flames, Shanbhogue et al. [18] found that the flame sheet response amplitude, determined as the amplitude of the Fourier transformed flame front fluctuation at the forcing frequency, is a nonlinear function of the downstream distance, as is illustrated in Fig. 1(d). The flame response in the vicinity of the bluff-body linearly increases with the downstream distance as flame holding mechanisms dominate the response. The flame response reaches its peak due to constructive interference between convection of vortical flow disturbances and propagation of coherent flame wrinkles along the flame front. Further downstream, in the absence of any disturbance generating mechanism, the flame response decays as the strength of vortical disturbances and coherent wrinkles diminish due to kinematic restoration and flame stretch processes in thermo-diffusively stable flames [22], [26].

We analyze the flame sheet response of a turbulent V-flame anchored on an oscillating bluff-body, and subjected to varying levels of turbulent intensity. We re-examine the dataset presented by Humphrey et al. [29]. The oscillating bluff-body allows us to induce narrowband disturbances without having to resort to forcing the acoustic field as was done in Shanbhogue et al. [18]. The primary motivation was to keep the boundary condition such that the amplitude of flame wrinkling in the linear, stretch-free analysis, remains constant with axial location, i.e., $L^{\prime }(y,t)=\epsilon \cos \left(2\pi f_{0}t\right)\Rightarrow |\widehat{L}(y,f_0)|=|\widehat{L}(y=0,f_0)|=\epsilon ,$ where ε is the forcing amplitude [26], [47]. This experimental configuration allows for relatively simple expressions for important quantities [26] as opposed to the acoustically forced flames, where the flame front is subjected to complex nonlinear modulations [48], [49].

Harmonic and broadband forcing were systematically carried out over a large frequency range ($f_0=200--1250$ Hz) and high turbulence levels ($u^{\prime }/{\overline{u}}_y\sim 8--36\%$). The flame–turbulence interaction in our study lies in the corrugated turbulent and thin reaction regimes [23], [50]. The largest Strouhal number based on the bluff-body diameter for our study is $S{t}_d=f_{0}d/{\overline{u}}_y=0.27$ for $d=0.81$ mm, ${\overline{u}}_y=3.69$ m/s and $f_0=1250$ Hz. Further, we consider a system where the effects of shear layer instability are minimized by incorporating a co-flow with the same nominal velocity as the main flow. So, any flow instability will only be due to the density gradient across the flame surface. Thus, for certain conditions, we elicit the flame response unaltered by self-excited fluctuations which the flow field would have undergone had there been shear layer instability.

In this study, we isolate the flame sheet response when the flame is subjected to varying forcing frequency, turbulence levels, and mean flow velocity. We show that the flame response has many highly nonlinear features different from what has been reported until now. We find that some of the simplifying assumptions required for carrying out linear analysis such as that of considering a constant amplitude flame response do not provide the complete picture of the flame response. Further, we determine the disturbance wavelengths that interfere to produce the nonlinear flame sheet response that we notice. We then evaluate the heat release response (HRR), which we assume to depend only on the fluctuations in the flame surface area. The flame surface area fluctuations are dependent on the flame geometry and the flame front fluctuations. Thus, we calculate the HRR at the component of forcing frequency by spatially integrating the flame sheet response along the flame length after accounting for the flame geometry. While calculating HRR, we consider the effect of flame asymmetry, which alters the flame geometry and, thus, the flame surface area. Specifically, flame structure – symmetric (varicose) or asymmetric (sinuous) – dictates the global HRR. In comparison to asymmetric flames, symmetric flames have a more significant contribution to the flame area fluctuations and, thus, to the overall HRR [15]. We also observe that the flame asymmetry very clearly manifests in the local HRR profile. Finally, we find the dependence of the integrated HRR on the turbulent intensity and harmonic forcing.

The rest of the paper is organized as follows. In Section 2, we discuss the experimental setup, the optical diagnostics used, and image processing techniques. In Section 3.1, we characterize the effect of turbulence on the amplitude spectrum of the flame sheet fluctuations. In Section 3.2, we isolate the effect of varying turbulence intensity, forcing frequencies and nominal flow velocities on the flame sheet amplitude response. In Section 3.3, we present the dependence of local and global flame response on the forcing frequency. In Section 3.4, we analyze the flame asymmetry. In Section 3.5, we consider the integrated local and global heat release rate response of the system. Finally, we present the conclusions of the study.