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Role of Darrieus–Landau instability in propagation of expanding turbulent flames

Published online by Cambridge University Press:  10 July 2018

Sheng Yang,
Abhishek SahaOpen the ORCID record for Abhishek Saha[Opens in a new window],
Zirui Liu and
Chung K. Law
Sheng Yang
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Abhishek Saha*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Zirui Liu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Chung K. Law
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USACenter for Combustion Energy, Tsinghua University, Beijing 100084, China
*
†Email address for correspondence: asaha010112@gmail.com
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Abstract

In this paper we study the essential role of Darrieus–Landau (DL), hydrodynamic, cellular flame-front instability in the propagation of expanding turbulent flames. First, we analyse and compare the characteristic time scales of flame wrinkling under the simultaneous actions of DL instability and turbulent eddies, based on which three turbulent flame propagation regimes are identified, namely, instability dominated, instability–turbulence interaction and turbulence dominated regimes. We then perform experiments over an extensive range of conditions, including high pressures, to promote and manipulate the DL instability. The results clearly demonstrate the increase in the acceleration exponent of the turbulent flame propagation as these three regimes are traversed from the weakest to the strongest, which are respectively similar to those of the laminar cellularly unstable flame and the turbulent flame without flame-front instability, and thus validating the scaling analysis. Finally, based on the scaling analysis and the experimental results, we propose a modification of the conventional turbulent flame regime diagram to account for the effects of DL instability.

JFM classification

Instability: InstabilityReacting Flows: Reacting FlowsReacting Flows: Turbulent reacting flows

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018, pp. 784 - 802
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Copyright
© 2018 Cambridge University Press 

1 Introduction

Turbulence facilitates the propagation of premixed flames in general by increasing the heat and mass transport rates as well as the surface area of the embedded flamelets by wrinkling them (Peters Reference Peters2000). Since Damköhler laid the fundamental concept of scaling of the turbulent flame speed, $S_{T}$ , extensive theoretical (Pope Reference Pope1985; Kerstein, Ashurst & Williams Reference Kerstein, Ashurst and Williams1988; Sivashinsky Reference Sivashinsky1988; Bray Reference Bray1990; Peters Reference Peters1999), experimental (Abdel-Gayed, Bradley & Lawes Reference Abdel-Gayed, Bradley and Lawes1987; Kido et al. Reference Kido, Nakahara, Nakashima and Hashimoto2002; Filatyev et al. Reference Filatyev, Driscoll, Carter and Donbar2005; Kobayashi et al. Reference Kobayashi, Seyama, Hagiwara and Ogami2005; Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012; Venkateswaran et al. Reference Venkateswaran, Marshall, Seitzman and Lieuwen2014) and computational (Chen & Im Reference Chen and Im1998; Bell et al. Reference Bell, Day, Shepherd, Johnson, Cheng, Grcar, Beckner and Lijewski2005; Creta & Matalon Reference Creta and Matalon2011) studies, summarized in various review articles (Bradley Reference Bradley1992; Peters Reference Peters2000; Lipatnikov & Chomiak Reference Lipatnikov and Chomiak2005; Driscoll Reference Driscoll2008), have been conducted to quantify the extent of the augmentation, frequently through the normalized turbulent flame speed, $\overline{S_{T}}=S_{T}/S_{L0}$ , where $S_{L0}$ is the mixture’s unstretched planar laminar flame speed, and to correlate it with the imposed turbulence properties. Among the various turbulent flame configurations employed in the experimental studies, the spark-ignited, spherically expanding flame in a fan-generated, turbulent environment within a combustion vessel possesses unique merit, in that the environment is close to being homogeneous and isotropic without a strong mean flow, and it is also amenable to high-pressure studies than the open flame configurations. Using such a configuration, Chaudhuri et al. (Reference Chaudhuri, Wu, Zhu and Law2012) recently developed a scaling between $\overline{S_{T}}$ and the flame Reynolds number, $Re_{T,f}$ , defined as $Re_{T,f}=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$ , where $U_{eff}$ is the effective turbulent velocity fluctuation experienced by the flame with the mean radius, $R_{f}$ ; and $S_{L0,u}$ and $\unicode[STIX]{x1D6FF}_{L0}$ are the corresponding unstretched upstream (unburned) laminar flame speed and flame thickness, respectively. It is to be noted that $Re_{T,f}$ , which is defined based on flame radius, $R_{f},$ and thus increases with the propagation of a spherical flame, is different from the classical turbulent Reynolds number which is defined based on the integral length scale and is fixed during flame propagation. The proposed exponential scaling, $\overline{S_{T}}\sim Re_{T,f}^{0.5}$ , represents the self-similar nature of the spherically expanding turbulent flames (Chaudhuri, Akkerman & Law Reference Chaudhuri, Akkerman and Law2011; Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012). Subsequently, a series of experiments (Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012; Chaudhuri, Wu & Law Reference Chaudhuri, Wu and Law2013; Wu et al. Reference Wu, Saha, Chaudhuri and Law2015; Jiang et al. Reference Jiang, Shy, Li, Huang and Nguyen2016) have been conducted to validate this scaling over a wide range of fuels, moderate-pressure (1–5 atm) range, and various turbulent intensities. The primary motivation of the present study is to extend these recent experimental and interpretive studies by incorporating the effects of the Darrieus–Landau (DL), hydrodynamic cellular flame-front instability (Darrieus Reference Darrieus1938; Landau Reference Landau1944) on the structure and propagation of turbulent flames, especially in identifying the potential further increase in the turbulent flame speed through the corresponding increase of the surface area of the embedded flamelets through the development of the instability cells. Mechanistically, while all premixed flames are predicted to be hydrodynamically unstable in the flame-sheet limit, the actual development of the DL cells can be inhibited for finite flame thickness in relation to the global, DL scale of the flame. Indeed, we (Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012, Reference Chaudhuri, Wu and Law2013; Wu et al. Reference Wu, Saha, Chaudhuri and Law2015) did not observe any DL cells for the equivalent laminar expanding flames, propagating in quiescence, under the same thermodynamic conditions over which the turbulent flames propagate in the presence of fan-generated turbulence. In fact, these turbulent experiments were purposefully designed to suppress the development of the DL cells so that the effects of pure turbulence can be unambiguously identified.

A readily implementable approach to induce the development of DL cells is to reduce the thickness of the flamelets by elevating the chamber pressure. Furthermore, with a sufficiently high, constant chamber pressure, the propensity for the cell development will continuously increase, as the overall size of the expanding flame increases, because of the corresponding decrease of the ratio of the flame thickness to the global flame size. In particular, our recent study (Yang et al. Reference Yang, Saha, Wu and Law2016) on expanding flames in quiescence and elevated pressures has identified three stages of flame propagation and cell development, namely an initial cell-free smooth propagation stage, followed by a transition stage with initiation of cell formation and rapid flame acceleration, which in turn leads to a final, saturated stage characterized by a power-law description of the flame propagation. These results also unambiguously demonstrate the phenomenon of self-acceleration of expanding laminar flames through the development of the DL cells.

Following the above study of the evolvement of the cellular laminar flames, it behoves us to assess the corresponding influence of the DL cells on turbulent flame propagation. The task is expected to be a challenging one because of the complex interaction of the wide spectra of scales characterizing the DL cells and the turbulent eddies. For example, experiments in various flame configurations (Kobayashi, Kawabata & Maruta Reference Kobayashi, Kawabata and Maruta1998; Savarianandam & Lawn Reference Savarianandam and Lawn2006; Bradley et al. Reference Bradley, Lawes, Liu and Mansour2013; Troiani, Creta & Matalon Reference Troiani, Creta and Matalon2015; Bauwens, Bergthorson & Dorofeev Reference Bauwens, Bergthorson and Dorofeev2017) suggested that DL instability can assert leading-order effects in mild turbulent environments, while computations (Akkerman, Bychkov & Eriksson Reference Akkerman, Bychkov and Eriksson2007; Fogla, Creta & Matalon Reference Fogla, Creta and Matalon2013, Reference Fogla, Creta and Matalon2015; Creta et al. Reference Creta, Lamioni, Lapenna and Troiani2016; Fogla, Creta & Matalon Reference Fogla, Creta and Matalon2017) showed the various degree of influence of DL instability depending on the level of the turbulence intensity and Markstein number. These two mechanisms leading to flame wrinkling have also been theoretically studied (Boughanem & Trouvé Reference Boughanem and Trouvé1998; Chaudhuri et al. Reference Chaudhuri, Akkerman and Law2011; Creta & Matalon Reference Creta and Matalon2011) in the limits of strong and weak turbulence, where one of the mechanisms is dominating. It is of course reasonable to expect that, since the DL instability and turbulence are both multi-scale in nature, there should also exist an intermediate, overlapping regime where both of them are of equal importance.

In light of the fundamental interest discussed above, and further recognizing the relevance of high-pressure combustion within internal-combustion engines, we have conducted a comprehensive study, phenomenologically and experimentally, on the role of the DL instability on turbulent flame propagation. Leading-order scaling analysis is performed on the spectra of the DL cells and turbulent eddies, leading to the identification of three regimes of interaction, for which acceleration exponents of the turbulent flames are determined. A modification of the classical regime diagram, which extends the previous experimental results that mostly fall within the flamelet regime, is also proposed to reflect the influence of DL instability on turbulent flames. We now present these investigations and their results.

2 Methodology

In order to compare the behaviours of the flames in different regimes proposed in the following sections, and to validate the leading-order scaling analysis, experiments with constant-pressure expanding flames have been conducted in both laminar and turbulent environments. The flame parameters, such as the unstretched laminar flame speed, thickness and temperature, were evaluated from the one-dimensional planar flame simulations using the Chemkin-Premix code (Kee et al. Reference Kee, Gracer, Miller and Meeks1985). A wide range of mixtures, pressures and flame temperatures were selected to cover conditions in various regimes.

2.1 Experimental specifications

Experiments were conducted in our well-vetted, fan-stirred dual-chamber vessel, shown in figure 1. Four orthogonally located fans were used to generate well-controlled nearly isotropic turbulent flow field, whose intensity has been calibrated by high-speed particle image velocimetry (PIV) reported in Chaudhuri, Saha & Law (Reference Chaudhuri, Saha and Law2015). Since we are interested in interaction between DL instability and turbulence, the experimental conditions were so chosen such that the effective Lewis number ( $Le$ ), defined as the ratio between thermal diffusivity to the controlling mass diffusivity, remains close to unity in order to avoid diffusional–thermal, stabilizing ( $Le>1$ ) or destabilizing ( $Le<1$ ) effects. Specifically, for hydrogen ( $\text{H}_{2}$ ) flames, $Le\approx 1$ condition was achieved by using stoichiometric $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixtures of unity equivalence ratio ( $\unicode[STIX]{x1D719}=1$ ). For hydrocarbon flames, a lean ( $\unicode[STIX]{x1D719}=0.8$ ) $\text{C}_{3}\text{H}_{8}/\text{O}_{2}/\text{Ar}$ mixture was selected as it is diffusionally neutral ( $Le\approx 1$ ) because of the similarity in the molecular weights between the deficient species ( $\text{C}_{3}\text{H}_{8}$ ) controlling mass diffusion and the abundant species (Ar) controlling thermal diffusion. Pressure and dilution ratio, $X_{inert}/(X_{inert}+X_{\text{O}_{2}})$ , where $X$ is the mole fraction, were varied in order to modulate the planar laminar flame speed and flame thickness of the mixture.

Figure 1.

Experimental set-up: (a) apparatus; (b) schematic configuration.

In order to investigate the role of DL instability, experiments were conducted not only over a wide range of pressure variations but also for very high pressures up to 20 atmospheres so as to systematically promote the development of DL cells. It is also noted that only the present constant-pressure, dual-chamber design can accommodate such high ambient pressures for the study of enclosed, expanding flames.

In the experiments, the instantaneous flame speeds were determined by first capturing the flame images using high-speed schlieren imaging at sampling rates up to 15 kHz, with resolution of $600\times 600~\text{pixel}^{2}$ . The captured images were post-processed using the edge-detecting technique in MATLAB to evaluate the enclosed area by the flame-front contour, $A_{f}$ , and the mean radius, $R_{f}$ . The instantaneous downstream flame speed, $S_{T}$ or $S_{L,b}$ , for the turbulent or laminar cases respectively, was then calculated by taking the time derivative of the mean radius, $\text{d}R_{f}/\text{d}t$ . It is noted that only data within $5~\text{mm}<R_{f}<20~\text{mm}$ were used to avoid the influences of ignition and chamber confinement (Burke et al. Reference Burke, Chen, Ju and Dryer2009). Multiple runs were conducted under the same conditions to reduce experimental scatter (shown in supplementary document is available at https://doi.org/10.1017/jfm.2018.426). The turbulent flow field has been characterized using high-speed PIV as reported in our previous publication (Chaudhuri et al. Reference Chaudhuri, Saha and Law2015). The effective root-mean-square (r.m.s.) velocity, termed as $U_{eff}$ , is weakly dependent on the flame radius, $U_{eff}\sim R_{f}^{0.33}$ . To obtain a single-valued representative r.m.s. velocity ( $U_{rms}$ ) of the flow field, $U_{eff}$ at $R_{f}=15~\text{mm}$ (average radius based on change in flame surface area) is used.

Table 1.

Experimental conditions and corresponding unburned ( $S_{L0,u}$ ) and burned ( $S_{L0,b}$ ) laminar flame speeds and flame thickness ( $\unicode[STIX]{x1D6FF}_{L0}$ ).

2.2 Computational specifications

The fundamental flame parameters, such as the unstretched laminar flame speeds, $S_{L0,u}$ (on the unburned side) and $S_{L0,b}$ (on the burned side), flame thickness, $\unicode[STIX]{x1D6FF}_{L0}$ and flame temperature, $T_{f}$ for the experimental conditions were computed using the PREMIX code of the Chemkin-Pro Software. The HP-Mech (Yang et al. Reference Yang, Yang, Wu, Ju and Law2017) and the USC-Mech II (Wang et al. Reference Wang, You, Joshi, Davis, Laskin, Egolfopoulos and Law2007) were used as the kinetic models for the $\text{H}_{2}$ and $\text{C}_{3}\text{H}_{8}$ flames respectively. The flame thickness was determined from the maximum temperature gradient, $\unicode[STIX]{x1D6FF}_{L0}=(T_{f}-T_{0})/(\text{d}T/\text{d}x)_{max}$ , where $T_{0}$ is the initial temperature and $(\text{d}T/\text{d}x)_{max}$ is the maximum temperature gradient. The calculated values for the experimental conditions are listed in table 1.

3 Flame speed scaling: isolated effects of hydrodynamic instability and turbulence

In this section we briefly summarize the recently developed flame speed scaling for expanding flames with separate DL instability and turbulence. In particular, the propagation of laminar, cellularly unstable, self-accelerating expanding flames are often described by a self-similar expression of $R_{f}\sim t^{\unicode[STIX]{x1D6FC}_{DL}}$ or $S_{L,b}=\text{d}R/\text{d}t\sim t^{1-1/\unicode[STIX]{x1D6FC}_{DL}}$ , where $\unicode[STIX]{x1D6FC}_{DL}$ is the acceleration exponent. In a recent work (Yang et al. Reference Yang, Saha, Wu and Law2016), the normalized flame speed, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ and a Péclet number, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$ were used to unify the data from various pressures, and as such the self-acceleration propagation can be expressed with a scaling law, $\overline{S_{L,b}}\sim Pe^{\unicode[STIX]{x1D6FD}_{DL}}$ , where $\unicode[STIX]{x1D6FD}_{DL}$ ( $=1-1/\unicode[STIX]{x1D6FC}_{DL}$ ) is a modified acceleration exponent, and is found to be around 0.23–0.28 ( $\unicode[STIX]{x1D6FC}_{DL}\approx 1.3{-}1.4$ ) for a large range of conditions. It is important to note that the inception of the DL cellular structure in an expanding flame occurs at a critical Péclet number ( $Pe_{c}$ ), such that for flames to be unstable, the condition of $Pe>Pe_{c}$ needs to be satisfied (Matalon Reference Matalon2007; Yang et al. Reference Yang, Saha, Wu and Law2016). On the other hand, the propagation of turbulent expanding flames without the presence of cellular flame-front instabilities was found to follow the scaling law (Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012),

(3.1) $$\begin{eqnarray}\displaystyle \overline{S_{T}}\sim Re_{T,f}^{1/2}=[(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})]^{1/2}. & & \displaystyle\end{eqnarray}$$

Recognizing that for the current set-up $U_{eff}\sim R_{f}^{0.33}$ (Chaudhuri et al. Reference Chaudhuri, Saha and Law2015) and that $S_{L0,u}$ and $\unicode[STIX]{x1D6FF}_{L0}$ are fixed for the same experimental condition (mixture, pressure and dilution ratio, etc.), the scaling law for turbulent flame propagation can also be expressed as $\overline{S_{T}}\sim Pe^{\unicode[STIX]{x1D6FD}_{T}}$ , where $\unicode[STIX]{x1D6FD}_{T}=0.67$ .

4 Hydrodynamic instability and turbulence interaction

We next present the scaling analysis for interacting DL instability and turbulence. Classical and recent theoretical works on this topic are largely based on mathematical analyses (Searby & Clavin Reference Searby and Clavin1986; Boughanem & Trouvé Reference Boughanem and Trouvé1998; Bychkov Reference Bychkov2003; Akkerman & Bychkov Reference Akkerman and Bychkov2005; Chaudhuri et al. Reference Chaudhuri, Akkerman and Law2011), which included such nonlinear effects as kinematic restoration, finite flame thickness and preferential diffusion, etc. Since our purpose is to capture the essence of the interaction between DL instability and turbulence, we shall focus and retain their leading-order effects by establishing the time scales through which the relative magnitudes of the two broad-spectrum processes can be assessed.

Recognizing that both these processes involve a wide spectrum of length and time scales, we compare the inverse of the time scales associated with the common length scales. For DL instability, the inverse time scale is the growth rate of the instability and can be expressed as $\unicode[STIX]{x1D714}_{DL}=1/\unicode[STIX]{x1D70F}_{DL}(k)$ , where $\unicode[STIX]{x1D70F}_{DL}$ is the characteristic time at wavenumber $k$ , the reciprocal of the corresponding length, i.e. $k\sim 1/l_{k}$ . Similarly for turbulence we consider the simplest form of the interaction between turbulence and a planar flame, namely wrinkling of the flame by a turbulent eddy of wavenumber $k$ . Since the only characteristic time here is the eddy turnover time, $\unicode[STIX]{x1D70F}_{T}(k)$ , and it is reasonable to assume that the flame surface growth rate due to this eddy is also of the order of the eddy turnover rate, $\unicode[STIX]{x1D714}_{T}(k)$ , which is the reciprocal of the characteristic time for the wavenumber $k$ , i.e. $\unicode[STIX]{x1D714}_{T}(k)=1/\unicode[STIX]{x1D70F}_{T}(k)$ . Consequently, we aim to compare $\unicode[STIX]{x1D714}_{DL}(k)$ and $\unicode[STIX]{x1D714}_{T}(k)$ at the same wavenumber to assess the relative significance of DL instability and turbulence in increasing the flame surface area and consequently the propagation speed. Such an approach has also been adopted in Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) with the nonlinear stabilizing effect of thermal diffusion.

The leading order of the growth rate for the DL instability at a wavenumber $k$ can be expressed as (Matalon Reference Matalon2007),

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{DL}(k)\approx S_{L}kX(\unicode[STIX]{x1D70E}), & \displaystyle\end{eqnarray}$$

where $S_{L}$ is the laminar flame speed in general, which more specifically can be represented by the unstretched, adiabatic, upstream, laminar flame speed, $S_{L0,u}$ , and $X(\unicode[STIX]{x1D70E})$ is the non-dimensional growth rate depending on the thermal expansion, $\unicode[STIX]{x1D70E}$ , as,

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle X(\unicode[STIX]{x1D70E})={\displaystyle \frac{-\unicode[STIX]{x1D70E}+\sqrt{\unicode[STIX]{x1D70E}^{3}+\unicode[STIX]{x1D70E}^{2}-\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x1D70E}+1}}. & \displaystyle\end{eqnarray}$$

For a typical variation of $\unicode[STIX]{x1D70E}$ , i.e. $\unicode[STIX]{x1D70E}=3{-}10$ , we have $X(\unicode[STIX]{x1D70E})=0.68{-}2.1$ , which can be considered to be $O(1)$ . Therefore, we set $X(\unicode[STIX]{x1D70E})=1$ to arrive at

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{DL}(k)\approx S_{L0,u}k. & \displaystyle\end{eqnarray}$$

Although for this analysis we have considered the growth rate of DL instability in a planar flame, in appendix A we have shown that the choice of flame geometry (spherical versus planar) would not affect the scaling dependence. In appendix A, we have also demonstrated that the diffusive stabilization effects are weak compared to the effect of thermal expansion.

For turbulent flames, assuming Kolmogorov turbulence, we can write the eddy turnover rate as,

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{T}(k)=1/\unicode[STIX]{x1D70F}_{T}(k)=u_{k}^{\prime }/l_{k}\sim u_{k}^{\prime }k, & \displaystyle\end{eqnarray}$$

where $u_{k}^{\prime }$ is the eddy velocity at wavenumber $k$ . Comparing these two growth rates leads to

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D714}_{DL}(k)}{\unicode[STIX]{x1D714}_{T}(k)}}\approx {\displaystyle \frac{S_{L0,u}}{u_{k}^{\prime }}}. & \displaystyle\end{eqnarray}$$

It is noted that for Kolmogorov turbulence, the spectrum of $k$ is bounded by the inverse of the Kolmogorov scale ( $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D702}}$ ) and the integral scale ( $\unicode[STIX]{x1D706}_{I}$ ). Consequently, $u_{k}^{\prime }$ is bounded by $u_{\unicode[STIX]{x1D702}}^{\prime }$ and $U_{rms}$ , namely the eddy velocities of Kolmogorov and integral scales respectively, with the former being always weaker than the latter: $u_{\unicode[STIX]{x1D702}}^{\prime }<U_{rms}$ . Based on (4.5), three possible regimes can therefore be envisioned in terms of the relative values of $S_{L0,u}$ , $u_{\unicode[STIX]{x1D702}}^{\prime }$ , and $U_{rms}$ , which will now be discussed and supported by experimental observations.

4.1 Regime I: instability dominated

In this regime, we have $S_{L0,u}>U_{rms}$ , where $U_{rms}$ is the turbulent fluctuating velocity at the largest length scale, hence the smallest wavenumber. This regime of $U_{rms}/S_{L0,u}<1$ coincides with the wrinkled flamelet regime in the turbulent flame regime diagram. Since $U_{rms}>u_{\unicode[STIX]{x1D702}}^{\prime }$ , the Karlovitz number, $Ka$ , defined as

(4.6) $$\begin{eqnarray}\displaystyle Ka={\displaystyle \frac{\unicode[STIX]{x1D70F}_{f}}{\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}}}=\left(\frac{\unicode[STIX]{x1D6FF}_{L0}}{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D702}}}\right)^{2}=\left(\frac{u_{\unicode[STIX]{x1D702}}^{\prime }}{S_{L0,u}}\right)^{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{f}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ are the time scales for the flame and Kolmogorov scale respectively, is less than unity. In this regime, thus, we have $\unicode[STIX]{x1D714}_{DL}(k)/\unicode[STIX]{x1D714}_{T}(k)>1$ for any $k$ in the turbulent spectrum. However, in order to properly assess the relative effects of the multi-scale DL instability and turbulence, the spans of their corresponding spectra need to be estimated as well, specifically in the context of spherically expanding flames. First, recognizing that the DL cell sizes for a spherically expanding flame can neither be larger than the flame radius ( $R_{f}$ ) nor smaller than the flame thickness ( $\unicode[STIX]{x1D6FF}_{L0}$ ), it can be argued that the wavenumbers for the DL instability are limited within the range of [ $1/R_{f}$ , $1/\unicode[STIX]{x1D6FF}_{L0}$ ], as also suggested by Matalon (Reference Matalon2007). Within this range, the growth rate of the DL cells is proportional to the corresponding wavenumber, as shown in (4.3).

Next we note that the spectrum for Kolmogorov turbulence is bounded by the wavenumber limits of [ $1/\unicode[STIX]{x1D706}_{I}$ , $1/\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D702}}$ ]. Furthermore, we can also write $(u_{k}^{\prime })^{3}/l_{k}\approx \unicode[STIX]{x1D700}$ , where $\unicode[STIX]{x1D700}$ is the dissipation rate which can be assumed to be constant (chapter 6 of Pope (Reference Pope2000)). This leads to $u_{k}^{\prime }\sim l_{k}^{1/3}\sim k^{-1/3}$ and as such the turbulent eddy turnover rate (growth rate) can be expressed as

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{T}(k)=1/\unicode[STIX]{x1D70F}_{T}(k)=u_{k}^{\prime }/l_{k}\sim k^{2/3}. & \displaystyle\end{eqnarray}$$

Furthermore, since in the present experimental set-up, $R_{f}$ is in the range of centimetres while $\unicode[STIX]{x1D706}_{I}$ is in the range of millimetres ( ${\sim}4~\text{mm}$ ), the smallest wavenumber in DL instability ( $1/R_{f}$ ) is smaller than that of turbulence ( $1/\unicode[STIX]{x1D706}_{I}$ ). Finally, by comparing the growth rates and limits of the wavenumbers (figure 2) for the two mechanisms of flame-front wrinkling in the context of this regime of $S_{L0,u}>U_{rms}$ , we can expect the DL instability to be stronger than turbulence and hence this regime is termed the instability dominated regime.

Figure 2.

Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$ , and turbulence, $\unicode[STIX]{x1D714}_{T}$ , at different wavenumbers in the instability dominated regime. The axes are plotted in logarithmic scales.

Figure 3.

Experimental results and conditions of laminar and turbulent flames of $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixture at ( $\unicode[STIX]{x1D719}=1$ ),  $T_{f}=2400~\text{K}$ , different pressures and turbulence intensities (fan speeds in r.p.m.): (a) normalized propagation speeds for laminar unstable flames, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ at different normalized flame radii, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$ , showing three different stages of the evolution of hydrodynamic instability; (b) experimental conditions for turbulent flames in the regime diagram; (c) normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ( $=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$ ); (d) normalized flame speeds, $\overline{S_{L,b}}$ and $\overline{S_{T}}$ , at different normalized flame radii, $Pe$ , showing the flame acceleration. The corresponding conditions are indicated by labels: pressure ( $P$ ) in atm, fan speed in r.p.m. and $U_{rms}$ in  $\text{ms}^{-1}$ .

Experimentally this condition was achieved by using stoichiometric $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixtures ( $\unicode[STIX]{x1D719}=1$ ) with a relatively high flame temperature ( $T_{f}=2400~\text{K}$ ). Flame images are shown in the supplementary material. First we analyse the propagation for laminar flames in figure 3(a) by comparing the normalized laminar flame speeds, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ with the scaled flame radii, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$ , for various pressures in the log–log plot. The figure clearly shows three distinctive stages of DL instability, mentioned in a recent work (Yang et al. Reference Yang, Saha, Wu and Law2016). It is also noted that the laminar flames at $3$ and 5 atm capture all the three stages of propagation while flames at $7$ and 10 atm show only the saturated stage of flame acceleration (Yang et al. Reference Yang, Saha, Wu and Law2016). Subsequently, the turbulent flame experiments are conducted at three pressures, $3$ , $5$ and 7 atm, to assess the interaction between these two mechanisms. These conditions with respect to the turbulent regime diagram are shown in figure 3(b) to confirm that turbulent experiments are indeed within the wrinkled flamelet regime, with some of them being close to the boundary with the corrugated flamelet regime. In figure 3(c), the normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ are plotted versus $Re_{T,f}$ . The slope of the data shows the dependence of $\overline{S_{T}}\sim Re_{T,f}^{0.27}$ (black line) which is clearly weaker than the scaling $\overline{S_{T}}\sim Re_{T,f}^{0.5}$ (red line) derived for cellularly stable flames (Chaudhuri et al. Reference Chaudhuri, Wu, Zhu and Law2012). To compare the acceleration exponents of these turbulent flames and those of the laminar flames, in figure 3(d), we plot $\overline{S_{L,b}}$ and $\overline{S_{T}}$ versus $Pe$ (log–log plot) for $3$ , $5$ and 7 atm at various fan speeds, which shows that the normalized flame speeds for both laminar and turbulent flames are almost parallel with a slope (modified acceleration exponent, $\unicode[STIX]{x1D6FD}$ ) of $0.3$ shown by the blue line, suggesting $\unicode[STIX]{x1D6FD}\approx \unicode[STIX]{x1D6FD}_{DL}<\unicode[STIX]{x1D6FD}_{T}$ . It is also noted that the normalized flame speed increases with the fan speed, or $U_{rms}$ , as expected since $\overline{S_{T}}\sim Re_{T,f}^{0.27}$ . Bauwens et al. (Reference Bauwens, Bergthorson and Dorofeev2017) also observed similar acceleration exponents for weakly turbulent flames under the influence of strong DL instability. Consequently, the experimental results confirm that in this regime, DL instability dominates over turbulence in controlling the flame propagation and its acceleration, hence verifying the scaling analysis.

It is also interesting to note that the turbulent flame speeds at $3$ and 5 atm show a slope of $0.30$ , which is similar to laminar flames at $Pe>Pe_{c}$ (figure 3 d) although part of the turbulent flames are in the range of $Pe<Pe_{c}$ . We have recognized that an external disturbance, such as turbulence, pre-triggers the transition from stable to unstable stage, a phenomenon which was also observed by Bauwens et al. (Reference Bauwens, Bergthorson and Dorofeev2017). This is why the entire propagation of the turbulent flame demonstrates similar slope.

4.2 Regime II: instability–turbulence interaction

Next we focus on the intermediate regime with $u_{\unicode[STIX]{x1D702}}^{\prime }<S_{L0,u}<U_{rms}$ , where DL instability dominates at higher $k$ with smaller $u_{k}^{\prime }$ , while turbulence dominates at lower $k$ with larger $u_{k}^{\prime }$ ; hence this regime is designated the instability–turbulence interaction regime. This relative dominance can be noticed when the growth rates are plotted for different wavenumbers (figure 4), as the relative wavenumber spans for DL instability and turbulence remain the same as those in the previous regime. Due to the dominance of turbulence and DL instability at separate (larger and smaller respectively) wavenumber ranges, it is expected that the flame propagation, in this regime, would be affected by both of the mechanisms. By definition, the upper and lower boundaries of this regime is defined by the two lines, $u_{\unicode[STIX]{x1D702}}^{\prime }=S_{L0,u}$ and $S_{L0,u}=U_{rms}$ , which are the upper $(Ka=(u_{\unicode[STIX]{x1D702}}^{\prime }/S_{L0,u})^{2}=1)$ limit and the lower ( $U_{rms}/S_{L0,u}=1$ ) limit of the corrugated flamelet regime.

Figure 4.

Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$ , and turbulence, $\unicode[STIX]{x1D714}_{T}$ , at different wavenumbers in the instability–turbulence interaction regime. The axes are plotted in logarithmic scales.

Figure 5.

Experimental results and conditions of laminar and turbulent flames of $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixture at ( $\unicode[STIX]{x1D719}=1$ ),  $T_{f}=1800~\text{K}$ , different pressures and turbulence intensities (fan speeds in r.p.m.): (a) normalized propagation speeds for laminar unstable flames, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ at different normalized flame radii, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$ , showing three different stages of the evolution of hydrodynamic instability; (b) experimental conditions for turbulent flames in the regime diagram; (c) normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ( $=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$ ); (d) normalized flame speeds, $\overline{S_{L,b}}$ and $\overline{S_{T}}$ , at different normalized flame radii, $Pe$ , showing flame acceleration. The corresponding conditions are indicated by labels: pressure ( $P$ ) in atm, fan speed in r.p.m. and $U_{rms}$ in  $\text{m}~\text{s}^{-1}$ .

To experimentally achieve the relevant conditions with the same mixture of $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ ( $\unicode[STIX]{x1D719}=1$ ), we reduced $T_{f}$ to 1800 K by increasing the dilution ratio, which in turn increased $U_{rms}/S_{L0,u}$ through the reduction in $S_{L0,u}$ . Flame images are shown in the supplementary material. The laminar flame experiments at different pressures (figure 5 a) demonstrate the similar three evolutionary stages as in the previous regime and confirm the presence of strong flame acceleration through DL instability at higher pressures ( $3$ , $5$ and 10 atm). Turbulent flame experiments, thus, were subsequently conducted at these pressures and their corresponding locations in the regime diagram are shown in figure 5(b). Although some of these data points with higher fan speed ( ${>}$ 1500 r.p.m.) marginally cross the boundary of this regime, we consider them to be representative conditions since the definition of these boundaries are rather qualitative in nature. The normalized turbulent flame speeds obtained for three different pressures and various fan speeds are shown in figure 5(c), which clearly demonstrates a dependence of $\overline{S_{T}}\sim Re_{T,f}^{0.33}$ , being weaker than $\overline{S_{T}}\sim Re_{T,f}^{0.5}$ for flames without cellular instabilities, but stronger than the exponent of $\overline{S_{T}}\sim Re_{T,f}^{0.27}$ observed in the instability dominated regime. Furthermore, we have compared the normalized flame speeds at 5 atm with $Pe$ for both laminar and turbulent flames (figure 5 d), finding the modified acceleration exponent ( $\unicode[STIX]{x1D6FD}\approx 0.43$ ) for turbulent flames is larger than that of the laminar flames with DL instabilities ( $\unicode[STIX]{x1D6FD}_{DL}\approx 0.3$ ), but lower than that of turbulent flames without instabilities ( $\unicode[STIX]{x1D6FD}_{T}=0.67$ ). Consequently, the results in figures 5(c) and 5(d) strongly suggest that in this regime, flame propagation is influenced by both DL instability and turbulence, and as such the scaling exponent takes a value between $\unicode[STIX]{x1D6FD}_{DL}$ and $\unicode[STIX]{x1D6FD}_{T}$ .

4.3 Regime III: turbulence dominated

The third regime is characterized by $S_{L0,u}<u_{\unicode[STIX]{x1D702}}^{\prime }$ . Since $u_{\unicode[STIX]{x1D702}}^{\prime }\leqslant u_{k}^{\prime }$ for any wavenumber ( $k$ ) in the turbulence spectrum, we expect turbulence to dominate in this regime. Since this regime satisfies the condition: $Ka=(u_{\unicode[STIX]{x1D702}}^{\prime }/S_{L0,u})^{2}>1$ , it coincides with the thickened flamelet and broken/distributed regimes of the turbulent flame regime diagram. Furthermore, in this regime we have a reversal in the smallest relative scales, i.e. $1/\unicode[STIX]{x1D6FF}_{L0}<1/\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D702}}$ (since $Ka>1$ ), although the relation between the largest length scales remains the same (i.e. $1/R_{f}<1/\unicode[STIX]{x1D706}_{I}$ ) compared to those of the previous two regimes. Consequently the spans of the wavenumbers for DL instability and turbulence now become only partially overlapping (figure 6) within the wavenumber regime, being reduced to [ $1/\unicode[STIX]{x1D706}_{I}$ , $1/\unicode[STIX]{x1D6FF}_{L0}$ ], while the growth rates for turbulence are larger as expected. Moreover, the eddies in the regime of $k>1/\unicode[STIX]{x1D6FF}_{L0}$ can penetrate into the inner structure of the flamelets, and further promote mixing to facilitate the overall flame propagation, while the DL cells in the range $k<1/\unicode[STIX]{x1D706}_{I}$ are not efficient in generating additional surface area with smaller growth rates than those at higher wavenumbers ( $k$ ), and as such may not contribute significantly to flame propagation. Consequently, turbulence dominates in this regime, which is therefore referred to as the turbulence dominated regime. It is noted that similar $Ka$ dependent transition boundaries between DL affected and turbulence dominated regimes have been reported by others (Boughanem & Trouvé Reference Boughanem and Trouvé1998; Chaudhuri et al. Reference Chaudhuri, Akkerman and Law2011). In particular, a comparison between the regimes proposed in the current study and in Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) has been detailed in the appendix B.

Figure 6.

Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$ , and turbulence, $\unicode[STIX]{x1D714}_{T}$ , at different wavenumbers in turbulence dominated regime. The axes are plotted in logarithmic scales.

Figure 7.

Experimental results and conditions of laminar and turbulent flames of $\text{C}_{3}\text{H}_{8}/\text{O}_{2}/\text{Ar}$ mixture at ( $\unicode[STIX]{x1D719}=0.8$ ),  $T_{f}=2150~\text{K}$ , different pressures and turbulence intensities (fan speeds in r.p.m.). (a) Experimental conditions for turbulent flames in the regime diagram. (b) Normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ( $=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$ ). (c) Normalized flame speeds, $\overline{S_{T}}$ , at different normalized flame radii, $Pe$ , showing the flame acceleration. The corresponding conditions are indicated by labels: pressure ( $P$ ) in atm, fan speed in r.p.m. and $U_{rms}$ in $\text{m}~\text{s}^{-1}$ .

Experimentally, conditions relevant to this regime were achieved with diffusionally neutral ( $Le\approx 1$ ) $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ ( $\unicode[STIX]{x1D719}=1$ ) mixtures with $T_{f}=1600~\text{K}$ , and $\text{C}_{3}\text{H}_{8}/\text{O}_{2}/\text{Ar}$ ( $\unicode[STIX]{x1D719}=0.8$ ) mixtures with $T_{f}=2150~\text{K}$ . Flame images are shown in the supplementary material. The conditions of turbulent flames and their corresponding positions with respect to the regime diagram are shown in figure 7(a). The normalized turbulent flame speeds across all pressures and fan speeds (figure 7 b) show the dependence, similar to what is suggested for cellularly stable flames (3.1) and as such confirm the minimal influence of DL instability on flame propagation. This has been further demonstrated in figure 7(c), which shows that the modified acceleration exponent, $\unicode[STIX]{x1D6FD}\approx 0.67$ , is the same with that for turbulent expanding flames ( $\unicode[STIX]{x1D6FD}_{T}$ ).

Figure 8.

Revised turbulent flame regime diagram recognizing the interaction between DL instability and turbulence.

5 Concluding remark

In summary, we have demonstrated the essential role of the cellular, Darrieus–Landau, hydrodynamic, instability on the propagation of expanding turbulent flames with conditions representing various regimes of the turbulent flames. The leading-order scaling analysis of the growth rates over the wavenumber spectra suggests three distinct regimes based on the relative role of DL instability cells and turbulence on flame propagation, which have been further validated experimentally with a wide range of conditions. These newly defined regimes, marked off in the turbulent flame regime diagram shown in figure 8, are:

(i) Instability dominated regime: This regime is characterized by $S_{L0,u}>U_{rms}$ , with DL instability controlling flame propagation. The regime superimposes with the wrinkled flamelet regime of the turbulent flame regime diagram.

(ii) Instability–turbulence interaction regime: This regime is characterized by $S_{L0,u}<U_{rms}$ and $Ka<1$ , with flame propagation controlled by the coupled effects of DL instability and turbulence. This regime coincides with the corrugated flamelet regime of the turbulent flame regime diagram.

(iii) Turbulence dominated regime: This regime is characterized by $Ka>1$ , and is dominated by the strong role of turbulence such that DL instability does not have significant influence on flame propagation. This regime covers the thickened flamelet regime and the broken/distributed reaction regime.

We close this exposition by noting that in this study we have considered only the leading-order effects of the DL instability and turbulence, for diffusionally neutral ( $Le\approx 1$ ) flames. Clearly more needs to be done, to capture the higher-order effects as well as the non-unity Lewis number, preferential diffusion effects.

Acknowledgements

The authors acknowledge financial support from the US National Science Foundation (CBET, grant no. 1510142).

Supplementary material

Supplementary material is available at https://doi.org/10.1017/jfm.2018.426.

Appendix A. Effect of flame geometry and diffusive terms on scaling analysis

The scaling analysis presented in this manuscript was intended for identifying the interaction of turbulence and DL instability for a generic idealistic flame configuration. This is why we chose the expression of DL instability in a planar flame configuration without the presence of the diffusive stabilizing effects. Although, admittedly, some diffusive effects are always present in a real system, in order to capture the pure DL instability in a simplistic form, we neglected all the stabilizing effects. Below we analyse how the choice of flame geometry (planar versus spherical) and inclusion of diffusive effects would have altered our analysis.

A.1 Planar versus spherical flame

The expression for growth rate of the cellular instability in a planar flame is given in Matalon (Reference Matalon2007)

(A 1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{DL}=S_{L}kX(\unicode[STIX]{x1D70E})-S_{L}\unicode[STIX]{x1D6FF}_{L}(B_{1}+Ze(Le_{eff}-1)B_{2}+PrB_{3})k^{2}, & \displaystyle\end{eqnarray}$$

where $X(\unicode[STIX]{x1D70E})=(-\unicode[STIX]{x1D70E}+\sqrt{\unicode[STIX]{x1D70E}^{3}+\unicode[STIX]{x1D70E}^{2}-\unicode[STIX]{x1D70E}})/(\unicode[STIX]{x1D70E}+1)$ , $\unicode[STIX]{x1D70E}$ is the thermal expansion ratio, $k$ the wavenumber and $B_{1}$ , $B_{2}$ and $B_{3}$ are all functions of $\unicode[STIX]{x1D70E}$ , Ze is the Zel’dovich number, $Pr$ the Prandtl number and $Le_{eff}$ the effective Lewis number of the mixture.

The analogous expression for an expanding flame was derived by Addabbo, Bechtold & Matalon (Reference Addabbo, Bechtold and Matalon2002) and was subsequently summarized in Matalon (Reference Matalon2007),

(A 2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}_{DL}={\displaystyle \frac{{\dot{R}}}{R}}\widetilde{X}-{\displaystyle \frac{{\dot{R}}}{R}}{\displaystyle \frac{\unicode[STIX]{x1D6FF}_{L}}{R}}(\widetilde{B_{1}}+Ze(Le_{eff}-1)\widetilde{B_{2}}+Pr\widetilde{B_{3}}), & \displaystyle\end{eqnarray}$$

where $\widetilde{X}$ , $\widetilde{B_{1}}$ , $\widetilde{B_{2}}$ and $\widetilde{B_{3}}$ are functions of $\unicode[STIX]{x1D70E}$ and $n$ ( $=2\unicode[STIX]{x03C0}Rk$ ) in complex forms due to the spherical geometry. Nevertheless, it can be shown that $\widetilde{X}\propto n\propto k$ and $\widetilde{B_{1}}$ , $\widetilde{B_{2}}$ and $\widetilde{B_{3}}\propto n^{2}\propto k^{2}$ (figure 9). Thus, one can conclude that the functional dependence on the wavenumber ( $k$ ) for the two expressions (A 1) and (A 2) is similar and as such the choice of flame geometry would not affect the scaling expression. In figure 10 we have compared the pure thermal expansion effect on the growth rates (the first terms on the right-hand side from (A 1) and (A 2)) and found that they are linearly proportional to each other.

Recognizing that it is almost impossible to reduce (A 2) to a simpler form because of the functional complexities involved in various terms, we used (A 1) to arrive at a scaling dependence.

Figure 9.

Functional dependence of (a) $\widetilde{X}$ , (b) $\widetilde{B_{1}}$ , (c) $\widetilde{B_{2}}$ and (d)  $\widetilde{B_{3}}$ with $n$ . It clearly shows that $\widetilde{X}\propto n\propto k$ and $\widetilde{B_{1}}$ , $\widetilde{B_{2}}$ and $\widetilde{B_{3}}\propto n^{2}\propto k^{2}$ .

Figure 10.

Comparison of pure thermal expansion effect (without diffusive stabilization effects) in spherical flame ( $({\dot{R}}/R)\widetilde{X}$ ) versus planar ( $S_{L}kX$ ) flame configurations.

A.2 Diffusive effects

Due to the stabilizing effects of the diffusive terms, their inclusion would limit the instability range. Specifically for expanding flame, it would introduce a critical Péclet number ( $Pe=R/\unicode[STIX]{x1D6FF}_{L}$ ) which separates the stable and unstable ranges of propagation. This is evident from laminar flame experiments reported in figures 3(a) and 5(a) of this manuscript and also in Yang et al. (Reference Yang, Saha, Wu and Law2016).

The diffusive effects are embedded in the terms, $B_{1}$ , $Ze(Le_{eff}-1)B_{2}$ and $PrB_{3}$ in (A 1) (similar terms are also there in (A 2)). Recognizing that in our analysis we have focused on only diffusionally neutral mixtures ( $Le_{eff}\approx 1$ ) and that contributions of other terms are rather weak, especially at higher pressures where flames are unstable even at small flame radii due to the reduced flame thickness, which also weakens diffusive effects, we have neglected these stabilization effects to simplify the scaling. It is evident from (A 1) and (A 2) that the stabilization effects are indeed weaker for flames with smaller thicknesses at higher pressures. This is why for turbulent experiments, we only compared data at pressure conditions such that $Pe>Pe_{c}$ , where laminar flames are unstable.

Appendix B. Comparison of proposed scaling analysis with Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011)

The detailed theoretical work presented in Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) and the simplified scaling proposed in the current work are both based on the comparison of growth rates of flame wrinkling by DL instability and turbulence in a planar flame configuration. However, there are some key differences between the regimes identified by these two studies.

Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) identified only one critical boundary which demarcates strong turbulence dominated regime and DL instability affected regime. Specifically, the analysis by Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) defined a parameter $\unicode[STIX]{x1D6E4}=\min (\unicode[STIX]{x1D714}_{T}/\unicode[STIX]{x1D714}_{DL})$ and $\unicode[STIX]{x1D6E4}=1$ line was used as a regime boundary. Here, $\unicode[STIX]{x1D714}_{T}$ and $\unicode[STIX]{x1D714}_{DL}$ are growth rates of flame wrinkling by pure turbulence and pure DL instability respectively. Note that Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) used the symbol $\unicode[STIX]{x1D6FD}$ to represent this ratio of growth rates. It is clear that for $\unicode[STIX]{x1D6E4}>1$ , at all wavenumbers $\unicode[STIX]{x1D714}_{T}>\unicode[STIX]{x1D714}_{DL}$ , hence turbulence dominates wrinkling. This regime of $\unicode[STIX]{x1D6E4}>1$ , thus, can be denoted as purely turbulence dominated regime. However, for $\unicode[STIX]{x1D6E4}<1$ , the opposite argument, i.e. at all wavenumbers $\unicode[STIX]{x1D714}_{DL}>\unicode[STIX]{x1D714}_{T}$ , cannot be made. This is why the domain of $\unicode[STIX]{x1D6E4}<1$ , cannot be denoted as purely DL instability dominated regime.

On the other hand, in the current work we showed that there exist two boundaries dividing the turbulent regime diagram into three domains; purely turbulence dominated regime, purely DL instability dominated regime and turbulence–DL instability coupled regime. Since, in our approach we focused on mixtures with $Le\approx 1$ and $Pe>Pe_{c}$ , effects of the diffusive terms are weak and as such, we could simplify the expression for $\unicode[STIX]{x1D714}_{T}/\unicode[STIX]{x1D714}_{DL}$ to identify these three different domains of interaction, namely, DL instability dominated (for all wavenumbers $\unicode[STIX]{x1D714}_{DL}>\unicode[STIX]{x1D714}_{T}$ ), turbulence dominated (for all wavenumbers $\unicode[STIX]{x1D714}_{T}>\unicode[STIX]{x1D714}_{DL}$ ) and DL instability–turbulence interaction (for some wavenumbers $\unicode[STIX]{x1D714}_{DL}>\unicode[STIX]{x1D714}_{T}$ and for some wavenumber $\unicode[STIX]{x1D714}_{T}>\unicode[STIX]{x1D714}_{DL}$ ) regimes.

From the analysis and detailed experiments reported in this study, it is clear that apart from the turbulence dominated and DL instability dominated regimes, there exists an intermediate regime, where both turbulence and instability influence the flame propagation. The detailed theoretical analysis by Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) could not identify such a regime. Although based on the simplified approach, the current study is able to identify all three possible regimes, which are strongly supported by the experiments.

Figure 11.

Experimental conditions for which the flame propagation is influenced by both turbulence and DL instability. The conditions are plotted with respect to regimes defined by Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) and present study.

For example, the experiments reported in figure 5 clearly suggest that flame propagation for these experiments is influenced by both turbulence and DL instability. Based on our analysis these conditions belong to the instability–turbulence interaction regime (very close to the transitional boundary), while analysis by Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) suggests that they are in the regime of strongly turbulence-controlled propagation ( $\unicode[STIX]{x1D6E4}>1$ ), as shown in figure 11. Clearly, the $\unicode[STIX]{x1D6E4}=1$ line does not capture the transition from strongly turbulence dominated regime to strongly instability dominated regime correctly. It is also noted that the major portion of $\unicode[STIX]{x1D6E4}=1$ line is parallel to the $Ka=1$ line in the turbulent flame regime diagram, as shown in figure 11. Recognizing that these lines represent the lower boundary of the turbulence dominated regime in Chaudhuri et al. (Reference Chaudhuri, Akkerman and Law2011) and the current work respectively, we can see that the leading-order scaling dependence of these two lines is inherently similar.

References

Abdel-Gayed, R. G., Bradley, D. & Lawes, M. 1987 Turbulent burning velocities: a general correlation in terms of straining rates. Proc. R. Soc. Lond. A 414 (1847), 389–413.Google Scholar
Addabbo, R., Bechtold, J. K. & Matalon, M. 2002 Wrinkling of spherically expanding flames. Proc. Combust. Inst. 29 (2), 1527–1535.Google Scholar
Akkerman, V. & Bychkov, V. 2005 Velocity of weakly turbulent flames of finite thickness. Combust. Theor. Model. 9 (2), 323–351.Google Scholar
Akkerman, V., Bychkov, V. & Eriksson, L.-E. 2007 Numerical study of turbulent flame velocity. Combust. Flame 151 (3), 452–471.Google Scholar
Bauwens, C. R., Bergthorson, J. M. & Dorofeev, S. B. 2017 On the interaction of the Darrieus–Landau instability with weak initial turbulence. Proc. Combust. Inst. 36 (2), 2815–2822.Google Scholar
Bell, J. B., Day, M. S., Shepherd, I. G., Johnson, M. R., Cheng, R. K., Grcar, J. F., Beckner, V. E. & Lijewski, M. J. 2005 Numerical simulation of a laboratory-scale turbulent v-flame. Proc. Natl Acad. Sci. USA 102 (29), 10006–10011.Google Scholar
Boughanem, H. & Trouvé, A. 1998 The domain of influence of flame instabilities in turbulent premixed combustion. Symp. (Int) Combust. 27 (1), 971–978.Google Scholar
Bradley, D. 1992 How fast can we burn? Proc. Combust. Inst. 24 (1), 247–262.Google Scholar
Bradley, D., Lawes, M., Liu, K. & Mansour, M. S. 2013 Measurements and correlations of turbulent burning velocities over wide ranges of fuels and elevated pressures. Proc. Combust. Inst. 34 (1), 1519–1526.Google Scholar
Bray, K. N. C. 1990 Studies of the turbulent burning velocity. Proc. R. Soc. Lond. A 431 (1882), 315–335.Google Scholar
Burke, M. P., Chen, Z., Ju, Y. & Dryer, F. L. 2009 Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames. Combust. Flame 156 (4), 771–779.Google Scholar
Bychkov, V. 2003 Importance of the darrieus-landau instability for strongly corrugated turbulent flames. Phys. Rev. E 68 (6), 066304.Google Scholar
Chaudhuri, S., Akkerman, V. & Law, C. K. 2011 Spectral formulation of turbulent flame speed with consideration of hydrodynamic instability. Phys. Rev. E 84 (2), 026322.Google Scholar
Chaudhuri, S., Saha, A. & Law, C. K. 2015 On flame–turbulence interaction in constant-pressure expanding flames. Proc. Combust. Inst. 35 (2), 1331–1339.Google Scholar
Chaudhuri, S., Wu, F. & Law, C. K. 2013 Scaling of turbulent flame speed for expanding flames with Markstein diffusion considerations. Phys. Rev. E 88 (3), 033005.Google Scholar
Chaudhuri, S., Wu, F., Zhu, D. & Law, C. K. 2012 Flame speed and self-similar propagation of expanding turbulent premixed flames. Phys. Rev. Lett. 108 (4), 044503.Google Scholar
Chen, J. H. & Im, H. G. 1998 Correlation of flame speed with stretch in turbulent premixed methane/air flames. Symp. (Int) Combust. 27 (1), 819–826.Google Scholar
Creta, F., Lamioni, R., Lapenna, P. E. & Troiani, G. 2016 Interplay of Darrieus–Landau instability and weak turbulence in premixed flame propagation. Phys. Rev. E 94, 053102.Google Scholar
Creta, F. & Matalon, M. 2011 Propagation of wrinkled turbulent flames in the context of hydrodynamic theory. J. Fluid Mech. 680, 225–264.Google Scholar
Darrieus, G. 1938 Propagation dun front de flamme. La Technique Moderne 30, 18.Google Scholar
Driscoll, J. F. 2008 Turbulent premixed combustion: flamelet structure and its effect on turbulent burning velocities. Prog. Energy Combust. Sci. 34 (1), 91–134.Google Scholar
Filatyev, S. A., Driscoll, J. F., Carter, C. D. & Donbar, J. M. 2005 Measured properties of turbulent premixed flames for model assessment, including burning velocities, stretch rates, and surface densities. Combust. Flame 141 (1), 1–21.Google Scholar
Fogla, N., Creta, F. & Matalon, M. 2013 Influence of the darrieus-landau instability on the propagation of planar turbulent flames. Proc. Combust. Inst. 34 (1), 1509–1517.Google Scholar
Fogla, N., Creta, F. & Matalon, M. 2015 Effect of folds and pockets on the topology and propagation of premixed turbulent flames. Combust. Flame 162 (7), 2758–2777.Google Scholar
Fogla, N., Creta, F. & Matalon, M. 2017 The turbulent flame speed for low-to-moderate turbulence intensities: hydrodynamic theory versus experiments. Combust. Flame 175, 155–169.Google Scholar
Jiang, L. J., Shy, S. S., Li, W. Y., Huang, H. M. & Nguyen, M. T. 2016 High-temperature, high-pressure burning velocities of expanding turbulent premixed flames and their comparison with Bunsen-type flames. Combust. Flame 172, 173–182.Google Scholar
Kee, R. J., Gracer, J. F., Miller, J. A. & Meeks, E.1985 Premix: a fortran program for modeling steady laminar one-dimensional premixed flames. Tech. Rep. Sandia National Laboratories, Report SAND85-8249.Google Scholar
Kerstein, A. R., Ashurst, W. T. & Williams, F. A. 1988 Field equation for interface propagation in an unsteady homogeneous flow field. Phys. Rev. A 37, 2728–2731.Google Scholar
Kido, H., Nakahara, M., Nakashima, K. & Hashimoto, J. 2002 Influence of local flame displacement velocity on turbulent burning velocity. Proc. Combust. Inst. 29 (2), 1855–1861.Google Scholar
Kobayashi, H., Kawabata, Y. & Maruta, K. 1998 Experimental study on general correlation of turbulent burning velocity at high pressure. Symp. (Int) Combust. 27 (1), 941–948.Google Scholar
Kobayashi, H., Seyama, K., Hagiwara, H. & Ogami, Y. 2005 Burning velocity correlation of methane/air turbulent premixed flames at high pressure and high temperature. Proc. Combust. Inst. 30 (1), 827–834.Google Scholar
Landau, L. D. 1944 On the theory of slow combustion. Acta Physicochim. USSR 19 (1), 77–85.Google Scholar
Lipatnikov, A. N. & Chomiak, J. 2005 Molecular transport effects on turbulent flame propagation and structure. Prog. Energy Combust. Sci. 31 (1), 1–73.Google Scholar
Matalon, M. 2007 Intrinsic flame instabilities in premixed and nonpremixed combustion. Annu. Rev. Fluid Mech. 39, 163–191.Google Scholar
Peters, N. 1999 The turbulent burning velocity for large-scale and small-scale turbulence. J. Fluid Mech. 384, 107–132.Google Scholar
Peters, N. 2000 Turbulent Combustion. Cambridge University Press.Google Scholar
Pope, S. B. 1985 Pdf methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11 (2), 119–192.Google Scholar
Pope, S. B. 2000 Turbulent Flows. chap. 6, Cambridge University Press.Google Scholar
Savarianandam, V. R. & Lawn, C. J. 2006 Burning velocity of premixed turbulent flames in the weakly wrinkled regime. Combust. Flame 146 (1), 1–18.Google Scholar
Searby, G. & Clavin, P. 1986 Weakly turbulent, wrinkled flames in premixed gases. Combust. Sci. Technol. 46 (3–6), 167–193.Google Scholar
Sivashinsky, G. I. 1988 Cascade-renormalization theory of turbulent flame speed. Combust. Sci. Technol. 62 (1–3), 77–96.Google Scholar
Troiani, G., Creta, F. & Matalon, M. 2015 Experimental investigation of Darrieus–Landau instability effects on turbulent premixed flames. Proc. Combust. Inst. 35 (2), 1451–1459.Google Scholar
Venkateswaran, P., Marshall, A. D., Seitzman, J. M. & Lieuwen, T. C. 2014 Turbulent consumption speeds of high hydrogen content fuels from 1–20 atm. Trans. ASME J. Engng Gas Turbines Power 136 (1), 011504.Google Scholar
Wang, H., You, X., Joshi, A., Davis, S., Laskin, A., Egolfopoulos, F. & Law, C. K.2007 USC mech version II. High-temperature combustion reaction model of H2/CO/C1-C4 compounds. Available at:http://ignis.usc.edu/Mechanisms/USC-Mech%20II/USC_Mech%20II.htm.Google Scholar
Wu, F., Saha, A., Chaudhuri, S. & Law, C. K. 2015 Propagation speeds of expanding turbulent flames of c 4 to c 8 n-alkanes at elevated pressures: experimental determination, fuel similarity, and stretch-affected local extinction. Proc. Combust. Inst. 35 (2), 1501–1508.Google Scholar
Yang, S., Saha, A., Wu, F. & Law, C. K. 2016 Morphology and self-acceleration of expanding laminar flames with flame-front cellular instabilities. Combust. Flame 171, 112–118.Google Scholar
Yang, S., Yang, X., Wu, F., Ju, Y. & Law, C. K. 2017 Laminar flame speeds and kinetic modeling of H2/O2/diluent mixtures at sub-atmospheric and elevated pressures. Proc. Combust. Inst. 36 (1), 491–498.Google Scholar
Figure 0

Figure 1.Experimental set-up: (a) apparatus; (b) schematic configuration.

Figure 1

Table 1.Experimental conditions and corresponding unburned ($S_{L0,u}$) and burned ($S_{L0,b}$) laminar flame speeds and flame thickness ($\unicode[STIX]{x1D6FF}_{L0}$).

Figure 2

Figure 2.Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$, and turbulence, $\unicode[STIX]{x1D714}_{T}$, at different wavenumbers in the instability dominated regime. The axes are plotted in logarithmic scales.

Figure 3

Figure 3.Experimental results and conditions of laminar and turbulent flames of $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixture at ($\unicode[STIX]{x1D719}=1$), $T_{f}=2400~\text{K}$, different pressures and turbulence intensities (fan speeds in r.p.m.): (a) normalized propagation speeds for laminar unstable flames, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ at different normalized flame radii, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$, showing three different stages of the evolution of hydrodynamic instability; (b) experimental conditions for turbulent flames in the regime diagram; (c) normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ($=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$); (d) normalized flame speeds, $\overline{S_{L,b}}$ and $\overline{S_{T}}$, at different normalized flame radii, $Pe$, showing the flame acceleration. The corresponding conditions are indicated by labels: pressure ($P$) in atm, fan speed in r.p.m. and $U_{rms}$ in $\text{ms}^{-1}$.

Figure 4

Figure 4.Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$, and turbulence, $\unicode[STIX]{x1D714}_{T}$, at different wavenumbers in the instability–turbulence interaction regime. The axes are plotted in logarithmic scales.

Figure 5

Figure 5.Experimental results and conditions of laminar and turbulent flames of $\text{H}_{2}/\text{O}_{2}/\text{N}_{2}$ mixture at ($\unicode[STIX]{x1D719}=1$), $T_{f}=1800~\text{K}$, different pressures and turbulence intensities (fan speeds in r.p.m.): (a) normalized propagation speeds for laminar unstable flames, $\overline{S_{L,b}}=S_{L,b}/S_{L0,b}$ at different normalized flame radii, $Pe=R_{f}/\unicode[STIX]{x1D6FF}_{L0}$, showing three different stages of the evolution of hydrodynamic instability; (b) experimental conditions for turbulent flames in the regime diagram; (c) normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ($=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$); (d) normalized flame speeds, $\overline{S_{L,b}}$ and $\overline{S_{T}}$, at different normalized flame radii, $Pe$, showing flame acceleration. The corresponding conditions are indicated by labels: pressure ($P$) in atm, fan speed in r.p.m. and $U_{rms}$ in $\text{m}~\text{s}^{-1}$.

Figure 6

Figure 6.Schematic of growth rates for DL instability, $\unicode[STIX]{x1D714}_{DL}$, and turbulence, $\unicode[STIX]{x1D714}_{T}$, at different wavenumbers in turbulence dominated regime. The axes are plotted in logarithmic scales.

Figure 7

Figure 7.Experimental results and conditions of laminar and turbulent flames of $\text{C}_{3}\text{H}_{8}/\text{O}_{2}/\text{Ar}$ mixture at ($\unicode[STIX]{x1D719}=0.8$), $T_{f}=2150~\text{K}$, different pressures and turbulence intensities (fan speeds in r.p.m.). (a) Experimental conditions for turbulent flames in the regime diagram. (b) Normalized turbulent flame speeds, $\overline{S_{T}}=S_{T}/S_{L0,b}$ at different $Re_{T,f}$ ($=(U_{eff}R_{f})/(S_{L0,u}\unicode[STIX]{x1D6FF}_{L0})$). (c) Normalized flame speeds, $\overline{S_{T}}$, at different normalized flame radii, $Pe$, showing the flame acceleration. The corresponding conditions are indicated by labels: pressure ($P$) in atm, fan speed in r.p.m. and $U_{rms}$ in $\text{m}~\text{s}^{-1}$.

Figure 8

Figure 8.Revised turbulent flame regime diagram recognizing the interaction between DL instability and turbulence.

Figure 9

Figure 9.Functional dependence of (a) $\widetilde{X}$, (b) $\widetilde{B_{1}}$, (c) $\widetilde{B_{2}}$ and (d) $\widetilde{B_{3}}$ with $n$. It clearly shows that $\widetilde{X}\propto n\propto k$ and $\widetilde{B_{1}}$, $\widetilde{B_{2}}$ and $\widetilde{B_{3}}\propto n^{2}\propto k^{2}$.

Figure 10

Figure 10.Comparison of pure thermal expansion effect (without diffusive stabilization effects) in spherical flame ($({\dot{R}}/R)\widetilde{X}$) versus planar ($S_{L}kX$) flame configurations.

Figure 11

Figure 11.Experimental conditions for which the flame propagation is influenced by both turbulence and DL instability. The conditions are plotted with respect to regimes defined by Chaudhuri et al. (2011) and present study.

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