Self-turbulization in cellularly unstable laminar flames

Abstract It has been suggested that a cellularly unstable laminar flame, which is freely propagating in unbounded space, can accelerate and evolve into a turbulent flame with the neighbouring flow exhibiting the basic characteristics of turbulence. Famously known as self-turbulization, this conceptual transition in the flow regime, which arises from local interactions between the propagating wrinkled flamefront and the flow, is critical in extreme events such as the deflagration-to-detonation transition (DDT) leading to supernova explosions. Recognizing that such a transition in the flow regime has not been conclusively demonstrated through experiments, in this work, we present experimental measurements of flow characteristics of flame-generated ‘turbulence’ for expanding cellular laminar flames. The energy spectra of such ‘turbulence’ at different stages of cellular instability are analysed. A subsequent scaling analysis points out that the observed energy spectra are driven by the fractal topology of the cellularly unstable flamefront.

propagation and flame wrinkling, with several (Fogla, Creta & Matalon 2015;Lamioni et al. 2018) discussing the flow patterns for cellular flames propagating in turbulence. Consequently, they are unrelated to the issues of self-turbulization.
Recognizing the lack of, and hence the need for, understanding the issue of self-turbulization, in this study we present an experimental investigation to provide useful information on the flow induced by cellular flames. Specifically, in the context of expanding cellular flames, we set out to answer the following three questions: (i) What are the primary characteristics of the flow induced by cellularly unstable flames? (ii) Because the flow is induced by the wrinkled flamefront, is there a connection between the cellular flamefront and the flow structure? (iii) Do cellular flames induce a turbulent flow?

Experiments
We conducted our experiments in a well-vetted, dual-chamber combustion vessel, in which propagation of a centrally ignited, outwardly propagating globally spherical premixed flame could be studied in a constant and high-pressure environment, triggering the Darrieus-Landau instability. The apparatus consists of two cylindrical chambers. The inner chamber of almost unity aspect ratio and filled with the test mixture is situated in an outer chamber of a much larger volume (inner diameter: 114 mm, outer diameter: 167 mm; length: 127 mm) and filled with nitrogen (N 2 ) having the same pressure as that of the test mixture. The flat ends of the inner chamber are sealed with thick quartz windows for optical access. The connection between the chambers is controlled by four valves (sliding sleeve with multiple holes) which are opened when the test mixture is spark ignited and allowed the expanding flame to come in contact with the inert gas in the outer chamber to quench. Thus, the expansion process mimicks an isobaric process owing to the small volume of the inner chamber compared to the outer chamber. Figure 1 shows the schematic of the apparatus. More details of the chamber can be found in our previous publications (Chaudhuri et al. 2012;Yang et al. 2016;Liu et al. 2020) and in the Supplementary material available at https://doi.org/10.1017/jfm.2021.330. The flame propagation was studied using two diagnostic techniques. High-speed Schlieren imaging was used to capture the flame morphology, the propagation speed and the acceleration exponent, as shown in figures 2(a) and 2(b). High-speed (8 kHz) 2-D particle image velocimetry (PIV) was used to measure the flow dynamics. The mixtures were seeded with a small amount of di-ethyl-hexyl-sebacat droplets (∼1-2 μm) before ignition.  (d) Vector field from particle image velocimetry. Colour represents magnitude of local velocity.
A planar 2-D laser sheet, which was generated using a high-speed Nd-YLF dual-pulsed laser and associated optics, was placed at the centre of the chamber. During propagation of the flame, the droplets evaporated in the burned side due to high temperature, which resulted in a clear flame boundary in the Mie scattering images captured by a high-speed camera orthogonally placed and synchronized with the laser sheet (shown in figure 2c). The flamefront in this study, therefore, is essentially the isothermal surface at the droplet boiling point, which does not vary significantly across the prevailing pressure condition. The mean flame radius R f is defined based on the area (A f ) enclosed by the flamefronts in the 2-D Mie scattering images, R f = A f /π. The DaVis PIV algorithm and in-house MATLAB codes were used to extract the velocity field around the flamefront (shown in figure 2d), which was then analysed to yield the specific flow characteristics. The uncertainty of the velocity measurement was evaluated to be less than 4 %. In this study, we solely focused on the effect of the Darrieus-Landau hydrodynamic cellular instability, and hence used diffusively neutral (Le ≈ 1) stoichiometric (equivalence ratio, φ = 1) hydrogen(H 2 )/oxygen(O 2 )/nitrogen(N 2 ) mixtures at a flame temperature of 2400 K to minimize stabilization/destabilization effects of diffusional-thermal instability. We also note that the effects of Rayleigh-Taylor instability were rather weak for our experimental conditions (Bell et al. 2004;Abarzhi et al. 2019) and hence, the reported flame dynamics solely arose from Darrieus-Landau instability. The fundamental flame properties, such as the laminar unstretched flame speed (S L,0 ) and adiabatic flame temperature (T f ), were calculated numerically using the CHEMKIN one-dimensional flame solver and a reaction model of H 2 -oxidation proposed by Burke et al. (2012). The experimental conditions and corresponding flame properties are listed in table 1. In this study, we defined the planar flame thickness based on diffusivity, δ L,0 = D T /S L,0 , where D T is the thermal diffusivity of the mixture. The instantaneous propagation speed of the expanding flame, S L,b = dR f /dt, was obtained from the time derivative of the mean flame radius, R f . To study a wide range of Pe, we experimented with a broad range (1-7 atm) of ambient pressures.

Global propagation
In our previous study (Yang et al. 2016), it was shown that the propagation of cellularly unstable expanding flames at various pressures can be unified when the normalized flame propagation speed is plotted with the normalized radius, or Péclet number, Pe = R f /δ L,0 , somewhat similar to turbulent flames (Chaudhuri et al. 2012). Figure 2(b) shows such a normalized plot for the current experiments, which identifies a three-stage behaviour with two critical Pe's. Specifically, the flamefront in the smooth expansion stage, Pe < Pe c1 ≈ 1000, is free from instabilities (inset of figure 2b) and hence the propagation is stretch controlled. As the cellular instability is triggered at Pe = Pe c1 , the flame shows sharp acceleration, and over a short Pe range, new cells start appearing and spreading over the entire flamefront (inset of figure 2b), which marks a transition stage (Pe c1 < Pe < Pe c2 ). At the saturated stage (Pe > Pe c2 ≈ 1900), the flamefront becomes fully covered with cells (inset of figure 2b) and the acceleration attains self-similarity. Yang et al. (2016) showed that for a wide range of conditions, the acceleration exponent (α) lies in the range 1.2-1.4 and the corresponding fractal excess (d = 1 − 1/α) is within 0.2-0.3. In a similar, but larger chamber, Huo et al. (2018Huo et al. ( , 2019 also observed the three-stage behaviour with a similar global acceleration exponent. However they noticed an oscillation in the mean propagation speed in the saturated stage. We note that such pulsation occurs for a larger flame and was not observed in our experiments. Furthermore, in this study, we will be analysing the energy spectra from local velocity fluctuation and, as such, oscillation in mean propagation speed or mean velocity is expected to have an insignificant effect on the reported statistics.

Flow statistics
Next, we focus our attention on the flow field ahead of the propagating cellular flamefront measured by PIV. To characterize the flow generated by the flame, we first explore the variation of the radial (U r (r, θ)) and azimuthal (U θ (r, θ)) components of the velocity in the radial direction. To present the radial variation of these velocities at various flame conditions, we take the azimuthal average of these velocities at any fixed r (averaged quantities denoted as · θ ) and normalize the averaged velocity with the flame propagation speed, S L,b (normalized quantities presented as (·)). In figure 3(a), we compare the averaged normalized radial velocity, U r θ , as function of distance from the flamefront, ΔR, for various Pe. For all Pe, U r θ is maximum adjacent to the flamefront and decay monotonically with ΔR as the effect of flame propagation weakens away from the flamefront. Interestingly, the averaged normalized radial velocity at the flamefront ( U r θ | ΔR=0 ) shows a non-monotonic dependence on Pe, in that at low Pe (smooth expansion stage, Pe < Pe c1 ) and high Pe (saturated stage, Pe > Pe c2 ), it is smaller compared to that at intermediate Pe (transition stage, Pe c1 < Pe < Pe c2 ), as shown in the Such dramatic increase, however, is attenuated at the saturated stage, as the cellular structure and flamefront morphology become saturated and the acceleration becomes 'steady', which in turn reduces U r θ | ΔR=0 to the original value. The averaged normalized azimuthal velocity, U θ θ , however, is much weaker than U r θ and does not show any correlated variation along the radial direction, as shown in figure 3(b). The flow structure generated by the flamefront is consumed by the propagating flamefront before it is convected to the far field and hence no particular correlation between near field and far field was observed. This can be further shown through the velocity fluctuations in the radial (u r ) and azimuthal (u θ ) directions. The RMS of u r and u θ monotonically decrease with ΔR, as shown in figure 3(c). Such monotonic decay suggests that the flow far away from the flame is dominated by the mean radial flow generated by thermal expansion and does not possess strong velocity fluctuations.
Recognizing that the flame-generated flow structures were mostly confined in the vicinity of the flamefront, we focus the rest of the analysis on the flow vectors conditioned on the wrinkled flamefront, and analysed them in a flame-coordinate system using the normal and tangential directions of the flamefront. Figure 4 shows the one-dimensional energy spectra φ nn , φ tt of the fluctuation velocity in the normal, u n , and tangential, u t , directions, respectively, measured along the flamefront for five values of Pe spanning all three stages of propagation. We observe that φ nn (k) shows a similar shape at the different stages of the propagation. Furthermore, the energy is large at small wavenumbers (large scales) and rapidly decayed at large wavenumbers (small scales). Although the shape of φ tt is similar to φ nn at low Pe, there are some characteristic changes with Pe ( figure 4b). First, the spectra shifts to higher energy values as Pe increases and the flame moves from the smooth expansion stage (Pe = 587) to the transition stage (Pe = 1235) to the saturated stage (Pe = 3005, 3552 and 4372) with the flamefront becoming progressively more wrinkled owing to the appearance of the cellular structure. Moreover, at the saturated stage, the spectra almost overlap (Pe = 3005, 3552 and 4372) at various Pe and as such, become independent of Pe. They also attains a power-law dependence, i.e. φ tt (k) ∼ k −β , where β was measured to be approximately 2.6 (the dashed line). The power-law behaviour is observed for length scales, 2π/k, roughly from the flame radius (R f ∼ O(10 mm)) to the smallest scales we can resolve in the experiments, which is O(0.1 mm) ∼ 20δ L,0 . We note that theoretical analyses (Pelce & Clavin 1982;Bychkov & Liberman 2000) have shown that the cut-off limit of Darrieus-Landau instability is approximately k cut−off ≈ 2π/(20δ L,0 ), which is close to the experimental resolution. Nevertheless, whether the power-law scaling holds for larger wavenumbers is still to be investigated. The turbulent Reynolds number based on the largest scale (Re t = u rms R f /ν) at the saturated stage was in the range of 1500-4000, where ν is the kinematic viscosity of unburned gas.
It should be noted that the flow statistics were measured in the meridional plane of the cellularly unstable expanding flame and as such, were 2-D in nature. However, these flames are statistically spherical (Chaudhuri, Wu & Law 2013;Chaudhuri, Saha & Law 2015) and as such, orientation of the measurement plane does not affect the statistics.

Scaling analysis
Next, we explore the origin of the power-law dependence observed in the flame-generated flow. Here, we note that in our experiments, the flame propagates in a quiescent environment and as such, the observed energy spectra of the flow adjacent to the flamefront is generated by the wrinkling of the flamefront. Consequently, we present a scaling analysis to connect the energy spectra with the cellular structure. In the Landau limit, every point on the flamefront propagates individually in the normal direction at the speed of S L,0 . Defining S w as the projected propagation speed on the mean spherical flamefront for every local infinitesimal flame segment, we can write S w /S L,0 = |∇G| = √ 1 + ∇g · ∇g, where G is the scalar describing the flamefront in the well-known G-equation (Peters 1999). We can also define G = R f + g − r, where g is the fluctuation of the flamefront from the mean surface for a mildly wrinkled flamefront, and r is the radial distance from the centre of the flame. Such a definition ensures G > 0 in the unburned side, G < 0 in the burned side and G = 0 on the flamefront, a convention commonly followed in the G-equation representation. Because the flow is induced by the flame, the kinetic energy of the flow (E) must arise from flame propagation, or E ∼ S 2 w . Considering the contribution from the flow velocity in the normal U n and tangential U t components, we can write (3.1) In the Landau limit, for adjacent fluid particles, U 2 n scales as S 2 L,0 , which, along with (3.1), suggests that U 2 t ∼ S 2 L,0 ∇g · ∇g. We notice that U 2 n is independent of g or the geometry of the wrinkled flamefront at all stages of propagation or Pe (figure 4a). Consequently, fluctuation of the normal velocity u n is mainly determined by the disturbance on the large scales in the system, while decaying rapidly on the small scales. This explains the similar energy spectra at all values of Pe, as the disturbance on the large scale is independent of the development of the cellular instability. For fluctuation of the tangential velocity u t at low Pe when the flame is free from instability, the energy spectra is the same as that from disturbance. However, as Pe increases, cellular instability induces strong tangential velocity structures and the influence of disturbance is suppressed. Now, we closely explore the relation between U 2 t and the flame geometry or g. First we note that the scaling U 2 t ∼ ∇g · ∇g suggests that the power spectral density (PSD) of U t should scale with the PSD of |∇g|, denoted as P g . Since PSD of the tangential velocity U t and its fluctuation u t = U t − U t is equivalent for k > 0, we have φ tt (k) ∼ P g (k). From the experiments, we measured g (shown in figure 5a) and evaluated |∇g| and P g (k), as shown in figure 5(b). At low Pe (smooth expansion stage), the flame edge (g) possesses less oscillation while at large Pe (saturated stage), it becomes highly oscillating (figure 5a). It is then interesting to note that P g (figure 5b) also shows a power-law dependence with wavenumber in the form of k −2.6 , which is very similar to φ tt , hence confirming our scaling.
Next, we attempt another scaling analysis to assess the theoretical approximation of β in the scaling of φ tt (k) ∼ P g (k) ∼ k −β . Studies (Liberman et al. 2004;Yu et al. 2015;Yang et al. 2016) have confirmed that cellularly unstable laminar flamefronts are fractal in nature. The perimeter, L, of the 2-D cross-section of such a wrinkled fractal expanding flame scales with the mean radius R f , as L ∼ R D f , where D is the fractal dimension of the flame edge. Here, L can be obtained by integrating the arc length of the flamefront R f + g(θ ) in the polar coordinate (Strang 1991), Because the cellularly unstable flamefronts are mildly wrinkled, i.e. R f g, and R f is independent of θ, we find , as it is the only term on the left-hand side of (3.4), which is a function of R f . As a result, we have The cellular structure on the flamefront is bounded by two length scales, the flame thickness scale (δ L,0 ) and the flame radius scale (R f ) (Matalon 2007), and as such, the spectrum of g, which describes the flamefront geometry, can be expressed as where k represents the wavenumber along the flamefront and ψ k is the phase angle. Assuming A(k), the amplitude at wavenumber k follows a power law A(k) ∼ k −γ . By differentiating it with respect to θ and substituting A(k) ∼ k −γ , we obtain The amplitude of dg/dθ at wavenumber k scales with k 1−γ , thus, PSD of dg/dθ has a k 2(1−γ ) dependence; in other words, β = −2(1 − γ ). Now rearranging (3.7) and substituting λ = kR f as a normalized wavenumber, we have λ 1−γ cos(λθ + ψ λ ) dλ. (3.8) It can be shown (see Supplementary material) that the integral in (3.8) converges to a constant value for any θ and ψ for γ > 1 at large Pe, and hence is independent of R f . This leads to |dg/(R f dθ)| ∼ R γ −2 f , and by comparing with (3.5), we find γ = D + 1 or β = 2D. Using the experimentally measured fractal dimension (D = 1 + d ≈ 1.25 ± 0.05) in the saturated stage for our flames (Yang et al. 2016), we expect β = 2.5 ± 0.1, which is close to the observed value of 2.6 shown in figures 4(b) and 5(b).

Concluding remarks
We conclude by revisiting the questions posed at the beginning. In terms of flow characteristics, we have shown that for an expanding cellularly (Darrieus-Landau) unstable flame, the far field flow is dominated by a radial mean flow, owing to thermal expansion, and the RMS of the velocity is rather weak. Close to the flame, however, the flow shows turbulence-like behaviour. In particular, the energy spectra from the tangential velocity component approach a power-law behaviour in the saturated stage of the propagation. The spectra from the normal component of velocity, however, remains unaltered between the stable and unstable flames. Further analysis based on the G-equation shows that the spectra from the tangential velocity arise from the wrinkling of the cellular flamefronts. We identify that the observed power-law dependence in the energy spectra in the flow is directly related to the fractal dimension of the wrinkled flamefront. We end this exposition with the understanding that the flow adjacent to a cellularly unstable flamefront does become 'turbulent'. Such turbulence, generated by the propagating wrinkled flamefront, has a strong directionality and hence, is anisotropic.