2. Experimental set-up

2.1. Facility

Our experimental facility (figure 1a) consists of a premixed $\text {CH}_4-$air V-flame stabilized on an oscillating flame holder, a typical configuration for the study of premixed flames (Petersen & Emmons Reference Petersen and Emmons1961). The flame holder, which is an electrically heated nichrome wire, is vibrated at a frequency of $f_f=1250$ Hz. The flow of $\text {CH}_4$ and air ensues out of a circular nozzle 10 mm below the flame holder, and turbulence is generated using a series of stator-rotor plates. The three-dimensional (3-D) flame surface, thus obtained, is shaped like a wedge (figure 1b). We measure the fluctuations of the flame edge and the underlying velocity field within a 2-D plane located at the mid-section of the wedge-shaped flame surface, using $\textrm {TiO}_2$ Mie scattering and high-speed particle image velocimetry (PIV). This experimental set-up has been employed previously for studying the effects of harmonic forcing and turbulence on premixed flames (Humphrey, Emerson & Lieuwen Reference Humphrey, Emerson and Lieuwen2018; Roy & Sujith Reference Roy and Sujith2019).

Figure 1. Turbulent V-flame facility. (a) Schematic of the combustor set-up. (b) Illustrative photograph of the V-flame. (c) Representative cropped and processed Mie-scattering snapshot depicting the flame surface. (d) Examples of the extracted instantaneous leading flame edge $\xi (y,t)$, along with the coordinate system. The dashed line represents the mean flame edge $\langle \xi (y)\rangle$ with respect to which the fluctuations $\xi ^\prime (y,t)$ are defined. The region highlighted by the box is used for determining the flow properties. Panels (a–c) adapted from Humphrey et al. (Reference Humphrey, Emerson and Lieuwen2018) with permission from Cambridge University Press.

2.2. Flame and flow characteristics

We consider two experimental flame configurations, F1 and F2, whose properties are listed in table 1. For both flames, the turbulence intensity is $u^\prime /\bar {u}_y \approx 0.1$, where $u^\prime$ is the root-mean-square (r.m.s.) of the velocity fluctuations and $\bar {u}_y$ is the mean longitudinal velocity. The Reynolds number $Re_\ell =\ell u^\prime /\nu$ of F1 is approximately 700 while that of F2 is nearly four times larger; the dissipative Kolmogorov length ($\eta$) and time ($\tau _\eta )$ scales are similar though for both flames (cf. table 1). Here, $\nu$ indicates the kinematic viscosity of the binary fuel-air mixture.

Table 1. Relevant physical properties of the two turbulent premixed flame configurations considered in this study. The laminar flame speed $s_L$ for the two cases was obtained using Chemkin Premix calculations (Humphrey Reference Humphrey2017), while the value of $Sc$ for methane–air premixed flames was obtained from Tamadonfar & Gülder (Reference Tamadonfar and Gülder2014). See the supplementary material for further details.

The flame speed $s_L$ is calculated using Chemkin Premix (Kee et al. Reference Kee2011) with detailed chemistry simulated through the GRIMech 3.0 mechanism (Smith et al. Reference Smith1999) at 300 K and 1 bar. The associated Karlovitz number $Ka =(\nu /Sc\, s_L)^2 \eta ^{-2} \sim 0.1$, for both configurations, which implies that the flames lie within the corrugated flamelet regime, close to the boundary with the thin reaction zone regime (Peters Reference Peters2000). Here, Sc indicates the Schmidt number. Thus, the flame front is continuous, enabling a well-defined description of the flame edge.

Turbulent eddies can distort the flame edge and produce wrinkles or corrugations, but on scales of the order of the Gibson scale $\ell _g = (s_L/u^\prime )^3\ell \approx 0.3$ mm or greater. This is because smaller eddies have velocities less than the laminar flame speed $s_L$ and so cannot distort the flame edge (Peters Reference Peters2000). Further information on the properties of the flame and the flow, as well as on how these are calculated, is provided in the supplementary material available at https://doi.org/10.1017/jfm.2023.63.

2.3. Window of interrogation

Our analysis of small-scale flame dynamics is based on measurements in a 2-D plane, within a sub-region of the entire domain (outlined by the grey rectangle in figure 1d). The extent of the sub-region in the longitudinal $y$-direction is chosen such that the flame fluctuations are not dominated by effects of flame anchoring and the oscillation of the flame holder, which is ensured for $y>\lambda _c$, where $\lambda _c=\bar {u}_y/f_f$ (notice the disappearance of the narrow-band peak from the power spectra in figure 3). Further, we disregard fluctuations at large downstream distances ($y>4\lambda _c$) where effects of large-scale flapping become significant (discussed further below in connection with figure 2a,b). The width of the sub-region in the transverse $x$-direction is restricted by the requirement that the measured velocity fluctuations exhibit nearly isotropic statistics. This is verified by ensuring that the cross-correlation $\langle u_x^\prime u_y^\prime \rangle$ remains small (see § 3 in the supplementary material).

Figure 2. Distinction between outer and inner intermittency. (a) Time series of the flame fluctuations $\xi ^\prime$ normalized by the standard deviation measured at $y=5\lambda _c$ (top panels) and $y=2\lambda _c$ (bottom panels). (b) The p.d.f.s of $\xi ^\prime$ at various longitudinal locations. (c) Time series of increments $\delta \xi ^\prime$, normalized by the standard deviation, for $\tau =0.03 \tau _\ell$ (top panels) and $\tau =6.67\tau _\ell$ (bottom panels), measured at $y=2\lambda _c$. (d) The p.d.f.s of the increment $\delta \xi ^\prime$ measured at $y=2\lambda _c$ for various values of the time lag. The data set in all panels corresponds to flame F1. As explained in the text, comparing panels (b,d) helps to clearly distinguish large-scale outer intermittency from small-scale inner intermittency. Here, the p.d.f.s have been shifted for clarity and the dashed lines represent $\mathcal {N}(0,1)$ Gaussian fits.

Our measurements in the $x$–$y$ plane give us access to the fluctuations of the flame edge in the direction normal to the mean flame edge (dashed line in figure 1d), as well as in the flow-aligned tangential direction. However, we do not have access to the fluctuations in the out-of-plane tangential direction ($z$-direction). We do not expect this omission to affect out key results, however, because within the sub-region of interest where the flow is approximately isotropic the only difference between these two tangential directions is the advection by the mean flow in the flow-aligned direction. We can account for this advection using Taylor's hypothesis ($u^\prime /\bar {u}_y \sim 0.1$) and thereby approximate the $z$-direction fluctuation statistics from data of the flow-aligned tangential fluctuations (Shin & Lieuwen Reference Shin and Lieuwen2013). This procedure is facilitated by the local homogeneity in the $z$-direction near the mid-section of the wedge-shaped flame surface where the measurement plane is located. In fact, many studies have carried out similar 2-D measurements for estimating important flame properties such as the fractal dimension of the flame surface (North & Santavicca Reference North and Santavicca1990; Smallwood et al. Reference Smallwood, Gülder, Snelling, Deschamps and Gökalp1995; Gülder et al. Reference Gülder, Smallwood, Wong, Snelling, Smith, Deschamps and Sautet2000).

3. Outer and inner intermittency: two distinct forms of extreme fluctuations

Let us begin by considering the fluctuations of the flame at various distances from the flame holder. At relatively large distances, $y/\lambda _c=5$ (where $\lambda _c=\bar {u}_y/f_f$), the flame propagation is erratic and the time series of $\xi ^\prime$ (top panel, figure 2a) exhibits an intermittent behaviour. Due to large-scale flapping, the flame undergoes abrupt excursions from the mean (bursts) at some time instances, while failing to propagate to the measurement location at other time instances (off events). The off events are marked by setting $\xi ^\prime = 0$, and so the corresponding normalized p.d.f. of $\xi ^\prime$ has a sharp peak at $\xi ^\prime = 0$ (figure 2b). Such a p.d.f. has a high kurtosis or flatness factor, $K= \langle {\xi ^\prime }^4 \rangle /\langle {\xi ^\prime }^2 \rangle ^2 =13.51$, and is typical of large-scale outer intermittency. Now, as we move closer to the flame holder, the flame becomes well maintained and outer intermittency is lost. Indeed, the p.d.f. of $\xi ^\prime$ for $y/\lambda _c=2$ (figure 2b, see also the bottom panel of figure 2a) has a kurtosis ($K = 3.47$) that is close to the Gaussian value of 3.

This near-Gaussian p.d.f. of $\xi ^\prime$, however, veils an intermittency of a different nature, which is revealed by examining the temporal increments of the flame position $\delta \xi ^\prime (\tau )=\xi ^\prime (t+\tau )-\xi ^\prime (t)$. The normalized p.d.f.s of this temporal structure factor, measured at $y/\lambda _c=2$, are presented in figure 2(d) for various values of $\tau /\tau _\ell$ ($\tau _\ell = \ell /u^\prime$ is the integral time scale). While the p.d.f. is near Gaussian for large $\tau$, it develops strongly flared tails for small $\tau$. The kurtosis for $\tau /\tau _\ell = 0.03$ is $K=55.13$, while for $\tau /\tau _\ell = 6.67$, it is $K=3.23$. Comparing the corresponding time series, shown in the bottom and top panels of figure 2(c) respectively, we see that $\delta \xi ^\prime (0.03\tau _\ell )$ intermittently undergoes large excursions – tens of times larger than the standard deviation – which are absent in the case of $\delta \xi ^\prime (6.67\tau _\ell )$. So, while the large-scale fluctuations at $y/\lambda _c=2$ are non-intermittent, the small-scale fluctuations exhibit extreme-value increments – a clear sign of inner intermittency.

4. Power-law scaling of the frequency spectrum

Before characterizing this intermittency further, it is instructive to examine the power spectrum of $\xi ^\prime$ in frequency space, which is closely related to the variation of the second moment of $\delta \xi ^\prime$ with $\tau$. The frequency spectrum is presented in figure 3 for three longitudinal locations. At $y/\lambda _c=1$, we see a minor imprint of the external vibration of the flame holder, in the form of a small peak at the forcing frequency $\omega _f/\omega _{\ell } = 5.79$, where $\omega _{\ell }=2 {\rm \pi}/\tau _{\ell }$. For $y/\lambda _c=1.5$ and 2, there is no trace of the external forcing, and the flame's fluctuations are dominated by its response to the turbulent flow. Interestingly, for frequencies beyond $\omega _{\ell }$, a self-similar power law is seen to emerge.

Figure 3. Temporal power spectral density $E(\tilde {\omega })$ measured at various longitudinal locations $y/\lambda _c$, for flame F1. The power spectrum varies as $E(\tilde {\omega })\sim \tilde {\omega }^{-\alpha }$ over an intermediate range of frequencies. The estimated values of the exponent $-\alpha$ at various $y/\lambda _c$ are shown in the inset for both flames F1 and F2. In both cases, the exponent is close to $-2$; the corresponding scaling behaviour is depicted by the solid line in the main panel. Estimates of the frequency corresponding to Gibson ($\tilde {\omega }_g$), Corrsin ($\tilde {\omega }_c$) and Kolmogorov ($\tilde {\omega }_\eta$) scales have been indicated by dashed lines.

To understand the origin of this power law, let us begin with the flame fluctuation spectrum in spatial wavenumber ($k$) space, which has been well studied. For a flame of finite thickness susceptible to diffusive effects ($Ka \sim {O}(1)$), in an isotropic and homogeneous turbulent flow, Kolmogorov's phenomenology (Peters Reference Peters1992; Chaudhuri et al. Reference Chaudhuri, Akkerman and Law2011) leads to a spectrum with a power-law behaviour,

(4.1)\begin{equation} \varGamma(k) \sim k^{{-}5/3}, \end{equation}

in the subrange $k_{\ell }< k< k_{c}$. Here $k_{\ell } = 2 {\rm \pi}/\ell$ corresponds to the large integral scale while $k_{c}=2 {\rm \pi}/\eta _c$ is the wavenumber of the Corrsin length scale $\eta _c=Sc^{-3/4}\eta$ ($Sc=\nu /\mathcal {D}_M$, where $\mathcal {D}_M$ is the Markstein diffusivity) after which diffusive effects within the flame become dominant. (The role of kinematic restoration is discussed below.) This subrange lies within the inertial range of the turbulent flow, $k_{\ell }< k< k_{\eta }$, where $k_{\eta }=2 {\rm \pi}/\eta$ corresponds to the viscous Kolmogorov length $\eta$ ($k_{\eta } > k_c$ as $Sc < 1$ for our flame). Notably, the scaling in (4.1) is the same as that for a passive scalar in the inertial–convective range (Oboukhov Reference Oboukhov1949; Corrsin Reference Corrsin1951; Davidson Reference Davidson2015), which is consistent with the fact that effects due to the flame's propagation do not play a role over this range of scales. In summary, the inertial-range velocity fluctuations educe an apparently self-similar response from the flame surface, which, however, is cut off by large-scale effects for $k < k_{\ell }$ and by diffusion within the flame for $k > k_c$.

Now, in order to translate this picture to the frequency domain, we assume that flame fluctuations with wavenumber $k$ are most strongly influenced by a turbulent eddy of size $2 {\rm \pi}/k$, whose typical velocity according to inertial-range scaling is $u^\prime \sim k^{-1/3}$. We can then relate the wavenumber of flame fluctuations to the frequency as $\omega \sim ku^\prime \sim k^{2/3}$. Then, using (4.1) and the fact that the frequency spectrum $E(\omega )$ must satisfy $\int E({\omega })\,{\rm d}{\omega }=\int \varGamma ({k})\,{\rm d}{k}$, we obtain

(4.2)\begin{equation} E(\omega) \sim {\omega}^{{-}2}. \end{equation}

The exponent of the power-law subrange of the frequency spectrum is presented in the inset of figure 3, and is seen to be close to this prediction of $-2$ for a range of longitudinal locations, $\lambda _c< y<3\lambda _c$. This figure also shows the frequencies $\omega _{\ell }$, $\omega _{c}$ and $\omega _{\eta }$ corresponding to the length scales $\ell$, ${\eta }_c$ and $\eta$, respectively. The power law is seen to commence after $\omega _{\ell }$, as expected, and then carry on for approximately a decade. However, near $\omega _c$, the spectrum displays a spurious flattening due to the limited temporal resolution of our measurements, which is insufficient to resolve the diffusive cutoff and subsequent stretched–exponential decay of the spectrum beyond $\omega _c$ (Peters Reference Peters1992; Chaudhuri et al. Reference Chaudhuri, Akkerman and Law2011).

Figure 3 also shows the frequency $\omega _g$, which corresponds to the Gibson length $\ell _g=(s_L/u^\prime )^3\ell$, at which the flame propagation speed becomes comparable to the turbulent velocity fluctuations. This scale could potentially cut off the power-law scaling due to kinematic restoration effects that act to smooth out turbulence-induced flame fluctuations (Lieuwen Reference Lieuwen2021). However, previous work indicates that, for flames of finite thickness, the effect of kinematic restoration on the spectra can be counter-acted by thermal expansion effects, so that the power-law behaviour persists until the Corrsin scale ($\omega _c$), after which it is terminated by diffusive effects within the flame (Gülder et al. Reference Gülder, Smallwood, Wong, Snelling, Smith, Deschamps and Sautet2000; Peters et al. Reference Peters, Wenzel and Williams2000; Shim et al. Reference Shim, Tanaka, Tanahashi and Miyauchi2011; Chatakonda et al. Reference Chatakonda, Hawkes, Aspden, Kerstein, Kolla and Chen2013).

5. Anomalous scaling of temporal increments

Let us now return to the issue of inner intermittency and its characterization. This is best done by examining the scaling of the structure functions $S_p(\tau )$, defined as the $p{\rm th}$ moment of the increment $\delta \xi ^\prime (\tau )$ (Frisch Reference Frisch1995; Falcon, Fauve & Laroche Reference Falcon, Fauve and Laroche2007)

(5.1)\begin{equation} S_p(\tau;y) \equiv \langle[\delta\xi^\prime(\tau;y)]^p\rangle=\langle[\xi^\prime(t+\tau;y)-\xi^\prime(t;y)]^p\rangle. \end{equation}

For the second-order structure function, which can be determined entirely from the power spectrum (Davidson Reference Davidson2015), we have $S_2 \sim (2{\rm \pi} /\tau ) E(2{\rm \pi} /\tau ) \sim \tau$. For the $p{\rm th}$ moment then, a naive expectation would be $S_p = \langle {[\delta \xi ^\prime ]}^p\rangle \sim \tau ^{p/2}$; this would be true if the flame fluctuations were non-intermittent and perfectly self-similar. However, the presence of extreme-value increments, evident in the flared-tail p.d.f.s of figure 2(d), causes the higher-order moments to have increasingly large values as $\tau$ decreases. So, for intermittent fluctuations, we expect $S_p \sim \tau ^{\zeta _p}$ with $\zeta _p$ becoming increasingly smaller than $p/2$ as $p$ increases.

This is exactly what we observe in figure 4(a), which presents the values of $\zeta _p$ as a function of $p$, up to the sixth order, for both flame configurations, at $y = 2\lambda _c$. Figure 4(b) illustrates the corresponding power-law scaling of $S_p$ for $\tau < \tau _{\ell }$. Equivalent results are obtained at other longitudinal locations in the interval $\lambda _c< y<3\lambda _c$, wherein the spectrum exhibited a power-law exponent close to $-2$ (figure 3). We also estimated $\zeta _p$ using the procedure of extended self-similarity (Benzi et al. Reference Benzi, Ciliberto, Tripiccione, Baudet, Massaioli and Succi1993), which takes advantage of the fact that $S_p/S_2$ scales as $\tau ^{\zeta _p/1}$ over an extended range of $\tau$, and obtained values nearly identical to those shown in figure 4(a). The dramatic departure of $\zeta _p$ from $p/2$, for $p$ beyond second order, makes evident the intensely intermittent nature of the small-scale fluctuations of the flame.

Figure 4. (a) The variation of the scaling exponents $\zeta _p$ with the order $p$ of the structure function, for both flames at $y/\lambda _c=2$. The dashed line indicates the non-intermittent limit of $\zeta _p=p/2$. The strong deviation of $\zeta _p$ from this limit for $p>2$, implies that the exponents scale anomalously and that the flame fluctuations are strongly intermittent. The error bars represent the standard deviation of the measured values obtained from different time series (see § S6 in the supplementary material). (b) Plots of the structure functions up to order $p=6$, compensated by the estimated scaling $\tau ^{-\zeta _p}$, which illustrates their scaling behaviour (for flame F1 measured at $y=2\lambda _c$). (c) The p.d.f.s of $\delta {\xi ^\prime }(\tau )$, for various values of $\tau$, multiplied with $\xi ^\prime _{rms}(\tau /\tau _{\ell })^{-\zeta _{\infty }}$, where $\zeta _{\infty }$ is the saturated value of $\zeta _{p}$ estimated from panel (a). The dashed lines are Gaussian fits for $\tau /\tau _\ell =0.03$ and $2.60$. The collapse of the tails of the p.d.f.s is striking, especially when compared with the behaviour of the Gaussian fits.

The saturation of the $\zeta _p$ exponents with increasing $p$, seen in figure 4(a), is reminiscent of the anomalous scaling behaviour of passive scalar turbulence (Celani et al. Reference Celani, Lanotte, Mazzino and Vergassola2000, Reference Celani, Lanotte, Mazzino and Vergassola2001; Watanabe & Gotoh Reference Watanabe and Gotoh2006), wherein the saturation arises due to steep ramp-cliff structures in the concentration field. The width of these structures decreases as the diffusivity is reduced, yet the amplitude of the concentration jump remains near the r.m.s. value of concentration fluctuations (Celani et al. Reference Celani, Lanotte, Mazzino and Vergassola2001).

In our case, figure 4(a) implies that, as the diffusive Corrsin scale is reduced (keeping $Ka$ and $Sc$ constant), the extreme-valued flame fluctuations, measured at a fixed $y$ position, would undergo a displacement of the order of the r.m.s. value, $\xi ^\prime _{rms}=\langle {\xi ^\prime }^2\rangle ^{1/2}$, in an ever-shortening time. To see this, note that, in the limit of large $p$, $S_p^{1/p} \sim \langle \delta {\xi ^\prime (\tau _{\ell })}^p\rangle ^{1/p} (\tau /\tau _{\ell })^{(\zeta _p/p)}$ is an estimate of the magnitude of extreme increments. So, given that $\zeta _p$ saturates as $p$ increases, we see that the magnitude of extreme increments does not decrease with $\tau$, but rather remains comparable to $\langle \delta {\xi ^\prime (\tau _{\ell })}^p\rangle ^{1/p} \sim \langle {\xi ^\prime }^p\rangle ^{1/p} \sim \xi ^\prime _{rms}$ ($\xi ^\prime$ has an approximately Gaussian distribution as seen in figure 2b). This behaviour is illustrated in figure 4(c), wherein the tails of the p.d.f.s of $\delta {\xi ^\prime }$, for various values of $\tau$, are seen to have the same width, and in fact collapse when the p.d.f.s are multiplied with $\xi ^\prime _{rms}(\tau /\tau _{\ell })^{-\zeta _{\infty }}$ ($\zeta _{\infty }$ is the saturated value estimated from figure 4a).

6. Origin of extreme-valued temporal increments

The extreme temporal increments implied by the saturating exponents in figure 4(a) have two possible causes: (i) rapid fluctuation events in which the flame advances and retreats very quickly, or (ii) advection of coherent spatial structures on the flame surface, such as cusps, past a fixed measurement location. The second scenario has been proposed as the cause of intermittency in temporal fluctuations of a free surface exhibiting gravity–capillary wave turbulence (Falcon et al. Reference Falcon, Fauve and Laroche2007). For flames, and propagating surfaces in general, cusp-like features are typical (Law & Sung Reference Law and Sung2000; Zheng, You & Yang Reference Zheng, You and Yang2017) and indeed appear quite frequently on the flame edge in our study (see figure 1c). Near such points, the flame edge typically traces out a large excursion in the transverse $x$ direction over a short distance in the longitudinal $y$ direction. So, when such a cusp-like structure is advected past the measurement location ($y=2\lambda _c$ in figure 4) by the mean flow, it will register as an extreme-valued increment of the flame position. The time scale of such events is estimated in § 2.4 to be approximately $10^{-4}$ s, which is strikingly similar to the time scale of the interval $\tau$ at which the p.d.f. of $\delta \xi ^\prime (\tau )$ begins to develop strongly flared tails (see figure 2(d) wherein $\tau = 0.03 \,\tau _\ell \approx 10^{-4}$ s). Indeed, an examination of the temporal evolution of the flame surface in tandem with the time trace of the temporal increment (presented in supplementary movie 1) strongly suggests that this scenario predominates and that the anomalous scaling of figure 4(a) is due to the advection of cusp-like structures along the flame edge.

As a quantitative check, we now calculate the curvature of the flame edge and examine whether extreme values of the temporal increment $\delta {\xi ^\prime }$ occur simultaneously with extreme values of curvature, which would correspond to cusps. Using the parametric representation of the flame edge $(x(s,t),y(s,t))$, where $s$ is the arc length, the curvature $\kappa$ is calculated as follows (Aris Reference Aris1990):

(6.1)\begin{equation} {\kappa}(s,t) = ({(\partial_{ss}x)^2+(\partial_{ss}y)^2})^{1/2}. \end{equation}

We construct the curve $(x(s,t),y(s,t))$ using fourth-order spline interpolation based on points spaced equally along the flame edge, such that the inter-point distance (${\rm d}s=\sqrt {{{\rm d} x}^2+{{\rm d} y}^2}=0.1$ mm) is greater than the pixel size $\Delta x$. This allows us to evaluate the derivatives in (6.1) and obtain the curvature as a function of the arc length $s$ (see Bentkamp et al. (Reference Bentkamp, Drivas, Lalescu and Wilczek2022), for a similar calculation for material loops in turbulence). Figure 5(a) illustrates the typical variation of the curvature along the flame edge when it has a cusp-like feature: we see that the cusp corresponds to a spike in the curvature profile. Figure 5(b) presents the stationary p.d.f. of curvature values sampled by the flame edge, within the interrogation window (box in figure 1d) and over time. The heavy tail of the distribution is a consequence of the extreme curvature values associated with cusp-like features of the flame surface.

Figure 5. (a) Illustration of the variation of the curvature $\kappa$ along the flame surface in the presence of a cusp-like structure. The curvature attains a large value at the cusp (right panel) and thus appears as a prominent spike in the plot of $\kappa$ as a function of the arc length $s$ (left panel). Note that the curvature has been non-dimensionalized using the flame thickness $\delta _F$. (b) Stationary p.d.f. of the curvature $\kappa$, shown for flames F1 and F2, exhibiting a flared tail which reflects the presence of cusp-like structures on the flame surface. The p.d.f.s are constructed by calculating the curvature along the flame edge contained within the window of interrogation (§ 2.3) and over time.

Having calculated the flame curvature, we next examine its correlation with the temporal increment of the flame fluctuations. Each measurement of the increment $\delta \xi ^\prime (\tau ;y^*)$ is associated with curvature values at two time instances, $\kappa (t;s_1)$ and $\kappa (t+\tau ;s_2)$, where the values of $s_1$ and $s_2$ are such that $y(s_1)=y(s_2)=y^*$ corresponds to the measurement location. Note that there will be more than one value of $s_1$, or $s_2$, when the flame edge becomes locally multi-valued. We consider the maximum of these curvature values $\max \{{\kappa }(t),{\kappa }(t+\tau )\}$ and depict its joint p.d.f. with the magnitude of the associated increment of the flame position $|\delta \xi ^\prime (\tau )|$ in figure 6, for various time intervals $\tau$. Clearly, the extreme values of the temporal increment which arise as $\tau$ decreases are strongly correlated with extreme values of the flame curvature. This is entirely consistent with our observation that large and rapid temporal fluctuations of the flame position are registered when cusp-like structures are advected past the measurement location.

Figure 6. Pseudo-colour plot of the joint p.d.f. of the increment $\delta \xi ^\prime (\tau ;y)$ and the maximum of the flame curvature values at times instances $t$ and $t+\tau$, as measured at $y = 2 \lambda _c$. The three panels correspond to different values of the time increment $\tau /\tau _\ell$: (a) 2.67, (b) 0.10 and (c) 0.03. The extreme-valued increments which appear as $\tau$ decreases show a clear correlation with extreme values of the curvature. These results correspond to flame F1 and are measured at $y=2\lambda _c$; similar results are obtained for flame F2 and for other measurement locations within $1< y/\lambda _c<3$. The coloured contours correspond to the logarithm of the joint p.d.f.

Cusp-like features and wrinkling in general can cause the flame edge to become a locally multi-valued function of $y$. This fact is accounted for in the calculation of curvature $\kappa (s,t)$ which is based on the arc-length parameterization, but not in the measurement of increments $\delta \xi ^\prime$ which is based on the single-valued leading edge $\xi (y,t)$. This raises the concern that multi-valued regions of the flame edge may appear as artificially abrupt variations in $\xi (y,t)$ which would in turn produce spurious large values of the increment. In the Appendix, we carry out two checks which show that our detection of inner intermittency is not an artefact of the treatment of the flame edge. First, we modify how we obtain the single-valued function $\xi (y,t)$: instead of taking the points closest to the $y$ axis, we now locally average the flame edge in regions where it is multi-valued. This procedure would smooth out artificially abrupt variations in $\xi (y,t)$. Second, we retain the leading edge profile, but eliminate all data points where the flame edge is multi-valued, i.e. we only calculate increments when $\xi (y)$ is single valued at both times, $t$ and $t+\tau$. For both cases, we find that anomalous scaling persists, which shows that the inner intermittency detected in this study is a genuine feature of the flame dynamics.

7. Concluding remarks

To summarize, we have seen that a well-maintained flame surface, devoid of large-scale bursts associated with outer intermittency, contains a subrange of scales wherein the flame surface merely responds to the fluctuations of the incident turbulent flow and exhibits power-law scaling that is entirely determined by the inertial-range scaling of the flow. However, this apparent simplicity comes with a caveat, which is the key result of our work: the flame fluctuations are intensely intermittent with structure functions that exhibit strongly anomalous scaling. The associated extreme events, which originate from cusp-like structures that are advected along the flame edge, have important implications for the modelling of turbulent premixed flames. For example, cusps with their extremely large values of flame curvature will affect the mean turbulent flame speed (Law Reference Law2010; Humphrey et al. Reference Humphrey, Emerson and Lieuwen2018; Dave & Chaudhuri Reference Dave and Chaudhuri2020). Furthermore, closure models of turbulent flame speeds and volumetric heat release rates, which depend on the fractal dimension of the flame surface, assume perfect self-similarity and so may be improved by accounting for inner intermittency (Charlette, Meneveau & Veynante Reference Charlette, Meneveau and Veynante2002; Gülder Reference Gülder2007; Roy & Sujith Reference Roy and Sujith2021).

It is intriguing to consider how the cusp-like features on the flame surface are related to the small-scale structures, such as ramp cliffs, of the underlying scalar fields. While ramp-cliff structures have been observed in direct numerical simulations of premixed combustion (Wang et al. Reference Wang, Tong, Barlow and Karpetis2007; Cai et al. Reference Cai, Wang, Tong, Barlow and Karpetis2009), their connection to intermittent fluctuations of the flame surface is unknown. More generally, the question of how the statistics of the flame surface are connected to those of the reacting scalar fields of combustion, possibly via a flame indicator function (Thiesset et al. Reference Thiesset, Maurice, Halter, Mazellier, Chauveau and Gökalp2016), deserves further study, especially in light of the extensive literature on the turbulent transport of conserved scalars (Warhaft Reference Warhaft2000; Falkovich & Sreenivasan Reference Falkovich and Sreenivasan2006); one would require simultaneous high-resolution measurements of the flame surface and the scalar fields, which, while beyond our current scope, is an important task for future work.

It is also interesting to note that the equation for the propagation of a thin premixed flame resembles the Kardar–Parisi–Zhang (KPZ) equation (Kerstein et al. Reference Kerstein, Ashurst and Williams1988; Kardar, Parisi & Zhang Reference Kardar, Parisi and Zhang1986) (in turn closely related to the Burgers equation (Bec & Khanin Reference Bec and Khanin2007)), whose dynamics in the presence of additive noise is well studied (Verma Reference Verma2000). However, a crucial difference arises due to the advection of the flame by the turbulent flow, which, if modelled as a random flow, appears as multiplicative noise. This results in a fundamentally distinct dynamics (Kerstein et al. Reference Kerstein, Ashurst and Williams1988; Yakhot Reference Yakhot1988), for example, the propagating flame attains a statistically stationary mean speed in contrast to the power-law growth predicted by the KPZ equation with additive noise. Thus, in light of the present results, it would be interesting to explore the intermittent properties of the KPZ equation with multiplicative and spatio-temporally correlated noise.

Finally, while we have focused on temporal measurements, the cusp-like structures of the flame surface will certainly give rise to inner intermittency in space, so that spatial structure functions obtained from a flame edge profile at a single snapshot in time should scale anomalously. Establishing this experimentally would require a very large flame in order to capture the required range of spatial scales, a challenge that will hopefully be taken up in the future. Other important questions raised by our work include how the inner intermittency of the self-similar range relates to the extreme-value statistics of the sub-diffusive dissipative scales (Hamlington et al. Reference Hamlington, Poludnenko and Oran2012; Chaudhuri et al. Reference Chaudhuri, Kolla, Dave, Hawkes, Chen and Law2017), and whether the scaling exponents for a premixed flame are universal, as they seem to be for a passive scalar (Watanabe & Gotoh Reference Watanabe and Gotoh2006).