4.1 Comparison with the experiment

The radial profiles of the mean temperature at $x/D=10$ , 24 and 30 are shown in figure 2(a). Along the radial direction, the mean temperature increases from the centreline, and then levels off in the combustion products. Along the streamwise direction, the mean temperature at the centreline increases progressively due to turbulent mixing and heat release. At $x/D=30$ , the reactants are almost completely consumed, and the mean temperature begins to approach that of the pilot. The mean temperature is predicted fairly well by the DNS at all locations. A minor discrepancy is found at $r/D=1{-}2$ and $x/D=10$ . In particular, the DNS slightly overestimates the mean temperature in this region. This could be attributed to a number of factors; for example, the pilot composition could be different from chemical equilibrium, or there might be heat loss from the flame through radiation in the near field, which is not accounted for in the simulation. Moreover, uncertainties in the Rayleigh scattering measurement are inevitable. Despite the near-field discrepancy in the mean temperature, it is seen that further downstream, i.e. at $x/D=24$ and 30, the DNS predictions agree with the experimental measurements very well.

Figure 2.

(a) Comparison of the mean temperature at $x/D=10$ , $x/D=24$ and $x/D=30$ , and (b) comparison of the mean and r.m.s. axial velocities at $x/D=4$ . The solid lines represent the DNS results and the symbols represent the experimental data.

Figure 3.

(a) Comparison of the normalised $[\text{CH}_{2}\text{O}]$ at $x/D=10$ , $x/D=24$ and $x/D=30$ . (b) Comparison of the normalised [OH] at $x/D=10$ , $x/D=24$ and $x/D=30$ . The solid lines represent the DNS results and the symbols represent the experimental data.

Figure 2(b) shows the radial profiles of the mean and r.m.s. axial velocities at $x/D=4$ , the only axial location where the LDA measurements were performed. Very good agreement is obtained between the DNS and the measurements for the mean axial velocity, confirming the validity of the DNS inlet velocity profile. Good agreement is also obtained for the r.m.s. velocity, especially in the shear region where the r.m.s. axial velocity peaks; minor discrepancy is found in the jet region. It should be noted that in the present turbulent jet flame, the shear-generated turbulence is dominant due to the existence of a large mean velocity gradient.

The PLIF signals of $\text{CH}_{2}\text{O}$ and OH were corrected to provide estimated species concentrations in the experiment (Zhou et al. Reference Zhou, Brackmann, Li, Wang, Petersson, Li, Aldén and Bai2015a ). Normalised species concentrations from the DNS and the experiment are compared, i.e. the species profiles are normalised by their peak values (normalisation is necessary because the experiments are not regarded to provide quantitative estimates). Figure 3 shows that good agreement of the normalised $[\text{CH}_{2}\text{O}]$ is obtained at all axial locations. The comparison of the normalised [OH] is also good in downstream locations. However, some discrepancies are found at $x/D=10$ in the oxidation layer. As mentioned earlier, the OH concentration of the pilot in the DNS is specified based on chemical equilibrium calculation, which might result in a lower value than the real OH concentration in the experiment and be responsible for the upstream discrepancies. This suggests that more accurate inlet boundary conditions from the experiment would be needed to improve the DNS predictions. Interestingly, the discrepancies of OH concentration in the oxidation layer were also found in a numerical and experimental study of laminar propane/air flames (Fayoux et al. Reference Fayoux, Zähringerc, Gicquela and Rolon2005).

Figure 4.

A typical instantaneous snapshot of (a) the simultaneous $\text{HCO/CH}_{2}\text{O/OH}$ experimental PLIF images and (b) the $\text{HCO/CH}_{2}\text{O/OH}$ molar concentration of the DNS.

Figure 4 shows instantaneous snapshots of species PLIF images of the experiment and species concentrations of the DNS. It should be noted that the species PLIF image intensities may be interpreted qualitatively to be proportional to the species molar concentrations. Panel (a) shows the simultaneous $\text{HCO/CH}_{2}\text{O/OH}$ PLIF images and panel (b) shows the $\text{HCO/CH}_{2}\text{O/OH}$ molar concentrations of the DNS. For consistency, the filtering from the experiment was also applied to the DNS species concentration fields. While these are random realisations and are not expected to agree one to one, the comparison exhibits qualitatively similar flame structure characteristics. For example, it is readily observed that $\text{CH}_{2}\text{O}$ and OH layers are distinct from each other in physical space, while HCO exists in the region between $\text{CH}_{2}\text{O}$ and OH. The DNS and the experiment exhibit similar flame wrinkling scales, indicating similar turbulence/chemistry interactions. There are pockets containing reactants pinching off the flame tip. Some of the pockets leave the image without burning completely. However, they are consumed further downstream in the computational domain. The $\text{CH}_{2}\text{O}$ and OH layers are seen to be broadened progressively along the streamwise direction, as turbulent eddies are able to penetrate the flame zones and disturb the flame structure. In addition, the layer merging also gives the appearance of a thickened reaction zone. The $\text{CH}_{2}\text{O}$ layer is closer to the reactant side where turbulent eddies are stronger, which may penetrate and broaden the $\text{CH}_{2}\text{O}$ layer more easily.

An important feature of high $Ka$ flames is that turbulent convection time scales can be smaller than species reaction time scales; even some short-lived species, such as HCO, can be perturbed by turbulent eddies and redistribute in space. Therefore, a spatial overlap and hence correlation of some species, which is not observed for flames in the flamelet regime, can be expected in high $Ka$ flames. The correlations of HCO and $\text{CH}_{2}\text{O}/\text{OH}$ are investigated in the present flame, and scatter plots of species concentrations in the DNS and species PLIF signals in the experiment at $x/D=24$ are shown in figure 5. The plots are normalised by the maximum value of the corresponding species concentration/signal. It is clear that, in contrast to flamelet behaviour (which would present a single curve), the species distributions are more scattered; moderate concentration of HCO can be found in high $\text{OH}/\text{CH}_{2}\text{O}$ concentration regions. It is also seen that the trends of the species correlations are similar in both the DNS and the experiment. It should be noted that the experimental results are more scattered in the plots, which is due to noise in the measurements, as discussed in Wang et al. (Reference Wang, Hawkes, Zhou, Chen, Li and Aldén2017).

Figure 5.

Normalised scatter plots of (a,b) [HCO] versus $[\text{CH}_{2}\text{O}]$ and (c,d) [HCO] versus [OH] at $x/D=24$ . The left-hand plots show the species concentrations of the DNS and the right-hand plots show the species PLIF signals of the experiment.

4.2 Flame stretch

The turbulence/flame interactions are examined using the DNS data. First, the flame stretch statistics are studied. Then, the scalar gradient statistics and flame thickness will be presented in § 4.3. In the present work, the progress variable, c, is defined based on the mass fraction of $\text{O}_{2}$ as

where $\text{Y}_{\text{O}_{2},u}$ is the $\text{O}_{2}$ mass fraction of the reactant while $\text{Y}_{\text{O}_{2},b}$ is the $\text{O}_{2}$ mass fraction of the pilot. According to this definition, c increases monotonically from zero in the reactant to unity in the pilot stream.

Equation (1.1) can be expanded to have the conventional form involving the tangential strain rate, $a_{t}$ , and the curvature stretch, $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , as follows (Pope Reference Pope1988; Candel & Poinsot Reference Candel and Poinsot1990):

where $S_{d}$ is the displacement speed and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ is the curvature. By definition, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ is positive (negative) when the centre of curvature is in the products (reactants). Here, the tangential strain rate, $a_{t}$ , is calculated as

where $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}$ is the dilatation and $a_{n}$ is the normal strain rate given by

Here, $n_{i}$ is the $i\text{th}$ component of the flame normal vector $\boldsymbol{n}$ , defined as

By definition, $\boldsymbol{n}$ points towards the reactants. The term $\unicode[STIX]{x1D61A}_{ij}$ is the strain rate tensor, defined as

The transport equation for the progress variable is given as

where $D_{c}$ is the diffusivity of the progress variable and $\dot{\unicode[STIX]{x1D714}}_{c}$ is the reaction rate of the progress variable. The progress variable isosurface propagates at the speed $S_{d}$ , which denotes the speed of the front relative to the flow normal to the front and is defined as (Gibson Reference Gibson1968; Pope Reference Pope1988)

The analysis in this section will exploit surface-averaged quantities. The fine-grained surface average of a quantity $Q$ is denoted as (Veynante & Vervisch Reference Veynante and Vervisch2002)

where $\unicode[STIX]{x1D6FF}$ is the delta function and $\unicode[STIX]{x1D6F4}^{\ast }$ is the fine-grained flame surface density,

The ensemble average, denoted as $\bar{(\cdots \,)}$ , is performed in time and the homogeneous (azimuthal) direction, so that $\unicode[STIX]{x1D6F4}^{\ast }$ and $\langle Q\rangle _{s}^{\ast }$ vary in the axial and radial directions. It should be noted that they are also functions of $c^{\ast }$ . These definitions can be extended to flames that are not thin (Veynante & Vervisch Reference Veynante and Vervisch2002), as in the present DNS flame. By integrating (4.10) over all isosurface levels, the generalised flame surface density, $\unicode[STIX]{x1D6F4}$ , is obtained,

The generalised surface average of $Q$ is denoted as

The distribution of $\unicode[STIX]{x1D6F4}$ in the $x$ – $r$ plane is shown in figure 6. It is readily observed that $\unicode[STIX]{x1D6F4}$ is high in the upstream region and the distribution is narrow. The magnitude of $\unicode[STIX]{x1D6F4}$ decreases and the distribution becomes wider as the flame develops. It should be noted that the integration of $\unicode[STIX]{x1D6F4}$ in the radial direction gives the flame area, which will be discussed shortly. Figure 6 also shows the generalised surface-averaged tangential strain rate, $\left\langle a_{t}\right\rangle _{s}$ , and curvature stretch, $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s}$ . It is seen that $\left\langle a_{t}\right\rangle _{s}$ is mostly positive, producing flame area, while $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s}$ is mostly negative, consuming flame area. This is consistent with studies of statistically 1D planar premixed flames in isotropic turbulence (Trouvé & Poinsot Reference Trouvé and Poinsot1994; Chakraborty & Cant Reference Chakraborty and Cant2005; Hawkes & Chen Reference Hawkes and Chen2005).

Figure 6.

The distributions of generalised flame surface density, generalised surface-averaged tangential strain rate and curvature stretch.

For the sake of a compact presentation, the generalised surface averages are also radially integrated such that they vary only with downstream distance. Considering the round jet geometry,

where $A(x)=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\unicode[STIX]{x1D6F4}r\,\text{d}r\unicode[STIX]{x0394}x$ is the flame area at the axial location $x$ .

Statistics such as PDF (probability density function), joint PDF and conditional means are also explored to reveal the underlying physics. Consistent with the generalised surface averages described above, the statistics are weighted by $|\unicode[STIX]{x1D735}c|$ , unless it is specifically stated otherwise. Three locations are selected to perform the analysis. Their axial distances are $x/D=8$ , 16 and 24, and their radial distances correspond to the locations where $\unicode[STIX]{x1D6F4}$ is the largest ( ${\sim}r/D=0.6$ ). In order to exclude regions where reaction is unimportant, the statistics are also conditioned on $0.5<c<0.9$ (this choice will be shown in § 4.3), except for conditional means which are directly conditioned on the progress variable.

The evolution of the integrated generalised surface-averaged flame stretch and its components along the streamwise direction is shown in figure 7(a). It is first noted that, consistent with the generalised surface-averaged results in figure 6, $\left\langle a_{t}\right\rangle _{s,x}$ tends to produce flame area while $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ tends to destroy flame area. The value of $\left\langle a_{t}\right\rangle _{s,x}$ is relatively small in the near field, which is due to the misalignment of the flame normal with the principal strain rates of the flow, as discussed in Wang, Hawkes & Chen (Reference Wang, Hawkes and Chen2016). The value of $\left\langle a_{t}\right\rangle _{s,x}$ increases and reaches its maximum at approximately $x/D=10$ . It decreases further downstream, consistent with the increasing characteristic turbulence time scales with downstream distance in this jet flame. Second, it is noted that the magnitude of each term significantly exceeds their sum, suggesting that a quasi-equilibrium exists for creation and destruction of flame area in this region. However, the overall balance of these terms changes with downstream distance. It is observed that, in the upstream region, $\left\langle a_{t}\right\rangle _{s,x}$ is larger than $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ , and thus the net effect of the flame stretch is the production of flame area. After $x/D=20$ , it is seen that $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ becomes increasingly dominant over $\left\langle a_{t}\right\rangle _{s,x}$ , leading to a net negative stretch, consistent with increasing flame annihilation.

Figure 7.

The axial evolution of (a) the integrated generalised surface-averaged flame stretch and its components and (b) the flame area.

The consequences of the flame stretch evolution with downstream distance for the flame area, $A(x)$ , are now examined. The evolution of flame area is presented in figure 7(b). It is seen that $A(x)$ first increases with downstream distance and peaks near $x/D=20$ . Further downstream, it decreases due to the depletion of reactants and the merging of flame segments. The evolution of the flame area is therefore quite consistent with that of the flame stretch. To the best of our knowledge, the spatial development of flame stretch and its correlation with flame area have not been reported before in the literature.

Figure 8.

The PDF of (a) the tangential strain rate, $a_{t}$ , and (b) the curvature stretch, $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , at $x/D=8$ , $x/D=16$ and $x/D=24$ .

To further understand the flame stretch components, figure 8 shows the PDF of the tangential strain rate, $a_{t}$ , and curvature stretch, $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , at various downstream locations. The PDF of $a_{t}$ is seen to be quite broad with a width that decreases with downstream distance, due to the increasing characteristic turbulence time scales with $x/D$ in this jet-type flow. The trend is consistent with that observed for increasing Damköhler numbers in DNS of statistically 1D planar premixed flames (Chakraborty & Swaminathan Reference Chakraborty and Swaminathan2007). The PDF of $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ has a quite different character, with most of the flame surface exhibiting low stretch values, while there is a very long tail with small amounts of flame surface exhibiting very large negative stretch. As noted in previous work on statistically 1D planar premixed flames (Hawkes & Chen Reference Hawkes and Chen2005), this trend is consistent with the concept of scale separation in turbulent flows, wherein significant dissipation (in this case, of the flame area) occurs in very small-scale features (in this case, features that have very large curvatures).

Figure 9.

The joint PDF of (a) the tangential strain rate and the curvature, and (b) the flame stretch and the curvature, at $x/D=24$ , where the conditional mean is denoted by the black solid line.

Strain rate and curvature are not independent quantities, however. To examine this, figure 9 shows the joint PDF of tangential strain rate and curvature (figure 9 a) and net flame stretch and curvature (figure 9 b), and the conditional means of each quantity on curvature at $x/D=24$ . Figure 9(a) shows that high positive tangential strain rate regions generally have low curvatures while high curvature regions generally have low strains, a finding that has been reported before for low Damköhler number statistically 1D planar premixed flames (Chakraborty & Cant Reference Chakraborty and Cant2005). Although the response of the curvature to strain depends on the alignment of their respective principal axes, intuition suggests that high tangential strain rate of a surface should tend to smooth wrinkles in the surface, which is exactly what is observed. The net flame stretch presents a somewhat different picture. Figure 9(b) shows that the strong net positive stretch regions consistently occur in regions that have low curvature magnitudes, which by comparison with figure 9(a) are deduced to be high positive strain regions. However, the high net negative stretch regions occur most often in regions of large curvature, the sign of the curvature being unimportant. Since curvature stretch is given by $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , but both large positive and large negative curvatures still lead to high negative stretch, it is deduced that the displacement speed and curvature must be strongly correlated.

To investigate the importance and nature of this correlation, the curvature stretch term, $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , is further studied. Previous modelling of flames in the flamelet regime has assumed that the displacement speed and the curvature are not correlated, and hence

However, in highly turbulent flames, this assumption is no longer appropriate, and the correlation must be accounted for. To quantify the importance of this correlation, the following decomposition of $\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\rangle _{s}$ is proposed:

where the first term gives the combined action of the mean flame speed with the mean curvature while the second accounts for flame speed–curvature correlations. Previous theoretical analysis for the thin reaction zones regime (Peters Reference Peters1999) suggested that these correlations could be approximated by breaking down the $S_{d}$ contributions into different components that have different curvature responses. Here, $S_{d}$ is decomposed into its reaction ( $S_{d,r}$ ), normal diffusion ( $S_{d,n}$ ) and curvature ( $S_{d,c}$ ) components as (Echekki & Chen Reference Echekki and Chen1999)

where $n$ is the distance along the flame normal direction. The sum of the reaction and normal diffusion components of the displacement speed is also known as kinematic restoration, $S_{d,r+n}$ (Peters Reference Peters1999). Therefore, $S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ can be recast as

Accordingly, $\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\rangle _{s}$ can also be written as

where both terms on the right-hand side may show correlations with curvature.

Following (4.13), the integrated generalised surface averages of the above-mentioned quantities are studied. The evolution of $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ and the relevant terms is shown in figure 10. It is first noted that the term due to correlation of flame speed with curvature, $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}-\left\langle S_{d}\right\rangle _{s,x}\left\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ , is strongly dominant over $\left\langle S_{d}\right\rangle _{s,x}\left\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ across almost the entire flame. It is also clear that $-\langle D_{c}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})^{2}\rangle _{s,x}$ is dominant over $\left\langle S_{d,r+n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ . This is because the curvature component of the displacement speed, $S_{d,c}$ , is always negatively correlated with curvature, contributing significantly to $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ . However, it is certainly not the case that the curvature correlation of $S_{d,r+n}$ can be ignored, and indeed in the downstream region, the contribution of $\left\langle S_{d,r+n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ to the surface-averaged flame stretch increases.

Figure 10.

The axial evolution of the integrated generalised surface-averaged curvature stretch term: $T_{0}$ , $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ ; $T_{1}$ , $\left\langle S_{d,r+n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ ; $T_{2}$ , $-\langle D_{c}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})^{2}\rangle _{s,x}$ ; $T_{3}$ , $\left\langle S_{d}\right\rangle _{s,x}\left\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ ; $T_{4}$ , $\left\langle S_{d}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}-\left\langle S_{d}\right\rangle _{s,x}\left\langle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}\right\rangle _{s,x}$ .

4.3 Flame thickness

In the following, we address the question: what is the influence of turbulence on the flame thickness and the corresponding scalar gradients? The thicknesses of several species layers were presented in figure 4. In particular, the $\text{CH}_{2}\text{O}$ and OH layers are increasingly broadened with increasing downstream distance. The flame structure is further investigated to show the flame thickness. Figure 11 displays the instantaneous distributions of the vorticity, heat release rate, and OH and CO mass fractions at the axial location $x/D=20$ . Of particular interest is the reaction zone, where chemical reactions occur and heat is released. The reaction zone is defined as the region where the heat release rate exceeds 35 % of its local maximum value, spanning approximately $c^{\ast }=0.5$ –0.9. The distance between $c^{\ast }=0.5$ and 0.9 for the unstrained laminar flame, denoted $l_{c}$ , is also shown for comparison. It is readily observed that turbulent eddies enter into and disturb the reaction zone. There are regions where the thickness of the reaction zone is an order of magnitude larger than its laminar counterpart. The distribution of CO is particularly disturbed by turbulence, a feature of the broken reaction zones regime. On the other hand, the distribution of OH is less affected, as OH has a higher concentration on the product side where turbulence is weaker due to the higher viscosity in the products. Following these observations, quantitative analyses of the turbulent flame thickness are performed next.

Figure 11.

Instantaneous distributions of the (a) vorticity ( $\text{s}^{-1}$ ), (b) heat release rate ( $\text{J}~\text{m}^{-3}~\text{s}$ ), (c) OH mass fraction and (d) CO mass fraction at $x/D=20$ . The isolines of the progress variable are delineated as follows: (black) $c^{\ast }=0.5$ , (red) $c^{\ast }=0.8$ and (white) $c^{\ast }=0.9$ . The nozzle diameter, $D$ , and the distance between $c^{\ast }=0.5$ and 0.9 in the unstrained laminar flame, $l_{c}$ , are also added for reference.

The thickness of the turbulent flame is discussed more quantitatively in terms of the scalar gradient magnitude, $|\unicode[STIX]{x1D735}c|$ , which represents a reciprocal flame thickness. Figure 12 shows $|\unicode[STIX]{x1D735}c|$ conditionally averaged on c at various downstream locations. (It should be noted that the results in figures 12–14 are not weighted by $|\unicode[STIX]{x1D735}c|$ .) The profile of $|\unicode[STIX]{x1D735}c|$ of the unstrained laminar premixed flame is also plotted. At $x/D=8$ , $|\unicode[STIX]{x1D735}c|$ of the DNS is higher than that of the unstrained laminar flame for $c<0.5$ . On the other hand, $|\unicode[STIX]{x1D735}c|$ of the DNS is lower than that of the laminar flame for $c>0.5$ . It is worth noting that the structure of the turbulent flame is expected to be different from that of the unstrained laminar flame due to the existence of pilot, mean shear and complex turbulence/flame interactions. A detailed comparison of the turbulent flame and strained laminar flames is a subject of future work. As the flame develops downstream, the mean strain relaxes while small-scale turbulence progressively penetrates the flame, modifying the inner layer of the flame at high local $Ka$ ( $100<Ka$ ). At $x/D=24$ , the flame is considerably broadened and the mean flame thickness is approximately $2\unicode[STIX]{x1D6FF}_{L}$ . Importantly, this thickening is also observed in the reaction zone.

Figure 12.

The surface density function $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the progress variable.

It is interesting to observe the large $|\unicode[STIX]{x1D735}c|$ region of the preheat zone upstream, which might at first consideration be an unexpected feature for high $Ka$ where on-average flame broadening is expected. In order to understand this behaviour, first the instantaneous/mean flame structure and then the joint PDF of $c$ and $|\unicode[STIX]{x1D735}c|$ are examined. A distance function, $L_{D}$ , is introduced, which represents the spatial distance with respect to the flame surface (Peters Reference Peters2000; Sankaran et al. Reference Sankaran, Hawkes, Yoo and Chen2015). The distance function $L_{D}$ is defined such that at each point in the computational domain it signifies the shortest normal distance to the flame surface, computed by solving the Eikonal equation,

The distance function $L_{D}$ is set to zero on the $c^{\ast }=0.8$ isosurface (this is the point of maximum heat release in a freely propagating laminar flame), negative in the reactants and positive in the products. Four representative instantaneous profiles of the progress variable versus the distance function at $x/D=8$ are shown in figure 13(a). It is seen that the instantaneous profiles are significantly perturbed by turbulence, and the progress variable does not increase monotonically with the distance function as it does in the laminar flame. This is because peninsular-like structures and/or pockets of partially reacted gas are formed in the preheat zone due to the interactions of small-scale turbulent eddies and flame, increasing the local scalar gradient and giving the appearance of a locally ‘thinner’ preheat zone as measured by the magnitude of the scalar gradient. In contrast, figure 13(b) shows the profiles of the progress variable conditionally averaged on the distance function. It is seen that the preheat zone of the mean flame structure is, however, thicker in the turbulent flame compared with the laminar flame. It should be noted that, due to strong turbulent mixing between the reactants and products, the mean progress variable of the reactants increases with downstream distance.

Figure 13.

(a) Four representative instantaneous profiles of the progress variable versus the distance function at $x/D=8$ . (b) The progress variable conditionally averaged on the distance function at $x/D=8$ , $x/D=16$ and $x/D=24$ .

Figure 14 shows the joint PDF of the progress variable and the surface density function at various downstream locations. It is readily seen that the distribution of the surface density function is broad, especially at $x/D=8$ , where the local flame can be very thin (as measured by the scalar gradient). It will be shown later that the local flame thinning (increasing scalar gradient) corresponds to large and positive tangential strain rate. It is suggested that the average of $|\unicode[STIX]{x1D735}c|$ is the result of the competition between large $|\unicode[STIX]{x1D735}c|$ induced by small-scale flame structures and small $|\unicode[STIX]{x1D735}c|$ due to the flame thickening. The former plays an important role in the preheat zone upstream, while the latter is dominant downstream as the flame is increasingly thickened.

Figure 14.

Joint PDF of the progress variable and the surface density function. The white line denotes the conditionally averaged surface density function of the DNS and the black line denotes the unstrained laminar premixed flame result.

As discussed earlier, the balance equation for $|\unicode[STIX]{x1D735}c|$ provides a framework for understanding the flame thinning/thickening as isoscalar surfaces are brought together or separated by turbulent straining and self-propagation. Following Sankaran et al. (Reference Sankaran, Hawkes, Chen, Lu and Law2007), equation (1.1) can be expanded as

Using the decomposition of the displacement speed, equation (4.20) can be recast as

In the following, the effects of the terms representing the normal strain rate ( $a_{n}$ ), kinematic restoration ( $\unicode[STIX]{x2202}S_{d,r+n}/\unicode[STIX]{x2202}n$ ) and curvature ( $\unicode[STIX]{x2202}S_{d,c}/\unicode[STIX]{x2202}n$ ) on the flame thickness are analysed. It should be noted here that positive-valued terms act as sinks for $|\unicode[STIX]{x1D735}c|$ , and vice versa.

Figure 15.

Budget of the source terms in the balance equation for $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the progress variable. The terms are non-dimensionalised using $\unicode[STIX]{x1D70F}_{L}$ . Solid line, $x/D=8$ ; dashed line, $x/D=16$ ; dotted line, $x/D=24$ .

Figure 15 shows the budget of various terms conditioned on c. It is obvious that the kinematic restoration term and the curvature term are positive, leading to lower scalar gradients and a thicker flame. On the other hand, the normal strain rate term is negative, contributing to a source for $|\unicode[STIX]{x1D735}c|$ and a locally thinner flame. Overall, the net balance is in favour of thickening of the flame across all values of $c$ . This result is important in that it reveals the mechanism for the flame thickening as the flame traverses downstream. In particular, the displacement speed of an isoscalar surface decreases monotonically from reactants to products, causing a larger distance between two isoscalar surfaces. Negative normal strain rate tends to bring the isoscalar surfaces together. However, its magnitude is small compared with the terms involving normal gradients of the flame speed ( $\unicode[STIX]{x2202}S_{d,r+n}/\unicode[STIX]{x2202}n+\unicode[STIX]{x2202}S_{d,c}/\unicode[STIX]{x2202}n$ ). This scenario is in contrast to the result of Dopazo et al. (Reference Dopazo, Cifuentes, Martín and Jiménez2015), who reported a positive normal strain rate term and a negative flame speed-gradient term in a low $Ka$ flame.

To further understand the influence of turbulence on disturbing the inner structure of the flame, the effects of the tangential strain rate, $a_{t}$ , and curvature, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ , on the flame thickness are investigated. Figure 16(a) shows $|\unicode[STIX]{x1D735}c|$ conditionally averaged on $a_{t}$ . It is clear that $|\unicode[STIX]{x1D735}c|$ increases with $a_{t}$ for positive $a_{t}$ . For negative $a_{t}$ , the scalar gradient magnitude is low and insensitive to $a_{t}$ . Figure 16(b) indicates that at positive $a_{t}$ , the normal strain rate term becomes the dominant source for $|\unicode[STIX]{x1D735}c|$ , which produces scalar gradients and is responsible for the local flame thinning. Meanwhile, at negative $a_{t}$ , the normal strain rate term and, to a lesser extent, the curvature term tend to dissipate scalar gradient and locally thicken the flame.

The influence of curvature is finally assessed. Figure 17(a) shows $|\unicode[STIX]{x1D735}c|$ conditionally averaged on curvature. In general, the flame is thicker as the curvature magnitude increases (both negative and positive). This observation is consistent with the DNS of Sankaran et al. (Reference Sankaran, Hawkes, Chen, Lu and Law2007) and the experiment of Tamadonfar & Gülder (Reference Tamadonfar and Gülder2015). The flames that are near zero curvature coincide with regions of highly negative normal strain rate (figure 17 b) and are locally thinned. The budget analysis of figure 17(b) indicates that the dominant source at large curvatures is the positive curvature term, which separates isoscalar surfaces and thickens the flame. At small curvatures, the normal strain rate term is seen to be counteracted by the kinematic restoration term.

Figure 16.

(a) The surface density function $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the tangential strain rate. (b) Budget of the source terms in the balance equation for $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the tangential strain rate. The terms are non-dimensionalised using $\unicode[STIX]{x1D70F}_{L}$ . Solid line, $x/D=8$ ; dashed line, $x/D=16$ ; dotted line, $x/D=24$ .

Figure 17.

(a) The surface density function $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the curvature. (b) Budget of the source terms in the balance equation for $|\unicode[STIX]{x1D735}c|$ conditionally averaged on the curvature. The terms are non-dimensionalised using $\unicode[STIX]{x1D70F}_{L}$ . Solid line, $x/D=8$ ; dashed line, $x/D=16$ ; dotted line, $x/D=24$ .