2.1 Combustor and operating conditions

Figure 1.

Experimental set-up showing (a) the swirl combustor and (b) the laser diagnostic set-up.

The swirl combustor used in this work (figure 1 a) was similar to that described by Meier et al. (Reference Meier, Weigand, Duan and Giezendanner-Thoben2007), but was operated with perfectly premixed reactants as in Caux-Brisebois et al. (Reference Caux-Brisebois, Steinberg, Arndt and Meier2014) and without the combustion chamber to prevent window contamination due to T-PIV seed particles. Ambient temperature reactants (methane (CH $_{4}$ ) and air) were metered using electromechanical mass flow controllers (Brooks) and mixed approximately 2 m upstream of the combustor plenum. Swirl was induced by a 12-vane radial swirler, with a measured swirl number of 0.55 at the exit of the $D_{e}=27.85~\text{mm}$ diameter nozzle. The inner surface of the nozzle was formed by a conical bluff body with a spherical tip.

The test conditions are summarized in table 1. Here, $\unicode[STIX]{x1D719}$ is the equivalence ratio, $U$ is the bulk-velocity (i.e. the ratio of the total volumetric flow rate and nozzle exit area) and the jet Reynolds number is $Re_{j}=UD_{e}/\unicode[STIX]{x1D708}_{r}$ . The laminar flame properties were calculated using Cantera with the GRI3.0 chemical mechanism. $Re_{T}$ was calculated based on $u^{\prime }$ at the centre of the nozzle exit. We note that neither $\unicode[STIX]{x1D702}$ nor $Ka$ are included in table 1 due to the difficulty in defining these in this highly inhomogeneous system. Estimates for $\unicode[STIX]{x1D702}$ based on $\ell \approx D_{e}/2$ and $\unicode[STIX]{x1D702}\approx \ell Re_{T}^{-3/4}$ yield $\unicode[STIX]{x1D702}\approx 35{-}45~\unicode[STIX]{x03BC}\text{m}$ .

Table 1.

Test conditions and turbulence properties for each case.

2.2 Diagnostics

Measurements were made using simultaneous T-PIV and $\text{CH}_{2}\text{O}$ PLIF (figure 1 b). A total of 1000 T-PIV and PLIF measurements were made at each reacting condition. Furthermore, an additional 1000 T-PIV measurements were made for the equivalent non-reacting flow condition for each case. In each non-reacting case, the air flow rate was increased appropriately to match the total volumetric flow rate of the air and fuel of the corresponding reacting case.

3.1 Overview

All of the analysis below is performed at the central plane of the T-PIV volume, corresponding to the locations of the PLIF measurements. We consider the Reynolds-averaged version of (1.1), viz.

(3.1) $$\begin{eqnarray}{\textstyle \frac{1}{2}}\overline{L}=\overline{I}-\overline{II}+\overline{III}+\overline{IV},\end{eqnarray}$$

where $\overline{L}=\overline{u_{j}\unicode[STIX]{x2202}_{j}\unicode[STIX]{x1D6FA}}$ represents the mean turbulent advection in this statistically steady flow. Because the flow was statistically stationary, the Reynolds averaging was performed by ensemble averaging. The mean vortex stretching $\overline{I}$ , dilatation $\overline{II}$ and advection $\overline{L}$ can be determined directly from well-resolved velocity measurements. However, $\overline{III}$ could not be measured, as it requires knowledge of both the pressure and density fields; our general approach will be to determine $\overline{III}$ as the difference between $\overline{L}$ and the other terms on the right-hand side of (3.1). We note here that the measurement of $\overline{IV}$ is challenging as it requires calculation of third spatial derivatives of the velocity field, as well as measurements of the fluid density and fluid viscosity, which are temperature dependent.

The analysis presented below therefore progresses in two steps. We first consider the sum $(\overline{III}+\overline{IV})_{L}=\frac{1}{2}\overline{L}-\overline{I}+\overline{II}$ . Here, the subscript $(\cdot )_{L}$ indicates that the term on the left is calculated by subtracting appropriate terms from the Lagrangian term. We compare the shapes of the $(\overline{III}+\overline{IV})_{L}$ curves in the reacting case to the $\overline{IV}_{L}=\frac{1}{2}\overline{L}-\overline{I}$ curves in the non-reacting case in order to describe qualitative differences. This analysis can be performed based on the T-PIV data alone. We then provide our best estimate of $\overline{IV}$ in the reacting cases by direct calculation using the T-PIV and PLIF data, thus allowing calculation of $\overline{III}_{L}=\frac{1}{2}\overline{L}-\overline{I}+\overline{II}-\overline{IV}$ . Uncertainty in all calculations is discussed below.

In order to compare (i) data at different axial locations $y$ in a given case, (ii) data from different cases and (iii) data from the non-reacting versus reacting conditions, it is necessary to reference the spatial location against physically meaningful locations in the system. To do so, we segment the data into 2 mm tall bands. Within each band, we plot the various quantities of interest against radial position $x$ , centred about the position $x_{0}$ that corresponds to the maximum $\overline{I}$ in that band. It was found that the profiles at different heights and conditions collapsed well in this coordinate system. To further facilitate comparison across different cases, we also present averages across all axial locations within a case. We denote these averaged results by $\langle \cdot \rangle$ .

3.2 Velocity uncertainty and resolution

Uncertainty in the measurement of the velocity field and its gradients arises due to both measurement noise and under-resolution. Uncertainty in the mean quantities of interest arises due to these sources and statistical convergence. Measurement noise is associated with a random uncertainty that reduces the convergence rate by artificially increasing the variance. The resultant uncertainty in the mean quantities that are calculated directly from the velocity measurements (e.g.  $\overline{I}$ ) was estimated by assuming a Gaussian process and are indicated by the symmetric error bars in the data presented below. The associated uncertainty in $(\cdot )_{L}$ terms was determined by summing the uncertainties of the terms on the right in quadrature.

Figure 2.

Joint probability density functions of (a) $u_{i}$ , (b) $\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ , (c)  $\unicode[STIX]{x2202}^{2}u_{i}/\unicode[STIX]{x2202}x_{j}^{2}$ and (d) $\unicode[STIX]{x2202}^{3}u_{i}/\unicode[STIX]{x2202}x_{j}^{3}$ as computed using 32 vx and 40 vx grids for case 1.

Under-resolution results in a systematic reduction in the measured gradients, which is also affected by the numerical methods used for smoothing and gradient calculation. We assessed the impact of resolution by calculating vector fields using interrogation volume lengths ranging from 48 vx to 24 vx and comparing the results. We note that there was a large increase in measurement noise at interrogation boxes smaller than 32 vx. The issue of measurement noise in the cross-correlated velocity fields associated with interrogation box size is discussed in greater detail below.

Figure 2 shows joint probability density functions of the velocity field and its gradients (up to third order) with interrogation boxes lengths of 40 vx and 32 vx for case 1. The higher-order derivatives were associated with higher noise, but there was no systematic under-prediction of the gradients at the lower resolution. Figure 3(a) shows the directly calculated terms in the average enstrophy transport budgets ( $\overline{L}$ , $\overline{I}$ , $\overline{II}$ , and $\overline{IV}$ ) for case 1 using the 40 vx and 32 vx interrogation boxes (the method by which $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ were calculated to determine $\overline{IV}$ is discussed below). The results from both resolutions agree within the convergence uncertainty. We therefore conclude that the measurements were well-resolved in case 1.

Figure 3.

Comparison of mean enstrophy transport terms demonstrating the degree of convergence with reducing interrogation box size in (a) case 1 and (b) case 3. Panel (c) shows the comparison between cases at 32 vx i. Shaded regions and the error bars represent the uncertainty in the mean due entirely to fluctuations (not systematic bias) for terms computed using 32 vx and 40 vx grids, respectively.

However, the same was not equally true for the more turbulent cases. Specifically, the measurements did not fully converge in the calculation of $\overline{IV}$ . This is demonstrated in figure 3(b), which shows $\overline{I}$ and $\overline{IV}$ calculated with 40 vx and 32 vx interrogation boxes for case 3. As can be seen, $\overline{I}$ did not change with the finer spatial resolution. However, the magnitude of $\overline{IV}$ increased with the finer resolution, indicating that the measurements were not fully resolved at 40 vx. As mentioned, it was not possible to further reduce the interrogation box due to noise issues, so it is not possible to determine whether the measurements were well-resolved at 32 vx. If not, this would result in a systematic under-estimation of the magnitude of $\overline{IV}$ and a corresponding under-estimation of $\overline{III}$ .

To estimate the bias uncertainty associated with possible under-estimation of $\overline{IV}$ , we speculate that, if fully resolved, the $\overline{I}$ and $\overline{IV}$ curves from the different cases should collapse when normalized by the maximum magnitude of $\overline{I}$ . Figure 3(c) shows $\overline{I}$ and $\overline{IV}$ from the different cases. Whereas the $\overline{I}$ curves collapse, the measured $\overline{IV}$ in case 1 has higher relative magnitude than in the other cases. Note that the collapse of $\overline{IV}$ for cases 2–4 does not indicate they are under-estimating by the same amount, as they are being normalized by the appropriate maximum $\overline{I}$ for each case.

Based on our assumption above, we estimate the bias uncertainty in the measurement of $\overline{IV}$ for cases 2–4 as the distance between the normalized $\overline{IV}$ curve for these cases and the $\overline{IV}$ curve for the fully resolved case 1, multiplied by the appropriate value of the maximum $\overline{I}$ for that case. The resultant asymmetric uncertainty bars are indicated in the figures in § 4.2.

While the resolution could potentially be increased by further decreasing the interrogation box size, this was limited by the need to have a sufficient number of particles in the interrogation box for accurate cross-correlation. A significant increase in random noise was observed when moving from 32 vx to 28 vx due to the reduced number of particles, and the particle seeding density could not be further increased due to unacceptable levels of ghost particles. Hence, an interrogation box length of 32 vx was selected for all further calculations.

3.3 Analysis of CH $_{2}$ O PLIF

The $\text{CH}_{2}\text{O}$ measurements were used to estimate the instantaneous progress variable field $c$ in a plane corresponding to the centre of the T-PIV volume ( $z=0$ ). This allowed both estimation of the local fluid density and viscosity needed to calculate $IV$ in the reacting cases, as well as characterization of the enstrophy transport relative to the mean flame brush.

It is common practice to use PLIF to determine a progress-variable-like parameter based on the infinitely thin flame assumption. That is, the PLIF data are used to separate the flow into unburnt ( $c=0$ ) and burnt ( $c=1$ ) regions; intermediate values of $c$ are not considered. This is most commonly done based on OH PLIF images, in which the OH PLIF signal rapidly increases in the thin oxidation layer of the flame (Steinberg et al. Reference Steinberg, Boxx, Arndt, Frank and Meier2011; Coriton, Steinberg & Frank Reference Coriton, Steinberg and Frank2014; Osborne et al. Reference Osborne, Ramji, Carter, Peltier, Hammack, Lee and Steinberg2016, Reference Osborne, Ramji, Carter and Steinberg2017; Wabel, Skiba & Driscoll Reference Wabel, Skiba and Driscoll2017a ; Wabel et al. Reference Wabel, Skiba, Temme and Driscoll2017b ).

However, using $\text{CH}_{2}\text{O}$ PLIF allows for a better estimation of the progress variable field. That is, $\text{CH}_{2}\text{O}$ occupies a region spanning the flame ‘preheat zone’ and is consumed by oxidation reactions. The region upstream of the $\text{CH}_{2}\text{O}$ therefore corresponds to $c\approx 0$ and the region downstream corresponds to $c\approx 1$ .

Figure 4.

Instantaneous $\text{CH}_{2}\text{O}$ distribution (a) at case 4, and the associated $c$ fields as calculated via (b) infinitely thin flame assumptions, (c) an intermediate $c=0.5$ assumption and (d) linearly-interpolated relative distance assumption. The solid red curve demarcates the region containing $\text{CH}_{2}\text{O}$ from the inner recirculation zone.

We used three methods of deducing the intervening values of $c$ from the PLIF fields; these methods are demonstrated in figure 4. In the first, we assigned all locations upstream of the most downstream edge of the $\text{CH}_{2}\text{O}$ PLIF a value of $c=0$ . This is analogous to the infinitely thin flame method using OH PLIF. In the second, we assigned the region between the upstream and downstream edges a value of $c=0.5$ . In the third, we assigned the value of $c$ at any point in the $\text{CH}_{2}\text{O}$ region using a linear fit between $c=0$ and $c=1$ based on the distance of that point from the nearest upstream and downstream $\text{CH}_{2}\text{O}$ edges.

Ensemble averaging was performed to obtain the $\overline{c}$ field, which is shown in figure 5(a) based on the third method. Figure 5(b) shows typical $\bar{c}$ profiles through the mean flame brush using the three different methods. The infinitely thin flame assumption generated a $\overline{c}$ distribution that is shifted towards the products by approximately 1 mm relative to the other methods. The other two methods showed nearly identical results. The results presented below are based on the third (distance-based) method.

Figure 5.

(a) Mean progress-variable field at case 4, calculated using the relative distance assumption; (b) $\overline{c}$ along $y=3~\text{mm}$ , as calculated using three estimation methods: infinitely thin flame (Bray, Libby & Moss Reference Bray, Libby and Moss1985; Libby Reference Libby1985), intermediate ( $c\,=\,0.5$ where $\text{CH}_{2}\text{O}$ is present), and linear interpolation of $c$ between two edges of the $\text{CH}_{2}\text{O}$ region. Shaded region represents uncertainty in $\overline{c}$ .

We use the instantaneous $c$ field to estimate the values of $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ needed for evaluation of $IV$ in the reacting cases. This is done by invoking a flamelet assumption across the instantaneous flame front. Dem et al. (Reference Dem, Stöhr, Arndt, Steinberg and Meier2014) performed Raman scattering measurements to determine the major species and temperature in the combustor used in the present study at conditions closely matching case 4 (the most turbulent case). They found that the thermo-chemical states – that is, the relationship between major species and temperature – within the inner shear layer corresponded closely to those of a laminar flame. For example, the measured instantaneous temperature at any given $\text{H}_{2}\text{O}$ mole fraction differed by less than 10 % from what would be expected in a laminar flame.

Hence, we assume that the instantaneous thermo-chemical state is a function of $c$ that can be determined from the corresponding laminar flame. The laminar flame profiles were determined using the same Cantera simulations used to determine $s_{L}^{0}$ and $\unicode[STIX]{x1D6FF}_{L}^{0}$ . Instantaneous $c$ fields were determined as described above, from which corresponding $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ fields were determined. Since the PLIF data are taken with a higher resolution than the TPIV, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ were averaged in each interrogation box to determine the value corresponding to each vector. To test the impact of this calculation on the final results, the enstrophy transport budgets for case 4 were performed again, but using the second method described above for determining the instantaneous $c$ field (figure 4 c); the choice of $c$ field had negligible impact on the enstrophy transport budgets.

4.1 Analysis of $\overline{I}$ , $\overline{II}$ and $\left(\overline{III}+\overline{IV}\right)_{L}$

In this section, we consider the distributions of $\overline{I}$ , $\overline{II}$ and $\left(\overline{III}+\overline{IV}\right)_{L}$ in the reacting cases. We also compare the reacting $\left(\overline{III}+\overline{IV}\right)_{L}$ to $\overline{IV}_{L}$ in the non-reacting cases.

Figure 6 shows the $\overline{I}$ , $-\overline{II}$ and $(\overline{III}+\overline{IV})_{L}$ fields from case 2; these are overlaid by contours of $\bar{c}$ indicating the position of the flame brush. Also shown are profiles of these terms along the $x$ -direction at several downstream locations $y$ . As expected, the production and dilatation terms are strictly positive and negative, respectively. In contrast, $(\overline{III}+\overline{IV})_{L}$ is negative towards the reactants, but obtains slightly positive values towards the products. A preliminary comparison of $\overline{IV}$ in figure 3(c) to $(\overline{III}+\overline{IV})_{L}$ shows that the maximum magnitude of the latter occurs further towards the reactants and the magnitude drops more steeply towards the products. These observations indicate that the baroclinic torque acts as a source of enstrophy towards the product side of the flame brush.

It is common practice in DNS of flames in HIT to plot budgets versus the mean progress variable, as opposed to spatial coordinates. However, as seen in figure 6, the budgets are not necessarily well-described by the position in the $\overline{c}$ field in this inhomogeneous system. For reference, figure 7 shows profiles of $\overline{I}$ , $-\overline{II}$ , $(\overline{III}+\overline{IV})_{L}$ at $y=3$  mm for case 2, all normalized by the maximum value of $\overline{I}$ , versus both physical space and mean progress variable. The trends are qualitatively similar, but the profiles are skewed towards the products in $\overline{c}$ space compared to physical space. The analysis below will be presented in physical space; the mean transport budgets also are presented as function of $\overline{c}$ for reference in appendix A.

Figure 6.

Reynolds-averaged enstrophy transport terms: $\overline{I}$ , $-\overline{II}$ and $(\overline{III}+\overline{IV})_{L}$ for case 2 on the central plane of the T-PIV volume ( $z=0~\text{mm}$ ). Contour lines of the $\overline{c}$ field are overlaid with enstrophy budget plots. The region containing the outer recirculation zone is masked because this area was not seeded for the T-PIV measurements. Radial $x$ profiles at various downstream locations $y$ for each term are shown to the right. Reference $x_{0}$ is the axial coordinate of maximum vortex stretching $\overline{I}_{max}$ at each $y$ .

The flame is stabilized in the inner shear layer between the inflowing reactants and the inner recirculation zone, resulting in a spatial overlap of the flame brush with the regions of high enstrophy production due to vortex stretching. The maximum enstrophy production due to vortex stretching occurs 3 mm above the exit plane and decreases further downstream due to weakening of the shear layer.

The mean dilatation acts to dissipate enstrophy across the flame ( $-\overline{II}<0$ ) and is therefore confined to the regions within the flame brush. A fluid element passing through the flame undergoes a change in volume by a factor of approximately five. Thus, the length scale of any vortical structures passing through the flame is correspondingly increased, attenuating vorticity. Unlike $\overline{I}$ , dilatation remains similar in magnitude – within measurement uncertainty – across a range of axial locations due to the fixed expansion across the flame.

The residual term $(\overline{III}+\overline{IV})_{L}$ , represents the cumulative effect of baroclinic torque and viscous dissipation/diffusion on the enstrophy transport. Note that, while dissipation is strictly negative, the diffusion may be of either sign. Holzner et al. (Reference Holzner, Liberzon, Nikitin, Luthi, Kinzelbach and Tsinober2008) and Watanabe et al. (Reference Watanabe, Sakai, Nagata, Ito and Hayase2014) observed that viscous diffusion of enstrophy had a positive contribution to enstrophy transport across the turbulent/non-turbulent boundary, while dissipation remained strictly negative. However, the relative magnitude of dissipation is usually greater than that of diffusion; $\overline{IV}$ was strictly negative in all cases studied here.

Figure 7.

Normalized enstrophy budget terms for case 2 at $y=3~\text{mm}$ . (a) Versus radial position $x-x_{0}$ . (b) Versus mean progress variable $\bar{c}$ . The error bars represent the convergence uncertainty in the mean.

Figure 8.

Comparison of normalized $(\overline{III}+\overline{IV})_{L}$ (solid red) in a reacting flow to normalized $\overline{IV}_{L}$ (dashed red) in a non-reacting flow in (a) case 1, (b) case 2, (c) case 3, (d) case 4. Vortex stretching in a reacting flow (solid black) is compared to vortex stretching in non-reacting flow (dashed black). The non-reacting flow profile was limited by the field of view.

As a first assessment of the impact of baroclinic torque, figure 8 compares the profiles of $(\overline{III}+\overline{IV})_{L}$ in the reacting cases to $\overline{IV}_{L}$ in the non-reacting cases. It would be expected that the reacting cases would exhibit greater dissipation in the products (i.e. at higher $x$ ) due to the increase in temperature and viscosity. However, $(\overline{III}+\overline{IV})_{L}$ in the reacting cases was greater than $\overline{IV}_{L}$ in the non-reacting cases. This is indicative of a positive mean baroclinic torque that is a source of enstrophy.

4.2 Baroclinic torque

We now aim to isolate the effects of baroclinic torque from dissipation, i.e. estimate $\overline{III}_{L}$ based on the measured data. The approach described in § 3.2 was used to approximate the magnitude of $\overline{IV}$ and its uncertainty.

Figure 9.

Reynolds-averaged $\overline{IV}$ and $\overline{III}_{L}$ for case 2 on the central plane of the T-PIV volume ( $z=0~\text{mm}$ ). Contour lines of the $\overline{c}$ field are overlaid with enstrophy budget plots. Corresponding radial $x$ profiles at various downstream locations $y$ are shown to the right.

Figure 9 shows the $\overline{III}_{L}$ and $\overline{IV}$ fields on the central plane of TPIV volume in case 2; radial profiles along select $y$ locations also are shown. Viscous dissipation is a sink of enstrophy ( $\overline{IV}<0$ ) within the shear layer. The maximum value of $\overline{IV}$ decreases with $y$ in a similar manner to $\overline{I}$ shown in figure 6. This trend was also observed at the higher turbulence intensities. Such a balance was also found to naturally hold for non-reacting HIT and HIT at high turbulence intensities by Bobbitt et al. (Reference Bobbitt, Lapointe and Blanquart2016).

The baroclinic torque $\overline{III}_{L}$ obtains small negative values on the reactant side of the flame brush and considerably larger positive values on the product side. This qualitatively agrees with the attenuation of reactant side vorticity and generation of product side vorticity seen in laminar flame/vortex interactions (Louch & Bray Reference Louch and Bray1998; Mueller et al. Reference Mueller, Driscoll, Reuss, Drake and Rosalik1998). However, in these interactions, the vorticity attenuation generally was greater than or comparable to the vorticity production. Here, we observe relatively weak attenuation and much stronger generation.

The sign of baroclinicity may be interpreted in terms of the alignment between density and pressure gradients. Density gradients are perpendicular to the local flame front, whereas the pressure gradients are determined by the simultaneous action of the turbulent vortices, combustor scale swirling flow and flame. The data demonstrate that the orientation and magnitude of the pressure field associated with the swirling flow results in significant baroclinic enstrophy production relative to what would occur in a non-swirling flow. Note that this indicates that the significance of baroclinic enstrophy production does not have a universal relationship with the turbulence and flame properties, but depends on the pressure field associated with the particular configuration of interest.

Figure 10.

Normalized radial profiles of the decoupled enstrophy transport terms in (a) case 1, (b) case 2, (c) case 3, (d) case 4. All radial profiles averaged over downstream locations $y$ .

The transition from $\overline{III}_{L}<0$ to $\overline{III}_{L}>0$ occurs around the location of maximum vortex stretching (i.e. $x=x_{0}$ ) at all downstream positions. The decreasing strength of the radial pressure gradient with axial distance results in a corresponding decrease in the peak magnitude of the baroclinic torque. Nevertheless, the significance of enstrophy production by baroclinic torque remains comparable to that of vortex stretching and dissipation, both of which also decrease with axial distance.

The radial profiles of $\langle \overline{III}_{L}\rangle$ and $\langle \overline{IV}\rangle$ (averaged over axial locations) are shown in figure 10 for the different cases. The uncertainties in $\overline{I}$ and $-\overline{II}$ are due to statistical convergence. The asymmetric uncertainty bars in $\langle \overline{III}_{L}\rangle$ and $\langle \overline{IV}\rangle$ combine the systematic under-prediction of $\overline{IV}$ and the convergence uncertainty. The systematic bias results in an under-estimation of the magnitude of $\langle \overline{IV}\rangle$ and a corresponding under-estimation of $\langle \overline{III}_{L}\rangle$ .

Also shown in figure 10 are normalized radial profiles of the advection of enstrophy by the mean flow ( $\overline{L}_{1}$ ) and the mean turbulent transport ( $\overline{L}_{2}$ ), which together constitute the Lagrangian derivative, viz.

(4.1) $$\begin{eqnarray}\overline{L}=\underbrace{\overline{u}_{i}\frac{\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D6FA}}}{\unicode[STIX]{x2202}x_{j}}}_{\overline{L}_{1}}+\underbrace{\overline{u_{i}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}^{\prime }}{\unicode[STIX]{x2202}x_{j}}}}_{\overline{L}_{2}}.\end{eqnarray}$$

Here, $u_{i}^{\prime }$ and $\unicode[STIX]{x1D6FA}^{\prime }$ represent the fluctuating components of velocity and enstrophy, respectively. Figure 10 shows that $\overline{L}_{2}$ is the main contributor to $\overline{L}$ inside the shear layer, whereas $\overline{L}_{1}$ is the main contributor inside the central recirculation zone. However, the relative magnitude of $\overline{L}_{1}$ and $\overline{L}_{2}$ to terms $\overline{I}$ through $\overline{IV}$ is small.

For all test cases, the baroclinic torque attenuates enstrophy at the earlier stages of combustion, but results in significant enstrophy production on the product side of the flame brush. Indeed, the baroclinic torque is the largest source of enstrophy at positions farthest towards the products, viz.  $x-x_{0}\gtrsim 2~\text{mm}$ . Moreover, the baroclinic torque remains significant relative to the other terms in the enstrophy transport equation across cases; there is little difference in the ratio of baroclinic torque to vortex stretching across cases once the uncertainty is considered. This observation also is true for the budgets as a function of $\bar{c}$ shown in appendix A.