3.1. An exothermic front in a quiescent fluid $(Ra_T < Ra_{T,c},Ra_s=0,\eta >0)$

Our first investigation provides insight into the fluid and front dynamics resulting from the propagation of an exothermic front in the absence of additional complexities such as solutal coupling or an externally driven flow field. This provides an important baseline understanding that will be useful in our subsequent studies when more complex couplings are present. We consider an exothermic front, $\eta >0$, propagating in a two-dimensional horizontal layer of an initially quiescent fluid, $Ra_T < Ra_{T,c}$, without solutal effects $Ra_s = 0$. The geometry used is shown in figure 1(a). Specifically, we use $Ra_T = 1000$ while varying $\eta$ over the range $1 \le \eta \le 100$. For these conditions, all fluid motion is caused by the heat release from the reaction. This generates a region of front-induced fluid motion that travels with the front and, after the front has passed, the fluid will return to a motionless state.

Figure 2 shows fronts for several values of $\eta$ where $\eta$ increases from (a–d), respectively. The fronts are travelling from left to right and the colour contours are of $c(x,z,t)$ where blue is pure reactants, red is pure products and the yellow–green region is the reaction zone. The finite width of the reaction zone is clearly evident. Fluid velocity vectors are shown by the black arrows, which illustrate the complex fluid motion generated in the region near the front. Figure 2(a–c) shows a close-up view of a portion of the domain, $2 \leq x\leq 10$, to highlight the front details. Figure 2(d) is for $\eta = 50$, which results in a much faster front and the portion shown is $8 \leq x \leq 16$. All fronts are shown at $t=6$ where the front initiation occurs at $t=0$. As a result, insight into the front velocity can be gained by the relative position of the fronts in figure 2(a–c), which indicates an increasing front speed with increasing $\eta$, as expected.

Figure 2. Exothermic fronts without solutal coupling propagating through a quiescent fluid ($Ra_T = 1000$, $Ra_s = 0$, $\eta > 0$). Colour contours of $c(x,z,t)$, where red is pure products and blue is pure reactants. Arrows are fluid velocity vectors; (a) $\eta = 1$, (b) $\eta = 10$, (c) $\eta = 20$, (d) $\eta = 50$. All images are at time $t = 6$ where the fronts were initiated at $t=0$. A zoomed-in view is shown to illustrate the front features: (a–c) show $2 \leq x \leq 10$ and (d) shows $8 \leq x \leq 16$. The maximum values of fluid velocity magnitudes present in (a–d) are $0.13$, $1.08$, $2.09$ and $3.85$.

For small and intermediate values of the heat release parameter, $\eta \lesssim 20$, which corresponds to figure 2(a–c), the front generates a pair of counter-rotating convection rolls that travel with the front and encompass the entire reaction zone (the region spanned in the $x$-direction by the yellow/green contours). Between the two convection rolls is an upflow due to the heat release of the reaction, which reduces the fluid density due to increased temperature. Fluid heated by the reaction within the reaction zone rises buoyantly toward the top wall. The fluid cools as it interacts with the cold top wall, which increases its density eventually causing it to fall. As $\eta$ increases, the magnitude of the fluid velocity, front velocity and mixing length increase. This causes the clockwise rotating convection roll (the right roll of the pair) to extend laterally toward larger values of $x$ as the front extends and deforms. As the magnitude of the upflow between the pair of convection rolls increases, reactants are advected vertically, creating a cusp-like feature in the front. The cusp is particularly striking in figure 2(c–d). The upflow draws in reactants which can yield isolated regions of reactants that become surrounded by products. These islands of reactants are rapidly consumed by the reaction. This event repeats periodically and results in a time dependence for the cusp region (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2022.375). We point out that this flow field is quite different than what occurs when only solutal coupling is included, which results in a single large convection roll that travels with the front (for example, see figure 2 of Mukherjee & Paul Reference Mukherjee and Paul2020).

Overall, the complexity of the front increases with increasing values of $\eta$. In figure 2(d) the cusp structure extends upward nearly to the top wall and the leading edge of the front is stretched considerably toward the right. The right convection roll of the roll pair created by the exothermic reaction no longer extends far to the right. A second pair of convection rolls has been formed on its right side, which results in the formation of a second smaller cusp feature. The leading edge of the front extends over the top regions of these convection rolls to yield a very stretched structure.

The spatio-temporal dynamics of the front for $\eta = 50$ is further illustrated by the space–time plot of figure 3(a), where we show colour contours of $c(x,z=1/2,t)$ as a function of $x$ and $t$. Red regions are pure products and blue regions are pure reactants. We have found that the mid-plane slice is insightful since it captures features that extend into the bulk of the fluid layer. Other choices of $z$ could be used to emphasize different dynamical features of the fronts if desired. The rounded peak features (red) are the first to complete the reaction. These peaked regions coincide with the initial formation of the cusp structure and the bottoms of the sharp trough regions (blue) coincide with the fully formed cusp. The horizontal dashed line at $t =6$ coincides with the front shown in figure 2(d) where the cusp structure is nearly fully formed.

Figure 3. The spatio-temporal dynamics of an exothermic front without solutal coupling propagating through an initially motionless fluid ($Ra_T = 1000$, $Ra_s = 0$, $\eta = 50$). (a) Space–time plot of $c(x,z=1/2,t)$, where red is products and blue is reactants. (b) The front velocity (solid line) as a function of time using the bulk-burning rate approach, see (3.1). The dashed line is the velocity of the leading edge of the front computed using a front-tracking approach.

The variation for the front velocity with time is shown in figure 3(b). The solid line is $v_f(t)$ when computed using the bulk-burning rate approach (Constantin et al. Reference Constantin, Kiselev, Oberman and Ryzhik2000) where

(3.1)\begin{equation} v_f(t) = \int_0^{1} {{\rm d}}z \int_0^{\varGamma} \frac{{{\rm d}}c}{{{\rm d}}t} {{\rm d}\kern0.7pt x}. \end{equation}

The bulk-burning rate is advantageous because it remains valid for complicated fronts, such as these, where front-tracking approaches can be difficult to implement and compute (Mukherjee & Paul Reference Mukherjee and Paul2019, Reference Mukherjee and Paul2020). An important feature of the bulk-burning rate approach is that it accounts for the front dynamics over the entire reaction zone, and not only a particular region that is tracked. For this front, the bulk-burning rate approach includes the region containing the dynamics of the cusp. It is insightful to compare $v_f(t)$ with the velocity of the leading edge of the front (the portion of the front furthest to the right) using a simple front-tracking approach. The velocity of the leading edge is shown by the dashed line, which reaches a steady value. This is in contrast to the time periodic front velocity shown by the solid line. The time average of $v_f(t)$ equals that of the leading edge for $t \gtrsim 4$. We will find the bulk-burning rate approach to determining the front velocity useful as we quantify more complex fronts.

The variation of the temperature field can be used to gain more insight into the dynamics of an exothermic front. The variation of $T(x,z,t)$ is shown in figure 4 for different values of $\eta$, where red is hot fluid and blue is cold. Fluid velocity vectors are shown using arrows and the front location is indicated by including the $c=1/2$ contour as a solid black line. Away from the reaction zone, where there is no fluid motion, the conduction temperature profile is evident by the linear transition from red to blue. For small values of heat release, such as figure 4(a), where $\eta = 1$, a small deviation from the linear temperature profile is evident at the upflow region in between the counter-rotating convection rolls. The front has also developed some curvature. For larger values of $\eta$, the deviation from the linear profile increases significantly and the shape of the front becomes more complex. As the front extends laterally for larger values of $\eta$, the region of heated fluid also increases as expected.

Figure 4. The variation of the fluid temperature for an exothermic front propagating though an initially motionless fluid, where (a) $\eta = 1$, (b) $\eta = 10$, (c) $\eta = 20$, (d) $\eta = 50$. Colour contours of $T(x,z,t)$, where red is hot fluid and blue is cold fluid. Arrows indicate fluid velocity vectors and the solid line is the isocontour at $c=1/2$ indicating the location of the front. The concentration fields for these fronts are shown in figure 2. All images are at $t = 6$. For (a–c) the maximum temperature is $T = 1$ and for (d) it is $T \gtrsim 1$ due to the strong exothermic reaction. Remaining parameters: $Ra_T =1000$, $Ra_s=0$.

The maximum temperature in the fluid layer for $\eta \lesssim 20$ is at the hot bottom wall. For larger values of $\eta$ the maximum temperature is away from the wall and is in the region where the front forms a cusp. For an exothermic reaction coupled to the flow field using the FKPP nonlinearity, as shown in (2.2), the production term $\eta c(1-c)$ is at the maximum at $c=1/2$. The front forms a fold in the region near the cusp, which results in a significant spatial region where $c \approx 1/2$. Therefore, in the cusp region there is significant heat release. Moreover, this is also the region with the strongest upflow, which draws in fresh reactants. Both of these effects combine to form a hotspot in the cusp region of the fluid. Once the reaction is completed in the cusp region, the front continues its advance to form another cusp and hot spot and the dynamical process repeats.

We next quantify the characteristic fluid velocity $U$, the front velocity $v_f$ and the mixing length $L_s$ as a function of the heat release $\eta$. For $U$ we use the maximum value of the fluid velocity in the domain. Here, $v_f$ is computed using (3.1) and $L_s$ is a measure of the width of the reaction zone and it is defined as the spatial extent in the $x$-direction over which the depth-averaged value of the concentration field is in the range $0.01 \leq \langle c(x,t) \rangle \leq 0.99$, where the angle brackets indicate the depth average (Rongy et al. Reference Rongy, Goyal, Meiburg and De Wit2007; Rongy & De Wit Reference Rongy and De Wit2009). All three of these measures are functions of time and they grow until reaching steady conditions for $t \gtrsim 4$. For larger values of the heat release, $\eta \gtrsim 10$, all of the measures display oscillations of magnitude ${O}(10^{-1})$ around a mean value that represents its time-averaged value. The mean values of the characteristic velocity, front velocity and mixing length will be denoted as $\bar {U}$, $\bar {v}_f$ and $\bar {L}_s$.

The variations of $\bar {U}$, $\bar {v}_f$ and $\bar {L}_s$ with $\eta$ are shown in figure 5, which reveals that all of these quantities monotonically increase with the amount of heat release. The trends in the data further suggest the presence of different regimes based upon the value of $\eta$. In light of this, we have divided the results into small, intermediate and large values of $\eta$. We use the following conventions in the figures: small $\eta$ is $0 \le \eta \le 1$ and uses circles (blue); intermediate $\eta$ is $1 < \eta \le 20$ and uses diamonds (green); and large values of $\eta$ is $\eta > 20$ and uses squares (red). We find that small and large values of $\eta$ are described well using scaling relationships and the intermediate values of $\eta$ appear to represent a transition region.

Figure 5. The variation of the asymptotic fluid velocity $\bar {U}$, front velocity $\bar {v}_f$ and mixing length $\bar {L}_s$ with $\eta$ for an exothermic front in a quiescent fluid $(Ra_T = 1000,\ Ra_s = 0, \eta > 0)$. Small, intermediate and large values of $\eta$ are represented using circles (blue), diamonds (green) and squares (red), respectively. (a) The variation of $\bar {U}$ with $\eta$. The solid line is $\bar {U} = 0.127 \eta$ and the dashed line is $\bar {U} = 0.465 \eta ^{1/2}$. (b) The variation of the front velocity with $\eta$, where $v_0=0.6$ is the analytical front velocity for $\eta =0$. The solid line is $( \bar {v}_f - v_0 )/v_0 = 0.151 \eta ^{3/2}$ and the dashed line is $(\bar {v}_f - v_0 )/v_0 = 0.503 \eta ^{1/2}$. (c) The variation of the mixing length with $\eta$, where $L_0=0.6$ is the analytical mixing length for $\eta =0$. The solid line is ${(\bar {L}_s - L_0)}/{L_0} = 0.49 \eta$ and the dashed line is ${(\bar {L}_s - L_0)}/{L_0} = 1.127 \eta ^{1/2}$.

Figure 5(a) shows the variation of $\bar {U}$ with $\eta$. For small $\eta$, $\bar {U}$ varies linearly with $\eta$, as indicated by the solid line, which is a curve fit of the form $\bar {U} = 0.127 \eta$. For large $\eta$, $U$ follows a square root scaling as indicated by the dashed line representing $\bar {U} = 0.465 \eta ^{1/2}$. For intermediate values of $\eta$ the trend suggests a transition between these two regimes of small and large heat release. A similar trend is observed for propagating fronts with only solutal feedback ($Ra_s \neq 0$, $\eta =0$, $Ra_T < Ra_{T,c}$), where $\bar {U}$ varies linearly for small values of $Ra_s$ and transitions to square root scaling for large values of $Ra_s$ although the flow structure for purely solutal feedback is a single convection roll that travels with the front (Mukherjee & Paul Reference Mukherjee and Paul2020).

The variation of $\bar {v}_f$ with $\eta$ is shown in figure 5(b). The front velocity is normalized by $v_0= 2 \sqrt {Le \xi }$, which represents the analytical result for the front velocity of a pure reaction diffusion system with FKPP kinetics (Kolmogorov et al. Reference Kolmogorov, Petrovskii and Piskunov1937). For our parameters this yields $v_0 = 0.6$, which represents a useful baseline value for comparison. For small $\eta$, the front velocity scales as $\eta ^{3/2}$ where the solid line is the curve fit $( \bar {v}_f - v_0 )/v_0 = 0.151 \eta ^{3/2}$. For large $\eta$, the front velocity transitions to a $\eta ^{1/2}$ scaling where the dashed line is the curve fit $( \bar {v}_f - v_0 )/v_0 = 0.503 \eta ^{1/2}$. The data in the transition regime are shown using the green diamonds.

Figure 5(c) shows the variation of $\bar {L}_s$ with $\eta$. The mixing length is normalized by the mixing length of a pure reaction diffusion front where $L_0 =0.6$. The mixing length follows the same scaling trends as the characteristic fluid velocity. For small $\eta$, the mixing length increases linearly with increasing $\eta$ where the solid line is the curve fit ${(\bar {L}_s - L_0)}/{L_0} = 0.49 \eta$. For large values of $\eta$, the square root dependence is shown by the dashed line which is the curve fit ${(\bar {L}_s - L_0)}/{L_0} = 1.127 \eta ^{1/2}$. The green diamonds again indicate a transition region between the scaling regimes found for small and large $\eta$.

3.2. An exothermic front with solutal feedback in a quiescent fluid $(Ra_T < Ra_{T,c},\ Ra_s\ne 0,\ \eta >0)$

We now consider the propagation of an exothermic front where the products and reactants have different densities in an initially motionless layer of fluid. We set $Ra_T = 1000$ and $\eta = 50$ and explore how the front dynamics varies with $Ra_s$. For these conditions, the presence of cooperative or antagonistic feedback will depend upon the sign of $Ra_s$. When $Ra_s > 0$, the products are less dense than the reactants, and this yields cooperative feedback since both the heat release and the density change due to the reaction are enhancing buoyant convection. When $Ra_s < 0$, the products are more dense than the reactants, and this yields antagonistic feedback because the solutal changes reduce buoyant convection while the exothermic reaction is enhancing buoyant convection. We explore a cooperative case where $Ra_s = 300$, and an antagonistic case where $Ra_s = -300$.

Figure 6 shows the fluid and front dynamics for (a) cooperative feedback and (b) antagonistic feedback using the plotting conventions of figure 2. In the cooperative case, the combined result of the solutal and thermal coupling yields a significantly extended leading edge of the front. The velocity of the leading edge of the front is steady in time. The extended leading edge of the front in figure 6(a) is coincident with a large clockwise convection roll that has been induced by the front. For $Ra_s > 0$, solutal coupling alone will generate a large clockwise convection roll that propagates with an extended front (cf. Rongy et al. Reference Rongy, Goyal, Meiburg and De Wit2007; Mukherjee & Paul Reference Mukherjee and Paul2020). Also, for $\eta > 0$, the heat release from the reaction generates a convection roll with clockwise rotation that travels with the leading edge of the front, as shown in figure 2(d). Therefore, in the case we are exploring here with $Ra_s > 0$ and $\eta > 0$, both of these effects contribute to the large clockwise convection roll that propagates with the leading edge on the right side of figure 6(a).

Figure 6. Exothermic fronts with solutal feedback propagating through an initially motionless layer of fluid. Colour contours of $c(x,z,t)$, where blue is pure reactants and red is pure products. Arrows are fluid velocity vectors. (a) Cooperative feedback: $Ra_T = 1000$, $Ra_s = 300$, $\eta = 50$. (b) Antagonistic feedback: $Ra_T = 1000$, $Ra_s = -300$, $\eta = 50$. The images are at $t = 5.97$ and a close-up view is shown for $8 \leq x \leq 17$. The characteristic fluid velocities are (a) $U=3.84$ and (b) $U=3.67$.

The extended leading edge of the front is followed, on its left side, by a dynamic cusp structure. The cusp structure can be identified as the region where the reactants extend further toward the top wall than in its immediate surroundings due to the upflow between two counter-rotating convection rolls. The dynamics of the cusp region is similar to what we found for an exothermic front without solutal feedback, for example see figure 2(d). As a result, we attribute this to the heat release from the reaction. The reactants inside the cusp are rapidly consumed and, as the front advances, these cusp structures are periodically formed. The cusps again contain a hotspot where the temperature of the fluid layer is at its maximum. It is interesting to note that Rongy & De Wit (Reference Rongy and De Wit2009) found that horizontally propagating cooperative fronts have a hotspot behind the front located at the bottom plate. In addition to exploring a different parameter regime than we do here for a cubic autocatalytic reaction kinetics, Rongy & De Wit (Reference Rongy and De Wit2009) also used thermally insulating boundary conditions at the top and bottom plates where the only source of heat was the exothermic reaction. This boundary condition is different from the problem we have studied where the top and bottom plates are perfect thermal conductors that are maintained at a constant temperature difference.

Figure 6(b) shows the fluid and front dynamics for antagonistic feedback. In this case, the spatial structure and dynamics of the leading edge of the front are significantly different. The leading edge of the front is again coincident with a convection roll of clockwise rotation. For solutal coupling alone with $Ra_s < 0$, the front generates a large extended surface that is coincident with a counter-clockwise convection roll (Rongy et al. Reference Rongy, Goyal, Meiburg and De Wit2007), which is contrary to the clockwise convection roll at the leading edge that is generated by the heat release. For the parameters we have used, $\eta = 50$ is large enough such that the resulting convection roll when both feedback mechanisms are present is of clockwise rotation even though $Ra_s < 0$ as shown in figure 6(b). Since the products of the exothermic reaction are more dense than the reactants, the leading edge extends only a short distance before falling down under the influence of gravity. During this motion, the leading edge traps fresh lighter reactants which are rapidly consumed. This is followed, to the left, by the pair of convection rolls generated by the exothermic reaction. The upflow between these rolls draws in fresh reactants to generate a plume structure containing reactants. The plume containing reactants is lighter than the surrounding products and extends nearly to the top wall before extending laterally. The isolated reactants are quickly consumed and the entire process repeats. This periodic rolling and tumbling motion of the leading edge has been observed experimentally (Nagypal, Bazsa & Epstein Reference Nagypal, Bazsa and Epstein1986; Rongy & De Wit Reference Rongy and De Wit2009; Rongy et al. Reference Rongy, Schuszter, Sinkó, Tóth, Horváth, Tóth and De Wit2009). Chemical oscillations and time varying convective dynamics have also been explored in detail for reactions with antagonistic coupling between solutal and surface tension contributions (Budroni, Rongy & De Wit Reference Budroni, Rongy and De Wit2012; Budroni, Upadhyay & Rongy Reference Budroni, Upadhyay and Rongy2019).

Figure 7 shows space–time plots of the fronts with (a) cooperative feedback and (b) antagonistic feedback, where red is pure products and blue is pure reactants. The reaction zone is located along the yellow–green region that extends from the upper left corner to the lower right corner. The slope of a line fitted along the reaction zone would yield an estimate of the inverse of the front velocity. It is evident from figure 7 that the front with cooperative feedback has a larger front velocity than the front with antagonistic feedback. The space–time plots also indicate clear differences in the front dynamics for these two cases, which we now discuss in more detail.

Figure 7. Space–time plots of fronts with (a) cooperative feedback and (b) antagonistic feedback through an initially motionless fluid. Colour contours of $c(x,z=1/2,t)$, where red is pure products and blue is pure reactants. (a) Cooperative feedback: $Ra_T = 1000$, $Ra_s = 300$, $\eta = 50$. (b) Antagonistic feedback: $Ra_T = 1000$, $Ra_s = -300$, $\eta = 50$. Front images at $t=5.97$ (dashed line) are shown in figure 6.

Figure 7(a) shows the space–time plot for the front with cooperative feedback. The extended leading edge of the front is steady in time and yields the linear shape on the right-hand side of the reaction zone. In the absence of solutal or thermal feedback, the front propagates with a constant velocity, which would yield a solid yellow–green line from the upper left to the lower right of the space–time plot, as shown in Mukherjee & Paul (Reference Mukherjee and Paul2020). However, the space–time plot of figure 7 shows repeating features in the reaction zone. These features are comprised of rounded peaks of products followed by troughs containing reactants. In a reference frame moving with the front, these repeating features represent the time-periodic dynamics that results from the cooperative coupling that is present. These repeating features are a result of the periodic dynamics of the cusp structure that is driven by the heat release from the reaction. A horizontal slice of the space–time plot would reveal the complex transition from products to reactants that occurs as a function of distance in the $x$ direction at that particular time. The dashed line at $t = 5.97$ corresponds to the front shown in figure 6(a). A vertical slice through the space–time plot would yield insight into the variation in time of the concentration at that particular location $x$. It is interesting to note that for certain locations $x$, this yields a brief temporal oscillation in the concentration. The peaks in the space–time plot correspond to the periodic formation of the cusp where the reaction is completed first. The time period of these peaks is $t\approx 0.8$. The periodic dynamics of the cooperative front is further illustrated in supplementary movie 2.

Figure 7(b) shows the space–time plot for the front with antagonistic feedback. The space–time plot indicates a sharper transition between the products and reactants. In addition, the right side of the reaction zone is not a linear feature as it was for cooperative feedback, it now forms an outline around the peaks containing products. This can be traced to the lack of an extended leading edge for fronts with antagonistic feedback. The dashed line at $t = 5.97$ corresponds to the front shown in figure 6(b). Vertical slices at carefully chosen locations on the $x$-axis would yield the presence of brief temporal oscillations of the concentration during the transition from reactants to products. Overall, the space–time plot for antagonistic feedback is quite sharp and jagged due to the absence of a steady extended leading edge. The leading edge of the front, in this case, consists of dense products that descend into the bulk of the fluid layer from the region near the top plate. This event repeats in time as the chemical reaction continues generating products that form the leading edge. The leading edge rises due to heat release until it cools due to the top wall, which contributes to its eventual descent. The peaks of the products coincide with the falling of the leading edge of the front to the point where it reaches the mid-plane and is therefore shown on the space–time plot. The peaks also indicate the locations where the reaction is completed first. The peaks have a time period of $t \approx 1.2$, which is larger than the period of the dynamics for cooperative feedback. The periodic dynamics for fronts with antagonistic feedback is driven predominantly by the periodic dynamics of the dense leading edge, which is driven by the competition between the solutal and thermal feedback. The periodic dynamics of the antagonistic front is further illustrated in supplementary movie 2.

The periodicity of the dynamics can be illustrated further by plotting the front velocity as a function of time as shown in figure 8(a). The front velocity for the cooperative case is larger as a result of the combined effect of the solutal and thermal coupling. For the antagonistic case, the denser products of the reaction result in slowing down the front. However, it is important to note that the front velocity in the absence of solutal and thermal feedback is $v_0=0.6$, which illustrates the significant front enhancement caused by either cooperative or antagonistic coupling when compared with this baseline case.

Figure 8. The time variation of the front velocity $v_f$ for exothermic fronts with solutal feedback in a quiescent fluid. (a) The variation $v_f$ with time $t$ for cooperative feedback (upper curve, red) and antagonistic feedback (lower curve, blue). (b) The same data plotted as a phase portrait for cooperative feedback (right, red) and antagonistic feedback (left, blue). The direction in which the phase portrait is traversed is indicated by the arrows. Cooperative feedback: $Ra_T = 1000$, $Ra_s = 300$, $\eta = 50$. Antagonistic feedback: $Ra_T = 1000$, $Ra_s = -300$, $\eta = 50$.

For cooperative feedback, $v_f(t)$ is periodic with a single peak with a time period of $t \approx 0.8$. The single peak is due to the dynamic cusp structure, caused by the heat release, which creates regions of isolated reactants that are rapidly consumed. For antagonistic feedback, this process is significantly enhanced by the periodic turnover of the leading edge, which yields a periodic signal for $v_f$ with two peaks. The turnover of the leading edge, in this case, significantly increases the amount of reactants in the isolated regions and results in the larger peak as the reactants are consumed. The smaller amplitude peak is due to the dynamics of the counter-rotating convection rolls and cusp structure and their role in the formation of the isolated regions of reactants. The time period of oscillation for antagonistic feedback is $t \approx 1.2$.

Figure 8(b) shows the phase portrait of the front velocity where we plot $d v_f/dt$ vs $v_f$. The curve on the right (red) is for cooperative feedback where the unimodal oscillations of $v_f(t)$ result in an oval shaped closed curve. The curve includes three orbits to illustrate the periodic dynamics. The time span used is $6.84 \le t \le 9.24$. The closed curve on the left (blue) is for antagonistic feedback. The bimodal oscillation of $v_f(t)$ results in the two-lobed shape of the phase portrait. The pinched region, between the lobes, is due to the dynamics of the dense leading edge descending into the bulk of the flow field. The phase portraits clearly reflect that the front with cooperative feedback is faster than the front with antagonistic feedback. In addition, the cooperative front undergoes smaller magnitude oscillations in both $d v_f/dt$ and $v_f$. This is because for the cooperative case, the leading edge is steady and the time varying dynamics is due to the cusp formed behind the front due to the heat release from the reaction. For antagonistic feedback, however, the motions of the leading edge of the front and the cusp region are periodic in time and both are evident in the phase portrait.

It is useful to compare the front velocities for several of the cases we have explored so far to quantify the front velocity enhancement due to the induced fluid motion from the solutal and thermal coupling. First, we note that the average front velocities for cooperative and antagonistic feedback are $\bar {v}_f=2.92$ and $\bar {v}_f=2.15$, respectively. In the absence of solutal feedback we found that the exothermic front had a velocity of $\bar {v}_f = 2.63$, as shown in figure 5(b), which is faster than the front with antagonistic feedback and slower than the front with cooperative feedback. This also suggests that for an exothermic front, cooperative feedback increases the front velocity whereas the antagonistic feedback reduces the front velocity.

3.3. An exothermic front with solutal feedback propagating through a cellular flow field $(Ra_T > Ra_{T,c},\ Ra_s\ne 0,\ \eta >0)$

We next study an exothermic front with solutal feedback that is propagating through a cellular flow field generated by Rayleigh–Bénard convection. Prior to initiating the front, we establish a chain of counter-rotating convection rolls over the entire fluid layer by numerically integrating (2.1)–(2.3) with the supercritical Rayleigh number $Ra_T = 3000$ and $Ra_s = \eta =0$. The cellular flow field is generated using hot sidewall boundary conditions such that $T(x=0)=T(x=\varGamma )=1$. This yields an upflow at the left and right sidewalls, which initiates the growth of convection rolls that eventually fill the fluid layer. For our domain with $\varGamma =30$, this procedure yielded $15$ pairs of time-independent counter-rotating convection rolls. The width of a single roll is approximately unity, as shown in figure 9. The characteristic fluid velocity of this field of convection rolls is $U = 11.29$. In determining $U$ we do not include the region near the hot sidewalls and use the spatial region $5 \leq x \leq 25$. After establishing the cellular flow field, we initiate the front at the left wall, which then propagates to the right. We use this cellular flow field to study a front with cooperative feedback ($Ra_s = 300$, $\eta =50$) and a front with antagonistic feedback ($Ra_s = -300$, $\eta =50$).

Figure 9. Counter-rotating convection rolls generated by Rayleigh–Bénard convection. This flow field is used to study fronts propagating though a cellular flow field. Colour contours are of the streamfunction, where red indicates counter-clockwise rotation and blue indicates clockwise rotation. The characteristic fluid velocity is $U = 11.29$. Simulation parameters: $\varGamma = 30$, $Ra_T=3000$.

Figure 10 shows fronts propagating through the cellular flow field where (a–c) are for a front with cooperative feedback and (d–f ) are for a front with antagonistic feedback. Colour contours of $c(x,z,t)$ are shown with arrows representing fluid velocity vectors. Figure 10(a–c) shows a front with cooperative feedback at three instances of time separated by approximately 0.5 time units, respectively. In figure 10(a), the cellular flow field is clearly evident in the velocity vectors of the pure reactants that are to the right of the front. The extended leading edge of the front is dynamic due to its interactions with the flow field. The leading edge is advected by the convection rolls during the front propagation. There are also smaller plumes composed of falling products that form behind the leading edge. These plumes result from the interactions of the heat release and the cellular flow to create disordered spatial structures. As the front advances, the cellular flow is temporarily annihilated by the clockwise rotating solutal roll coincident with the leading edge of the front. The cellular flow re-emerges behind the reaction zone after the front has passed. The re-established cellular flow is clearly evident at the far left of figure 10(c). There are also isolated patches of reactants (blue) that form between the plumes during the reaction.

Figure 10. Exothermic fronts propagating through a cellular flow field generated by Rayleigh–Bénard convection. Colour contours of $c(x,z,t)$, where red is pure products and blue is pure reactants. Arrows are fluid velocity vectors. A close-up view is shown for $10 \leq x \leq 20$ to emphasize the front. (a–c) Cooperative feedback: $Ra_T=3000$, $Ra_s= 300$, $\eta =50$. (d–f ) Antagonistic feedback: $Ra_T=3000$, $Ra_s= -300$, $\eta =50$. Panels (a–c) and (d–f ) are separated by approximately 0.5 time units to illustrate the dynamics.

Figure 10(d–f ) shows the front with antagonistic feedback travelling through the cellular flow field, where each panel is separated in time by approximately 0.5 time units. In this case, the leading edge of the front consists of heavier products, which descend while scooping reactants that become surrounded by products. The leading edge of the front interacts with the cellular flow in a complicated way resulting in a spatially disordered structure with intricate features. The spatial extent of the leading edge is smaller than what was found for cooperative feedback. The front annihilates the cellular flow as it propagates while creating secondary cellular flow structures in its immediate wake. The cellular flow field of Rayleigh–Bénard convection re-emerges after the front passes through which can be seen on the far left of the fluid layer in figure 10(f ).

Space–time plots of the fronts propagating through the cellular flow field are shown in figure 11 using the same conventions as figure 7. The solid (dashed) lines indicate the locations of the centre of the Rayleigh–Bénard convection rolls with clockwise (counter-clockwise) rotation before the front propagates through. The centre of a convection roll is identified as the $x$ location where $T(z=1/2) = 1/2$.

Figure 11. Space–time plots of fronts propagating through a cellular flow field. Colour contours are of $c(x,z=1/2,t)$, where red is products and blue is reactants. The locations of the centres of the Rayleigh–Bénard convection rolls are indicated by a solid (dashed) line for rolls with clockwise (counter-clockwise) rotation. (a) Cooperative feedback: $\eta = 50$, $Ra_s = 300$, and $Ra_T = 3000$. Fronts are shown in figure 10(a–c). (b) Antagonistic feedback: $\eta = 50$, $Ra_s = -300$, and $Ra_T = 3000$. Fronts are shown in figure 10(d–f ).

Figure 11(a) shows the space–time plot for a front with cooperative feedback propagating through a cellular flow field. The space–time plot is highly disordered with several interesting features. The islands of products along the reaction zone occur when the extended leading edge of the front crosses below the mid-plane. This crossing is due to the presence of a clockwise rotating convection roll, which advects the leading edge downwards even though the leading edge consists of lighter products. The leading edge is coincident with a clockwise rotating solutal roll as indicated in figure 10(a–c). This clockwise rotating roll becomes strengthened when it absorbs a clockwise rotating thermal convection roll. Similarly, it gets weakened when it absorbs a counter-clockwise rotating thermal convection roll. The islands of products represent the regions in the space–time plot where the reaction is first completed. These regions coincide with a clockwise rotating thermal convection roll pulling the leading edge downward to complete the reaction. This is evident from the space–time plot since these features predominantly occur on the right of the centres of the clockwise rotating rolls (solid lines).

The space–time plot for a front with antagonistic feedback that is propagating through the cellular flow field is shown in figure 11(b). In this case, the space–time plot is more disordered than what was found for cooperative feedback. The front with antagonistic feedback, in the absence of a cellular flow field, is coincident with a clockwise rotating convection roll at the leading edge as shown in figure 6(b). The clockwise thermal convection rolls strengthen this leading clockwise front-induced roll and helps to advance the front forward. This mechanism is similar to the dynamics found for a front with cooperative feedback in a cellular flow field. However, the leading edge in the antagonistic case is composed of denser products, which descends significantly trapping reactants in the process.

Although the dynamics shown in figure 11 contains features and structures that occur repeatedly in time, the dynamics is not periodic. This is evident from the space–time plots, where the disordered features such as fingers and isolated patches of products do not precisely repeat. It is apparent that the interactions of the front with the cellular flow field breaks the periodicity of the dynamics. This should be contrasted with fronts that propagate through a cellular flow field without heat release or solutal feedback which show a time-periodic dynamics (for example, see figure 12(d) in Mukherjee & Paul Reference Mukherjee and Paul2020). This indicates that the feedback between the reaction and the cellular flow field results in the aperiodic dynamics.

The variation of the front velocity with time for the cooperative and antagonistic fronts propagating though a cellular flow field is shown in figure 12(a). The upper curve (red) is $v_f(t)$ with cooperative feedback and the lower curve (blue) is $v_f(t)$ with antagonistic feedback. The front velocity for cooperative feedback is faster than that of antagonistic feedback on average. It is evident that neither of the front velocities repeat in time.

Figure 12. (a) The variation of the front velocity with time for fronts propagating through a cellular flow field with cooperative feedback (red, upper) or antagonistic feedback (blue, lower). (b) The phase portrait for cooperative feedback for $2 \le t \le 6$. (c) The phase portrait for antagonistic feedback for $2 \le t \le 8$. Arrows indicate the direction of the trajectory. Cooperative feedback: $Ra_s = 300$, $Ra_T = 3000$, $\eta = 50$. Antagonistic feedback: $Ra_s = -300$, $Ra_T = 3000$, $\eta = 50$.

The aperiodic nature of $v_f(t)$ is illustrated further by the phase portraits shown in figure 12(b,c). The direction in which the trajectory traverses the phase portrait is indicated by the arrows. The phase portrait for a front with cooperative feedback is shown in figure 12(b) and is quite disordered. The clockwise orbit of the trajectory reveals a noisy signature of the oval structure shown in figure 8(b) for a front with cooperative feedback propagating through an initially motionless fluid. For the front with antagonistic feedback, the trajectory also reveals a noisy signature of the orbit shown in figure 8(b). There is a two-lobed structure with a pinched region between the lobes due to the dynamics of the dense leading edge of the front. To explore if the dynamics ever settles to a periodic solution at long times, we performed a numerical simulation in a larger domain with an aspect ratio of $\varGamma = 60$. Even in this larger domain, the dynamics remained aperiodic for the duration of the front propagation. The interactions of the front with a cellular flow field leads to an aperiodic dynamics for both cooperative and antagonistic feedback. The intricate spatio-temporal dynamics of the cooperative and antagonistic fronts in the presence of background convection is further illustrated in supplementary movie 3.

In figure 13 we gather together the time-averaged diagnostics of the characteristic fluid velocity $\bar {U}$, front velocity $\bar {v}_f$ and the length of the reaction zone $\bar {L}_s$ for an exothermic front, $\eta =50$, that is propagating through a two-dimensional fluid layer. Figure 13 presents a comparison of the results for fronts propagating through an initially motionless fluid, $Ra_T = 1000$, with fronts propagating through a cellular flow field, $Ra_T = 3000$. Fronts with cooperative coupling, $Ra_s = 300$, are squares (red) and fronts with antagonistic coupling, $Ra_s=-300$, are circles (blue). The error bars on the symbols indicate the standard deviation of the time varying values about the mean.

Figure 13. A comparison of the time-averaged values of the (a) characteristic fluid velocity, (b) front velocity and (c) reaction zone length of an exothermic front with $\eta =50$ travelling through a two-dimensional layer of fluid that is initially motionless ($Ra_T = 1000$) or contains a cellular flow field ($Ra_T = 3000$). Squares (red) are for cooperative coupling with $Ra_s = 300$, circles (blue) are for antagonistic coupling with $Ra_s = -300$. The error bars represent the magnitude of the standard deviation of the time variation about the mean value.

Figure 13(a) illustrates the variation of the characteristic fluid velocity. When the fluid layer is initially motionless, all fluid motion is due to the front, and it is evident that the fluid velocity induced by the front is larger for the case of cooperative coupling. However, for the case of cellular flow, the value of $\bar {U}$ is not significantly affected by the presence of the front or the type of coupling. This indicates that the characteristic fluid velocity is dominated by the cellular flow field for these conditions. We note that the characteristic fluid velocity for the cellular flow, in the absence of the front, is $\bar {U} = 11.29$ as shown in figure 9.

The time-averaged values of the front velocities are plotted in figure 13(b). It is evident that the front with cooperative coupling is always faster than the front with antagonistic coupling. Furthermore, the fronts are significantly faster when propagating through the cellular flow field. The oscillations of $v_f(t)$ about the mean value $\bar {v}_f$ are larger for the case when the front propagates through the cellular flow field. This is due to the complex interactions between the front and the counter-rotating convection rolls of the cellular flow. Figure 13(c) shows the time average of the reaction zone length, or mixing length. The reaction zone length exhibits a similar trend to that of $\bar {v}_f$ where $\bar {L}_s$ is always larger for the front with cooperative feedback and $\bar {L}_s$ increases in the presence of a cellular flow field.

3.4. Fronts travelling through a chaotic three-dimensional layer of fluid

We now investigate an exothermic front that propagates through a three-dimensional convective flow field exhibiting spatio-temporal chaos generated by Rayleigh–Bénard convection. We use the cylindrical domain shown in figure 1(b) with an aspect ratio of $\varGamma =40$ that contains a fluid with Prandtl number $Pr = 1$. The thermal Rayleigh number is set to $Ra_T = 6000$, which yields the state of spiral defect chaos (Morris et al. Reference Morris, Bodenschatz, Cannell and Ahlers1993). Spiral defect chaos consists of a dynamic pattern of convection rolls that contains many defect structures including spirals, targets and dislocations. Previous numerical investigations have explored the chaotic nature of this flow field in quantitative detail (cf. Paul et al. Reference Paul, Einarsson, Fischer and Cross2007; Karimi & Paul Reference Karimi and Paul2012). The flow field is chaotic as indicated by a positive leading Lyapunov exponent. We first integrate (2.1)–(2.3) for $t \approx \varGamma ^{2}$ to allow for initial transients to decay and to establish the chaotic flow field. It is within this flow field that we initiate a front at the centre of the domain and then study its propagation outward toward the sidewall. The time-averaged value of the characteristic fluid velocity for these conditions is $\bar {U} = 28.63$, where $\bar {U}$ is computed at the mid-plane ($z = 1/2$) of the convection domain.

The scale of the computations for the three-dimensional study is significantly larger than for our investigation of fronts through a two-dimensional layer of fluid. The complexity of the front increases, and its thickness decreases, with decreasing values of the Lewis number. The necessity to resolve the small scale features of the front dominate the calculation for these conditions, and we found that it was prohibitive to conduct the three-dimensional study using $Le = 0.01$. We found that it was possible to proceed with $Le= 0.1$. For the exothermic reaction we use $\eta = 18$, which yields a maximum temperature of the fluid layer of approximately unity. For the solutal Rayleigh number we use $Ra_s = 6000$ for cooperative feedback and $Ra_s = -6000$ for antagonistic coupling. This larger value of $Ra_s$ is to ensure that the front-induced fluid velocity from the solutal coupling is comparable to the fluid velocity of the spiral defect chaos flow field. We initiate the front at the centre of the domain using a Gaussian initial condition for the radial variation of the initial concentration field.

Figure 14 illustrates fronts propagating through a chaotic flow field for the different conditions that we have quantified. In all panels, a mid-plane slice is shown at $z = 1/2$ with colour contours of the concentration where red is products and blue is reactants. The black solid lines are contours of the temperature at $T = 1/2$, which yields the location of the centre of the convection rolls. As a result, the black solid lines provide insight into the underlying pattern of convection rolls. The front is initiated at $t=0$ and all panels show the front at $t = 3.4$. We emphasize that all fronts were initiated from the exact same initial flow field.

Figure 14. The propagation of a front through a three-dimensional chaotic flow field. The front is initiated at the centre at $t=0$. Colour contours are shown of $c(x,y,z=1/2,t)$ at $t = 3.4$, where red is products and blue is reactants. Black lines are contours of $T = 1/2$, indicating the location of the centre of the convection rolls. (a) Without solutal coupling and without heat release: $Ra_s=\eta =0$. (b) Solutal coupling and without heat release: $Ra_s = 6000$, $\eta =0$. (c) Without solutal coupling and with heat release: $Ra_s = 0$, $\eta = 18$. (d) Cooperative feedback: $Ra_s = 6000$, $\eta = 18$. (e) Antagonistic feedback: $Ra_s = -6000$, $\eta = 18$. In all panels: $Ra_T = 6000$, $Le = 0.1$.

Figure 14(a) shows a front without solutal feedback and without heat release ($Ra_s= \eta = 0$). In this case without front-induced fluid motion, the chaotic fluid motion of Rayleigh–Beńard convection alters the front interface as it travels radially outward. The interactions with the chaotic flow field create small scale features, which wrinkle the front that effectively increases the surface area for the reaction to occur. A detailed study of fronts travelling through a chaotic flow field without solutal feedback and without heat release can be found in Mukherjee & Paul (Reference Mukherjee and Paul2019). Figure 14(a) provides a useful baseline case upon which to compare the fronts with more complex interactions that are shown in figure 14(b–d).

Figure 14(b) shows a front with solutal coupling and without heat release ($Ra_s = 6000$, $\eta = 0$). The solutal coupling yields a large solutally driven toroidal convection roll that encompasses the leading edge of the front and travels with the front. This results in an extended front interface, which provides a greater surface area for the reaction to occur. This is indicated by the prominent yellow/green ring around the front, which yields an increased front speed when compared with figure 14(a). The absence of black contour lines in the reaction zone indicate that the solutally driven convection roll has annihilated the Rayleigh–Bénard convection rolls that were present prior to the approach of the front. Behind the propagating front, the Rayleigh–Bénard convection rolls re-emerge in a target pattern of concentric convection rolls. This target pattern formed behind the front is unstable to spiral defect chaos, which is eventually reestablished for long times.

Figure 14(c) illustrates the front without solutal coupling and with heat release ($Ra_s = 0$, $\eta = 18$). In this case, the heat release from the reaction generates a pair of counter-rotating convection rolls that travel with the front. The heat release from the reaction, and the travelling pair of convection rolls, causes some distortion and annihilation of the Rayleigh–Bénard convection rolls that are present, which can be seen upon comparing with the black contour lines of figure 14(a). Behind the front, a chaotic fluid state is formed and there is no intermediate pattern of concentric rolls. The front velocity with heat release is slightly larger than the baseline front shown in figure 14(a). The reaction zone consists of fluid at a higher temperature than the surrounding fluid due to the heat release, which yields a jagged front interface. Overall, the front interface is more circular, on average, than the front of figure 14(a).

Figure 14(d) shows the front with cooperative feedback where $Ra_s=6000$ and $\eta =18$. It is useful to compare this result with the case of only solutal coupling shown in figure 14(b) and the case of only heat release shown in figure 14(c). It is clear that the solutally driven convection roll is present in figure 14(d) and that this roll annihilates the Rayleigh–Bénard convection rolls as the front propagates outward. Behind the front is a pattern of concentric rolls that eventually evolve to spiral defect chaos. Furthermore, the front velocity for figure 14(d) is larger than the front velocities for either figure 14(b,c). This indicates that the combined cooperative effect of the solutal coupling and the heat release contribute to an increased front velocity.

The front with antagonistic feedback is shown in figure 14(e) where $Ra_s=-6000$ and $\eta =18$. The front has an extended leading edge due to the solutal coupling. However, the products are more dense than the reactants, which decreases the length of the reaction zone when compared with the cooperative case shown in figure 14(d). The front is followed by concentric convection rolls, which are formed by the solutal convection roll as it travels with the front. The front velocity with antagonistic coupling is less than the front velocity for the cooperative case shown in figure 14(d) and it is larger than the front velocity when only heat release is included as shown in figure 14(c). This supports the conclusion, for the parameters we have explored, that any induced fluid motion by the front will increase the front velocity, with respect to the front velocity $v_0$ in the absence of induced fluid motion, where a cooperative contribution is more effective than an antagonistic contribution.

The temperature of the fluid layer at mid-depth is shown in figure 15 for the fronts shown in figure 14. In figure 15, red is hot rising fluid ($T=1$) and blue is cold descending fluid ($T=0$). Figure 15(a) shows the temperature field without any influence of the front where the spiral defect chaos state of the fluid is evident. Figure 15(b) shows the fluid temperature field for a front without heat release and with solutal coupling. The large green/yellow ring is due to the solutal convection roll that is annihilating the Rayleigh–Bénard convection rolls while travelling outward with the front. Since there is no heat release due to the reaction for this case, the temperature of the solutal roll at mid-depth is $T\approx 1/2$.

Figure 15. The fluid temperature of a chaotic flow field containing a propagating front. Red is hot rising fluid ($T = 1$) and blue is cold descending fluid ($T = 0$). (a) A front without solutal coupling and without heat release ($Ra_s = \eta = 0$). In this case, the temperature field is unaltered by the front and the state of spiral defect chaos is clearly evident. (b) A front with solutal coupling and without heat release ($Ra_s = 6000$, $\eta =0$). (c) A front without solutal coupling and with heat release ($Ra_s = 6000$, $\eta = 18$). (d) A front with cooperative feedback ($Ra_s = 6000$, $\eta = 18$). (e) A front with antagonistic feedback ($Ra_s = 6000$, $\eta = 18$). In all panels $Ra_T = 6000$ and $Le = 0.1$.

The temperature field for the front without solutal coupling and with heat release is shown in figure 15(c). The heat release from the reaction generates a ring of elevated temperatures that travel with the front. It is also clear that the heat release due to the reaction has altered the chaotic fluid motion in the reaction zone. The temperature field for the front with cooperative feedback is shown in figure 15(d). The large solutal convection roll travelling with the front is composed of warm fluid due to the heat release from the reaction. The temperature field for the front with antagonistic feedback is shown in figure 15(e). The solutal convection roll is composed of warm fluid due to the heat release from the reaction.

The flow field of a striped pattern with defects can be quantified effectively using the local wavenumber $q$, which can be computed using the approach of Egolf, Melnikov & Bodenschatz (Reference Egolf, Melnikov and Bodenschatz1998). We compute the local wavenumber using the temperature variation shown in figure 15. This yields a value of $q$ for every location in space at the mid-depth slice, which allows the quantification of how a pattern changes spatially. Figure 16 shows the spatial variation of $q$ where colour contours are for $1 \le q \le 4$ where blue is small wavenumber and red is large wavenumber.

Figure 16. The spatial variation of the local wavenumber $q$ of the fluid patterns. Colour contours are of $q$, where blue is small and red is large over the range $1 \leq q \leq 4$. (a) The spiral defect chaos flow field ($Ra_s = \eta = 0$). (b) A front with solutal coupling and without heat release ($Ra_s = 6000$, $\eta =0$). (c) A front without solutal coupling and with heat release ($Ra_s = 0$, $\eta = 18$). (d) A front with cooperative feedback ($Ra_s = 6000$, $\eta = 18$). (e) A front with antagonistic feedback ($Ra_s =- 6000$, $\eta = 18$). In all panels, $Ra_T = 6000$ and $Le = 0.1$.

Figure 16(a) shows $q$ when the front does not affect the flow field ($Ra_s = \eta =0$). Therefore, this image shows the wavenumber variation for a flow field of spiral defect chaos. The wavenumber varies significantly over space with regions of small and large wavenumber associated with different topological feature of the flow field. The spatial average of the local wavenumber of this flow field is $\langle {q} \rangle =2.36$.

Figure 16(b) shows $q$ for a front with solutal coupling and without heat release. The large solutal convection roll yields the ring of small wavenumber (blue) that travels with the front. The concentric convection rolls that form behind the front are at large wavenumber (red). The spatial average of the local wavenumber selected by the concentric rolls is $\langle q \rangle =3.93$, which is significantly larger than the average wavenumber of the spiral defect chaos state. The instability of this transient field of concentric rolls is already evident by the deviations from a perfect target pattern. This large value of the wavenumber lies outside of the stability balloon at these conditions (Bodenschatz et al. Reference Bodenschatz, Pesch and Ahlers2000). Figure 16(c) shows $q$ for a front without solutal coupling and with heat release. In this case, the spatial variation of $q$ is less significant and $\langle q \rangle =2.38$. The smaller effect of the heat release on the flow field pattern is evident by the ring of wavenumber values where $q \approx 2.5$ near the front. It would be interesting to explore the influence of the heat release on the fluid dynamics for larger values of the heat release parameter $\eta$. However, we have not explored this possibility further.

The variation of $\langle q \rangle$ for a front with cooperative and antagonistic feedback are shown in figure 16(d,e), respectively. It is clear that the solutally generated roll yields a ring of small wavenumber (blue) that travels with the front and annihilates Rayleigh–Bénard convection rolls as it progresses. Behind the front is a field of concentric convection rolls of high wavenumber (red).

Concentric rolls are formed behind the front only when solutal coupling is present for the parameter values we have explored as shown in figure 16(b,d,e). In these cases with solutal coupling, a clockwise rotating solutal convection roll (in the $r$–$z$ plane) is formed that travels with the front. The leading edge of the solutal roll is a downflow, which interacts directly with the chaotic flow field that is ahead of the front. For our parameters, the leading edge of the solutal roll annihilates the chaotic flow field. The trailing edge of the solutal roll is an upflow. This upflow initiates the formation of Rayleigh–Bénard convection rolls since $Ra_T > Ra_{T,c}$. The azimuthal symmetry of the upflow causes these convection rolls to emerge as concentric rolls. The concentric convection rolls are also unstable, which leads to the fluid returning to a state of spiral defect chaos. At the instant of time shown in figure 16, the concentric rolls exhibit signs of instability as they evolve back toward the state of spiral defect chaos. It is interesting to note that the wavenumber of the concentric rolls behind the front does not vary significantly for the three cases we explored with solutal coupling.

The concentric convection rolls are formed behind a moving upflow that is travelling with the front with velocity $v_f$. The average wavenumber of the concentric rolls is approximately 1.5 times larger than the wavenumber of the spiral defect chaos state. We highlight that the wavenumber selected by the concentric rolls behind the front $\langle q \rangle \approx 3.75$ is smaller than the wavenumber of maximum linear growth $\langle q \rangle \approx 5.04$ (Dominguez-Lerma, Ahlers & Cannell Reference Dominguez-Lerma, Ahlers and Cannell1984) and larger than the wavenumber selected by a target pattern $\langle q \rangle \approx 2.8$ (Buell & Catton Reference Buell and Catton1986). The wavenumber selection of this complex pattern forming front would be interesting to explore in more detail.