3.1 The flame temperature

The steady energy equation is

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}={\displaystyle \frac{1}{RePr}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\left(\tilde{\unicode[STIX]{x1D706}}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right)+\tilde{\dot{\unicode[STIX]{x1D714}}}, & \displaystyle\end{eqnarray}$$

and the reaction progress variable transport equation is

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}{\displaystyle \frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}\tilde{x}_{1}}}=\tilde{\dot{\unicode[STIX]{x1D714}}}+{\displaystyle \frac{1}{ReSc}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\left(\tilde{\unicode[STIX]{x1D70C}}\tilde{D}{\displaystyle \frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right). & \displaystyle\end{eqnarray}$$

Integrating (3.4) from $\tilde{x}_{1,-}$ to $\tilde{x}_{1,+}$ in the flame zone, one obtains

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}c_{b}-\tilde{\unicode[STIX]{x1D70C}}_{u}\tilde{u} _{1,u}c_{u}=\int _{\tilde{x}_{1,-}}^{\tilde{x}_{1,+}}\tilde{\dot{\unicode[STIX]{x1D714}}}\,\text{d}\tilde{x}_{1}. & \displaystyle\end{eqnarray}$$

For (3.3), after integration in the preheat zone (from $\tilde{x}_{1,-}$ to $\tilde{x}_{1,0}$ ) and in the reaction zone (from $\tilde{x}_{1,0}$ to $\tilde{x}_{1,+}$ ), we then have

(3.6) $$\begin{eqnarray}\displaystyle \left.\tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}\tilde{T}\right|_{\tilde{x}_{1,0}}-\tilde{\unicode[STIX]{x1D70C}}_{u}\tilde{u} _{1,u}\tilde{T}_{u}={\displaystyle \frac{\tilde{\unicode[STIX]{x1D706}}}{RePr}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right|_{\tilde{x}_{1,0}}, & & \displaystyle\end{eqnarray}$$

and

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}\left(\tilde{T}_{b}-\left.\tilde{T}\right|_{\tilde{x}_{1,0}}\right)=\int _{\tilde{x}_{1,0}}^{\tilde{x}_{1,+}}\tilde{\dot{\unicode[STIX]{x1D714}}}\,\text{d}\tilde{x}_{1}+{\displaystyle \frac{\tilde{\unicode[STIX]{x1D706}}}{RePr}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right|_{\tilde{x}_{1,+}}-{\displaystyle \frac{\tilde{\unicode[STIX]{x1D706}}}{RePr}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right|_{\tilde{x}_{1,0}}. & \displaystyle\end{eqnarray}$$

Combining (3.5), (3.6) and (3.7), together with the boundary conditions $\tilde{T}_{u}=0$ and $c_{u}=0$ , leads to

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}\tilde{T}_{b}={\displaystyle \frac{\tilde{\unicode[STIX]{x1D706}}{\tilde{T}_{b}}^{\prime }}{RePr}}+\tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}c_{b}, & \displaystyle\end{eqnarray}$$

where ${\tilde{T}_{b}}^{\prime }=\unicode[STIX]{x2202}\tilde{T}/\unicode[STIX]{x2202}\tilde{x}_{1}|_{\tilde{x}_{1,+}}$ .

3.2 The flow field behind the flame

For the axially symmetric counter-flow, the upstream velocity ahead of the flame can be expressed as

(3.9a,b ) $$\begin{eqnarray}\displaystyle \tilde{u} _{1}(\tilde{x}_{1})=-2\unicode[STIX]{x1D700}\left(\tilde{x}_{1}-\left(\tilde{x}_{1,-}+{\displaystyle \frac{1}{2\unicode[STIX]{x1D700}}}\right)\right),\quad \tilde{u} _{r}\left(\tilde{x}_{r}\right)=\unicode[STIX]{x1D700}\tilde{x}_{r}. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D700}$ is the strain rate non-dimensionalized by $S_{L}^{0}/L$ , and $\tilde{u} _{r}$ is the radial velocity of such an axisymmetric flow.

In the downstream region behind the flame, the continuity relation is

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}+{\displaystyle \frac{1}{\tilde{x}_{r}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}_{r}}}\left(\tilde{x}_{r}\tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{r}\right)=0. & \displaystyle\end{eqnarray}$$

Eteng, Ludford & Matalon (Reference Eteng, Ludford and Matalon1986) studied the vorticity induced by the flame interacting with an adiabatic wall. Differently, in the present analysis, because of the variable density $\unicode[STIX]{x1D70C}$ , the stream-function $\unicode[STIX]{x1D713}$ is defined as

(3.11a,b ) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}=-{\displaystyle \frac{1}{\tilde{x}_{r}}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}\tilde{x}_{r}}},\quad \tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{r}={\displaystyle \frac{1}{\tilde{x}_{r}}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}, & & \displaystyle\end{eqnarray}$$

and accordingly the vorticity is

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D71B}\equiv {\displaystyle \frac{\unicode[STIX]{x2202}\tilde{u} _{r}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}-{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{u} _{1}}{\unicode[STIX]{x2202}\tilde{x}_{r}}}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\left({\displaystyle \frac{1}{\tilde{\unicode[STIX]{x1D70C}}\tilde{x}_{r}}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right)+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}_{r}}}\left({\displaystyle \frac{1}{\tilde{\unicode[STIX]{x1D70C}}\tilde{x}_{r}}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}\tilde{x}_{r}}}\right). & \displaystyle\end{eqnarray}$$

The jump conditions across the flame read

(3.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\left(\tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}\right)_{u}=\left(\tilde{\unicode[STIX]{x1D70C}}\tilde{u} _{1}\right)_{b},\\[2.0pt] \left(\tilde{u} _{r}\right)_{u}=\left(\tilde{u} _{r}\right)_{b},\\[2.0pt] \left(\tilde{p}+\tilde{\unicode[STIX]{x1D70C}}{\tilde{u} _{1}}^{2}\right)_{u}=\left(\tilde{p}+\tilde{\unicode[STIX]{x1D70C}}{\tilde{u} _{1}}^{2}\right)_{b}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Following a similar idea (Eteng et al. Reference Eteng, Ludford and Matalon1986), combining the above jump conditions with the momentum equation in the $\tilde{x}_{r}$ direction on both sides of the flame, one is able to determine the pressure gradient along $\tilde{x}_{r}$ . Consequently, the flame-induced vorticity for the present non-adiabatic boundary case is given by

(3.14) $$\begin{eqnarray}\displaystyle \left.\unicode[STIX]{x1D71B}\right|_{\tilde{x}_{1,+}}=\left.{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{u} _{r}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right|_{\tilde{x}_{1,+}}=\unicode[STIX]{x1D700}^{2}{\displaystyle \frac{\tilde{\unicode[STIX]{x1D70C}}_{u}-\tilde{\unicode[STIX]{x1D70C}}_{b}}{\tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}}}\tilde{x}_{r}. & & \displaystyle\end{eqnarray}$$

Under the adiabatic wall condition, Eteng et al. (Reference Eteng, Ludford and Matalon1986) concluded that for the thin flame, to the leading order, $w/\tilde{x}_{r}$ remains constant along $\tilde{x}_{1}$ . For the present non-adiabatic case, such a relation also approximately holds. In other words,

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D71B}}{\tilde{x}_{r}}}=\unicode[STIX]{x1D700}^{2}{\displaystyle \frac{\tilde{\unicode[STIX]{x1D70C}}_{u}-\tilde{\unicode[STIX]{x1D70C}}_{b}}{\tilde{\unicode[STIX]{x1D70C}}_{b}\tilde{u} _{1,b}}},\quad \text{for}~\tilde{x}_{1}\geqslant \tilde{x}_{1,+}. & \displaystyle\end{eqnarray}$$

The stream-function is re-expressed as

(3.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}={\tilde{x}_{r}}^{2}\tilde{\unicode[STIX]{x1D70C}}\left(\tilde{x}_{1}\right)F\left(\tilde{x}_{1}\right),\quad \text{for}~\tilde{x}_{1}\geqslant \tilde{x}_{1,+}. & & \displaystyle\end{eqnarray}$$

Combining (3.12) and (3.15) with the isobaric condition $\tilde{\unicode[STIX]{x1D70C}}(1+\unicode[STIX]{x1D70F}\tilde{T})=1$ and (3.1), we then obtain

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle F^{\prime \prime }-{\displaystyle \frac{\unicode[STIX]{x1D70F}\tilde{T}^{\prime }}{1+\unicode[STIX]{x1D70F}\tilde{T}}}F^{\prime }+\left(\left({\displaystyle \frac{\unicode[STIX]{x1D70F}\tilde{T}^{\prime }}{1+\unicode[STIX]{x1D70F}\tilde{T}}}\right)^{2}-{\displaystyle \frac{\unicode[STIX]{x1D70F}\tilde{T}^{\prime \prime }}{1+\unicode[STIX]{x1D70F}\tilde{T}}}\right)F=\unicode[STIX]{x1D700}^{2}{\displaystyle \frac{\unicode[STIX]{x1D70F}\tilde{T}_{b}}{\left(1+\unicode[STIX]{x1D70F}\tilde{T}_{b}\right)\tilde{S}_{L}}}, & \displaystyle\end{eqnarray}$$

with the following boundary conditions

(3.18a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle F(0)=0,\quad F\left(\tilde{x}_{1,+}\right)=-{\displaystyle \frac{\tilde{S}_{L}\left(1+\unicode[STIX]{x1D70F}\tilde{T}_{b}\right)}{2}},\quad F^{\prime }\left(\tilde{x}_{1,+}\right)=\unicode[STIX]{x1D700}-{\displaystyle \frac{\tilde{S}_{L}\unicode[STIX]{x1D70F}{\tilde{T}_{b}}^{\prime }}{2}}. & \displaystyle\end{eqnarray}$$

From (3.16), the streamwise velocity $\tilde{u} _{1}(\tilde{x}_{1})=-2F(\tilde{x}_{1})$ , for $\tilde{x}_{1}\geqslant \tilde{x}_{1,+}$ . Thus the temperature equation (3.3) in the downstream region without the chemical source becomes

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle -{\displaystyle \frac{2F\left(\tilde{x}_{1}\right)}{1+\unicode[STIX]{x1D70F}\tilde{T}}}{\displaystyle \frac{\text{d}\tilde{T}}{\text{d}\tilde{x}_{1}}}={\displaystyle \frac{\tilde{\unicode[STIX]{x1D706}}}{RePr}}{\displaystyle \frac{\text{d}^{2}\tilde{T}}{\text{d}{\tilde{x}_{1}}^{2}}},\quad \text{for}~\tilde{x}_{1}\geqslant \tilde{x}_{1,+}. & \displaystyle\end{eqnarray}$$

To close this set of equations, the dependence of $\tilde{S}_{L}$ on $\tilde{T}_{b}$ is needed a priori as a flame eigen property. A possible choice, for instance, is the theoretical relation (Glassman & Yetter Reference Glassman and Yetter2008):

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{S}_{L}=\left\{\exp \left[-{\displaystyle \frac{\unicode[STIX]{x1D6FD}\left(1-\tilde{T}_{b}\right)}{1-\unicode[STIX]{x1D6FC}\left(1-\tilde{T}_{b}\right)}}\right]\right\}^{0.5}. & \displaystyle\end{eqnarray}$$

It should be mentioned that the strain rate $\unicode[STIX]{x1D700}$ is a free control parameter, representing the different inlet boundary conditions of the upstream flow.

3.3 Model solution

The coupled equations (3.17), (3.19) with the corresponding boundary conditions (3.8), (3.18), specified wall temperature $\tilde{T}_{w}$ and the flame speed relation (3.20), can be iteratively solved. In the present analysis, the flame surface is defined as some specific isosurface $c$ , and accordingly the flame temperature is chosen on the corresponding $c$ isosurface. Parameters $\unicode[STIX]{x1D6FF}_{c=0.85}$ and $\unicode[STIX]{x1D6FF}_{c=0.95}$ denote the distance from the wall to the flame front at the $c=0.85$ isosurface and $c=0.95$ isosurface, respectively. Numerically, $\unicode[STIX]{x1D6FF}_{c=0.85}$ and $\unicode[STIX]{x1D6FF}_{c=0.95}$ are almost identical because of the thin flame thickness.

Figure 4(a) shows, at different strain rates $\unicode[STIX]{x1D700}$ , the dependence of $\tilde{T}_{c=0.95}$ on $\unicode[STIX]{x1D6FF}_{c=0.95}$ for case A ( $\tilde{T}_{w}=0.0$ ) and case B ( $\tilde{T}_{w}=0.5$ ). When far away from the wall, $\tilde{T}_{c=0.95}$ remains close to the adiabatic flame temperature. When approaching the wall, the flame senses the influence of the cold boundary at the Péclet number $P=\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}=8$ , and consequently $\tilde{T}_{c=0.95}$ starts to decrease. If $\unicode[STIX]{x1D6FF}_{c=0.95}$ decreases further, the higher wall heat loss makes the flame temperature decrease more rapidly. Eventually, in case A, the flame quenches at a critical Péclet number $P_{q}=\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}=3.3$ , beyond which there is no flame solution. However, in case B, although heat loss does exist, heat release from the chemical reactions is self-sustainable to maintain the flame solution until $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ is very close to zero with $\tilde{T}_{c=0.95}\approx 0.7$ .

Figure 4. Model solution showing the relation between the flame–wall distance and (a) flame temperature and (b) wall heat flux.

Figure 4(b) shows the model solutions of the wall heat flux $Q_{w}$ in the non-dimensional form:

(3.21) $$\begin{eqnarray}\displaystyle {Q_{w}}^{+}=Q_{w}/\unicode[STIX]{x1D70C}_{u}S_{L}^{0}C_{p,u}\left(T_{ad}-T_{u}\right)={\displaystyle \frac{1}{RePr}}\left(-\tilde{\unicode[STIX]{x1D706}}\left.{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{T}}{\unicode[STIX]{x2202}\tilde{x}_{1}}}\right|_{w}\right). & & \displaystyle\end{eqnarray}$$

Let $Q_{w,q}^{+}$ represent the critical value at the quenching state. For case A ( $\tilde{T}_{w}=0.0$ ), $Q_{w}^{+}$ peaks at $P\approx 4.5$ with a value of approximately $Q_{w}^{+}\approx 0.34$ , which is close to the critical $Q_{w,q}^{+}=0.34$ at $P_{q}=\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}=3.3$ . Similar results have also been obtained in unsteady laminar quenching analyses (Poinsot et al. Reference Poinsot, Haworth and Bruneaux1993; Lai & Chakraborty Reference Lai and Chakraborty2016a ), showing a close peak $Q_{w}^{+}$ and the corresponding Péclet number, although the flow configurations are quite different. When the Péclet number drops below the critical Péclet number $P_{q}$ , the flame quenches and the wall heat flux plummets to zero. For case B, $Q_{w}^{+}$ peaks with a much smaller value at a smaller Péclet number, which can be explained by the smaller temperature difference between the wall and the flame. The tendency for both cases is similar when the flame is far away from the wall, the flame temperature is almost constant and the normalized heat flux $Q_{w}^{+}$ decreases when $P$ increases. The turbulent counterpart from DNS is discussed in the following section.

4.1 Flame temperature

Figure 6 shows for the three cases at the same instant the flame fronts at $c=0.85$ , which are coloured by the local non-dimensional temperature. Except for the near-wall part, the flame topology for the three cases is similar. For case C with an adiabatic wall, the flame temperature is almost constant. When the wall temperature is not very low, as for case B ( $\tilde{T}_{w}=0.5$ ), only a small part of the flame is influenced by the wall because of the weak heat loss.

Figure 7 shows for three cases the probability density function (PDF) of $\unicode[STIX]{x1D6FF}_{c=0.85}$ , which in three-dimensional space is the distance along the $x_{1}$ direction from a local frame to the wall. In more general curvilinear coordinate systems, such a distance can be treated as the minimum flame–wall distance. It can be seen that from case C to case A, the PDF peak moves toward smaller $\unicode[STIX]{x1D6FF}_{c=0.85}$ when the wall heat loss increases, which is consistent with the graph in figure 5; whereas the PDF peak value becomes smaller and the PDF shows wider variation, suggesting that larger wall heat loss makes the flames more spatially distributed.

Figure 6. Spatial structure of the flame isosurface ( $c=0.85$ ) coloured by the local temperature for (a) case A, (b) case B and (c) case C.

Figure 7. The PDF of the normalized flame–wall normal distance $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}$ for the three cases.

Figures 8(a) and 8(b) present the joint PDF between $\tilde{T}_{c=0.95}$ and the normalized flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ for cases B and C, respectively. The temperature distribution in case C is consistent with the spatial temperature distribution in figure 6(c) (i.e. $\tilde{T}_{c=0.95}$ is almost equal to the $c$ value of $0.95$ ). Conversely, for case B with an isothermal wall boundary, the temperature $\tilde{T}_{c=0.95}$ varies from a value close to the adiabatic value to the wall temperature $\tilde{T}_{w}=0.5$ . For case A, the local flame quenching leads to entrainment of the fresh reactant toward the cold wall. As shown schematically in figure 9(a), the complex turbulent flame in the near-wall region consists of two parts, the head-on flame part and the entrained flame part, and the two parts can behave differently. The flame normal vector, which is defined as $\boldsymbol{n}=-\unicode[STIX]{x1D735}c/|\unicode[STIX]{x1D735}c|$ , points from the burnt side toward the unburnt side. The head-on flame part and entrained flame part differ in the relative orientation between $\boldsymbol{n}$ and the wall normal vector $\boldsymbol{N}$ (i.e. $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}>0$ for the head-on case, while $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{N}<0$ for the entrained case). Quantitatively, the PDFs of $\cos \langle \boldsymbol{n},\boldsymbol{N}\rangle$ are shown in figure 9(b). For all of the cases, the positive $\cos \langle \boldsymbol{n},\boldsymbol{N}\rangle$ component is much larger than the negative $\cos \langle \boldsymbol{n},\boldsymbol{N}\rangle$ component, implying that the flames are more likely to be head on. When the wall heat flux decreases from case A to case C, the negative $\cos \langle \boldsymbol{n},\boldsymbol{N}\rangle$ part shrinks because flame quenching/breakup is likely responsible for the entrained flames.

The entire flame front statistics for case A are shown in figure 8(c), while figure 8(d) and figure 8(e) show the results only for the entrained part and the head-on part, respectively. When the flame front is close to the cold wall, the entrained flame temperature $\tilde{T}_{c=0.95}$ fluctuates much more than that of the head-on flames, implying more complex underlying physics, which will be discussed in the subsequent work. The following analysis will be restricted to the head-on flame part.

Figure 8. Joint PDFs between the burnt gas temperature $\tilde{T}_{c=0.95}$ and the wall distance $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ for (a) case B, (b) case C, (c) case A, (d) case A including only the entrained flame part and (e) case A including only the head-on flame part. The influence zone and quenching zone are marked by the dashed lines.

Figure 9. (a) Schematic of the flame–wall interaction. Based on the orientation between the flame normal vector and the wall normal vector, the turbulent flame consists of the head-on flame part (solid line) and the entrained flame part (dotted line); (b) PDF of the relative orientation $\cos \langle \boldsymbol{n},\boldsymbol{N}\rangle$ between the flame normal $\boldsymbol{n}$ and the wall normal $\boldsymbol{N}$ for the three cases.

According to Poinsot et al. (Reference Poinsot, Haworth and Bruneaux1993), the FWI zone can be divided into two sub-zones as follows.

(i) ‘Influence zone’: the region in which the flame is influenced by the cold wall boundary, but no quenching has yet occurred. Figure 8(e) suggests that if set to a low enough flame temperature ( $\tilde{T}_{c=0.95}=0.7$ ) as the quenching state, the quenching distance $P_{q}=\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ is approximately $2.66$ . The present critical Péclet number is comparable to the results by Poinsot et al. (Reference Poinsot, Haworth and Bruneaux1993) ( $P_{q}=3.4$ ) and Lai & Chakraborty (Reference Lai and Chakraborty2016a ) ( $P_{q}=2.8$ ). Meanwhile, when $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}>8$ , the flame temperature is almost constant because of the negligible heat loss to the cold boundary. Thus, the influence zone can be quantified as $2.66<\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}\leqslant 8$ .

(ii) ‘Quenching zone’: the region from the cold wall to the quenching point (i.e. $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}\leqslant 2.66$ ). Inside the quenching zone, because of negligible heat release, the flame temperature decreases almost linearly from $0.7$ to $0.0$ (i.e. the wall temperature).

As expected, big differences are observed between the model solutions in figure 4 and the turbulent statistical results in figure 8, particularly for the small wall distance and low temperature region. Because of the laminar condition in model analysis, the turbulent flame physics cannot be reasonably captured. The $\tilde{S}_{L}(\tilde{T}_{b})$ relation in (3.20) suggests that when the flame front approaches the cold wall, the flame speed $\tilde{S}_{L}$ decays exponentially with the flame temperature $\tilde{T}_{b}$ . The result is different for the turbulent case because the flame front is convected by turbulent eddies. Even at small flame–wall distances with a low flame temperature, the flame speed can still be large enough once the flame is locally pushed close to the cold wall by randomly moving eddies. Such a scenario can be verified numerically. First, the definition of the flame speed under turbulent condition needs to be considered. From the flamelet point of view, the local laminar flame front is thin, and different $c$ isosurfaces move consistently, from which the flame speed can be defined as the relative normal speed between the incoming flow and the flame front, as a whole. In the present analysis, the chemical reaction rate decreases with increasing chemical time scale as the flame approaches the cold wall, and locally the flame structure may be more complex by entrainment of small eddies (Gruber et al. Reference Gruber, Sankaran, Hawkes and Chen2010). To reasonably estimate the flame speed with contributions from different $c$ isosurfaces, the following overall integration along $\boldsymbol{n}$ through the flame zone is introduced as

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \tilde{S}_{L}\rangle ={\displaystyle \frac{\displaystyle \int \tilde{S}_{d}\tilde{\unicode[STIX]{x1D70C}}|\tilde{\unicode[STIX]{x1D735}}c|\text{d}\boldsymbol{n}}{\tilde{\unicode[STIX]{x1D70C}}_{u}\displaystyle \int |\tilde{\unicode[STIX]{x1D735}}c|\text{d}\boldsymbol{n}}}={\displaystyle \frac{\displaystyle \int \left(\tilde{\dot{\unicode[STIX]{x1D714}}}+{\displaystyle \frac{1}{ReSc}}\tilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\left(\tilde{\unicode[STIX]{x1D70C}}\tilde{D}\tilde{\unicode[STIX]{x1D735}}c\right)\right)\!\text{d}\boldsymbol{n}}{\tilde{\unicode[STIX]{x1D70C}}_{u}\displaystyle \int |\tilde{\unicode[STIX]{x1D735}}c|\text{d}\boldsymbol{n}}}, & \displaystyle\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D735}}=L\unicode[STIX]{x1D735}$ and $\tilde{S}_{d}$ is the displacement speed of a $c$ isosurface. The geometrical meaning of $\langle \tilde{S}_{L}\rangle$ is the mean (density-weighted) displacement speed weighted by $|\unicode[STIX]{x1D735}c|$ , i.e. the surface area-weighted displacement speed. It is worth noting that because of the complex near-wall flame behaviour and flame structure, as shown in figure 9, the integration path $\text{d}\boldsymbol{n}$ in (4.1) is confined within the flame zone under consideration. Therefore, the integration $\int |\tilde{\unicode[STIX]{x1D735}}c|\,\text{d}\boldsymbol{n}$ does not need to extend from $c=0$ to $c=1$ . By the same token, the integration $\int (\tilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }(\tilde{\unicode[STIX]{x1D70C}}\tilde{D}\tilde{\unicode[STIX]{x1D735}}c))\,\text{d}\boldsymbol{n}$ does not need to vanish.

Figure 10 presents plots from the DNS data, which show how the mean flame speed $\langle \tilde{S}_{L}\rangle$ varies with the flame temperature on $c=0.95$ for three cases. Interestingly, for case C there is only one state point (i.e. the laminar flame speed solution at the adiabatic flame temperature). For cases A and B, as expected, under turbulent conditions, $\langle \tilde{S}_{L}\rangle$ is still reasonably large even when the flame temperature drops to the cold wall temperature. This result is of essential importance in accounting for the turbulent FWI physics. Therefore, based on the figure 10 results, $\tilde{S}_{L}$ is modelled as

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{S}_{L}=c_{1}+c_{2}\left\{\exp \left[-{\displaystyle \frac{\unicode[STIX]{x1D6FD}\left(1-\tilde{T}_{b}\right)}{1-\unicode[STIX]{x1D6FC}\left(1-\tilde{T}_{b}\right)}}\right]\right\}^{0.5}, & \displaystyle\end{eqnarray}$$

where $c_{1}$ and $c_{2}$ satisfy $c_{1}+c_{2}=1$ . Although a degree of empiricism is involved in the derivation of (4.2), this equation satisfies the asymptotic requirements and such an expression is as valuable as any new experimental parameterization which describes a physical phenomenon yet to be analysed in detail. As shown in figure 11, by controlling $c_{1}$ and $c_{2}$ , the dependence of $\tilde{S}_{L}$ on $\tilde{T}_{b}$ can reasonably be modelled. It seems that $c_{1},c_{2}=0.4,0.6$ is a representative match.

Figure 10. Dependence of mean flame speed $\langle \tilde{S}_{L}\rangle$ on the burnt temperature $\tilde{T}_{c=0.95}$ for the three cases.

Figure 11. The modelled $\tilde{S}_{L}$ and $\tilde{T}_{b}$ relation. By controlling the $c_{1}$ and $c_{2}$ parameters, the flame speed remains reasonably large even at small flame temperature.

As has been discussed, the model relation (4.2) addresses a larger flame speed at a small wall distance, because the flame can be locally pushed close to the cold wall by randomly moving eddies. With the updated $\tilde{S}_{L}$ , the model in  § 3 is similarly solved. The results for case A and case B are presented in figures 12(a) and 12(b), respectively. Even quantitatively, the DNS results can reasonably be reproduced using different $c_{1}$ and $c_{2}$ combinations, representing fluctuations in turbulence. Compared to the laminar solutions shown in figure 4(a), the main difference lies in the solution in the small wall distance region.

Figure 12. Satisfactory agreement between the model solutions and (a) figure 8(e) for case A with $\tilde{T}_{w}=0.0$ , and (b) figure 8(a) for case B with $\tilde{T}_{w}=0.5$ .

4.2 Wall heat flux

From the consideration of practical importance, the wall heat flux is a critical quantity for the combustor performance and its lifespan, particularly for the development of small-sized gas turbines. Meanwhile, as mentioned earlier, the heat flux through the wall is the direct reason for flame quenching. Figure 13(a,b) shows the instantaneous wall heat flux maps for the isothermal wall boundary cases. Because of different wall temperatures, case A and case B present quite different maps, in both the heat flux magnitude and the map structure. The distributions of the averaged wall heat flux evaluated over two throughpass times ( $=2L/U$ ) for cases A and B are shown in figure 13(c,d), respectively. At least for the present case where the wall is not large, the average heat flux remains almost uniform on the wall.

Figure 13. Instantaneous wall heat flux map for (a) case A, (b) case B; averaged wall heat flux map for (c) case A, and (d) case B. The wall heat flux is averaged over two throughpass times for (c) and (d).

To understand the physical reasons for this, figure 14 presents the joint PDF between the normalized wall heat flux ${Q_{w}}^{+}$ and the normalized flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ for case A and case B. Overall, ${Q_{w}}^{+}$ increases when the flame–wall normal distance decreases. The magnitude of ${Q_{w}}^{+}$ for case A remains approximately in the range 0.2–0.4, which is much larger than the value in case B. In case A, flame quenching leads ${Q_{w}}^{+}$ to peak at the critical Péclet number $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}=2.66$ . In the quenching zone, the tail part of the joint PDF shows that ${Q_{w}}^{+}$ decreases when $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ decreases. Flame quenching is relatively much weaker in case B, and thus ${Q_{w}}^{+}$ peaks at a smaller value, and accordingly the joint PDF tail is clearly smaller.

Figure 14. Joint PDFs between the wall heat flux ${Q_{w}}^{+}$ and the flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ for (a) case A and (b) case B.

Such ${Q_{w}}^{+}$ results can also be predicted from the model solution. In a similar vein as in figure 11, by choosing different $\tilde{S}_{L}(\tilde{T}_{b})$ functions, the calculated ${Q_{w}}^{+}$ and $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ dependence is shown in figure 15(a) for case A. In the quenching zone, if the near-wall flame speed is not very large, i.e. $c_{1}\leqslant 0.2$ , the wall heat flux after quenching decreases significantly, while for $c_{1}\geqslant 0.4$ the tendency is different. Overall, the model solution matches the DNS results well, in terms of both the heat flux peak and the corresponding distance $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ by varying $c_{1},c_{2}$ . For case B, the match is good as well, as shown in figure 15(b).

Figure 15. The model solution of the ${Q_{w}}^{+}$ and $\unicode[STIX]{x1D6FF}_{c=0.95}/\unicode[STIX]{x1D6FF}_{z}$ dependence for (a) case A and (b) case B.

The wall heat flux is also related to the strain rate, especially along the wall normal direction. Figure 16 presents the joint PDF between ${Q_{w}}^{+}$ and $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{1}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}|_{w}$ , the normalized strain rate along the wall normal direction. Because of the overall counter-flow configuration, $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{1}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}|_{w}$ is predominantly negative. Larger negative $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{1}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}|_{w}$ means stronger impingement of turbulent eddies toward the wall, which naturally gives rise to larger ${Q_{w}}^{+}$ , while positive $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{1}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}|_{w}$ means that the turbulent eddies move away from the wall, leading to smaller ${Q_{w}}^{+}$ . The results for cases A and B are qualitatively similar.

Figure 16. Joint PDFs between the normalized wall heat loss ${Q_{w}}^{+}$ and the normalized strain rate along the wall normal direction $\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}x_{1}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}|_{w}$ , for (a) case A and (b) case B.

4.3 Near-wall behaviour of flame–turbulence interaction

Because of the interference from the wall boundary, flame–turbulence interaction in the near-wall region is important as well in studying FWI. First, the flame dilatation and flame tangential strain rate, both of which are closely related to flame quenching, will be addressed. Figure 17 shows the joint PDFs between the normalized flame dilatation $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{i}\times \unicode[STIX]{x1D6FF}_{th}/S_{L}^{0}$ on the $c=0.85$ isosurface and the normalized flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}$ for the three cases, which overall are quite different.

Physically, the flame dilatation is determined by two counteracting effects, heat generation due to combustion and heat loss to the cold wall boundary. For case C without heat loss to the adiabatic wall, the flame dilatation assumes almost constant positive values due to the net heat release due to combustion. For cases A and B, the joint PDFs are more complex inside the influence zone. Specifically for case A, as the wall is approached, the normalized dilatation $\unicode[STIX]{x1D6E5}$ continuously decreases to approximately $-0.5$ and then increases to zero, i.e. the dilatation value on the cold wall. The Péclet number where the minimum $\unicode[STIX]{x1D6E5}$ appears is smaller than the critical $P_{q}=2.66$ . For case B, because of much weaker flame quenching, the minimum normalized $\unicode[STIX]{x1D6E5}$ is much smaller (approximately $-0.1$ ). It is also interesting to mention that the zero $\unicode[STIX]{x1D6E5}$ point, at which the heat generation and heat loss roughly balance, occurs at approximately $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}=3.5$ for case A and approximately $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}=2.5$ for case B.

Figure 17. Joint PDFs between the dilatation at $c=0.85$ and the flame–wall distance for (a) case A, (b) case B and (c) case C.

Figure 18 presents the joint PDFs between the normalized flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}$ and the normalized flame tangential strain rate $s_{t}\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}=(\unicode[STIX]{x1D6FF}_{ij}-n_{i}n_{i})\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ $\times \unicode[STIX]{x1D6FF}_{Z}/S_{L}^{0}$ under different wall boundary conditions. For case C, the flame tangential strain rate is mainly influenced by the turbulent straining. The gentle increase of $s_{t}$ when $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}$ decreases, as shown in figure 18, is explained as follows. The flame front can be convected close to the wall by turbulent eddies. The more energetic a turbulent eddy is, the closer the flame will be pushed toward the wall. Thus, the flame is more squeezed for higher values of tangential strain rate. For case A, in the influence zone, the wall heat flux reduces the fluid temperature and increases the density, which will then reduce the dilatation term $\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ in the tangential strain rate. Therefore, $s_{t}$ remains almost invariant when $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}>2.66$ , as indicated in figure 18(a). Inside the quenching zone, where flames need to be strongly pushed to get closer to the wall ( $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}<2.66$ ), the flame tangential strain rate becomes much higher. The results for case B lies between those of case A and case C.

Figure 18. Joint PDFs between the flame tangential strain rate at $c=0.85$ and the flame–wall distance for (a) case A, (b) case B and (c) case C.

As a topic of broad interest, the flame structure is influenced by FWI for at least two reasons. First, the geometric boundary confinement will change the local flame structure; second, the thermal boundary condition will change the flame chemistry and thus the flame structure. Figure 19 demonstrates the distribution of the reaction progress variable $c$ and the vorticity component $w_{2}$ (along the $x_{2}$ direction) on the cross-cut plane at $x_{2}=L/2$ for three cases. The black and white lines represent the positive and negative $w_{2}$ isolines. Similar to the result from Poinsot et al. (Reference Poinsot, Haworth and Bruneaux1993), the vortex pair plays a dominant role in the evolution of the flame front. The positive vorticity component $w_{2}$ pushes the flame close to the wall, while the negative $w_{2}$ extracts the flame away from the wall. Specifically, generation of the flame bubble is strongly influenced by the joint action of the vorticity pair and the wall boundary condition.

Figure 19. The correlation between the vorticity component in the $x_{2}$ direction and the flame wrinkling at the cross-cut plane ( $x_{2}=L/2$ ) for (a) case A, (b) case B and (c) case C. White and black lines indicate positive and negative values of $w_{2}$ , respectively.

To quantify such a flame wrinkling feature, figure 20 shows the joint PDF between the velocity component $u_{1}$ and the flame curvature $K=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ . Overall, these two quantities are slightly negatively correlated, which can be explained from the predominance of the negative curvature (concave toward the reactants) when the flame front is close to the wall. On the greatest part of the spatial flames, $u_{1}$ is positive. The additional tail in case A with relatively small $u_{1}$ is the main difference from the other cases. In case A, the tail in figure 20 looks more prominent because of a higher likelihood of flame quenching near the wall, where $u_{1}$ is small.

Figure 20. The joint PDF between the velocity component $u_{1,c=0.85}$ and the flame front curvature $K$ for (a) case A, (b) case B and (c) case C.

Flame quenching will also change the magnitude of the reactive scalar gradient. It is useful to note that in figure 19, especially in case A, the scalar gradient decreases once flame quenching occurs. To investigate the scalar gradient statistics, figure 21 shows the relation between the local non-dimensional reaction progress variable gradient $\left|\unicode[STIX]{x1D735}c\right|\unicode[STIX]{x1D6FF}_{z}$ at $c=0.85$ isosurface and the normalized distance of the flame from the wall. Clearly, the results for case A and case B differ from that of case C because of the influence of the non-adiabatic wall. In figure 21(c) for case C, the scalar gradient magnitude exhibits scatter around a value which remains almost unchanged with the variation of the distance from the wall. For cases A and B, in the early influence zone, the scalar gradient magnitude does not decrease much for $5<\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}<8$ . If the flame moves deeply inside with $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}\leqslant 5$ , $\left|\unicode[STIX]{x1D735}c\right|\unicode[STIX]{x1D6FF}_{z}$ starts to decrease significantly, implying a rapid decrease of the scalar gradient. As the distance from the wall is reduced, the chemical reaction rate becomes smaller. Therefore, under the action of turbulence, e.g. entrainment and random convection, the flame may undergo a regime change from the typical ‘thin flamelet’ regime to a ‘thickened wrinkled’ regime (Gruber et al. Reference Gruber, Sankaran, Hawkes and Chen2010). Although the wall boundary conditions for cases A and B are different, the joint PDFs in figure 21 are fairly similar. At the present flow and wall temperature conditions, quantitatively the scalar gradient magnitude can be four times smaller than that outside the influence zone.

Figure 21. The joint PDF between the non-dimensional gradient $\left|\unicode[STIX]{x1D735}c\right|\unicode[STIX]{x1D6FF}_{z}$ of the progress variable at $c=0.85$ and the flame–wall distance $\unicode[STIX]{x1D6FF}_{c=0.85}/\unicode[STIX]{x1D6FF}_{z}$ for (a) case A, (b) case B and (c) case C.