3.2.1. Edge-stabilized wake flame

The edge-stabilized wake flame is conjectured to have a similar flame structure to that of a branched-edge flame that is stabilized over a Bunsen burner (Law Reference Law2010). The relative flow around the edge-stabilized wake flame remains attached in the Reynolds number range of $5 < R{e_{local}} < 20$ without any recirculation in the wake. Hence, it is reasonable to consider the wake flow as concentric fuel and air streams. The flame edge consists of a premixed flame front which tends to propagate upstream and is stabilized due to the local balance of the relative flow velocity scale (${v_{rel}}$) and flame propagation speed ($ {\textrm{S}_L}$). Due to the presence of the annular air stream and the central fuel stream, a gradient of fuel concentration exists along the contour of the premixed flame front. Thus, the equivalence ratio ($\boldsymbol{\phi }$) continuously changes along the radius from $\boldsymbol{\phi } = {\bf 0}$ on the outer edge (oxidizer end) to $\boldsymbol{\phi } \gg {\bf 1}$ at the centre. The inner fuel-rich branch of the flame front has $\boldsymbol{\phi } \ge {\bf 1}$, whereas the outer fuel-lean branch has $\boldsymbol{\phi } < {\bf 1}$ (Ko & Chung Reference Ko and Chung1999; Vadlamudi et al. Reference Vadlamudi, Thirumalaikumaran and Basu2021), see figure 3(a).

Figure 3. (a) Schematic of edge-stabilized wake flame and edge-flame stabilization. (b) Normalized flame height ($h/d$) v/s normalized relative flow velocity scale at the droplet (${\tilde{v}_{rel}}\sim {v_{rel}}/S_L^o$). The error bars represent an uncertainty of ±10 %.

However, the portion of the flame which is away from the flame edge is stabilized through the local balance between the flame speed and the component of the flow velocity at the droplet perpendicular to the flame front, as shown in figure 3(a). Hence, balancing the local flame speed (${S_L}$) and flow velocity scale relative to the flame (${v_{rel}}$)

(3.1)\begin{gather}{v_{rel}}\,\textrm{sin}\,\theta = {S_L},\end{gather}

(3.2)\begin{gather}{v_{rel}} = \frac{{{S_L}}}{{\textrm{sin}\,\theta }},\end{gather}

(3.3)\begin{gather}{v_{rel}} = {S_L}\sqrt {1 + {{\left( {\frac{{{h}}}{{d/2}}} \right)}^2}} ,\end{gather}

(3.4)\begin{gather}{v_{rel}}\sim \frac{h}{d},\end{gather}

(3.5)\begin{gather}{v_{rel}} = {c_1}\frac{h}{d} + {c_2}.\end{gather}

After simplifying (3.1), the relation between ${v_{rel}}$ and $h/d$ is obtained as (3.3), which can be approximately expressed as a simplified scaling expression (3.4), assuming linear approximation in the range of $0.35 < h/d < 0.8$, where the edge-stabilized wake flame is observed and ${d_f}\sim d$ in this range. Further details on the linear approximation can be found in the supplementary material (figure S3). It is to be noted that, beyond $Re > 20$, the characteristics of the flow around the droplet change by establishing axisymmetric vortices in the rear stagnation region. Hence, the flame response will also change for $Re > 20$ (corresponding normalized local flame height: $h/d > 0.8$). For simplicity, the correlation in (3.4) can be rewritten as a linear equation, as shown in (3.5), using coefficients that can be found empirically from experimental data from our previous study for the edge-stabilized wake flame (Vadlamudi et al. Reference Vadlamudi, Thirumalaikumaran and Basu2021). The flame height variation (h) with change in local flow velocity scale relative to the flame (${v_{rel}}$) is found to match with the linear correlation shown in (3.5).

Using the previous experimental data (Vadlamudi et al. Reference Vadlamudi, Thirumalaikumaran and Basu2021), the instantaneous flame height characteristics for the whole range of v_rel were measured from the binary images obtained by performing Otsu's thresholding on the flame images, as explained in § 2. For this particular set of experiments, since the flame is stabilized in the far wake, the droplet velocity cannot be used for evaluating the relative bulk flow velocity perceived by the flame (v_rel). The flame dynamics is modulated due to the bulk relative flow velocity scale (v_rel) experienced by the flame as a result of the net external flow imposition. Furthermore, the bulk velocity scale of the relative flow (v_rel) experienced by the wake flame is the direct consequence of the flame descent speed through the quiescent environment in the laboratory reference frame. Thus, the instantaneous relative velocity scale of the bulk flow (v_rel) is computed by spatially tracking the instantaneous location of the flame base with time in the laboratory reference frame that is obtained from Otsu's thresholding for the freely falling droplet combustion data of previous experiments without co-flow (Vadlamudi et al.). It is to be noted that the v_rel measured is just the scale of the relative velocity of the bulk flow and is a parameter used to characterize the instantaneous flow perceived by the flame. However, the effect of the flow field around the droplet on mixing is considered in the subsequent sections that discuss the flame stabilization mechanisms. Instantaneous droplet diameter is measured from the simultaneous shadowgraphy images. Thus obtained normalized flame height ($h/d$) from the previous data (Vadlamudi et al. Reference Vadlamudi, Thirumalaikumaran and Basu2021) is plotted against relative flow velocity scale at the droplet (${v_{rel}}$), which is normalized using the laminar flame speed ($S_L^o$), as shown in figure 3(b).

The plot shows a linear correlation between $h/d$ and the ${\tilde{v}_{rel}}$ i.e. ${v_{rel}}$ normalized with the laminar flame speed ($S_L^o$), and can be used to estimate the coefficients of the scaling relation shown in (3.5). The obtained empirical relation is presented in (3.6).

(3.6)\begin{equation}\frac{{{v_{rel,EdgeStb}}}}{{S_L^o}} = 0.44\frac{h}{d} + 0.1.\end{equation}

3.2.2. Edge-stabilized wake flame stabilization mechanism

The edge-flame stabilization mechanism has been investigated in this section. The schematic of the heat and mass transfer near the droplet wake and the flame (assumed planar for simplicity) is shown in figure 4(a). Since the edge-stabilized flame in the current experiments stabilizes in the range 9 < Re < 18, the airflow around the droplet is characterized by an attached wake flow (with no recirculation eddies in the wake), similar to the flow over a sphere for 0 < Re < 20, (Kalra, T.R. Reference Kalra and U1971) (see figure 4b). The flow in the wake can be approximated to be analogous to co-axial jets, wherein the inner core stream is fuel, and the outer annular stream is the oxidizer. Figure 4(c) shows the schematic of the co-axial jets analogy and shows the entrainment of the outer jet into the inner jet. The entrained mass increases as the axial distance ($x$) increases. Assuming an infinite rate chemistry, this now reduces to a species diffusion problem (mixing problem). Our previous schlieren experiments showed that the fuel vapour wake emanates only from the region near the rear stagnation point. This suggests that the vaporization only occurs from the small region (∼2 % of droplet surface area) that is exposed to the wake flame. Since only 2 % of the total droplet surface area (${A_{total}}$) is exposed to the wake flame, the fuel stream is assumed to be originating only from the portion of the droplet with surface area, ${A_o}$ (where ${A_o}\sim 0.02{A_{total}}$) (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020). The amount of air entrainment into the fuel stream is calculated using the co-axial jet analogy, which is used to acquire the theoretical value of the equivalence ratio along the axial direction ($x$).

Figure 4. (a) The schematic of the heat and mass transfer near the droplet wake and the flame (assumed planar for simplicity). (b) The schematic of expanding jets shows the air entrainment into the fuel stream in the droplet wake. (c) Schematic of co-axial jets depicting entrainment.

Ricou & Spalding (Reference Ricou and Spalding1961) showed that, for entraining fuel jets, the ratio of the mass flux (at a given axial distance) to the initial fuel mass flux at the jet exit is given by (3.7)

(3.7)\begin{equation}\frac{1}{{{Y_F}}} = \frac{{\dot{m}_F + {{\dot{m}}_{Ox}}}}{\dot{m}_F}\sim \left( {\frac{x}{{{d_o}}} } \right){\left( {\frac{{{\rho_{out}}}}{{{\rho_{in}}}}} \right)^{1/2}},\end{equation}

where the outer flow is the oxidizer (air), and the inner flow is the fuel vapour. For the current co-axial jet approximation, the axial variation of the equivalence ratio can be estimated from the axial variation of fuel-air ratios obtained using the entrainment relations of co-axial jets from the literature (Villermaux & Rehab Reference Villermaux and Rehab2000; Schumaker & Driscoll Reference Schumaker and Driscoll2008; Bennewitz et al. Reference Bennewitz, Schumaker, Lietz and Kastengren2021). The fuel mole fraction (${X_F}$) of the jet at a given axial location ($x$) can be scaled using the relation given below (3.8):

(3.8)\begin{equation}{X_F}\sim {\left( {\frac{x}{{{d_o}}}{M^{1/2}}} \right)^{ - 1}}\sim {\left( {\frac{x}{{{d_o}}}{{\left( {\frac{{{\rho_{out}}}}{{{\rho_{in}}}}} \right)}^{1/2}}\left( {\frac{{{v_{out}}}}{{{v_{in}}}}} \right)} \right)^{ - 1}},\end{equation}

where ${d_o}$ is the diameter corresponding to the droplet surface area ${A_o}$, assuming ${A_o}$ to be a flat circle (since ${A_o} \ll {A_{total}}$) and M is the momentum flux ratio of outer and inner jets

(3.9)\begin{equation}M = \left( {\frac{{{\rho_{out}}}}{{{\rho_{in}}}}} \right){\left( {\frac{{{v_{out}}}}{{{v_{in}}}}} \right)^2},\end{equation}

where ${v_{out}}$ is considered to be equal to the average surrounding relative flow velocity (${v_{rel}}$) where edge-stabilized wake flames are observed.

The velocity scale of the fuel vapour stream (${v_{in}}$) can be estimated by evaluating the droplet evaporation rate. Since it has been established that fuel vaporization only occurs from the small surface area (${A_o}$) on the droplet surface near the rare stagnation point, the pool-fire energy balance model by Yip, Haelssig & Pegg (Reference Yip, Haelssig and Pegg2021) can be used to estimate the burning rate of the fuel droplet (see figure 3a). A global energy balance around the evaporating droplet can be written down as heat transfer due to net conduction $({\dot{Q}_{cond}})$, convection $({\dot{Q}_{conv}})$, radiation $({\dot{Q}_{r,f - d}})$ from the flame to the liquid pool (${A_o}$) and the radiative heat loss to the surroundings $({\dot{Q}_{r,d - a}})$, as shown below

(3.10)\begin{equation}{\dot{m}_F}(\Delta {H_v}) = {\dot{Q}_{cond}} + {\dot{Q}_{conv}} + {\dot{Q}_{r,f - d}} + {\dot{Q}_{r,d - a}},\end{equation}

where ${\dot{m}_F}$ is the fuel evaporation rate and $\Delta {H_v}$ is the enthalpy of vaporization of the fuel. Using the Fourier's heat conduction equation, ${\dot{Q}_{cond}}\sim k{A_o}(({T_{ad}} - {T_s})/{d_{st}})$ is obtained to be $0.026\ \textrm{W}$.

The radiative heat transfer term is estimated using ${\dot{Q}_{r,f - d}}\sim \sigma {F_{df}}{A_o}{\varepsilon _D}(T_{ad}^4 - T_s^4)$, where $\sigma $ is the Stefan–Boltzmann constant, ${\varepsilon _D}$ is the emissivity of the liquid fuel and ${F_{df}}$ (view factor) accounts for a fraction of radiative energy from the droplet that is interacting with the flame. The view factor ${F_{df}}$ is evaluated assuming a flame with area ${A_f}$ and the exposed droplet surface area ${A_o}$ as parallel discs separated by the stand-off distance (${d_{\boldsymbol{st}}}$). The view factor (${F_{df}}$) is then given as

(3.11)\begin{equation}{F_{df}} = \frac{1}{2}\left[ {S - {{\left[ {{S^2} - 4{{\left( {\frac{{{d_f}}}{{{d_O}}}} \right)}^2}} \right]}^{1/2}}} \right],\quad \textrm{where}~S = \frac{{d_o^2 + d_f^2 + 4d_{st}^2}}{{d_o^2}}.\end{equation}

Then, ${\dot{Q}_{r,f - d}}$ is obtained to be 0.072 W, and similarly, the heat loss from the droplet is obtained to be ${\dot{Q}_{r,d - a}}\sim \sigma {A_o}{\varepsilon _D}(T_\infty ^4 - T_s^4) ={-} 0.00034\ \textrm{W}$.

The formulation of the pure convective heat flux on the surface of the pool in steady pool fires by Spalding (Reference Spalding1953) can be extended to evaluate ${\dot{Q}_{conv}}$, as shown below

(3.12) \begin{equation}{\dot{Q}_{conv}} = {A_o}\frac{{\mathop{h}\limits^{\prime}}}{{{C_p}}}\left[ { \frac{{\Delta {H_c}({\chi_a} - {\chi_r})}}{{{\chi_a} r}} - {C_p}({T_s} - {T_\infty })} \right]\frac{\lambda }{{{e^\lambda } - 1}}.\end{equation}

Where $\mathop{h}\limits^{\prime}$ is the coefficient of convective heat transfer, ${C_p}$ is the specific heat capacity of the ambient, $\Delta {H_c}$ is the enthalpy of combustion of fuel, ${\chi _a}$ is the combustion efficiency, ${\chi _r}$ is the radiative fraction, r is the stoichiometric air-to-fuel mass ratio and $\lambda = {\dot{m}_F}{C_p}/\mathop h\limits^{\prime}{A_o}$. For the convective heat transfer from a flat pool, the convective heat transfer coefficient $\mathop h\limits^{\prime} = k(Nu)/{d_o}$ can be evaluated using the Nusselt number (Nu) correlation that is expressed in terms of Rayleigh number ($Ra$), for the natural convection over a horizontal plate, i.e.

(3.13)\begin{equation}Nu = 0.54{(Ra)^{1/4}}\sim 5.44,\quad \textrm{where}~Ra = \frac{{g\beta (\Delta T)d_o^3}}{{{\nu ^2}}}Pr.\end{equation}

Then the convective heat transfer coefficient ($\mathop h\limits^{\prime} $) is substituted in the (3.12), to get

(3.14)\begin{equation}{\dot{Q}_{conv}} = 0.2316\frac{\lambda }{{{e^\lambda } - 1}},\quad \textrm{where}~\lambda = 7755389.8{\dot{m}_F}.\end{equation}

Substituting for all the terms in (3.10), an expression is obtained with ${\dot{m}_F}$ as the only unknown

(3.15)\begin{equation}{\dot{m}_F} = (2.725 \times {10^{ - 7}}) + (6.444 \times {10^{ - 7}})\left( {\frac{\lambda }{{{e^\lambda } - 1}}} \right).\end{equation}

Solving the above equation iteratively, the rate of evaporation at the droplet surface $({\dot{m}_F})$ is evaluated.

Using the evaporation mass flux $({\dot{m}_F})$, the order of magnitude of the vapour velocity scale (${v_{in}}$) is obtained to be ${\sim} 0.21\ \textrm{m}\ {\textrm{s}^{ - 1}}$. The value of ${v_{in}}$ obtained is found to be of a similar order of magnitude as the natural convection velocity scale (${v_{NC}}\sim \sqrt {g\beta \Delta Td} $).

Finally, the corresponding fuel–air mass ratio at a given axial distance in terms of the fuel mass fraction (${Y_F}$) is calculated using (3.16)

(3.16)\begin{equation}\frac{{{{\dot{m}}_F}}}{{{{\dot{m}}_{Ox}}}} = \frac{1}{{{Y_F} - 1}}.\end{equation}

The mean equivalence ratio (${\boldsymbol{\phi }_m}$) at a given axial distance (x), is then estimated as the ratio of the fuel–air ratio at a given location (x) to the stoichiometric fuel–air ratio

(3.17)\begin{equation}{\boldsymbol{\phi }_m} = \frac{{\dfrac{{{{\dot{m}}_F}}}{{{{\dot{m}}_{Ox}}}}}}{{{{\left( {\dfrac{{{{\dot{m}}_F}}}{{{{\dot{m}}_{Ox}}}}} \right)}_{St}}}}.\end{equation}

Here, ${\boldsymbol{\phi }_{\boldsymbol{m}}}$ is observed to reduce as the axial distance from the droplet surface increases. The axial variation of the equivalence ratio is plotted in figure 5(a). Using this, the value of the mean equivalence ratio (${\boldsymbol{\phi }_m}$) at the axial location of the flame base ($x = {d_{\boldsymbol{st}}}$) can be evaluated. This will be used further to evaluate the radial variation of the equivalence ratio necessary for gain further insights into edge-flame stabilization.

Figure 5. (a) Mean equivalence ratio (${\boldsymbol{\phi }_{\boldsymbol{m}}}$) variation plotted against normalized axial distance ($x/d$). (b) The schematic depiction of the radial and axial coordinates with respect to the edge-stabilized flame. (c) Radial variation of equivalence ratio ($\boldsymbol{\phi }$) and the variation of corresponding flame temperatures ${T_g}$ and ${T_{eq}}$ plotted against normalized radial distance ($\tilde{r} \sim r/{r_o}$) at the axial flame location.

3.2.3. Radial variation of the local equivalence ratio $(\hat{\boldsymbol{\phi }})$

The equivalence ratio $({\boldsymbol{\phi }_{\boldsymbol{m}}})$ estimated at a given axial distance (x) in the previous section is the mean value of $\boldsymbol{\phi }$ in that plane situated at an axial distance (x). However, the radial profile of the local equivalence ratio $(\hat{\boldsymbol{\phi }})$ within that plane is dependent on both the centreline equivalence ratio $({\hat{\boldsymbol{\phi }}_{\boldsymbol{r} \to {\bf 0}}} = \boldsymbol{\phi}_c)$ and the far-field equivalence ratio $({\hat{\boldsymbol{\phi }}_{\boldsymbol{r} \to {\boldsymbol{r}_\infty }}} = {\boldsymbol{\phi }_\infty })$. It is known that the far-field equivalence ratio ${\boldsymbol{\phi }_\infty }\mathop \to \limits^{\boldsymbol{ }} \boldsymbol{ }0$ due to the absence of fuel in the free stream. Since the length of the inner potential core is less than ${d_{\boldsymbol{st}}}$ for the velocity ratio (Larsson et al. Reference Larsson, Lycksam, Lundström and Marjavaara2020), the centreline composition at the axial location ($x = {d_{\boldsymbol{st}}}$) will not be pure fuel. Thus, the local centreline equivalence ratio is unknown and will be estimated in this section. Since, experimentally, the radial location of the base of the edge flame is known ($r = {r_{\boldsymbol{f}}}$), it is reasonable to assume the corresponding equivalence ratio at the flame base to be stoichiometric $({\hat{\boldsymbol{\phi }}_{\boldsymbol{r} = {\boldsymbol{r}_{\boldsymbol{f}}}}} = {\boldsymbol{\phi }_{\boldsymbol{f}}} = 1)$. A Gaussian profile is shown to be a reasonable fit for the radial mixing profiles (Hotz et al. Reference Hotz, Haas, Wachter, Fleck and Kolb2023) since the variation of the mixture fraction of the fuel vapour is axisymmetric, and it peaks at the centre and approaches zero asymptotically on either side in the free stream. Since the $\hat{\boldsymbol{\phi }}$ value at two radial locations (i.e. $r = {r_{\boldsymbol{f}}}\ \textrm{and}~{r_\infty }$) is known, the three-point generated Gaussian function (Li, Valentine & Rana Reference Li, Valentine and Rana1999) can be used to generate the radial variation of $\hat{\boldsymbol{\phi }}$ in the plane (at $x = {d_{\boldsymbol{st}}}$), by assuming values for ${\boldsymbol{\phi }_{\boldsymbol{c}}}$ (centreline equivalence ratio). The three-point generated Gaussian function equation with respect to the three points $({r_c},{\boldsymbol{\phi }_c})$, $({r_f},{\boldsymbol{\phi }_f})$ and $({r_\infty },{\boldsymbol{\phi }_\infty })$ is shown below

(3.18a)\begin{equation}\hat{\boldsymbol{\phi }}(\boldsymbol{r}) = \frac{A}{{\sqrt {2{\rm \pi} {\sigma ^2}} }}\,\textrm{exp}\left[ { - \frac{{{{(r - \mu )}^2}}}{{2{\sigma^2}}}} \right],\end{equation}

where

(3.18b)\begin{gather}A = \sqrt {2{\rm \pi} {\sigma ^2}} {({\boldsymbol{\phi }_c}{\boldsymbol{\phi }_f}{\boldsymbol{\phi }_\infty })^{1/3}}\,\textrm{exp}\left[ {\frac{{{{({r_c} - \mu )}^2} + {{({r_f} - \mu )}^2} + {{({r_\infty } - \mu )}^2}}}{{6{\sigma^2}}}} \right],\end{gather}

(3.18c,d)\begin{gather}\sigma = \sqrt {\frac{{({r_c} - {r_\infty })({r_f} - {r_c})({r_\infty } - {r_f})}}{{2\,\textrm{ln}[\boldsymbol{\phi }_c^{{r_\infty } - {r_f}}\boldsymbol{\phi }_f^{{r_c} - {r_\infty }}\boldsymbol{\phi }_\infty ^{{r_f} - {r_c}}]}}} ,\quad \mu = \frac{{\textrm{ln}[\boldsymbol{\phi }_c^{r_\infty ^2 - r_f^2}\boldsymbol{\phi }_f^{r_c^2 - r_\infty ^2}\boldsymbol{\phi }_\infty ^{r_f^2 - r_c^2}]}}{{\textrm{ln}[\boldsymbol{\phi }_c^{{r_\infty } - {r_f}}\boldsymbol{\phi }_f^{{r_c} - {r_\infty }}\boldsymbol{\phi }_\infty ^{{r_f} - {r_c}}]}},\end{gather}

where $\mu ,\sigma $ are the mean and standard deviation of the Gaussian profile. The radial profiles of $\hat{\boldsymbol{\phi }}$ were obtained for different values of ${\boldsymbol{\phi }_c} \in (0,6)$. Then, the average value of the stoichiometric ratio (${\boldsymbol{\phi }_{avg,{d_{\boldsymbol{st}}}}}$) is evaluated for each of the profiles that are obtained for different values of ${\boldsymbol{\phi }_c} \in (0,6)$ using (3.18) (for the plane $x = {d_{st}}$, marked by dotted blue line in figure 5b). Thus, the correct value of the centreline equivalence ratio $({\hat{\boldsymbol{\phi }}_{\boldsymbol{r} \to {\bf 0}}} = {\boldsymbol{\phi }_{c,{d_{\boldsymbol{st}}}}})$ for $x = {d_{st}}$ is evaluated by equating the average radial equivalence ratio $({\boldsymbol{\phi }_{avg,{d_{\boldsymbol{st}}}}})$ values obtained in this section with the already established mean equivalence ratio corresponding to the axial location, $x = {d_{st}}$ $({\boldsymbol{\phi }_{m,(x = {d_{\boldsymbol{st}}})}})$, which was evaluated in the previous section (see (3.17)) using the coaxial jet formulation.

Thus, after obtaining the correct value for ${\boldsymbol{\phi }_c}$, the radial variation of $\hat{\boldsymbol{\phi }}$ in the axial plane $x = {d_{st}}$ is obtained in the form of a Gaussian profile and is plotted in figure 5(c) against the radial distance normalized by the radius of the fuel jet i.e. $\tilde{r}\sim r/{r_{\boldsymbol{o}}}$. The plot shows that the equivalence ratio $(\hat{\boldsymbol{\phi }})$ peaks at the centre and decreases in the radially outward direction, asymptotically reaching zero in the free stream. The radial location ($r = {r_{\boldsymbol{f}}}$) corresponding to $\hat{\boldsymbol{\phi }} = 1$ is the location of the flame base obtained experimentally, where the local flame propagation speed (${S_E}$) is maximum, and it matches the flow velocity at the droplet, leading to flame stabilization (as shown in figure 3a). At the flame base, a premixed edge-stabilized flame propagates upstream with respect to the upstream flow, containing two branches: fuel rich and fuel lean. As shown in figure 5(c), from the flame base location ($r = {r_{\boldsymbol{f}}}$), the equivalence ratio $(\hat{\boldsymbol{\phi }})$ becomes less than 1 and decreases in the radially outward direction, forming a fuel-lean region, whereas, in the radially inward direction, the local equivalence ratio $(\hat{\boldsymbol{\phi }})$ becomes more than unity and increases radially inward from the flame base location ($r = {r_{\boldsymbol{f}}}$). This leads to a gradient of the local equivalence ratio $(\hat{\boldsymbol{\phi }})$ along the flame branch on either side of the stoichiometric location $(\hat{\boldsymbol{\phi }} = 1)$, forming a fuel-rich branch on the $\hat{\boldsymbol{\phi }} > 1$ side ($r < {r_{\boldsymbol{f}}}$) and fuel-lean branch on the $\hat{\boldsymbol{\phi }} < 1$ side ($r > {r_{\boldsymbol{f}}}$). This agrees with the literature on edge-stabilized flames, where the flame front propagates upstream against the flow-field having a mixture stratification (Chung Reference Chung2007).

3.2.4. Flame temperature estimation using global reaction (simplistic model)

The equivalence ratio ($\boldsymbol{\phi }$) can now be used to calculate the theoretical flame temperature (${T_g}$). The stoichiometric equation has been modified to accommodate both the excess air ($\boldsymbol{\phi } < 1$) and excess fuel ($\boldsymbol{\phi } > 1$) conditions. The global stoichiometric equation for the combustion of dodecane is presented in (3.19). An extra ‘a’ moles of air are added to the stoichiometric equation to represent fuel-lean combustion, obtaining (3.20). Similarly, for fuel-rich combustion, ‘${f_e}$‘ moles of fuel are added to the stoichiometric equation. For simplicity, CO formation and its dissociation reactions are neglected in the fuel-rich combustion case and is shown in (3.21)

(3.19)\begin{gather}{\textrm{C}_{12}}{\textrm{H}_{26}} + \frac{{37}}{2}({\textrm{O}_2} + 3.76{\textrm{N}_2}) \to 12\textrm{C}{\textrm{O}_2} + 13{\textrm{H}_2}\textrm{O} + \left( {\frac{{37}}{2}} \right)3.76{\textrm{N}_2},\end{gather}

(3.20)\begin{gather}{\textrm{C}_{12}}{\textrm{H}_{26}} + \left( {\frac{{37}}{2} + a} \right)({\textrm{O}_2} + 3.76{\textrm{N}_2}) \to 12\textrm{C}{\textrm{O}_2} + 13{\textrm{H}_2}\textrm{O} + \left( {\frac{{37}}{2} + a} \right)3.76{\textrm{N}_2} + a{\textrm{O}_2},\end{gather}

(3.21)\begin{gather}(\,{f_e} + 1){\textrm{C}_{12}}{\textrm{H}_{26}} + \left( {\frac{{37}}{2}} \right)({\textrm{O}_2} + 3.76{\textrm{N}_2}) \to 12\textrm{C}{\textrm{O}_2} + 13{\textrm{H}_2}\textrm{O} + \left( {\frac{{37}}{2}} \right)3.76{\textrm{N}_2}\nonumber\\ + \ \,{f_e} {\textrm{C}_{12}}{\textrm{H}_{26}},\end{gather}

(3.22)\begin{gather}(\,{f_e} + 1){\textrm{C}_{12}}{\textrm{H}_{26}} + \left( {\frac{{37}}{2} + a} \right)({\textrm{O}_2} + 3.76{\textrm{N}_2}) \to 12\textrm{C}{\textrm{O}_2} + 13{\textrm{H}_2}\textrm{O} + \left( {\frac{{37}}{2} + a} \right)3.76{\textrm{N}_2}\nonumber\\ +\ \, a{\textrm{O}_2} + {f_e}{\textrm{C}_{12}}{\textrm{H}_{26}}.\end{gather}

A combined representative chemical equation is considered, as shown in (3.22), where ${f_e} = 0$, $a = 0$ and $a = {f_e} = 0$ represents fuel-lean, fuel-rich and stoichiometric conditions, respectively.

A global energy balance of the above equation set (a simplified approach) can be used to estimate the approximate flame temperature (${T_g}$). The resulting equation is shown in (3.23)

(3.23) \begin{align} {T_g} = \frac{\begin{array}{l}(\alpha + 1)h_{f,DD}^o - 12h_{f,C{O_2}}^o - 13h_{f,{H_2}O}^o + {C_{p,DD}}({T_{wb}} - {T_\infty })\\ \quad +\ {T_\infty }\left({12{C_{p,C{O_2}}} + 13{C_{p,{H_2}O}} + \left( {\dfrac{{37}}{2} + a} \right)3.76{C_{p,{N_2}}} + a{C_{p,{O_2}}} + \alpha {C_{p,DD}}} \right)\end{array}} {12{C_{p,C{O_2}}} + 13{C_{p,{H_2}O}} + \left( {\dfrac{{37}}{2} + a} \right)3.76{C_{p,{N_2}}} + a{C_{p,{O_2}}} + \alpha {C_{p,DD}}},\end{align}

where $a = {\textstyle{{37} \over 2}}[(\alpha + 1)/\phi - 1]$ and ${f_e} = 0$ (for the fuel-lean case); ${f_e} = (1 + 2a/37)\phi $ and $a = 0$ (for the fuel-rich case); ${f_e} = 0$ and $a = 0$ (for the stoichiometric case). Using (3.22), ${T_g}$ has been calculated for different values of $\boldsymbol{\phi }$ in the radial direction.

3.2.5. Equilibrium flame temperature estimation using a reduced reaction mechanism

To supplement the temperature estimation from the aforementioned simplistic model, a detailed equilibrium estimation was performed considering the reduced mechanism of n-dodecane in the Chemkin software. The reduced mechanism includes the intermediate species, radicals as well as the dissociation of heavy molecules. The skeleton model is based on the lumped POLIMI (an open source reaction kinetics mechanism by researchers at Politecnico di Milano) pyrolysis and oxidation mechanism of hydrocarbon fuels (Ranzi et al. Reference Ranzi, Frassoldati, Stagni, Pelucchi, Cuoci and Faravelli2014), and the model used consists of 130 species and 2323 reactions (Ranzi et al. Reference Ranzi, Frassoldati, Stagni, Pelucchi, Cuoci and Faravelli2014; Stagni et al. Reference Stagni, Cuoci, Frassoldati, Faravelli and Ranzi2014, Reference Stagni, Frassoldati, Cuoci, Faravelli and Ranzi2016).

The equilibrium flame temperature (${T_{eq}}$) is calculated at different equivalence ratios and is plotted against the radial distance alongside the flame temperature estimate (${T_g}$) from the simplified model in figure 5(c). The ${T_{eq}}$ and ${T_g}$ values are found to be in a similar range at the stoichiometric conditions ($\boldsymbol{\phi } = 1$).

The radial variation of the flame temperature (both ${T_g}$ and ${T_{eq}}$) (see figure 5c) peaks near the flame base location ($r = {r_{\boldsymbol{f}}}$), corresponding to the stoichiometric condition, i.e. $\hat{\boldsymbol{\phi }} = 1$. The flame temperature is observed to be lower on either side of the flame base location ($r = {r_{\boldsymbol{f}}}$), hence leading to lower flame propagation velocities on either side, both on the fuel-rich ($r < {r_{\boldsymbol{f}}}$) and fuel-lean ($r > {r_{\boldsymbol{f}}}$) branches. Due to the lower flame propagation speeds on the fuel-rich and fuel-lean branches, the flame self-adjusts its orientation based on the local $\hat{\boldsymbol{\phi }}$ value, as shown in figures 3(a) and 5(b,c), which is also observed experimentally. The concentration gradient at the flame location $(\textrm{d}{Y_F}/\textrm{d}{\tilde{r}_{\boldsymbol{\phi } = {\bf 1}}})$ can be obtained from the radial profile of the equivalence ratio variation, which can be used to evaluate the edge-flame propagation speed (${S_E}$) using the scaling ${S_E}/S_{L_{\boldsymbol{\phi } = {\bf 1}}}^o\sim 1/(\textrm{d}{Y_F}/\textrm{d}\tilde{r})$, for any given fuel using the aforementioned correlation if the local edge-flame orientation is accurately known, as shown by Chung (Reference Chung2007). However, such measurements are beyond the scope of current experiments.

3.2.6. Bluff-body-stabilized wake flame

The relative flow around the bluff-body-stabilized wake flame is characterized by an axisymmetric vortical structure near the rear stagnation region in the Reynolds number range observed in current experiments $(20 < R{e_{local}} < 60)$. The open-brush bluff-body-stabilized wake flame is stabilized at the recirculation zone near the rear stagnation point of the fuel droplet (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020). The flame is fully premixed due to the presence of the recirculation zone that enhances the mixing of fuel and oxidizer. In wake configuration, only around 2 % of the total droplet surface area is exposed to the flame. It can be assumed that the fuel will vaporize only from the small circular region (${A_o}$) of diameter ${d_o}$, near the rear stagnation point (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020) (see figure 6a). The flame is observed to have a quiescent open-brush-like shape similar to that reported in our previous studies (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020), where the burning droplet has a long laminar wake with no detectable flow fluctuations. This can be explained using the vorticity transport equation given below

(3.24)\begin{equation}\frac{{\textrm{D}\omega }}{{\textrm{D}t}} = (\omega \boldsymbol{\cdot }\boldsymbol{\nabla })v - \omega (\boldsymbol{\nabla }\boldsymbol{\cdot }v) + \frac{{\boldsymbol{\nabla }\rho \times \boldsymbol{\nabla }P}}{{{\rho ^2}}} + \vartheta {\nabla ^2}\omega ,\end{equation}

where $\omega $ is the vorticity, v is the relative flow velocity, $\rho $ is the fluid density and P is pressure. The first term on the right-hand side of (3.24) is the vortex stretching term, the second term is the gas expansion term (which can be neglected due to the incompressible flow assumption), the third term represents the vorticity generation due to baroclinic torque and the last term contributes for the viscous dissipation effects. For axisymmetric, incompressible and isobaric conditions, all the terms except the last viscous dissipation term can be assumed to be negligible. It is to be noted that the wake of a burning droplet is at elevated temperatures due to the presence of the flame, which leads to a substantial increase in kinematic viscosity ($\vartheta $) by an order of ${\sim} O({10^1})$. The high-temperature flame present in the droplet wake modifies the flow field in the wake region due to the hot exhaust gases. This open-brush-shaped flame closely resembles a laminar premixed jet flame (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020), exhibiting an open tip with a premixed structure due to intense mixing in the recirculation zone. Hence, assuming a point momentum source, the round jet analogy (Schlichting Reference Schlichting1933; Tyler Landfried et al. Reference Tyler Landfried, Jana and Kimber2015) is applied, which suggests that the jet half-width is proportional to $R{e^{ - 1}}$ and it also increases linearly with the axial distance ($x$). This is mathematically expressed as follows:

(3.25)\begin{equation}\frac{{{d_f}(x = {x_1})}}{{{d_o}}} = \frac{{C{x_1}}}{{Re}},\end{equation}

where ${d_o}$ is the initial jet width at the rear stagnation point, ${d_f}$ is the jet width at an axial distance of ${x_1}$ from the rear stagnation point and $Re$ is the Reynolds number of the relative flow based on ${d_o}$. Equation (3.25) suggests that, for a given $Re$, the flame width increases linearly in the axial direction. Consider flame heights ${h_1}$, ${h_2}$ and the corresponding flame widths ${d_{f,1}}$, ${d_{f,2}}$ at two different velocities ${v_{rel,1}}$ and ${v_{rel,2}}$, respectively. Substituting the values in (3.25) and taking the ratio of the two equations yields

(3.26)\begin{equation}\frac{{{d_{f,1}}}}{{{d_{f,2}}}} = \frac{{{h_1}}}{{{h_2}}}\frac{{R{e_2}}}{{R{e_1}}}\,\mathop \Rightarrow \limits^{ } \,\frac{{{d_{f,1}}}}{{{d_{f,2}}}}\frac{{{h_2}}}{{{h_1}}} = \frac{{{v_{rel,2}}}}{{{v_{rel,1}}}}.\end{equation}

Equation (3.26) shows that the flame width is inversely proportional to the relative flow velocity, whereas the flame height is directly proportional to the relative flow velocity. The flame height can be normalized using the instantaneous droplet diameter ($d$). Hence, the relation between the normalized flame height ($h/d$) with the relative flow velocity is obtained to be

(3.27)\begin{equation}{v_{rel}}\sim \frac{h}{d}.\end{equation}

For simplicity, this proportionality can be rewritten as a linear equation using empirical constants ${s_1}$ and ${s_2}$ as presented in (3.28)

(3.28)\begin{equation}{v_{rel}} = {s_1}\left( {\frac{h}{d}} \right) + {s_2}.\end{equation}

From our previous experimental data (Pandey et al. Reference Pandey, Basu, Vadlamudi, Potnis and Chattopadhyay2020), the instantaneous flame height characteristics for the whole range of v_rel were obtained using Otsu's thresholding on the flame images. Thus, the obtained relative flow velocity is normalized with the laminar flame speed (${v_{rel}}/S_L^o$) and is plotted against the normalized flame height $(h/d)$ in figure 6(c). The error bars in the plot correspond to the flame height variation (< ±10 %) within each bin containing multiple data points from multiple runs considered in a similar range of relative velocity or flame height value. The plot shows a linear correlation between $h/d$ and the ${v_{rel}}$ normalized with the laminar flame speed ($S_L^o$). The data can be used to estimate the coefficients of (3.28). The obtained empirical relation is presented in (3.29)

(3.29)\begin{equation}\frac{{{v_{rel,bluffbody}}}}{{S_L^o}} = 2.22\frac{h}{d} + 0.16.\end{equation}

Figure 6. (a) Schematic of bluff-body-stabilized wake flame. The flame is represented by the blue colour, and the yellow line represents the edge of the recirculation zone. The schematic of flow inside the recirculation zone is shown using arrows (blue for air and red for the fuel stream). (b) Schematic showing heat transfer into and out of the recirculation zone using arrow marks. (c) Normalized flame height ($h/d$) v/s normalized relative flow velocity at the droplet (${\tilde{v}_{rel}} \sim {v_{rel}}/S_L^o$). The error bars represent an uncertainty of ±10 %. (d) The instantaneous velocity experienced by the droplet (${v_{rel, inst}}$) plotted against time after exit (t) at co-flow velocities 2.2 and4 m s⁻¹ for ${\textrm{v}_o} = \textrm{const}\textrm{.}$ (${v_{o,i}}$) and ${v_o}$ variable along the axial distance ($z$) from the co-flow tube.

3.2.7. Bluff-body wake flame stabilization mechanism

In current experiments, for the bluff-body-stabilized wake flame, the relative velocity between the surrounding air and the droplet varies between 0.65 and 2 m s⁻¹. The corresponding Re based on the droplet diameter (d) varies between 29 and 100. In this Re range (20 < Re < 130), airflow past a sphere (droplet) establishes a steady symmetrical vortical structure in the droplet wake (Taneda Reference Taneda1956; Goldburg & Florsheim Reference Goldburg and Florsheim1966; Kalra, T.R. Reference Kalra and U1971; Sakamoto & Haniu Reference Sakamoto and Haniu1990). Beyond this, low-frequency wave-like oscillations ensue (for Re < 270) (Taneda Reference Taneda1956; Sakamoto & Haniu Reference Sakamoto and Haniu1990). The regime considered in the current experiments falls well within the range where steady recirculation vortices are expected in the droplet wake.

Results from the experimental and numerical studies (Taneda Reference Taneda1956; Kalra, T.R. Reference Kalra and U1971; Fornberg Reference Fornberg1988; Dandy & Dwyer Reference Dandy and Dwyer1990; Lee Reference Lee2000) show that the wake length from the droplet centre varies linearly with log(Re) (supplementary figure S4). Thus, the wake length is expected to vary between 4.8 and 3 mm, corresponding to the Re range. At a given instance, a quasi-steady approximation can be used to estimate the wake flame stand-off distance (${d_{st}}$) (figure 6a). The outer air vortical structures in the wake region force the vaporized fuel stream to curl and form a counter-rotating vortex. The interaction between the air and fuel vortex ensures the mixing of the constituent reactants resulting in a premixed wake flame stabilized near the recirculation zone. Vortical structures inside the recirculation zone promote the exchange of mass, momentum and energy with the entraining air stream flow.

Following a similar procedure as proposed by Kundu et al. (Reference Kundu, Banerjee and Bhaduri1977), a simplified theoretical model for two-dimensional bluff-body-stabilized premixed flames is used to estimate the flame stand-off distance (${d_{st}}$).

The mass exchange rate at the recirculation zone boundary (${\dot{m}_e}$) is obtained as follows:

(3.30)\begin{equation}{\dot{m}_e}\sim \frac{{{\rho _r}{V_R}}}{{{\tau _r}}},\end{equation}

where ${\rho _r}$ is the average density of the fuel–air mixture inside the recirculation zone whose volume is ${V_R}$ (3.31) and ${\tau _r}$ is the mean residence time of the flow inside the volume

(3.31)\begin{equation}{V_R} = {F_{RV}} \cdot L \cdot b.\end{equation}

Where ${F_{RV}}$ is the shape factor, and L, b are the dimensions of the recirculation zone. The mean residence time can be estimated as follows (Kundu et al. Reference Kundu, Banerjee and Bhaduri1977):

(3.32)\begin{equation}{\tau _r} = 104\frac{b}{{{U_\infty }}},\end{equation}

where ${U_\infty }$ is the mean velocity of the premixed reactants into the flame front. The relation is valid for a wide range of bluff-body dimensions and equivalence ratios and is almost unchanged for bluff bodies with different shapes (Bovina Reference Bovina1958). Using the above relations (3.31 and 3.32), (3.30) can be simplified as

(3.33)\begin{equation}{\dot{m}_e}\sim \frac{{{\rho _r}{F_{RV}}{U_\infty }}}{K}L.\end{equation}

If the flame is stabilized at an axial distance of $x = {d_{st}}$ in the droplet wake, it divides the lateral surface of the recirculation zone into two zones, one which interacts with the products of combustion (figure 6b: zone B) and the other that interacts with the ambient (figure 6b: zone A). Hence, using the enthalpy exchange, the heat gained by the recirculation zone (at Zone B) can be estimated as

(3.34)\begin{equation}{\dot{q}_{in,r}}\sim\frac{{{\rho _r}{F_{RV}}{U_\infty }}}{K}(L - {d_{st}})({C_{pf}}{T_f} - {C_{pr}}{T_r}),\end{equation}

where ${C_{pf}}$, ${T_f}$ and ${C_{pr}}$, ${T_r}$ are the specific heats and the temperature of the products stream and premixed reactants inside the recirculation zone, respectively. Likewise, the heat lost by the recirculation zone (at zone A) can be estimated as

(3.35)\begin{equation}{\dot{q}_{out,r}}\sim\frac{{{\rho _r}{F_{RV}}{U_\infty }}}{K}({d_{st}})({C_{pr}}{T_r} - {C_{p\infty }}{T_\infty }).\end{equation}

Where ${C_{p\infty }}$ and ${T_\infty }$ are the outer ambient properties.

For a quasi-steady flame, assuming heat balance inside the recirculation zone, i.e. equating (3.34) and (3.35), yields an estimate for the standoff distance given as

(3.36)\begin{equation}\frac{{{d_{st}}}}{L}\sim \frac{{{C_{pf}}{T_f} - {C_{pr}}{T_r}}}{{{C_{pf}}{T_f} - {C_{p\infty }}{T_\infty }}}.\end{equation}

Considering the premixed reactants to be at the average value of flame temperature and wet-bulb temperature of the fuel and assuming stoichiometric premixture conditions for simplicity, ${d_{st}}$ can be evaluated using (3.36) as

(3.37)\begin{equation}\frac{{{d_{st}}}}{L}\sim 0.54.\end{equation}

The recirculation zone length ($L$) can be estimated by subtracting the droplet radius from the aforementioned wake length (from the droplet centre). Using (3.37), the range of recirculation zone lengths is found to be 3.3 to 1.5 mm, and the stand-off distance is between 1.8 and 0.85 mm (based on (3.37)) for the Re range 29 < Re < 100, which is in good agreement with the experiments.

3.5.1. Transition 1: edge-stabilized wake flame to open-brush wake flame $(20 < R{e_{local}} < 30)$

It is to be noted that the bluff-body wake flame to edge-stabilized flame transition occurs in the non-equilibrium regime. Whereas the reverse, edge-stabilized flame to bluff-body flame, transition occurs after the flame has already achieved equilibrium.

For ${v_{o,i}}\sim 3\ \textrm{m}\ {\textrm{s}^{ - 1}}$ case (${\tilde{v}_{o,i}}\sim 4.8$), during edge-stabilized to bluff-body flame transition (equilibrated regime), the edge-stabilized flame height initially starts to increase in response to the increasing instantaneous relative velocity of the flow (${v_{rel,inst}}$) due to gravitational acceleration between 63 and 67 ms (figure 2b). However, between 67 and 79 ms ($\tau \sim 670\unicode{x2013} 790$), the flame height starts to decrease until it reaches a minimal value. During this process, the flame transitions into an open-brush shape. This decreasing phase of flame height before transition demonstrates the re-adjustment of the flame from one stabilization mechanism to another. The temporal flame intensity variation plot in figure 8(a) reflects a similar trend as that of the flame height variation for transition 1.

Figure 8. Flame luminosity variation normalized using time-averaged luminosity value (${I_f}/{I_{avg}}$) plotted against normalized time ($\tau \sim t/{t_r}$) for the corresponding flame images for ${v_{o,i}}\sim 3\ \textrm{m}\ {\textrm{s}^{ - 1}}$ during the transition of: (a) transition 1, edge-stabilized wake flame (yellow) to bluff-body-stabilized wake flame (blue),(b) transition 1: backwards, open-brush flame (blue) to edge-stabilized flame (yellow). The transition zone is shown in green colour. (c) Variation of normalized flame luminosity and flame images for ${v_{o,i}}\sim 4\ \textrm{m}\ {\textrm{s}^{ - 1}}$ during the enveloping event (transition 2: edge-stabilized to envelop) plotted against $\tau \sim t/{t_r}$. (d) Equivalence ratio ($\boldsymbol{\phi }$) obtained for different flow velocities and the equivalence ratio computed at the pure diffusion limit (${\boldsymbol{\phi }_{\boldsymbol{diff}}}$), all plotted against varying axial flame stand-off distance which is normalized using droplet diameter (x/d).(e) Schematic of the mass diffusion of fuel and air for the pure diffusion flame limit.

In the case of the reverse transition i.e. bluff-body wake flame to edge-stabilized flame (non-equilibrated regime), a slight increase in the normalized flame height ${(h/d)}$ is observed between 26 and 34 ms ($\tau \sim 260\unicode{x2013} 340$) during which it transitions to an edge flame, and then it starts to decrease further before enveloping (figure 2b). The temporal flame intensity variation plot in figure 8(b) reflects a similar trend to that of the flame height variation for transition 1 in the reverse direction. In both cases, the flame height variation and the flame luminosity variation are in agreement with each other's trends, as the net flame luminosity indirectly depends on the flame dimensions.

In edge-stabilized flame, the flame is stabilized at the flame base, as explained in § 3.2.1, due to the balance between the local flame speed and flow velocity at the droplet. However, once the instantaneous upstream velocity (${v_{rel,inst}}$) exceeds the edge-flame speed (${S_E}$), the edge-stabilized flame cannot be stabilized, which may result in a blowoff. Nevertheless, current experiments show that the flame alters its stabilization mechanism to remain stabilized at a higher ${v_{rel,inst}}$, in the droplet wake, beyond $Re > 30$.

It is known that the flow over a sphere shows an attached flow for $0 < Re < 20$, the flow detaches with symmetric wake eddies for $20 < Re < 130,$ produces slight periodic oscillations for $130 < Re < 300$ and shows vortex shedding for $Re > 300 $ (Taneda Reference Taneda1956; Goldburg & Florsheim Reference Goldburg and Florsheim1966; Kalra, T.R. Reference Kalra and U1971; Sakamoto & Haniu Reference Sakamoto and Haniu1990). The value of $Re$ is calculated for the droplet in the current experiments to check the flow characteristics and the Re range for the edge-stabilized flame is $9 < Re < 18$ and $29 < Re < 100$ for the bluff-body-stabilized flame configuration. The wake is characterized by an attached flow without eddies (see figure 4b) in the case of the edge-stabilized flame ($9 < Re < 18$), and it is expected to have a recirculation zone in the case of the bluff-body-stabilized flame ($29 < Re < 100$), see figure 6(a).

Hence, in case of transition 1, when the Reynolds number reaches the critical value $Re\sim 20\unicode{x2013} 30$, the flow over the droplet detaches from the wake to form recirculating eddies near the rear stagnation zone. This disturbs the attached flow field in the wake, which was responsible for generating the equivalence ratio gradient field necessary for edge stabilization. Thus, due to the formation of the rear stagnation vortical structures for ($Re > 30$), the flame shape alters from a dome-shaped edge-stabilized to a bluff-body-stabilized configuration (open-brush-shaped wake flame).

Hence, during transition 1, the edge-flame height suddenly starts to decrease, re-adjusting to flow field at higher Reynolds numbers (see figure 2b) wherein the stabilization mechanism itself is altered to a bluff-body wake stabilization after $\tau \sim 790$. During the reverse of transition 1, as the open-brush flame transitions back to edge-stabilized flame in the non-equilibrium regime, the stabilization mechanism changes from bluff-body stabilization to edge-flame stabilization as the Reynolds number becomes $Re < 20$, which results in an attached flow. Edge-stabilized flame gets established in the absence of rear stagnation vortical structures corresponding to the instantaneous flow velocity (${v_{rel,inst}}$).

3.5.2. Transition 2: edge-stabilized wake flame to enveloped flame $(0 < R{e_{local}} < 5)$

It is known that, as the velocity is reduced, the edge-stabilized flame comes closer to the droplet, forming an enveloped flame. The numerical simulation conducted by Chiu (Reference Chiu2000) has also confirmed that, as Re is reduced, the wake flame transitions into a side flame (partially enveloping), which finally envelops the droplet below a critical Re. It is known that the enveloped flame is a pure diffusion flame $(R{e_{local}} \to 0)$, whereas the edge-stabilized flame primarily has a premixed flame front (Chen & Lin Reference Chen and Lin2012). Hence, while the edge-stabilized flame is trying to envelop the droplet, the extent of the premixed mode decreases, and the extent of the diffusion mode of mass transfer increases. Finally, when it is fully enveloped, it becomes a pure diffusion flame where fuel diffuses from the inside, and the oxidizer diffuses from the outside. As discussed in § 3.1, for ${v_{o,i}}\sim 4\ \textrm{m}\ {\textrm{s}^{ - 1}}$ $({\tilde{v}_{o,i}}\sim 6.3)$, the edge-stabilized wake flame $(5 < R{e_{local}} < 20)$ tries to envelop the droplet around 26 ms ($\tau \sim 260$), thus fully transitioning into an enveloped flame around 30 ms ($\tau \sim 300$). The similar trend is reflected in the flame luminosity plot, as shown in figure 8(c), where ${I_f}$ starts to increase substantially and peaks at 30 ms ($\tau \sim 300$) corresponding to the fully enveloped state of the droplet flame. It is to be noted that the flame luminosity is found to be maximum in the fully enveloped state compared with the wake flame configuration.

From the experiments, the relative flow velocity (${v_{rel}}$) is found to be in the range of 0.2–0.4 m s⁻¹ $(0.31 < {\tilde{v}_{o,i}} < 0.64)$, for the edge-stabilized wake flame. Following a similar procedure to § 3.2.1, the equivalence ratio (${\boldsymbol{\phi }_{\boldsymbol{m}}}$) is calculated using (3.7–3.17) for different values of ${v_{rel}}$ in the mentioned range for varying values of flame stand-off distance along the normalized axial direction ($x/d$) and is plotted in figure 8(d). This shows that the equivalence ratio plot and the axial stoichiometric location (${x_{\boldsymbol{\phi } = {\bf 1}}}$) come closer to the droplet as ${v_{rel}}$ decreases. Considering the pure diffusion limit, the axial variation of the equivalence ratio (${\boldsymbol{\phi }_{\boldsymbol{diff}}}$) has to be estimated and then compared with the equivalence ratio corresponding to different ${v_{rel}}$ (for the premixed mode).

For the pure diffusion flame case, since the flame tends to envelop the droplet, the schematic depicted in figure 8(e) has been considered. It shows that only fuel and oxidizer diffusion is considered for this case. It is assumed that the flame is partially enveloping the droplet, subtending an angle of $60^\circ$ at the droplet centre, for simplicity (see figure 8e). The area of the droplet and flame are assumed to be circular, whose diameter is calculated geometrically using the angle subtended. The equivalence ratio for pure diffusion condition (${\boldsymbol{\phi }_{\boldsymbol{diff}}}$) is calculated using (3.17), whose fuel–air ratio is obtained based on the fuel diffusion mass flow rate (${\dot{m}_{F,diff}}$) and oxidizer diffusion mass flow rate (${\dot{m}_{Ox,diff}}$) given by (3.43) (see figure 8e)

(3.43a)\begin{gather}{\dot{m}_{F,diff}}\sim {\rho _F}\frac{\mathrm{\mathfrak{D}}}{{{d_{st}}}}\frac{{\rm \pi} }{4}d_{o,e}^2,\end{gather}

(3.43b)\begin{gather}{\dot{m}_{Ox,diff}}\sim {\rho _{Ox}}\frac{\mathrm{\mathfrak{D}}}{{{t_{flame}}}}\left( {\frac{{\rm \pi} }{4}d_{f,e}^2} \right) + {\rho _{Ox}}\frac{\mathrm{\mathfrak{D}}}{{{d_{o,e}}/2}}{\rm \pi} \frac{{{d_{o,e}} + {d_{f,e}}}}{2}{d_{st}},\end{gather}

where ${t_{flame}}$ is the length scale corresponding to flame thickness (Vancoillie et al. Reference Vancoillie, Demuynck, Galle, Verhelst and van Oijen2012), and $\mathrm{\mathfrak{D}}$ is the mass diffusion coefficient. Since the fuel has to travel the stand-off distance (${d_{st}}$) to reach the flame, ${d_{st}}$ is used as the length scale, whereas the oxidizer is freely available all around the flame, so the oxidizer just has to diffuse through a length scale comparable to the flame thickness (${t_{flame}}$).

Figure 8(d) shows that the axial variation of the equivalence ratio corresponding to the diffusion limit (${\boldsymbol{\phi }_{\boldsymbol{diff}}}$) and the ${\boldsymbol{\phi }_{\boldsymbol{m}}}$ variation for ${v_{out}} \sim 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}$ (${\boldsymbol{\phi }_{{v_{out}} = 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}}}$), both match near the stoichiometric conditions, i.e. ${\boldsymbol{\phi }_{{v_{out}} = 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}}} \approx {\boldsymbol{\phi }_{\boldsymbol{diff}}}$, where the flame gets stabilized. As it is hypothesized that, while the edge-stabilized flame transitions to an enveloped flame, the premixed flame front gradually diminishes, leaving behind the diffusion flame alone. The variations of the equivalence ratio for the two cases matching with each other, i.e. ${\boldsymbol{\phi }_{{v_{out}} = 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}}} \approx {\boldsymbol{\phi }_{\boldsymbol{diff}}}$, suggests that the velocity scale corresponding to the case of ${v_{out}} = 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}$ is of the same order as that of the pure diffusion limit (${v_{NC}}\sim 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}$). This is conjectured to be the transition condition between edge-stabilized flame and enveloped flame. Figure 8(d) shows that the transition between the edge-stabilized and diffusion flames approximately occurs at ${v_{out}} = 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}$. To verify this conclusion, the average value of $h/d$ ratio for the edge-stabilized wake flame corresponding to the beginning of the enveloping event ($\tau \sim 260$; figure 8c) is extracted from the experimental data $(h/{d_{before~envelop}} \le 0.45)$. When this value is substituted into the ${v_{rel}}~\&~h/d$ relation for the edge-stabilized flame (3.6), the value of ${v_{rel}}$ is obtained to be ${v_{rel,h/d = 0.45}} = 0.188\ \textrm{m}\ {\textrm{s}^{ - 1}}$ $({\tilde{v}_{rel,h/d = 0.45}}\sim 0.3)$. This value is in very close agreement with the ${v_{rel}}\sim 0.2\ \textrm{m}\ {\textrm{s}^{ - 1}}$ $({\tilde{v}_{rel}}\sim 0.31)$ value obtained using the equivalence ratio method.