3.1. Flow characterisation

After the system is triggered, the blast wave generated travels away from the electrodes (origin) towards the combusting pendant droplet coaxially and reaches the flame location at t = t_s. The blast wave is visualised through schlieren imaging as it passes by the droplet flame. The flame dynamics during the interaction is observed to vary drastically with the shock tube configuration (open field vs focused) and charging voltages. Also, M_s is observed to decay temporally as the blast wave propagates against the quiescent ambient medium. However, all cases are characterised by the value of the Mach number (M_s) near the location of the droplet that is measured along the centreline. The open-field blast wave showed the lowest M_s values as the blast wave expanded outwards in all directions from the point of the wire blast. However, a higher M_s is obtained in the case of blast focusing (with shock tube), where the blast wave is planar inside the shock tube flow channel (Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016) and starts to expand only after exiting the shock tube. Additionally, after the blast wave exits the shock tube and propagates downstream, an induced flow is observed behind the blast wave, which exits the shock tube and interacts with the combusting droplet after a time delay (see figure 2).

Figure 2. (a) Time series of the schlieren imaging of the blast wave propagation near the shock tube exit. (b) Schematic of the blast wave propagation near the shock tube exit.

Figure 2(a) shows the time series of the flow exiting the shock tube channel with an expanding blast wave. Figure 2(b) gives a schematic of the flow. As the planar propagating blast wave inside the shock tube exits into the quiescent ambient, the blast wave expands outward similar to a radially expanding blast wave, exhibiting significant curvature as it propagates. This radial expansion results in a continuous temporal decay of the shock strength and M_s as the blast wave propagates further. Since the blast wave is almost planar at that instant as it exits the shock tube, the shock Mach number at the exit (M_{s ,e}) is measured and normal shock relations can be used to evaluate the characteristics of the jet upstream of the blast wave. Table 1 gives the details of the initial upstream flow that is exiting the shock tube. On the other hand, in the case of the open-field configuration, a cylindrical blast wave propagates radially outward away from the copper wire having Mach numbers (M_s) of 1.03, 1.05, 1.055, 1.065 for the charging voltages of 5, 7, 7.8 and 10 kV, respectively. Thus, the flow imposed by the blast wave in either of the configurations has to be characterised to understand the interaction with the droplet flame.

Table 1. Pressure ratio, jet velocity and Reynolds number upstream of the blast wave corresponding to different focused cases having different M _s at the droplet.

Using a similar methodology as Bach & Lee (Reference Bach and Lee1970) (which is valid for the M_s range of current experiments, i.e. 1.02 < M_s < 1.6), a power-law density profile can be assumed behind the blast wave, and the velocity profile can be obtained from the mass conservation differential equation.

Initially, we define the non-dimensional parameters for velocity and density as follows:

(3.1)\begin{gather}\phi (\xi ,\eta ) = \frac{{u(r,t)}}{{{{\dot{R}}_s}(t)}},\end{gather}

(3.2)\begin{gather}\psi (\xi ,\eta ) = \frac{{\rho (r,t)}}{{{\rho _o}}},\end{gather}

where, $\theta (\eta ) = {R_s}{\ddot{R}_s}/{\dot{R}_s}$, $\; \xi = r/{R_s}(t)$, $\eta = c_o^2/\dot{R}_s^2 = 1/M_s^2$, $y(\eta ) = {({R_s}/{R_o})^{j + 1}}$.

The conservation of mass for unsteady one-dimensional adiabatic motion of a perfect gas behind the expanding blast wave is considered

(3.3)\begin{equation}(\phi - \xi )\frac{{\partial \psi }}{{\partial \xi }} + \psi \frac{{\partial \phi }}{{\partial \xi }} + j\phi \frac{\psi }{\xi } = 2\theta \eta \partial \psi /\partial \eta .\end{equation}

Here, R_o is the characteristic explosion length, R_s is the instantaneous shock radius, u is the velocity field, ρ is the density field and c_o is the speed of sound. The boundary conditions at the shock front ξ = 1 are obtained from the standard normal shock relationship given by

(3.4)\begin{gather}\phi (1,\eta ) = \frac{{2(1 - \eta )}}{{\gamma + 1}},\end{gather}

(3.5)\begin{gather}\psi (1,\eta ) = \frac{{\gamma + 1}}{{\gamma - 1 + 2\eta }}.\end{gather}

Thus, following Bach & Lee's (Reference Bach and Lee1970) formulation, the density profile can be assumed as follows:

(3.6)\begin{equation}\psi (\xi ,\eta ) = \psi (1,\eta ){\xi ^{q(\eta )}}.\end{equation}

The value of the exponent ‘q’ can be obtained using the mass integral obtained from the conservation of the mass enclosed by the blast wave at a given instant

(3.7)\begin{equation}\int_0^1 {\psi {\xi ^j}\,\textrm{d}\xi } = \frac{1}{{j + 1}},\end{equation}

where the values j = 0, 1, 2 are for planar, cylindrical and spherical blast waves, respectively. Thus, the exponent of the density power law is obtained to be

(3.8)\begin{equation}q(\eta ) = (j + 1)[\psi (1,\eta ) - 1].\end{equation}

The density profile can be obtained from (3.5), (3.6) and (3.8) at any given instant for a specific Mach number (M_s). Thus, the mass conservation equation (3.3) can be rewritten using the density profile as follows:

(3.9)\begin{equation}\frac{{\partial \phi }}{{\partial \xi }} + (q + j)\left( {\frac{\phi }{\xi }} \right) = q + \frac{{2\theta \eta }}{{\psi (1,\eta )}}[1 + (j + 1)\psi (1,\eta )\,\textrm{ln}\,\xi )]\frac{{\partial \psi (1,\eta )}}{{\partial \eta }}.\end{equation}

Substituting the boundary condition at the origin as $\phi (0,\eta ) = 0$, the velocity profile is obtained as

(3.10)\begin{equation}\phi = \phi (1,\eta )\xi (1 - \varTheta \,\ln \,\xi ),\end{equation}

where

(3.11)\begin{equation}\varTheta = \frac{{ - 2\theta \eta }}{{\phi (1,\eta )\psi (1,\eta )}}\frac{{\partial \psi (1,\eta )}}{{\partial \eta }}.\end{equation}

The $\theta $ vs $\eta $ relation can be obtained by a similar methodology as shown by Bach & Lee (Reference Bach and Lee1970), and then (3.10) can be used to obtain the temporally decaying velocity profile at a given location ‘r’ behind the blast wave for a given M_s.

The shock trajectory can be experimentally obtained by tracking the shock location along the centreline using high-speed schlieren imaging. In the case of the open-field blast wave, the shock is observed to propagate away from the wire location cylindrically outward and this transitions into an ellipsoidal shape later (Chiu, Lee & Knystautas Reference Chiu, Lee and Knystautas1977). Hence, the wire location can be considered as the origin (r = 0) for the open-field blast wave and it can be considered as a cylindrical blast wave at the droplet location as within the time scale of ~O(10⁻¹) ms, the curvature effects at the edges during the transition to ellipsoidal shape would not reach the centreline cylindrical blast flow. Hence, the cylindrical assumption can be used for the open-field case. The spherical blast wave assumption can also be considered at the droplet location with respect to copper wire dimensions $({L_t} + {L_d} \gg w)$. Nevertheless, neither cylindrical nor spherical formulations can fully capture the three-dimensional effects of the ellipsoidal shape and the transition from cylindrical shape. On the other hand, when the shock tube focusing is used, the blast wave is focused inside the shock tube along its length, which gets modulated as a planar blast wave as it propagates inside the shock tube (see figure 3a). However, when the blast wave exits the tube, it expands radially outward at the tube opening into the ambient atmosphere, similar to a blast wave. Experimentally, it is evident that the shape of the blast wave transitions from planar to cylindrical as it exits the rectangular shock tube opening. This is evident from the temporal variation of the radius of curvature (R_s) of the blast wave, which is observed to approach the instantaneous centreline distance (x) from the shock tube exit as the blast wave expands outward (see figure 3l). This implies that the blast wave tends to expand outward cylindrically while approximately maintaining the centre of curvature at the shock tube exit location. That means the planar blast wave inside the shock tube transitions into a cylindrical blast wave with the shock tube exit location as its centre of curvature (origin). Hence, for simplicity, this expanding cylindrical blast wave can be assumed to be similar to a blast wave that has originated from the shock tube exit location as its virtual origin (r = 0) and is expanding radially outward into the ambient atmosphere, as shown in figure 3(a). Thus, this expanding blast wave is assumed to be equivalent to the hypothetical cylindrical blast wave (j = 1) originating from the shock tube opening, and the instant when the blast wave exits the shock tube opening is considered to be the reference time ($t_{v} = 0$, virtual time), as shown in figure 3(a). As shown in figure 3(a), the length of the shock tube is L_t, and the droplet is placed at a distance of L_d from the shock tube opening. Since the droplet position is maintained to be the same for both open and focused cases, the location of the droplet in the open-field case is r = L_t + L_d (with r = 0 at the copper wire location).

Figure 3. (a) Schematic of the blast wave propagation in open-field and focused configurations depicting the dimensions, droplet location and the cylindrical blast wave virtual origin (r = 0), reference time (${t_v} = 0$) for both cases. (b,d) Variation of the Mach number along its trajectory (radial distance from origin, r = 0) for focused and open-field blast wave cases, respectively. Here, ‘kV’ refers to the charging voltage for blast wave generation and ‘B’, ‘S’, ‘Open’ denote ‘big channel (2 cm × 10 cm c/s)’, ‘small channel (2 cm × 4 cm c/s)’ and ‘open-field’ configurations, respectively. (c,e) Temporal variation of the Mach number for focused and open-field blast wave cases, respectively. ( f–i) Theoretical and experimental plots of Mach number (M_s) vs virtual time (${t_v}$) of the blast waves generated for 4 kV big channel, 6 kV, 8 kV small channel and 5 kV, 10 kV open-field configuration, respectively. (j) Log–log plot of shock radius vs time. (k) Temporal variation of the Mach number (M_s) for different cases plotted against time, t (in ms) with t = 0 at the time of the explosion. (l) Temporal variation of the deviation of instantaneous axial distance from shock tube exit (x) and instantaneous radius of curvature (R_s) of the blast wave that is exiting the shock tube. The error bars represent the maximum error arising due to the shock location measurement variation due to pixel resolution.

Figure 3(b,d) shows the variation of Mach number along the shock trajectory location (mm) measured experimentally as the shock propagates for focused (i.e. big channel (B) and small channel (S)) and open-field (Open) cases, respectively, designated based on different charging voltages (kV) and shock tube configuration. Higher charging voltages (kV) and smaller shock tube dimensions result in higher Mach numbers (M_s), and the open-field cases exhibited the lowest Mach numbers (M_s < 1.08) for the same kilovolts (kV). The droplet location L_d (in case of focused) and L_t + L_d (for open-field configuration) are indicated in figure 3(b,d) with the help of a vertical orange dotted line. This implies the droplet and flame starts to experience the imposed flow of the blast wave only to the right side of this line. Figure 3(c,e) shows the temporal variation of the Mach number (M_s) (measured experimentally) plotted against virtual time (${t_v}$, ms). The theoretical blast wave trajectory can be obtained using the following expression (Bach & Lee Reference Bach and Lee1970):

(3.12)\begin{equation}\frac{{{c_o}t}}{{{R_o}}} ={-} \frac{1}{2}\int_0^\eta {\frac{{{y^{1/(j + 1)}}\,\textrm{d}\eta }}{{\theta {\eta ^{1/2}}}}} .\end{equation}

Where $\theta (\eta )$ is the shock decay coefficient and $y(\eta ) = {({R_s}/{R_o})^{j + 1}}$ is the dimensionless instantaneous shock radius.

The parameters $\theta (\eta )$ and $y(\eta )$ were obtained by numerically solving the pair of first-order differential equations that are given by Bach & Lee (Reference Bach and Lee1970), which are obtained by substituting the profiles of different properties into the energy integral. The value of the characteristic explosion length (R _o) is dependent on the initial energy input to the blast wave explosion. However, in the current experiments, the assumption of mass and energy conservation of Bach & Lee (Reference Bach and Lee1970) is not fully valid due to the entrainment effects that start to occur during the later stages of the shock propagation (>1 ms, from the explosion). Hence, the exact value of the equivalent explosion energy cannot be obtained for the current experiments. This limitation can be circumvented by using the shock arrival time at a specific location (r). Using the experimental value of the Mach number (M_s) at a specific location as a reference, the theoretical value of M_s is iteratively evaluated at that location using different values of R_o, and the value of R_o is obtained corresponding to a given experimental run. Thus, following Bach & Lee (Reference Bach and Lee1970) and Chandra et al. (Reference Chandra, Sharma, Basu and Kumar2023), the theoretical shock arrival time (t_{arr ,Th}) is evaluated by substituting the value of R_o in (3.12), which is in agreement with the experimentally obtained shock arrival time (t_{arr ,exp}) at the droplet location (see supplementary figure S1).

Thus, the shock trajectories, i.e. R_s(t) and Mach number (M_s) evolution, have been evaluated theoretically for the corresponding values of R_o using j = 1 for an open-field blast wave (cylindrical assumption at the droplet location). This has been plotted (solid lines) in figure 3(i) for the open-field blast wave cases (j = 1), with origin (r = 0) at the location of copper wire and time t = 0 at the time of the explosion. The experimentally measured shock trajectory evolution has also been plotted and is found to be in good agreement with the theory in the initial stages (~0–1 ms after the explosion). Similarly, the theoretical shock trajectories have also been obtained for the cylindrical blast waves (j = 1) exiting the shock tube (focused cases), taking the reference time of the hypothetical explosion (${t_v} = 0$) at the shock tube exit, which is assumed to create a similar blast wave that expands cylindrically outward, with the virtual origin at the shock tube exit (r = 0). Similarly, the theoretical shock trajectory of the planar blast wave (j = 0) travelling inside the shock tube from the time of explosion (t = 0) of the copper wire has been obtained using the arrival time of the planar shock at the shock tube opening. The experimentally measured shock trajectories (points), and the theoretically obtained shock trajectories (solid lines) are plotted in figure 3( f–h) for different focused cases against the virtual time ${t_v},\; in\;ms$ (where ${t_v} = 0$ is the instant the shock location is at the shock tube exit).

The experimental shock trajectories are in good agreement with the theoretical estimates (both planar and cylindrical regimes), showing the coefficient of determination obtained to be ${R^2}\sim 0.88{-}0.97$ for the deviation of the theoretical shock trajectory and experimental measurement for all cases except 4kV_B, see figure 3( f–h). The deviation in the 4kV_B case can be attributed to the experimental limitation of lack of contrast in the schlieren images (absence of distinct shock boundary) resulting in higher error in shock location measurement for the 4kV_B case. However, the normalised root mean squared error of the data points of 4kV_B from theoretical prediction is found to be less than 1.5 %, which is within the acceptable range. From the plots, it can be clearly observed that the experimental shock trajectory initially follows the theoretical estimate of the planar shock trajectory (from the time of explosion) and then gradually shifts and deviates to follow the cylindrical shock trajectory estimate (with the shock tube exit as the virtual origin). This establishes the previously hypothesised planar to cylindrical (centre at shock tube exit) transition of the propagating blast wave after exiting the shock tube opening.

The temporal variation of the shock location is tracked using high-speed schlieren imaging and is plotted in figure 3(k) for all the cases. Figure 3(j) shows that all the plots merge into one single straight line in the log–log plot of shock radius (R_s) vs time (${t_v}$) and the slope of the line is found to be near unity. It is shown that the scaling for the strong blast wave propagation is ${R_s}\sim t_v^{2/5}$ (exponent ~2/5) and for the weak acoustic limit, the weak blast wave propagation scaling is linear ${R_s}\sim t$ with (exponent ~1) (Wei & Hargather Reference Wei and Hargather2021; Díaz & Rigby Reference Díaz and Rigby2022). This suggests that all the blast waves observed in the current experiments are near the weak blast wave acoustic limit (exponent near unity). However, the slopes of the individual cases (in the log–log plot) are found to vary between 0.89 and 0.99. This shows that the current experiment is in the transition regime between the strong blast wave and acoustic limit. The plots in figure 3(b–i) show the minimal variation in M_s for open-field cases, suggesting that they are near the acoustic limit. In contrast, the focused blast wave cases showed significant temporal variation of M_s.

3.2. Interaction with a combusting droplet

As mentioned before, the droplet is placed coaxially at a distance of ${L_t}\; + \; {L_d}$ from the copper wire, i.e. ${L_d}$ from shock tube opening in both the open-field and focused configurations. After the pendant droplet is ignited and shock set-up is triggered, the blast wave expands radially outward and interacts with the droplet flame in the open-field case, whereas in the focused case, the blast is directed along the shock tube, and interacts with the droplet flame after it exits the shock tube. Figures 4(a,b) and 4(c–f) show the flame interaction with the blast wave generated in the open-field and focused configurations for different charging voltages. Since M_s varies temporally, the instantaneous value of M_s when the blast wave passes the droplet location is considered as the reference value to characterise different cases and is mentioned in figure 4, corresponding to each case. The subsequent discussion will categorise the different configurations of the shock tube at varying charging voltages as follows: ‘B’, ‘S’ and ‘Open’ for big, small sections and open-field configurations, respectively, accompanied by their corresponding kilovolt (kV) values. For instance, the notation ‘5kV_Open’ denotes a charging voltage of 5 kV in an open-field configuration.

Figure 4. Time series of the schlieren imaging depicting the overall interaction of the droplet flame with the shock flow for open-field configuration with charging voltage (a) 7 kV, (b) 10 kV; for bigger shock tube channel (2 cm × 10 cm c/s) with charging voltages (c) 4 kV, (d) 7 kV; and for the smaller shock tube channel (2 cm × 4 cm c/s) with charging voltages (e) 6 kV, ( f) 8 kV. Note: the schlieren snapshots taken at 7000 fps are chosen to be displayed to better represent the flame dynamics, extinction (due to brighter flame) for cases (b) and (c). Rest of the snapshots are obtained at 75 000 fps.

Along with the blast wave visualisation and density gradient contrast corresponding to the plume of hot gases around the flame, the bright flame tip is also visible in the schlieren images. Figure 4(a,b) depicts the time series of the schlieren images, showing the blast wave propagation and flame response for the open-field configuration at two charging voltages. The Mach numbers (M_s) for the open-field configuration are less than 1.1 and the flame response is observed to occur over a longer time scale of order ${\sim} \; O({10^0} {-} {10^1})\;\textrm{ms}$. The flame is observed to lift-off in both the cases; however, as shown in figure 4(b), total extinction occurs at higher charging voltage, i.e. 10 kV, and partial extinction followed by reignition is observed in the case of 7 kV and lower.

The time series presented in figure 4(c–f) are the high-speed schlieren snapshots during the interaction of the droplet flame with the focused blast wave (using the shock tube). Figures 4(c,d) and 4(e, f) correspond to the focused cases using the shock tube with bigger (2 cm × 10 cm c/s) and smaller (2 cm × 4 cm c/s) cross-sectional dimensions, respectively. At the same charging voltage, higher M_s is obtained for the smaller section compared with the larger section. Except for the 4kV_B case, the Mach numbers obtained for all the focused cases are greater than 1.1. Furthermore, for similar M_s values, the flame response behaviour for the focused cases is consistent with that of the open-field configuration. This can be observed in the 4kV_B (big section) focused case with M_s < 1.1, where the flame responds over a longer time scales of order ${\sim} \; O({10^{ - 1}} {-} {10^1})\;\textrm{ms}$ similar to the open-field configuration, see figure 4(c). However, the flame response is within the period of the shock decay ($t\sim 0{-}1\;\textrm{ms}$, time from explosion) for higher M_s (> 1.1). In all the focused cases figure 4(c–f), a compressible vortex ring (CVR) is observed to form and interact with the droplet and flame after some delay. This interaction is observed to result in droplet breakup, which will be discussed in the later sections.

From the blast wave literature (Sedov Reference Sedov1957; Goldstine & von Neuman Reference Goldstine and von Neumann1963; Bach & Lee Reference Bach and Lee1970), it is known that the velocity profile at a given location is imposed by a propagating blast wave decaying temporally. The decay of the velocity profile behind the blast wave occurs until a time scale of $\mathrm{\sim }1\;\textrm{ms}$ (from the time of explosion), after which the velocity (${v_s}$) decays down to zero and even to negative values (of lower magnitude). Interestingly, for the open-field configuration (see figure 4a,b and Movie 1, Movie 2), it can be observed that the time of the total flame interaction occurs in the range $\mathrm{\sim }O({10^0} {-} {10^1})\;\textrm{ms}$, which is one order slower when compared with the shock decay time period of $t\sim (0.6{-}1)\;\textrm{ms}$ (from the time of explosion). During the time period of $t\sim 0.6{-}1\;\textrm{ms}$ corresponding to the decaying velocity profile imposed by the blast wave, the droplet flame is initially lifted off away from the droplet surface and it subsequently recedes towards the droplet. This lift-off is the consequence of the interaction of the blast wave profile with the droplet flame and, subsequently, the flame starts to recede and reattach to the droplet as the decaying velocity profile (${v_s}$) decays and approaches zero near the droplet. However, even beyond this time period of influence of the decay profile, ${v_s}$ ($t > 1\;\textrm{ms}$), see figure 4(a,b), where the velocity should have fully decayed to near zero, the flame is observed to exhibit a gradual lift off, albeit at a slower time scales of $\mathrm{\sim }O({10^0} {-} {10^2})\;\textrm{ms}$. This can be attributed to a slower induced flow velocity ($v_{ind}$) at the time scales of $\mathrm{\sim }O({10^0} {-} {10^2})\;\textrm{ms}$, which may occur as a result of the entrainment effects at the edge of the expanding blast wave and thus deviate from the local velocity variation obtained from the theoretical formulation in § 3.1 (mass conservation is violated). The static pressure profile behind the blast wave (P) also decays temporally and decays below the ambient pressure locally at t = t_d (see supplementary figure S2). These negative pressures (i.e. P < P_atm) behind the blast wave can be attributed to the occurrence of air entrainment effects. Thus, for the time t < t_d, the decaying profile of the blast wave is dominant and after t > t_d, the magnitude of the local velocity (${v_s}$) becomes small and negative pressures occur at the droplet location, drawing the bulk-induced flow ($v_{ind}$) near the droplet. It is to be noted that the pressure is not experimentally measured but the pressure variation obtained using the theoretical modelling is used to infer the pressure variation.

However, the effects of $v_{ind}$ are only experienced at the droplet location after some delay beyond a certain time period (after t = t_d), depending on the velocity scale of the induced flow ($v_{ind}$). Based on the response time scales of the slower flame response for t > t_d, it can be concluded that the velocity scale of the induced flow ($v_{ind}$) is significantly slower compared with the shock flow. Thus, the induced flow will affect the droplet flame only after it travels from the blast wave edge and reaches the droplet location. Thus, it can be concluded that the velocity decay profile (${v_s}$) behind the blast wave is only valid during the initial stages of the interaction before the induced flow ($v_{ind}$) reaches the droplet location. This induced flow ($v_{ind}$) is responsible for the slower flame lift-off beyond $t > 1\;\textrm{ms}$, as shown in figure 4(a,b). In all cases, under the influence of the imposed flow, the fuel vapour plume itself is swept downstream (see figure 4) in response to either the velocity profile at the droplet due to the blast wave velocity profile (${v_s}$) or the induced velocity scale ($v_{ind}$). This advection of the fuel vapour plume corresponds to the aforementioned flame lift-off phenomenon.

The entrainment effects (near the shock tube mounting at the cover plate) are also present in the case of shock tube focusing, and along with this, the induced flow ($v_{ind}$) encounters the ambient atmosphere as it exits the shock tube. This results in the curling of the induced flow ($v_{ind}$), forming vortical structures, as shown in figure 4(c–f) and Movies 3–6. These vortical structures show distinct contrast in the schlieren images, making them clearly visible (dark patches). This can be attributed to the compressible nature of these vortical structures. Thus, these vortical structures will be referred to as CVRs hereafter. The velocity scale ($v_{ind}$) of the CVR is also relatively slower compared with the shock flow, which is evident from the CVR arrival time at the droplet. The time from the explosion is normalised using the pressure decay time below ambient at the droplet (t = t_d) given by $\tau \; = \; t/{t_d}$. Thus, the blast wave reaches and interacts with the droplet flame at $\tau = {\tau _s}$ and the static pressure at the droplet decays below ambient at $\tau = 1$.

Figure 5 shows the overall schematic of the different phenomena simultaneously occurring during the interaction of the blast wave and the droplet flame. As shown at the top of figure 5, the entire process of the interaction with the droplet flame occurs in two phases. The first stage is the blast wave–flame interaction corresponding to the time scale of ${\tau _s} < \tau < 1$ (approximately $t\sim 0.6{-}1\;\textrm{ms}$ after explosion). The second stage ($\tau > 1$) comprises the induced flow–flame interaction ($t\sim O({10^0} {-} {10^1})\;\textrm{ms}$ after explosion) and the simultaneous induced flow CVR–droplet interaction ($t\sim O({10^0})\;\textrm{ms}$ after explosion). During the blast wave–flame interaction, the droplet flame responds directly to the propagating blast wave; thus, the droplet flame initially ($\tau < 1$) interacts with the decaying velocity profile (${v_s}$) that is imposed by the blast wave (Goldstine & von Neuman Reference Goldstine and von Neumann1963; Bach & Lee Reference Bach and Lee1970). Later, for $\tau > 1$, when the induced flow ($v_{ind}$) behind the blast wave reaches the test section, both the droplet as well as the droplet flame interact with the induced flow simultaneously. Since the blast wave effects have fully decayed beyond $\tau > 1$, the flame response is only due to the induced flow ($v_{ind}$). The order of magnitude of the induced flow ($v_{ind}$) can be reasonably assumed to be of the same order as that of the lift-off speed of the flame base ($v_{b,lft}$) during this second stage of interaction.

Figure 5. The schematic of temporal variation of the velocity during different stages of the interaction (top) plotted against normalised time, $\tau = t/{t_d}$. Schematic of the simultaneous response of the flame and droplet to the blast wave profile (${v_s}$) and induced flow ($v_{ind}$) at different stages of interaction. The coloured subfigures are the instantaneous snapshots obtained from high-speed flame imaging.

It is to be noted that, even though the induced flow–flame interaction regime and CVR–droplet interaction regimes are shown side by side separately in figure 5, these two interactions occur around a similar time scale ($t\sim O({10^0} {-} {10^1})\;\textrm{ms}$ after explosion). The green region in the plot shown at figure 5 (top) shows the trend of the temporal variation of velocity (${v_s}$) at a given location as the blast wave propagates. When the blast wave propagates past the droplet, locally, an instantaneous velocity peak (${v_{peak}}$) is experienced due to the discontinuity imposed by the blast wave. Then, the local velocity due to the blast wave profile (${v_s}$) decays as the blast wave propagates further downstream (Bach & Lee Reference Bach and Lee1970, Goldstine & von Neuman Reference Goldstine and von Neumann1955). However, an induced flow is entrained behind the blast wave, which reaches the droplet location after some delay during the later stages of the interaction, i.e. $\tau > 1$ (i.e. $t\sim O({10^0} {-} {10^1})\;\textrm{ms}$ after the explosion). This induced flow ($v_{ind}$) is depicted in the indicative plot of the velocity variation in figure 5 (top). The green, yellow and blue backgrounds in the plot represent the shock–flame interaction, induced flow–flame interaction (in the absence of shock tube) and droplet–induced flow CVR interaction (with shock tube), respectively. The approximate time scales corresponding to each of the interactions are mentioned in the figure. The schematic of the simultaneous response of both the flame and droplet during the different stages of interaction is depicted in figure 5.

Broadly the response of the flame and droplet to the imposed flow is divided into two regimes: low shock strengths (M_s < 1.1) and high shock strengths (M_s > 1.1), as shown in figure 5. Among the different experimental cases, all the open-field cases and the focused case: 4kV_B (charging voltage: 4 kV, big section, 2 cm × 10 cm c/s), fall under the M_s < 1.1 behaviour. The rest of the higher shock strengths in the focused case follow the high shock strength (M_s > 1.1) behaviour. In the low shock strength regime (M_s < 1.1), the flame is observed to be sustained beyond the initial blast wave–flame interaction stage ($\tau > 1$, figure 5, top). Whereas, for the M_s > 1.1 regime, the flame is observed to fully extinguish during the interaction with the decaying velocity profile (${v_s}$) imposed by the blast wave ($\tau < 1$), i.e. green zone (figure 5, top). This is depicted under the flame–shock interaction regime in figure 5. Within the M_s < 1.1 regime, the flame behaviour can be further divided based on whether flame extinction occurs or not.

For M_s < 1.04, during the initial blast wave profile (${v_s}$) interaction, $\tau < 1$ (green zone, figure 5), the flame lift-off is observed to be minimal (no forward extinction). However, for M_s > 1.04, the flame exhibits local extinction due to the critical strain rate at the forward stagnation point as a result of the externally imposed flow during this interaction stage (Pandey et al. Reference Pandey, Basu, Krishan and Vadlamudi2021). This is hereby referred to as ‘forward extinction’. In both the cases for the M_s < 1.1 regime, the flame is not extinguished during this initial stage. Subsequently, the flame starts to recede towards the droplet as soon as the velocity profile (${v_s}$) at the flame decays and approaches zero around $t\sim 1\;\textrm{ms}$ (after the explosion), as depicted at the junction of the green and yellow regions in the plot shown in figure 4 (top). Subsequently, in the time period of $\tau > 1$ (yellow zone, figure 5), the slower induced flow ($v_{ind}$) reaches and interacts with the droplet flame, resulting in flame forward extinction and lift-off in the M_s < 1.1 regime. In the case of M_s < 1.04, the induced flow velocity ($v_{ind}$) results in minimal lift-off (h_lft) of the flame base accompanied by flame tip stretching and shedding, followed by reattachment of the flame onto the droplet surface and enveloping around the time scale of $\mathrm{\sim }O({10^2})\;\textrm{ms}$ (depicted with brown background in figure 5). For M_s > 1.04, the flame tip shedding or stretching is not observed and the flame base undergoes slower but continuous lift-off, achieving higher lift-off (h_lft > 2d) compared with the M_s < 1.04 regime. Furthermore, for 1.04 < M_s < 1.06, the continuous flame lift-off leads to partial extinction followed by subsequent reignition of the droplet around $t\sim O({10^2})\;\textrm{ms}$. However, for M_s > 1.06, full extinction without reignition is observed. All these subregimes are portrayed in the global schematic shown in figure 5. Consistency in flame behaviour based on Mach number (M_s) is observed, where the focused case 4kV_B (M_s = 1.07) exhibited full extinction similar to 10kV_Open (M_s = 1.065) as M_s in both cases falls under the same subregime of M_s > 1.06. Simultaneously, the droplet also interacts with the induced flow ($v_{ind}$), undergoing minimal deformation for the open configuration but does not exhibit any breakup due to low values of $v_{ind}$ (We ~ 0.05–5).

For M_s > 1.1, due to higher M_s, the velocity scale (${v_s}$) associated with the decay profile behind the blast wave is higher. Thus, the flame lift-off is more dominant in this regime, occurring at faster time scales of similar order to that of the shock decay time period ($t\sim 0.6{-}1\;\textrm{ms}$) for $\tau < 1$. Thus, the flame undergoes rapid continuous lift-off in response to ${v_s}$ imposed by the blast wave, ultimately leading to imminent extinction during the initial interaction stage, i.e. $\tau < 1$ (green zone, figure 5). After extinction, the hot gases are swept downstream and are observed to curl into a vortex ring (depicted in the droplet–flow interaction of figure 5) and advect downstream due to the vorticity generation effect of the blast wave interaction due to RM instability (Picone & Boris Reference Picone and Boris1988; Ju et al. Reference Ju, Shimano and Inoue1998), as shown in figure 4(c) (rightmost image). Experimentally, it has been observed that the flame fully extinguishes before the induced flow CVR interacts with the droplet flame. Simultaneously, while the flame is interacting with the decaying profile (${v_s}$) of the blast wave ($\tau < 1$), the droplet also interacts with the ${v_s}$ profile and subsequently with the induced flow CVR ($v_{ind}$) while undergoing either shear-induced stripping or Rayleigh–Taylor (RT) piercing. In all cases, in response to the decaying velocity of the blast wave (${v_s}$) for $\tau < 1$, the droplet exhibits continuous temporal deformation (increase in major axis length that is perpendicular to the flow direction). Additionally, perturbations on the windward surface of the droplet due to KH instability are observed for the M_s > 1.3 subregime due to the decaying velocity (${v_s}$). Later for $\tau > 1$, after some delay when the induced flow CVR ($v_{ind}$) reaches the droplet and interacts with it, shear-induced stripping is observed due to further growth of KH waves. The induced flow CVR ($v_{ind}$) is observed to reach and interact with the droplet quicker as M_s is increased. For the M_s > 1.3 subregime, after the CVR (${v_s}$) interacts with the droplet (We_CVR > 100), the KH waves grow and travel towards the equatorial plane, resulting in a sheet formation and a subsequent sheet breakup following the shear-stripping mechanism. However, in the case of the M_s < 1.3 subregime, the KH waves are not formed due to lower velocity scales ($v_{ind}$), i.e. 0 < We_CVR < 90; however, the droplet deformation continues, leading to RT piercing followed by bag breakup. This RT piercing breakup mode is experimentally observed to be approximately one order slower compared with the shear-induced stripping mode. It is to be noted that, for the case of 4 kV charging with a big section (M_s ~ 1.07), even though the flame follows a similar trend as shown in figure 5 for the M_s < 1.1 regime, due to the presence of the shock tube and focusing of the induced flow ($v_{ind}$) in the form of a CVR, the CVR interaction with the droplet results in RT piercing breakup based on the Weber number of the CVR incident on the droplet.

3.2.1. Shock–flame interaction (${\tau _s} < \tau < 1$)

This section will primarily discuss the interaction of the droplet flame for $\tau < 1$ (faster time scales) with the decay velocity profile (${v_s}$) imposed by the blast wave (green zone, figure 5). For all the cases, the droplet flame is observed to immediately start to respond to the propagating blast wave when it reaches the droplet location at $\tau = {\tau _s}$ (see figure 4). Figure 6 shows the temporal variation of the flame dimensions (in mm) during the interaction. The green background represents the first stage of interaction between the blast wave and the droplet flame. The orange data points represent the distance between the flame tip and the droplet, whereas the blue data points represent the distance between the flame base and the droplet (flame stand-off distance). This implies that the flame is vertically present in between the orange and blue data points, with droplet location on the x-axis. The red dotted line depicts the approximate instantaneous location of the propagating blast wave. In figure 6, the shock strength increases from left to right due to the increase in charging voltage for the same shock tube configuration. Figure 6(a–f) shows the temporal flame dimension variation for the open field, big channel shock tube (2 cm × 10 cm c/s) and small channel shock tube (2 cm × 4 cm c/s), respectively.

Figure 6. Temporal variation of the flame dimensions, i.e. flame tip (orange) and base (blue) distances (mm) from the droplet surface during the initial interaction between blast velocity profile, ${v_s}$ (green region) and later interaction between the induced flow, $v_{ind}$ (yellow region) plotted for (a,b) open field, (c,d) bigger channel (2 cm × 10 cm c/s) and (e, f) smaller channel (2 cm × 4 cm c/s) at different charging voltages. The flame dimensions are plotted against normalised time ($\tau $). The simultaneous shock and vortex locations are shown in the plots with red and green dotted lines, respectively, and the corresponding values M_s are mentioned for each case.

During the first stage of shock–flame interaction (${\tau _s} < \tau < 1$), in the case of the open-field blast wave with M_s (< 1.04) (figure 6a), the flame lift-off from the droplet surface remains minimal, exhibiting flame tip stretching in the form of a quick jittery motion. As the blast wave passes by, the flame stretching is observed in the range ${\tau _s} < \tau < 1$ (almost immediately as the shock propagates) in response to the decaying velocity profile (${v_s}$) imposed by the blast wave. No significant delay is observed between the flame response and the instant of shock interaction. Subsequently, the flame tip recedes towards the droplet (flame length reduces) as the velocity profile (${v_s}$) decays at the flame, approaching near zero towards the end of the first stage interaction (green zone, figure 6). This is evident from the plot in figure 6(a), which shows a local peak in the flame tip distance (orange) in the green region (shock–flame interaction phase) corresponding to the flame tip stretching. Later, the drop in flame tip distance (orange) shows the receding of the flame tip distance after the shock passes. The plot also shows no significant flame lift-off (forward extinction) for this case (blue) in the green region. However, in case of a higher Mach number (M_s ~ 1.065) (figure 6a), the flame undergoes local extinction at the forward stagnation point due to critical strain rate as a result of the externally imposed flow, followed by significant flame lift-off (h_lft > 3d). Similar to the low M_s case, the flame tip exhibits stretching followed by receding of the flame base and tip towards the droplet in response to the temporally decaying ${v_s}$ at the flame location. This flame tip stretching and receding is reflected in the plot in figure 6(b) in the form of a momentary spike in flame tip distance (orange) that is accompanied by the drastic increase in flame base lift-off (blue). The receding of the flame tip and base distance corresponds to the flame trying to reattach to the droplet after the blast velocity profile (${v_s}$) decays significantly, approaching zero around $\tau \sim 1$ (at the end of the green zone).

As mentioned before, higher confinement (focusing) of the blast wave results in higher M_s. The green zones in figure 6(c–f) shows the plots of the temporal variation of the flame dimensions during ${\tau _s} < \tau < 1$ (shock–flame interaction phase) for bigger and smaller shock tube channels, respectively. The subfigures in figure 6 (flame images with dark background) are the instantaneous snapshots obtained from high-speed flame imaging. Since the 4kV_B case (M_s ~ 1.07) (bigger channel) has a similar M_s as that of 10kV_Open (M_s ~ 1.065), a similar flame dynamics is observed in both cases. The flame shows forward extinction as well as significant lift-off (h_lft > 3d) and flame tip stretching. The same can be seen in figure 6(c), which shows an initial rise in flame tip (orange) and flame base (blue) distances corresponding to flame stretching and flame lift-off, respectively, in the green region (shock–flame interaction phase, i.e. ${\tau _s} < \tau < 1$).

When the charging voltage is further increased in the big channel (2 cm × 10 cm c/s), the flame lift-off during $\tau < 1$ (green zone) becomes more drastic as higher M_s (> 1.1) values are achieved. The flame lift-off distance (h_lft) becomes more than five times the droplet diameter after interaction with the blast wave, leading to a blowout. The flame lift-off is observed to continuously increase in response to the imposed ${v_s}$. The same is reflected in the plot in figure 6(d), where the flame base distance (blue) continuously increases drastically (h_lft > 10d) in the green region (${\tau _s} < \tau < 1$) before the imminent blowout. The rate of increase in flame tip and flame base distances is observed to increase as the shock strength is increased. This entire phenomenon is observed to occur directly in response to the imposed decaying velocity profile (${v_s}$) at the flame (${\tau _s} < \tau < 1$, green region). For shock focusing with a smaller channel (2 cm × 4 cm c/s), higher M_s values are achieved at the same charging voltages (M_s > 1.1). Thus, the flame dynamics observed is qualitatively similar to the M_s > 1.1 cases of the bigger channel and the flame response is found to occur at even shorter time scales due to higher velocity scales (${v_s}$), as seen in figure 6(e, f). In both the focused cases, for M_s > 1.1, full extinction of the flame is observed during this initial stage of interaction (${\tau _s} < \tau < 1$).

3.2.2. Theoretical estimate of the flame response to the decaying velocity profile (${v_s}$)

The initial flame base lift-off (advection of the flame base in the downstream direction) in different cases in the time period of ${\tau _s} < \tau < 1$ has been observed to be as a result of the interaction of velocity profile (${v_s}$) with the droplet flame. Thus, the temporal variation of ${v_s}$ at the droplet location needs to be evaluated during this stage.

Thus, from the blast wave formulation (§ 3.1), (3.10) can be used to obtain the velocity variation at a given location (r) at any given instant if the instantaneous shock radius (R_s) and Mach number (M_s) are known. Since it has already been established previously that the shock trajectory and M_s are in good agreement with the theory (§ 3.1 and figure 3f–i), the instantaneous values R_s(t), M_s(t) are measured experimentally, and the local velocity, ${v_s}$(t), at the droplet location, i.e. r = L_t + L_d (for open-field case) or r = L_d (for focused case) can be evaluated using (3.10). The blast wave location is tracked along the centreline using Otsu's thresholding (in built in ImageJ software) on the schlieren images (§ 2) to obtain the shock radius (R_s). Thus, the instantaneous velocity variation (${v_s}$) at the droplet flame location, which is responsible for the advection of the flame base, needs to be evaluated. The instantaneous flame base location (h_lft) has been extracted from the high-speed flame imaging using Otsu's thresholding and the advection velocity ($v_{b,lft}$) of the flame base is evaluated. The temporal variation of the local velocity at the droplet due to the blast wave profile (${v_s}$_,Th) based on theoretical shock trajectory (explained in § 3.1) is also evaluated.

The local velocity (${v_s}$) calculated using instantaneous R_s and M_s (from experiments), the temporal variation of the local velocity based on theoretical shock trajectory (${v_s}$_,Th) and the instantaneous flame base advection (lift-off) velocity ($v_{b,lft}$) are plotted in figure 7(a–f) for different cases. As seen in figure 7(a–f) (green region), the flame base advection velocity ($v_{b,lft}$) is obtained to be in decent agreement with the decaying velocity profile (${v_s}$) obtained from the blast wave formulation in the initial stages of the shock–flame interaction (${\tau _s} < \tau < 1$). As the time progresses ($\tau > 1$), the induced flow ($v_{ind}$) leads to the deviation of the actual flow velocity (corresponding to the flame response) from the theoretical velocity profile (${v_s}$) of the blast wave.

Figure 7. (a–f) Temporal variation of the local velocity (${v_s}$) near the flame (based on instantaneous shock parameters), temporal variation of the theoretically obtained local velocity near the flame location (${v_s}$_,Th) and instantaneous flame base advection velocity ($v_{b,lft}$) plotted against time, t from the explosion for different cases 4kV_B (Movie 3), 8kV_B (Movie 4). 4kV_S (Movie 5), 8kV_S (Movie 6), 5kV_Open (Movie 1), 10kV_Open (Movie 2). The subfigures are the flame snapshots of OH* chemiluminescence depicting flame topology at a given time instant (for reference). (g) Schematic of droplet flame showing flame tip (system I) and flame base (system II), (h) effect of externally imposed flow on flame tip (shedding mechanism), (i) effect of externally imposed flow on flame base (extinction criteria). The error bars represent the maximum error arising due to the shock location measurement variation due to pixel resolution.

3.2.4. Shedding criteria of the flame

Experimentally, it has been observed that, for the current droplet flame considered, the flame shedding only occurs at the low Mach number (M_s < 1.04) for $\tau > 1$ during the interaction with $v_{ind}$, which corresponds to the 5kV_Open case (no shock tube). The shedding events are observed to occur on a time scale of order t ~ O(10¹ ms) where the velocity scale due to the blast wave profile (${v_s}$) has completely decayed, and the induced flow ($v_{ind}$ ~ O(10¹) m s⁻¹) is the dominant velocity scale present during the shedding phenomena. Thus, compressible effects are minimal corresponding to the low velocity scales of $v_{ind}$ on this time scale of $\tau > 1$, where flame shedding is observed. The schematic of the flame-shedding mechanism is depicted in figure 7(h) (left), where, due to the buoyancy-induced instability, the hot gases around the flame accelerate along the flame length near the shear layer, leading to continuous vortex rollup. This leads to a continuous feeding of circulation from the locally perturbed region near the droplet (fuel source) along the flame length, which eventually reaches a critical value leading to flame puffing or shedding. As shown by Xia & Zhang (Reference Xia and Zhang2018), in the context of diffusion flames ($Ri \to \infty $), the gravitational (buoyancy) term is pivotal for the vorticity generation. Thus, the vorticity transport equation considered near the flame tip (system I) can be written as follows:

(3.13)\begin{equation}\frac{{\textrm{D}\omega }}{{\textrm{D}t}} = \; \frac{{{\rho _a}}}{{{\rho ^2}}}(\boldsymbol{\nabla }\rho \times g) + \nu {\nabla ^2}\omega ,\end{equation}

where $\omega $ is the vorticity, $\rho $ is the local density at a given location, ${\rho _a}$ is the air density and $\nu $ is fluid viscosity. The last term on the right-hand side of the above equation represents the vorticity diffusion term and can be neglected when compared with the first term (baroclinic term due to buoyancy) (Xia & Zhang Reference Xia and Zhang2018). Proceeding in a similar approach as provided by Xia & Zhang (Reference Xia and Zhang2018), the temporal variation of the circulation in the control volume enclosing the shear boundary, as shown in figure 7(h) (right), is given by the summation of the initial circulation strength fed due to the local velocities on either side of the shear layer and the buoyancy-induced vorticity buildup, shown below

(3.14)\begin{equation}\frac{{\textrm{d}\varGamma }}{{\textrm{d}t}} = {\rho _a}g\left( {\frac{1}{{{\rho_a}}} - \frac{1}{{{\rho_f}}}} \right)\Delta h + \frac{{\textrm{d}{\varGamma _{initial}}}}{{\textrm{d}t}}.\end{equation}

In the above equation, ${\rho _f}$ represents the density of the hot gases inside the flame, $\mathrm{\Delta }h$ represents the length scale associated with the control volume, $\varGamma $ represents circulation in the control volume at a given height and ${\varGamma _{initial}}$ represents the initial circulation present in the system due to the local velocity. Hence, for the droplet interacting with flow imposed by the blast wave in the current experiments, $\textrm{d}{\varGamma _{initial}} = ( - {v_{NC}} + {v_{ind}})\,\textrm{d}h\; $, as shown in figure 7(h), because $v_{ind}$ is the dominant velocity scale imposed. Since ${v_{ind}}$ has been observed to be the dominant convective velocity experimentally, dh can be scaled as $\textrm{d}h\sim ({v_{ind}} + {v_{NC}})\,\textrm{d}t$. Thus, $\textrm{d}{\varGamma _{initial}} = {( - {v_{NC}} + {v_{ind}})^2}\,\textrm{d}t$. Here, $v_{ind}$ is the induced flow velocity, which can be experimentally obtained.

Thus, the circulation buildup equation becomes

(3.15)\begin{equation}\frac{{\textrm{d}\varGamma }}{{\textrm{d}t}} = {\rho _a}g\left( {\frac{1}{{{\rho_a}}} - \frac{1}{{{\rho_f}}}} \right)\Delta h + v_{ind}^2 - v_{NC}^2.\end{equation}

Thus, the rate of circulation buildup depends on the buoyancy-induced instability and the externally imposed flow (${v_{ind}}$). The magnitude of $v_{ind}$ for the 5kV_Open case is of similar order as the velocity scale observed by Pandey et al. (Reference Pandey, Basu, Krishan and Vadlamudi2021). Thus, using the scaling for buoyant flickering, i.e. ${t_{shd}}\sim \sqrt {h/g} $ (Pandey et al. Reference Pandey, Basu, Krishan and Vadlamudi2021), substituting in the circulation buildup equation and integrating the (3.15) on both sides over one shedding cycle (where the critical circulation is reached for shedding to ensue)

(3.16)\begin{equation}{\varGamma _{critical}} = \frac{{{\rho _a}{g^{1/2}}k}}{3}\left( {\frac{1}{{{\rho_a}}} - \frac{1}{{{\rho_f}}}} \right)h_{sh}^{3/2} + k(v_{ind}^2 - v_{NC}^2){g^{ - 1/2}}h_{sh}^{1/2}.\end{equation}

Thus, flame shedding occurs at a height of ${h_{sh}}$ where the circulation buildup will reach the critical value (${\varGamma _{critical}}$) and the time scale of this shedding is ${t_{shd}}$. Experimentally, for 5kV_Open, the time scale of the flame shedding is observed to be around ~15 ms, which is of the order of ~O(10¹) ms during the interaction with $v_{ind}$ ($\tau > 1$). This is evident from figure 6(a) in the form of a spike in the flame tip distance (orange) in the yellow region. However, in all the other open-field cases where the flame extinction is observed, the time scale of extinction is found to be of the order of ~O(10⁰) ms, which is one order faster compared with the flame-shedding time scale. Thus, there is not sufficient time for the circulation buildup to reach the critical value (that ensues shedding) before system II (flame base) reaches system I (flame tip), resulting in an extinction event due to the higher advection velocity of system II at higher M_s.

3.2.5. Extinction criteria of the flame

Figure 8(a) shows the temporal variation of the net local velocity ($v_{comb}$) estimated at the droplet location as the blast wave propagates radially outward, plotted for different cases, where $v_{comb}$ = ${v_s}$ + $v_{ind}$. The initial spike in velocity corresponds to the velocity imposed at the droplet (${v_s}$) in the range ${\tau _s} < \tau < 1$ and the secondary spike/discontinuity in the velocity corresponds to the arrival of the induced flow vortex having a velocity scale of $v_{ind}$ (obtained experimentally) that arrives at the droplet after some delay ($\tau > 1$). The droplet flame response to the externally imposed flow is observed to manifest at the flame base (lift off) and flame tip (shedding) independently. Thus, as shown in figure 7(g), the droplet flame tip and flame base are considered to be two independent systems: system I (flame tip) and system II (flame base). Figure 7(i) depicts the advection of the flame base (system II) towards system I. This advection (lift-off) of system II is dependent on the external flow imposed on the droplet flame. The solid green line figure 7(a–f) shows the theoretically obtained temporal variation of the decaying local velocity (${v_s}$_,Th) imposed by the blast wave at the droplet flame location. The flame lift-off (h_lft) is influenced by only the blast wave profile for M_s > 1.1, however, for M_s < 1.1, the induced flow ($v_{ind}$) also affects the flame base advection velocity ($v_{b,lft}$). Thus, the combined velocity ($v_{comb}$) of the decay profile at the droplet flame location (${v_s}$_,Th) and the subsequent induced flow velocity ($v_{ind}$) determine the overall advection of the flame base (system II)

(3.17)\begin{equation}{v_{comb}}(t) = {v_{s,Th}}(t) + {v_{ind,inst}}(t).\end{equation}

Figure 8. (a) Temporal variation of the total velocity at the droplet location that is obtained by the summation of the instantaneous velocity estimated using the blast wave formulation and the induced flow. (b,c) Temporal variation of the flame tip and base dimensions depicting the partial extinction, reignition for 7kV_Open, and full extinction for 10kV_Open. The subfigures are the instantaneous snapshots obtained from high-speed flame imaging. (d) Extinction criteria: lift-off ratio vs time for different cases. (e) Time series snapshots of schlieren flow visualisation showing the shape deformation of the windward density interface of the plume of hot gases around the droplet flame during interaction with shock. ( f) Schematic of blast wave and the windward density interface of the hot gas plume around the droplet flame depicting the density and imposed pressure gradient directions that governs the baroclinic torque: RMI. (g) Schematic of the shape deformation phenomenon of the windward density interface of the hot gas plume during interaction with the blast wave.

Since it has been established that the flame base lift-off velocity is a direct response to the local velocity at the flame (see figure 7a–f), the cumulative area under the plot $v_{comb}$(t) vs time gives the total distance travelled by the flame base (h_{lft ,max}). Experimentally, it has been observed that flame extinction occurs when the advection velocity of system II (flame base) is sufficient to traverse the flame length and reach system I and, thus, interact with it. This interaction between system II and system I results in a critical strain rate, leading to flame extinction. Thus, the advection of system II towards downstream, reaching system I (flame tip), is the criterion for extinction. Thus, the ratio of h_lft and h_tip (flame tip distance) is plotted against time in figure 8(d). Figure 8(d) shows that the lift-off ratio (h_lft/h_tip) gradually increases with time as the flame interacts with the total local velocity imposed by the external flow. The vertical dotted line in figure 8(d) represents the approximate time instant when the velocity profile behind the blast wave decays and approaches zero ($\tau > 1$). The left side of the vertical dotted line represents the initial shock–flame interaction (${\tau _s} < \tau < 1$) and the right side of the dotted line represents the induced flow–flame interaction ($\tau > 1$). The horizontal dotted line corresponds to the lift-off ratio approaching unity. It can be observed that, for focused blast wave cases at higher M_s > 1.1, the lift-off ratio (h_lft/h_tip) crosses unity during the initial interaction with the decaying velocity profile. This implies that, for these cases, system II interacts with system I during the initial shock–flame interaction (${\tau _s} < \tau < 1$), which is in agreement with the experiments.

On the contrary, for lower M_s < 1.1, the lift-off ratio (h_lft/h_tip) does not approach unity during the initial interaction (${\tau _s} < \tau < 1$). That means the flame is sustained beyond $\tau \sim 1$, and it subsequently starts to interact with the induced flow ($v_{ind}$) for $\tau > 1$. During the induced flow–flame interaction ($\tau > 1$), the lift-off ratio (h_lft/h_tip) is observed to reach unity for M_s > 1.04 (except 5kV_Open). This suggests that the flame extinction occurs during the interaction with the induced flow ($v_{ind}$) for lower Mach number cases except for the 5kV_open case, for which the lift-off ratio does not reach unity. This is consistent with the experimental observation of the absence of extinction occurrence in 5kV_Open. Furthermore, the lift-off ratio of 10kV_open (M_s ~ 1.065) approached unity with a steeper slope compared with 7kV_open (M_s ~ 1.05), 8kV_open (M_s ~ 1.055). This suggests that, even though the extinction occurs in all the cases, the velocity ($v_{b,lft}$) of the flame base (system II) advection is faster in the case of 10kV_open compared with 7kV_open and 8kV_open. This higher velocity of system II advection can be attributed to the occurrence of full extinction (due to the critical strain rate) in the case of 10kV_open (see figure 8c). Additionally, it can be hypothesised that the shallow slope of 7kV_open and 8kV_open indicates a lower advection velocity of system II resulting in lower strain rates during its interaction with system I (when the lift-off ratio reaches unity). This might be attributed to the occurrence of partial extinction in these two cases (7kV_Open, 8kV_Open), which resulted in reignition and reattachment of the flame subsequently (see figure 8b).

3.2.6. The RM instability during flame–shock interaction

Figure 8(e) shows the schlieren snapshot time series of the response of the hot plume (around the flame) during the interaction with the blast wave (${\tau _s} < \tau < 1$) for the case of 8kV_S. It is evident from the images that, during the initial blast wave interaction, the hot plume is swept downstream, which corresponds to the flame base lift-off event. Along with this, it can also be observed that the topology of the hot plume and the flame alter during the interaction with the blast wave. As shown in figure 7(a–d) (subfigures), the flame is observed to deform, forming a vortical structure for (M_s > 1.1) during the initial stage of interaction with the blast wave, which can be attributed to the vorticity generation in the flame region as a result of shock propagation (Picone & Boris Reference Picone and Boris1988; Ju et al. Reference Ju, Shimano and Inoue1998). The vorticity is generated in the flame region due to RM instability (RMI; a special case of RT instability) that occurs when the local pressure gradient (imposed by the shock) is misaligned with the density gradient across the plume interface. The pressure gradient is caused by the acceleration field (gravity) in RT instability, whereas, in the case of RMI, the pressure gradient is caused by the propagating shock wave (Zhou et al. Reference Zhou2021). Considering the vorticity transport equation at the windward side of the interface of the hot plume around the flame (see figure 8f)

(3.18)\begin{equation}\frac{{\textrm{D}\omega }}{{\textrm{D}t}} = u\boldsymbol{\cdot }\boldsymbol{\nabla }\omega - \omega \boldsymbol{\cdot }\boldsymbol{\nabla }u + \frac{{\boldsymbol{\nabla }\rho \times \boldsymbol{\nabla }P}}{{{\rho ^2}}} + \nu {\nabla ^2}\omega .\end{equation}

The third term on the right-hand side represents the baroclinic component of the vorticity transport equation that occurs due to the misalignment of the density and pressure gradients and predominantly contributes to the vorticity generation. Furthermore, this baroclinic term is shown to be most pertinent to the immediate discussion of RMI by Zhou et al. (Reference Zhou2021) compared with the first term (vortex stretching), second term (vorticity dilation) and the last term (vorticity dissipation) on the right-hand side even during shock wave interaction. Thus, this baroclinic vorticity term is activated when pressure and density gradients are misaligned, which deposits vorticity on the density interface upon passage of the shock wave. This vorticity causes the perturbations on the density interface to deform, which leads to RMI perturbation growth. Thus, depending on the curvature and geometry of the flame region (blob of density inhomogeneity), baroclinic vorticity is generated in the system, which eventually rolls up into vortex filaments or vortex rings (Picone & Boris Reference Picone and Boris1988), as shown in figure 7(a–d) (subfigures). As consequence of this RMI-induced vorticity generation, the plume around the flame also exhibits shape deformation during its interaction with the blast wave.

The general behaviour of RMI perturbation growth at an interface strongly depends on the Atwood number (A), which signifies the difference in density of the two mediums on either side of an interface and it is given by

(3.19)\begin{equation}A = \frac{{{\rho _2} - {\rho _1}}}{{{\rho _2} + {\rho _1}}},\end{equation}

where ${\rho _1}$ and ${\rho _2}$ are the densities of the media on either side of the interface and the shock wave is considered to propagate from medium 1 (${\rho _1}$) to medium 2 (${\rho _2}$). If A > 0, then the interface perturbation spikes simply grow towards the lower-density material. However, for A < 0, phase inversion is observed where the perturbations begin to deform, forming inverted spikes that grow in the opposite direction into the low-density side (Sterbentz et al. Reference Sterbentz, Jekel, White, Aubry, Lorenzana and Belof2022).

When the shock propagates downstream across the interface, it interacts with the flame region containing a lower-density hot plume (around the flame). In the current experiments, the shock enters from a denser medium (air) to a rare medium (plume of hot gases), thus the Atwood number A < 0, which leads to phase inversion of the perturbations. It is also reflected in the experiments as shown in figure 8(e), where the shape of the windward density interface of the hot plume is initially convex upstream (downwards) and inverts, exhibiting a concavity (facing upstream) after the initial interaction with the blast wave (see figure 4c–f and supplementary figure S3). This phenomenon of the reversal of the interface curvature is due to RMI which occurs when the shock wave interacts with a density interface propagating from a denser (unburnt gases) to a rarer (burnt gases) medium, i.e. A < 0 (La Flèche Reference La Flèche2018; Sterbentz et al. Reference Sterbentz, Jekel, White, Aubry, Lorenzana and Belof2022; Yang et al. Reference Yang, Lin, Li and Lai2023). The schematic of this instability is depicted in figure 8(g), showing the phase inversion of the curvature of the density interface leading to the formation of a cusp-like structure. This cusp-like structure further grows into the lighter fluid side and subsequently curls up, forming vortical structures near the flame base. This leads to the aforementioned alteration in the flame topology forming vortical structures (as shown in figure 7a–d, subfigures) as the consequence of the RMI-induced vorticity generation.

Kramer et al. (Reference Kramer, Pullin, Meiron and Pantano2010) investigated the single-mode RMI and showed that, for weak and intermediate shocks, the impulsive model of the RT instability is adequate for the prediction of the perturbation growth rate. Hence, this is applicable to the current experiments where the blast wave is near the weak shock limit. The growth of the amplitude (a) due to RT instability of the single-mode perturbation on a discontinuous interface is given by (Zhou Reference Zhou2017; Zhou et al. Reference Zhou2021)

(3.20)\begin{equation}\frac{{{\textrm{d}^2}a}}{{\textrm{d}{t^2}}} = gkAa.\end{equation}

Where k = 2${\rm \pi}$/λ is the wavenumber, ‘g’ is acceleration and ‘a’ is perturbation amplitude.

Following Zhou (Reference Zhou2017) and Zhou et al. (Reference Zhou2021), for a varying acceleration, i.e. $g(t) = \Delta u\delta (t)$, (using the Dirac delta function) the initial impulse corresponding to the velocity jump imparted by the shock is given by $\int {g(t)\,\textrm{d}t} = \Delta u$, which can be integrated to obtain the following equation for perturbation growth during shock interaction that is valid for the specific type of interaction observed in the current experiments:

(3.21)\begin{equation}\frac{{\textrm{d}a}}{{\textrm{d}t}} = kA\Delta u{a_o}.\end{equation}

Where ‘a_o’ is the initial amplitude of the perturbation, ‘$\Delta u$’ is the velocity jump imposed by the blast wave, i.e. (~$v_{peak}$) and ‘A’ is the post-shock condition Atwood number across the density interface.

3.3. Effect of shock focusing using a shock tube

Unlike the open-field case, in focused cases (with shock tube) the induced flow behind the blast wave has to exit from the shock tube into the surrounding ambient. This causes the flow to curl as it exits the shock tube, forming vortical structures. Researchers like Qin et al. (Reference Qin, Xiang, Lin and Liu2020), Zare-Behtash, Kontis & Gongora-Orozco (Reference Zare-Behtash, Kontis and Gongora-Orozco2008), Zare-Behtash et al. (Reference Zare-Behtash, Kontis, Gongora-Orozco and Takayama2009), Zhang et al. (Reference Zhang, Chen, Li and Jiang2014) and Ahmad, Hasan & Sanghi (Reference Ahmad, Hasan and Sanghi2020) investigated the CVRs exiting as shock tube. The formation dynamics of the CVRs has been investigated for different types of CVRs: shock-free, with embedded shock and with secondary vortices. It has been shown that the CVRs roll up and grow in size and pinch off from the shock tube exit as they propagate downstream. In the current experiments, the CVR is a consequence of a blast wave generation (using the wire-explosion technique) due to which the current CVR observed needs to be characterised experimentally. From the experiments, it is observed that, at lower M_s (lower charging voltage), the vortex formed tends to dissipate quickly as it propagates downstream. The induced flow vortex has to travel L_d ~ 35 mm outside the shock tube in order to interact with the droplet flame. This induced flow vortex is visually noticeable in the experimental images in figure 4(c–f). Similar to the blast wave, the induced flow vortex is observed to travel faster in the case of the smaller shock tube channel (2 cm × 4 cm c/s) compared with the bigger channel (2 cm × 10 cm c/s). It is to be noted that, unlike the literature, the CVR behind the blast wave in the current experiments did not exhibit any shock-cell structures in its trailing wake that are observed behind shock waves in the literature (Ahmad et al. Reference Ahmad, Hasan and Sanghi2020). This behaviour qualitatively matches with the observations of experiments by Chan et al. (Reference Chan, Giannuzzi, Kabir, Hargather and Doig2016), where the distinct shock-cell structures are observed in the CVR wake in case of a shock tube exhaust of a compressed-air-driven shock wave but no visible shock-cell structures are observed in the CVR wake in the case of a shock tube exhaust of an explosively driven shock wave.

As the charging voltage is increased in the case of a bigger channel (M_s > 1.1), the induced flow vortex behind the blast wave becomes more distinct, as shown in figure 4(d). However, by the time the induced flow reaches the droplet, the flame blowout would have already occurred during the initial shock–flame interaction phase due to higher M_s. Similar blowout during initial shock–flame interaction is observed in the case of a smaller channel as well, due to the higher focusing effect resulting in higher M_s (M_s > 1.1). This can be seen in figures 6(d) and 6(c) where the flame extinction has already occurred in the green region (shock–flame interaction) when the flame base (blue) is advected downstream rapidly to interact with the flame tip (orange). In all the cases, as the flame extinguishes, the hot gases at the flame rise up, and they curl to form a toroidal structure due to RT instability (see figure 4).

3.3.1. Induced flow vortex characterisation

As explained before, the induced flow begins to curl as it exits the shock tube channel forming a vortical structure. The vortical structures are visibly noticeable in the schlieren imaging, suggesting density variation in the vortical flow behind the blast wave. Thus, it can be concluded that it is a compressible vortex. Figure 9(a) shows the schematic of the flow exiting the open end of the shock tube. It can be seen in figure 9(a) that the blast wave first exits the shock tube followed by the induced flow that travels at lower velocity scales compared with the blast wave. As discussed before, the blast wave has a decaying velocity profile behind it which is depicted in the schematic. The propagating blast wave imposes a velocity jump (${v_s}$) or discontinuity (compared with the ambient downstream of it), and the velocity monotonically decays in amplitude behind the blast wave until the induced flow vortex arrives at the location.

Figure 9. (a) Schematic showing the blast wave, induced flow behind it and the CVR originating from the shock tube exit. Time series of the schlieren imaging of the shock flow for different cases: (b) 4kV_B, (c) 8kV_B, (e) 4kV_S, ( f) 8kV_S, (d) schematic of the three-dimensional evolution of the CVR exiting the shock tube.

Figure 9(b) shows the time series of the flow exiting the shock tube for low shock strength, i.e. 4kV_B case (M_s ~ 1.07). The shock exiting the shock tube initially is planar near the exit. However, as it propagates downstream, the blast wave attains curvature as it expands radially outward away from the shock tube. This is the planar to cylindrical transition of the blast wave depicted in figure 3(a), which has been explained in detail in § 3.1. The induced flow follows the blast wave and exits the shock tube with some delay, which is evident from the slow-moving vortex at the shock tube exit. The vortex ring is not strong enough and its translation velocity is significantly slower compared with the blast wave for the 4kV_B case. Furthermore, the vortex starts to dissipate before reaching the droplet (see figure 4c).

However, in the case of higher charging voltages with the bigger channel, i.e. 8kV_B case (figure 9c), the Mach number is higher, and the blast wave shows higher contrast, indicating a higher jump in the fluid property at the shock front. The CVR is more pronounced, exhibiting higher contrast, and the CVR translation velocity is also higher. As the induced flow exits the shock tube, expansion fan structures are formed due to flow expansion, as shown in figure 9(c). A trailing jet can be observed in the case of higher charging voltages. In the case of the smaller channel (figure 9e, f), at a similar Mach number, the translational velocity of the vortex is observed to be higher compared with the bigger channel.

At lower charging voltages, the CVR appears as a thin vortex loop, indicating a very thin localised region having compressible effects near the vortex core similar to figure 9( f). Multiple corrugations and oscillations were observed in the vortex ring as it travelled downstream. As the charging voltage is increased with the smaller channel, the velocity scales become significantly higher and the CVR appears even more pronounced with higher contrast. For high charging voltages, the CVR is observed to have embedded-shock structures. The figure also shows the presence of multiple smaller secondary vortices at the CVR (see figure 9e). Figure 9(d) shows the temporal schematic of the evolution of the three-dimensional macro-features of the CVR, which are also observed in the time series shown in figure 9( f). The further details of the CVR evolution are provided in the supplementary material.

3.4. Flow–droplet interaction ($\tau > 1$)

The droplet dynamics during the interaction with the shock tube exhaust flow occurs in two stages. During the initial stage of the interaction, continuous droplet deformation is observed, which occurs in the time scale of the decay of the velocity imposed by the blast wave (${v_s}$), i.e. ${\tau _s} < \tau < 1$, as shown in figure 10(a) (green region). Due to the local velocity imposed by the blast wave (${v_s}$), a high-pressure region is developed at the forward stagnation point of the droplet (polar location). This difference between the stagnation pressure at the polar location and the static pressure at the equator of the droplet, i.e. ($\mathrm{\Delta }P = {P_p}\; - \; {P_{eq}}$) causes deformation of the droplet into an oblate shape, which results in a continuous increase in the droplet length scale in the equatorial plane. The surface tension forces on the droplet surface resist this deformation. Hence, the non-dimensional parameter that indicates the competing forces involved in deformation, i.e. the deformation Weber number, can be written as $W{e_d}\sim \Delta P \cdot {R_e}/\sigma$, which is the ratio of the deforming force and the restoring surface tension force, where R_e is the equatorial radius of curvature (Sharma et al. Reference Sharma, Chandra, Basu and Kumar2023a). Hence, as the droplet starts to deform, the surface tension forces are overpowered by the deforming forces, as indicated by the increasing Weber number with deformation. Thus, the droplet continues to deform, leading to flattening into an oblate shape leading to bag-type breakup, as shown in figure 10(a,b) (green region).

Figure 10. (a) Time series plot depicting two stages of the interaction with the droplet: (i) Decaying velocity profile behind the blast wave interacts with the droplet (green region) and (ii) CVR formed from the induced flow exiting the shock tube interacting with the droplet (purple region). The simultaneous time series images of the droplet shadowgraphy (top) and schlieren (bottom) are shown in each of the interaction stages (with time stamps) for three different cases on the x-axis: 4kV_S, 6kV_B and 8kV_S, respectively. (b) Schematic of the droplet breakup modes at different Weber numbers. Green region represents the blast wave decay profile (${v_s}$) interaction and purple region represents the induced flow interaction ($v_{ind}$).

Furthermore, during the same time scale (${\tau _s} < \tau < 1$), simultaneously alongside this continuous deformation, KH instability-induced perturbations were also observed on the windward surface of the droplet at higher M_s (higher instantaneous Weber numbers based on ${v_s}$), as shown in figures 5 and 10(b) (green region). These perturbations temporally grow into KH waves along the windward surface of the droplet, leading to transport of the fluid towards the periphery in the equatorial plane. This results in the formation of a sheet at the equatorial location. A similar mechanism has been reported by Sharma et al. (Reference Sharma, Singh, Rao, Kumar and Basu2021), which has been observed in the current experiments, as shown in figure 10(a) (green region). It is to be noted that, since the quartz rod is holding the droplet in pendant mode, due to the presence of the rod, the droplet breakup will not be the same as that of the contactless droplet. However, the qualitative response of the droplet dynamics will remain similar so long as the droplet dynamics does not enter the wake of the quartz rod. Since in the initial deformation process, the KH wave formation and sheet formation are observed to occur upstream of the quartz rod, the droplet dynamics observed is qualitatively representative of that of contactless droplets.

3.5. Regime map

A regime map is depicted in figure 11, showing the overall picture of the flame and droplet dynamics during the interaction between a combusting droplet and shock tube exhaust flow. The regime map depicts three stages of the interaction phenomena in the current experiments, where the effect of M_s is depicted for different cases for all three stages of interaction. In figure 11, the cyan background is used to represent the open-field blast wave case (without shock tube), whereas the red background is used to represent the shock tube focusing cases. Data points are plotted for different charging voltages and shock tube configurations with respect to M_s. The representative images for those cases depicting the flame response are included in figure 11 accordingly. A time axis is shown in z-direction, depicting the time scale of occurrence of each stage of the interactions. The data points plotted in the regime map have different symbols with different colours to represent specific types of phenomena that occur at a specific stage of interaction. The regime map is broadly divided into two zones along the time axis: (i) interaction with the shock (${\tau _s} < \tau < 1$) and (ii) interaction with the induced flow ($\tau > 1$). The direct shock wave interaction is only seen with the droplet flame, whereas the droplet itself remains unaffected by it in the case of the open-field configuration. However, for the focused cases, the droplet shows deformation for all M_s and simultaneous KH wave perturbation growth is also observed in the case of high M_s. Furthermore, the induced flow in the open-field case affects only the flame and the induced flow CVR that forms at the shock tube exit, and only interacts with the droplet and not with the flame as the flame blowout would have already occurred for focused cases by the time the CVR arrives. The induced flow–droplet interaction is plotted with respect to the Weber number based on the velocity scale of the induced flow ($v_{ind}$) instead of M_s, in figure 11. For focused cases, the induced flow velocity scale is the same as the velocity scale associated with the CVR, i.e. $v_{CVR}$.

Figure 11. Regime map depicting phenomena in different configurations (open-field and two shock tube focusing cases) in three interaction regimes: shock–flame, induced flow–flame and induced flow CVR–droplet interactions with a time axis in the z-direction. The flame interaction regimes are plotted against M_s whereas the droplet interaction is plotted against the Weber number based on the induced flow and initial droplet diameter. The cyan region represents the open-field and the red region represents the shock tube focusing configurations, respectively. All the different phenomena occurring in different regimes are represented by different symbols and colour codes as shown in the legend.