## 3 Results and Discussion
### Effect of Ignition Energy on Ignition Kernel Delay Time in Quiescence  
To quantify the ignition kernel delay time as a function of ignition energy during the early development of flame kernel for successful ignition, the time evolutions of flame kernel radii (\(R\)) are recorded by using high-speed schlieren imaging (Figure 3). \(R=(A/\pi)^{0.5}\) and \(A\) is the area enclosed by the developing flame kernel front. As such, flame speeds (\(\mathrm{d}R/\mathrm{d}t\)) of the developing flame kernels versus \(R\) can be obtained. Figure 4a presents six data sets of (\(\mathrm{d}R/\mathrm{d}t\)) versus \(R\) at different \(\mathrm{E_{ig}}\) varied from one half to twice of \(\mathrm{MIE_{L}}\) (\(\frac{1}{2}\mathrm{MIE_{L}}\leq\mathrm{E_{ig}}\leq 2\mathrm{MIE_{L}}\), where \(\mathrm{MIE_{L}}=3.4\) mJ) for successful ignition cases. After the discharge, all six laminar flame speeds (\(\mathrm{d}R/\mathrm{d}t\)) first decrease drastically to a minimum value of (\(\mathrm{d}R/\mathrm{d}t\))\({}_{\mathrm{min}}\) corresponding to a critical flame radius, which is termed as \(R_{\mathrm{Lmin}}\) (see Figure 4a). If the kernel's heat release generated by chemical reaction can be sustained to be greater than heat losses to electrodes and ambient after \(\tau_{\mathrm{delay}}\) determined at \(R_{\mathrm{Lmin}}\) with the minimum (\(\mathrm{d}R/\mathrm{d}t\)), successful ignition should occur. As such, the self-sustained laminar flame propagation becomes possible, by which \(\mathrm{d}R/\mathrm{d}t\) starts to increase and then the flame speed asymptotically approximates to its laminar burning velocity when \(R>15\) mm in the present lean n-butane/air flames (Figure 4a). Therefore, \(\tau_{\mathrm{delay}}\) is the time interval from the initiation of the spark to the time that the flame speed reaches its minimum at \(R_{\mathrm{Lmin}}\). \(R_{\mathrm{Lmin}}\) is found to be about \(5\) mm in the present study, regardless of different values of \(\mathrm{E_{ig}}\) applied (Figure 4a). As shown in the inset of Figure 4a, an increase of \(\mathrm{E_{ig}}\) from \(0.5\)\(\mathrm{MIE_{L}}\) to \(2\)\(\mathrm{MIE_{L}}\) promotes the early development of the flame kernel growth rate with an increase of (\(\mathrm{d}R/\mathrm{d}t\))\({}_{\mathrm{min}}\), resulting in a reduction of \(\tau_{\mathrm{delay}}\).  
Figure 3: Time evolutions of high-speed schlieren images of the lean n-butane/air mixture at \(\phi=0.7\) with \(Le\approx 2.1>>1\) at \(d_{\mathrm{gap}}=2.0\) mm in quiescence and at various turbulent conditions.  
Figure 4b shows the ignition kernel delay time measured at \(R_{\rm Lmin}\approx 5\) mm that is plotted against a normalized ignition energy (\(\rm E_{ig}/MIE_{L}\)), revealing a transition of \(\tau_{\rm delay}\) by which the decreasing slope of \(\tau_{\rm delay}\) with \(\rm E_{ig}/MIE_{L}\) changes from rapid to gradual when \(\rm E_{ig}/MIE_{L}\) reaches its critical value of unity. Note that the reverse result is found for that of \(\rm(d\)\(R\)/\(\rm d\)\(t\))\({}_{\rm min}\). In the pre-transition, when \(0.6<\rm E_{ig}/MIE_{L}<1\), values of \(\tau_{\rm delay}\) decrease from 7.5 ms to 5 ms, but a much smaller decrease of \(\tau_{\rm delay}\) from about 4.6 ms to 4.2 ms is observed when \(1<\rm E_{ig}/MIE_{L}\leq 2.15\) in the post-transition. This is attributed to the reverse behavior of the drastic and modest increase of \(\rm(d\)\(R\)/\(\rm d\)\(t\))\({}_{\rm min}\) before and after the transition occurring at the same critical value of (\(\rm E_{ig}/MIE_{L}\))\({}_{\rm c}\approx 1.1\). The marginal effect of higher \(\rm E_{ig}/MIE_{L}>1\) on \(\rm(d\)\(R\)/\(\rm d\)\(t\))\({}_{\rm min}\) and \(\tau_{\rm delay}\) is probably due to the saturation of the refreshed mixture induced by recirculation vortices inside the initial kernel [31,32] that may limit the increase of active radicals for the laminar case; the increase of heat conduction losses to electrodes with increasing \(\rm E_{ig}\)[19] could be another possible factor.