## 3 Results and Discussion  
### Laminar and Turbulent Minimum Ignition Energies of Lean n-Butane/Air Mixture  
Figure 2a shows the ignition probability versus ignition trials with various values of E\({}_{\rm ig}\) of the lean n-butane/air mixture at \(\phi\) = 0.7 in quiescence, as a typical example for demonstrating the statistical determination of MIE at 50% ignitability. Each datum of MIE\({}_{\rm L}\) and MIE\({}_{\rm T}\) is determined by more than 30 runs over a range of E\({}_{\rm ig}\) with either \"Go\" (open circle symbols) or \"No Go\" (cross symbols), as indicated in Figure 2a. There is an overlapping regime within which \"Go\" and \"No Go\" co-exist. As indicated in Figure 2a, the newly obtained datum of MIE\({}_{\rm L}\) for the present lean n-C\({}_{4}\)H\({}_{10}\)/air mixture at \(d_{\rm gap}\) = 2.0 mm is 3.4 mJ. The reader is directed to Ref. [7] for a detailed methodology on MIE determination using the logistic regression method.  
Figure 2: (a) A typical example for determining MIE at 50% ignitability (=E\({}_{\rm ig,50\%}\)) using the logistic regression method. (b) The present MIE\({}_{\rm T}\) data as a function of \(u^{\prime}\), showing a regular MIE transition occurring at a critical \(u^{\prime}_{\rm c}\approx 0.92\) m/s. In addition, also plotted for comparison are previous data measured at small \(d_{\rm gap}\) = 0.8 mm for the same lean n-butane/air mixture, revealing an irregular MIE transition with \(TFI\) when \(u^{\prime}\) < \(u^{\prime}_{\rm c}\approx 0.92\) m/s [24].  
In the presence of turbulence at large \(d_{\rm gap}\) = 2.0 mm, as shown in Figure 2b, newly measured values of MIET are respectively 3.96 mJ/9.37 mJ/17.3 mJ/30.5 mJ at \(u^{\prime}\approx\) 0.46 m/s/0.92 m/s/1.4 m/s/2.1 m/s, where values of MIET are all larger than MIEL = 3.4 mJ. This confirms that that there is no _TFI_ at \(d_{\rm gap}\) = 2.0 mm, even when the mixture's Lewis number is sufficiently greater than unity (\(Le\approx\) 2.1 >> 1). At large \(d_{\rm gap}\) = 2.0 mm, turbulence always renders ignition more difficult. A sufficiently large Eig is required to overcome heat losses induced by electrodes and turbulence for successful ignition. In the pre-transition when \(u^{\prime}<u^{\prime}c\), MIET increases modestly with increasing \(u^{\prime}\), while such increase becomes much more drastic when in the post-transition \(u^{\prime}>u^{\prime}c\). As previously shown on the inset of Figure 5 in Ref. [24], at small \(d_{\rm gap}\) = 0.8 mm, it was found that MIET first decreases with increasing \(u^{\prime}\) where MIET < MIET \(\approx\) 23 mJ, showing _TFI_ with the minimum value of MIET \(\approx\) 19.8 mJ occurring also at \(u^{\prime}c\approx\) 0.92 m/s. When \(u^{\prime}>u^{\prime}c\), turbulence re-claims its totally dominating role where MIET increases rapidly to be greater than MIEL \(\approx\) 23 mJ. Again, the phenomenon of _TFI_ only occurs at sufficiently large \(Le>>\) 1 and at sufficiently small \(d_{\rm gap}\) = 0.8 mm. This is attributed to the competition between differential diffusion (the effect of \(Le\)) and turbulent dissipation (the effect of \(u^{\prime}\)), similar to that discussed in our previous studies for lean PRF95/ air mixture with \(Le\approx\) 2.95 >> 1 [23] or rich hydrogen/air mixture at \(\phi\) = 5.1 with \(Le\approx\) 2.3 >> 1 [28]. In the present study, except at the largest \(u^{\prime}\) = 2.1 m/s, where almost the same value of MIET \(\approx\) 31 mJ is found for both small and large \(d_{\rm gap}\), all other values of MIEL and MIET at \(d_{\rm gap}\) = 2.0 mm are smaller than those of \(d_{\rm gap}\) = 0.8 mm (Figure 2b). As such, the use of large \(d_{\rm gap}\) = 2.0 mm is better than small \(d_{\rm gap}\) = 0.8 mm because the former can have a higher ignition probability than the latter at the same \(u^{\prime}\) up to 2 m/s.