## 1 Introduction  
It is known that lean premixed turbulent combustion can enhance the thermal efficiency of spark ignition (SI) engines through the increase of the specific heat ratio [1] and the reduction of heat-transfer losses due to smaller adiabatic flame temperature with lower NO\({}_{\rm{x}}\) emissions [2, 3]. However, one of the challenges to apply lean burn technology is thersig problem that often relates to a longer ignition kernel delay time (\(\tau_{\rm{delay}}\)) [1, 4, 5, 6] or a slower development of initial flame kernel at the early stage of combustion, which may worsen cycle-to-cycle variations in SI engines. A higher root-mean-square (r.m.s.) turbulence fluctuation velocity (\(u^{\prime}\)) could be useful in accelerating the initial flame kernel development and thus shortening \(\tau_{\rm{delay}}\). However, the increase of \(u^{\prime}\) usually requires a higher ignition energy (E\({}_{\rm{ig}}\)) or minimum ignition energy (MIE) for successful ignition [7]. As such, a clear understanding of the relations among \(\tau_{\rm delay}\), MIE, and \(u^{\prime}\) during the combustion process is essential to the further development of high-thermal efficiency SI engines operated at lean-burn turbulent conditions. This motivates the present study. To achieve a self-sustained flame propagation, many researchers have investigated a critical flame radius (\(R_{\rm c}\)) for spherical flames (e.g., [8; 9; 10]) or for spray combustion flames [11]. When the heat release from chemical reaction is greater than the heat dissipation, the initial kernel radius could develop and grow beyond the required \(R_{\rm c}\), and then successful ignition with self-sustained flame propagation could occur [11]. Therefore, the ignition kernel delay time, which is determined as the time interval from the spark initiation to the time required to reach \(R_{\rm c}\), is an effective means to evaluate the physicochemical properties of combustible mixtures [11; 12; 13; 14]. Although many ignition studies are available in the literature (e.g., [PERSON]. [15; 16; 17]), most of these studies are for the autoignition delay time. Hence, \(\tau_{\rm delay}\) data at the early stage of flame propagation are still sparse. Based on both experimental and numerical studies of laminar spherical flames, [PERSON] et al. [8] and [PERSON] et al. [9; 10] reported that an increase of \(\rm E_{\rm ig}\) could enhance the early flame propagation speed, resulting in a shorter \(\tau_{\rm delay}\). The larger \(\rm E_{\rm ig}\), the higher concentration of active radicals inside the spark gap [18] and/or the higher chemical power release [19]. Further, using two-dimensional direct numerical simulation, [PERSON] et al. [13] investigated the influence of turbulent strain rate (\(u^{\prime}/l\)) on \(\tau_{\rm delay}\) of the lean n-C\({}_{\rm 7}\)H\({}_{16}\)/air mixture at an equivalence ratio of \(\phi=0.5\), where \(u^{\prime}\) and \(l\) were the r.m.s. turbulent fluctuating velocity and the integral length scale of turbulence. By assuming the initial peak temperature \(T_{\rm peak,ini}=1500\) K and \(R_{\rm c}=0.5\) mm for all turbulent cases, they predicted an increase of \(\tau_{\rm delay}\) of proportional to \((u^{\prime}/l)^{2}\). However, the engine result [20] indicated that the combination of the increase of \(\rm E_{\rm ig}\) and the enhancement of in-cylinder turbulence level can lead to a shorter inflammation time from the spark timing to a crank angle of 5%-mass fraction burnt (CA5) and a shorter main combustion duration time (CA10 to CA90), allowing SI engines to be operated at an extra-lean flammability limit. The result in [20] suggested that the determination of \(\tau_{\rm delay}\) at various \(\rm E_{\rm ig}\) (or MIE) and \(u^{\prime}\) may be important for a better design of SI engines.