The effect of turbulence on the end-gas auto-ignition process of different primary reference fuels (PRFs) under engine-relevant conditions is numerically investigated through 3-D LES (large eddy simulation) and 1-D LEM (linear eddy model) simulations. Spontaneous ignition in the end-gas and pressure oscillations under the knock condition of a downsized spark ignition engine are qualitatively captured through the LES simulation. In order to take a further insight into the effect of full-scale turbulence-chemistry interaction on the auto-ignition process of the end-gas, 1-D LEM simulations are performed in a 1-D domain composed of a homogeneous core region and a linearly thermal distributed boundary layer. The effects of turbulent fluctuating velocity, fuel composition, turbulence length scale and initial temperature on the ignition process are investigated in the present study. It is found that the turbulence has a non-monotonic influence in the auto-ignition (AI) formations and expansion process of the PRF/air mixtures. Increased turbulence intensity delays the formation of the initial AI and shrinks the kernel size at the onset of AI due to the enhanced heat and radicals dissipation. However, after the first AI kernel forms, the intensified turbulence accelerates the overall combustion as a result of expanded preheating zone and more homogeneous unburned mixtures caused by turbulent stirring. In addition, the dissipation and homogenization effects of turbulence on the high-octane fuel are more prominent than on the low-octane fuel. The analysis of cases with different turbulent integral length scales reveals that the dissipation effect of the turbulence with the similar length scale of the nascent AI kernel is more prominent while the small-scale turbulence is more apt to shift the combustion mode after the first auto-ignition flame forms. At a high initial temperature of 900 K falling in the NTC regime, the first AI kernel of PRF10 appears in the boundary layer with a lower temperature instead of the core region. As the turbulence intensifies, the effect of NTC on the end-gas auto-ignition vanishes.