Numerical simulations of turbulent jet ignition (TJI) and combustion in a rapid compression machine (RCM) are conducted by a hybrid Eulerian-Lagrangian large eddy simulation/filtered mass density function (LES/FMDF) computational model. TJI is a novel method for initiating combustion in ultra-lean mixtures and often involves one or several hot combustion product turbulent jets, rapidly propagating from a pre-chamber (PCh) to a main chamber (MCh). An immersed boundary method is developed and used together with LES to handle complex geometries and to decrease the complexity and computational cost of the Monte Carlo (MC) particle operations, while maintaining the high accuracy of the hybrid LES/FMDF model. Analysis of numerical data suggests three main combustion phases in the RCM-TJI: (i) cold fuel jet, (ii) turbulent hot product jet, and (iii) reverse fuel-air/product jet. The effects of various parameters (e.g., the igniter location, mixture composition, and wall heat transfer) on these phases are studied numerically. It is found that the turbulent jet features and the MCh combustion are very much dependent on the PCh ignition details. Igniting the PCh at the lower locations close to the nozzle forces the PCh charge to fully participate in the PCh combustion and prevents the unburned fuel leaking to the MCh. It also leads to longer discharge of the PCh hot products into the MCh with more uniform jet velocity, enhancing the MCh combustion. The results predicted by LES/FMDF are found to be comparable with the available experimental data, both qualitatively and quantitatively. (C) 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.