Fully three-dimensional computational fluid dynamic simulations with detailed combustion chemistry of a turbulent jet ignition system installed in a rapid compression machine are presented. The turbulent jet ignition system is a prechamber-initiated combustion system intended to allow lean-burn combustion in spark ignition internal-combustion engines. In the presented configuration, the turbulent jet ignition prechamber has a volume that is 2% of the volume of the main combustion chamber in the rapid compression machine and is separated from the main chamber by a nozzle containing a single orifice. Four simulations with orifice diameters of 1.0 mm, 1.5 mm, 2.0 mm, and 3.0 mm respectively are presented in order to demonstrate the effect of the orifice diameter on the combustion behavior of the turbulent jet ignition process. Data generated by the simulations is shown including combustion chamber pressures, temperature fields, jet velocities and mass fraction burn durations. From the combustion pressure trace, the jet velocity, and other field data, five distinct phases of the turbulent jet ignition process are identified. These phases are called the compression phase, the prechamber combustion initiation phase, the cold jet phase, the hot jet phase, and the flow reversal phase. The four simulations show that the orifice diameter of 1.5 mm provides the fastest ignition and the fastest overall combustion as reflected in the 0-10% and 10-90% mass fraction burn duration data generated. Meanwhile, the simulation for the orifice diameter of 1.0 mm produces the highest jet velocity and has the shortest delay between the spark and the exit of a jet of hot gases into the main chamber but produces a slower burn duration than the simulation for the larger orifice diameter of 1.5 mm. The simulations for orifice diameters of 2.0 mm and 3.0 mm demonstrate that the combustion speed is reduced as the orifice diameter increases above 1.5 mm. Finally, a discussion is given which examines the implications that the results generated have in regard to implementation of the turbulent jet ignition system in an internal-combustion engine.