In our previous studies, we have developed natural gas engines operating under lean conditions to improve thermal efficiency and emission characteristics. We applied a sub-chamber injection system to our engines, in which natural gas is directly injected into a combustion sub-chamber in order to completely separate stoichiometric mixture in the sub-chamber from ultra-lean mixture in the main chamber. The results obtained from engine tests demonstrated excellent performance of our engines in view point of efficiency and NOx emissions. However, we have poor knowledge of mixture distributions in the combustion chambers to understand the mechanism of the improvements. The aim of this paper is to clarify the mixture formation in combustion chambers by means of numerical simulations in the combustion chamber with and without the sub-chamber at a variety of operating conditions. The commercial software of Fluent 16.0 was used to conduct RANS simulations in an axial 2D numerical domain considering movements of piston. Non-reactive flow in the combustion chamber was simulated during the compression period at an engine speed of 2000 rpm. The turbulence model employed here is standard k-e model. A/F is set with a lean condition of 30. The results obtained from the numerical simulations demonstrates higher equivalence ratio in the sub-chamber than that in the main chamber, which extends the lean limit at engine operations. Furthermore, existing probability of mixture with low equivalence ratio is higher than that without the sub-chamber, which is an evidence of lower NOx emissions from test engines incorporating the sub-chamber. On the other hand, high equivalence ratio mixture remains in the squish area of the main chamber when the sub-chamber is installed. The high HC emissions from the test engine are probably caused by the unburned fuels in the squish area.