Under-expanded jets are typically formed when the high-pressure fuel is injected into the combustion chamber under ambient conditions. In the case of rocket engines, under-expanded jets are the primary cause of the traveling shock wave formations immediately outside of the nozzle exit. The formulations of these unsteady flow structures, and its development into fully turbulent steady state flows are of interest to designers. Many PLIF experiments on the temporal development of jet flows revealed the existence of highly under-expanded flow, moderately under-expanded and subsonic flow regions, during ‘the injection process’. The under-expanded flow field features consist of the main vortex pair, the vortex-induced shock waves, shock-vortex interactions, and Kelvin-Helmholtz type-structures, their evolution and their interactions until the flow transforms into a fully developed turbulent profile [1–2]. However, these visualization experiments revealed limited or no information of the instability features of the jet, and methods of enhancing their influence. The instabilities within the jets are directly responsible for the turbulence transition process. Further, these instabilities lead directly to the development of fully turbulent flows, the much-needed mixing process within the combustion chamber. The ‘Integro-Differential Scheme’ (IDS) investigation conducted herein, investigated the detailed unsteady flow structures within these jet flows. Further, this numerical study revealed the existing of the Kelvin-Helmholtz instability structures, their evolutions, and their interactions. Further, the IDS unsteady simulations were compared to existing LES and PLIF studies, and they were found to reproduce all the expected flow field features. In the process of investigating the under-expanded jet flow, the IDS numeral method demonstrated that it is capable of fully reproducing the fundamental flow physics within the under-expanded jets. The next step in this study is the manipulation of the flow physics in efforts of enhance mixing.