Axial compressors are used in a wide variety of aerodynamic applications and are one of the most important components in aero-engines. The operability of compressors is however limited at low-mass flow rates by fluid dynamic instabilities such as stall and surge. These instabilities can lead to engine failure and loss of engine power which can compromise the aircraft safety and reliability. Therefore, a better understanding of how stall occurs and the causes behind its inception is extremely important. In the vicinity of the stall line, the flow field is inherently unsteady due to the interactions between adjacent rows of blades, formation of separation cells, and the viscous effects including shock-boundary layer interaction. Accurate modeling of these phenomena requires a proper set of stable and accurate boundary conditions at the rotorstator interface that conserve mass, momentum and energy, while eliminating false reflections. As a part of this effort, an existing 3D Navier-Stokes analysis for modeling single stage compressors has been modified to model multi-stage axial compressors and turbines. Several rotor-stator interface boundary conditions have been implemented. These have been evaluated for the first stage (a stator and a rotor) of the two stage fuel turbine on the space shuttle main engine (SSME). Their effectiveness in conserving global properties such as mass, momentum, and energy across the interface, while yielding good performance predictions has been evaluated. While all the methods gave satisfactory results, a characteristic based approach and an unsteady sliding mesh approach are found to work best. Accurate modeling of the formation of stall cells requires the use of advanced turbulence models. As a part of this effort, a new advanced turbulence model called Hybrid RANS/KES (HRKES) has been developed and implemented. This model solves Menter's k--SST model near walls and switches to a Kinetic Eddy Simulation (KES) model away from walls. The KES model solves directly for local turbulent kinetic energy and local turbulent length scales, alleviating the grid spacing dependency of the length scales found in other Detached Eddy Simulation (DES) and Hybrid RANS/LES (HRLES) models. Within the HRKES model, combinations of two different blending functions have been evaluated for blending the near wall model to the KES model. The use of realizability constraints to bound the KES model parameters has also been studied for several internal and external flows. The current methodology is used in the prediction of the performance map for the NASA Stage 35 compressor configuration as a representative of a modern compressor stage. The present approach is found to satisfactory predict the onset of stall. It is found that the rotor blade tip leakage vortex and its interaction with the shock wave is mainly the reason behind the stall inception in this compressor stage.