Chapter 2 provides a comprehensive overview of the fundamental equations and methodologies essential for simulating compressible flows within compressors using Detached-Eddy Simulation (DES) and its variants. It begins by outlining the governing Navier-Stokes equations, emphasizing their role in capturing mass, momentum, and energy conservation in fluid dynamics. The chapter then delves into the discretization techniques for both time and space, highlighting high-precision schemes that balance accuracy with computational efficiency.A significant portion is dedicated to explaining the DES methodology, which integrates Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) approaches to effectively model complex turbulent flows. The discussion includes various enhancements to the original DES method, such as Delayed DES (DDES) and Enhanced DDES (EDDES), which address challenges like grid dependency and accurate turbulence representation in different flow regions.Furthermore, the chapter explores the factors influencing the fidelity of DES methods, including grid resolution and spatio-temporal discretization accuracy, and presents validation cases that demonstrate the effectiveness of DES in capturing intricate flow phenomena like vortex shedding and flow separation. By addressing both the theoretical foundations and practical considerations, Chapter 2 lays the groundwork for applying high-fidelity numerical simulations to optimize compressor design and performance.