Compressible turbulent flow over a modern gas turbine blade is modeled in this thesis using Large Eddy Simulation (LES) methods. CFD simulation of turbo-machinery flows is a challenging problem due to the high Reynolds and Mach numbers of such flows. Reynolds Averages Navier Stokes (RANS) methods that are currently used in industry to model such turbulent flows do not give satisfactory results specifically at off-design conditions and therefore there is a need to further improve the turbulence modeling in the CFD simulations. Herein, different variations of the LES method are investigated to simulate this compressible turbulent flow. LES methodologies consist of Smagorinsky, Dynamic Smagorinsky, and Implicit LES accompanied by Yoshizawa and Van Driest relations. In this work, a 3D unstructured tetrahedral Navier-Stokes solver is applied using a mixed finite-volume-finite-element method. LES terms are discretized using the finite-element method. Parallel computation is performed according to MPI standards.  LES pressure distribution results have significant differences with the 3D RANS results. Much more discrepancy is expected in velocity profiles, shear stresses, and heat transfer characteristics. Considering pressure distribution results and compared to the currently used RANS models, LES results are fairly superior especially in the off-design conditions. LES results of this work are more superior to RANS results in the regions close to leading edge which contains very high strain in fluid elements and it is not satisfactorily resolvable using RANS. RANS results of [66] are superior to LES results of this work for the mid suction side at which the turbulence structures are not resolved well in LES where a separation leads to constant pressure distribution which is captured as an attached flow in LES. This might be due to not introducing fluctuations at LES inlet and also to not having enough mesh resolution. Both LES and RANS results have difficulty resolving the shock/boundary layer interaction on the rear suction side of the blade. This might also be cured in LES by improving the mesh at that region or using adaptation methods or enhancing the shock capturing characteristics of the flux calculation method. The deficiencies of the LES simulation are discussed and possible cures and future works are elaborated.  This work aims at accomplishing the first step and providing a solid basis for future works on an intellectually challenging topic which is of high importance in academia and industry: Turbulence Modeling