A dynamic Large Eddy Simulation (LES) Predictor-Corrector Split (PCS) h-adaptive Finite Element Method (FEM) for solving both incompressible and compressible flows has been developed. This particular PCS h-adaptive FEM makes it possible to solve both incompressible and compressible flows in one numerical system. Development of the PCS formulation is well described in Carrington et al [1]. The additional effort to incorporate an LES turbulent model with PCS h-adaptive FEM was needed to more accurately account for turbulence modeling capabilities. In our formulation, the Vreman dynamic LES approach by Lau [2] is able to solve compressible and incompressible fluid flow without any wall damping function or ad-hoc clipping to prevent an unstable (negative) eddy viscosity, unlike the Smagorinsky subgrid model (SM). The underlying discretization is an h-adaptive finite element formulation for the general flow momentum transport equation along with other transport equations for multiphase flow. This method provides measurement of the actual error in the discretization, and can adjust spatial accuracy to minimize the error to some specified amount. The dynamic characteristics of the pressure and velocity fields of unsteady incompressible and compressible turbulent wakes behind a circular cylinder are investigated numerically and analyzed physically.