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Numerical solutions of compressible turbulent flow remains a challenge. Because of the wide range of temporal and spatial length scales, different mathematical techniques have been implemented to resolve these complex flows. Large Eddy Simulation (LES) is one of the most promising techniques; however, some understanding of the expected turbulence must be known a priory and computed using sub-grid scale models. Unfortunately, there is no guarantee that the selected model will effectively capture the fine-scale turbulence. Mathematical handling of discontinuities such as shock waves typically increase the complexity and difficulty of compressible flow computations. To alleviate this problem, Essentially Non-Oscillatory (ENO) schemes equipped with Riemann solvers are widely used. The numerical algorithm presented in this work represents a new approach based on a consistent averaging procedure that solves an integral form of the Navier Stokes Equations. This method leads to a set of differential-algebraic equations that are solved numerically using spatial averaging. We present several computations demonstrating the flow physics capturing capabilities of the new scheme. We investigate 2D solutions of the stratified Kelvin-Helmholtz instability shear layer, the Taylor-Green Vortex, and the Riemann problem. This is our first attempt to provide a thorough study of inviscid and viscous flows. Our primary goal is to quantify the dissipative behavior, resolution characteristics, and shock-capturing capabilities of the proposed scheme. To this end, we qualitatively compared the numerical solution to reference data. Quantitatively, we use statistical techniques such as measurement of the kinetic energy spectrum and the conservation of total energy and enstrophy in the flow.