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Computational fluid dynamics (CFD) is becoming an increasingly important tool for engineering analysis. In CFD simulations obtaining grid independent solutions without excessive cost is desirable. Hence, there is a need to evaluate higher order accurate discretization schemes (both spatial and temporal). Moreover, many journals require at least second-order accuracy for publication. Also, there is a need to maximize the efficiency of Navier-Stokes solvers. Efficiency is both a function of how the flow field is calculated and how effectively the discretized equations are solved. All of these issues are important in the emerging field of Large Eddy Simulation (LES) where very fine grids are needed to accurately resolve evolution of highly transient turbulent flows. A time accurate Navier-Stokes solver was written that followed a variety of segregated procedures. The code allowed for first- and second-order time advancement. Also, three different upwinding schemes were implemented for discretizing the convective terms including representative methods that were first-, second-, and third-order accurate. Iterative solution methods based on the preconditioned conjugate gradient method were incorporated into this code and evaluated. The code was first validated on several different benchmark problems. It was then used to simulate the unsteady flow in an axisymmetric isothermal engine cylinder. The SIMPLEC method was found to be the most efficient solver among those tested. Also, ICCG was the most efficient solver for symmetric matrices while ILUBiCGSTAB was preferred for nonsymmetric matrices. From benchmarking, it was determined that a third-order upwinding scheme known as Extended Linear Upwind Differencing (ELUD) is the most accurate of those schemes tested and is not severely oscillatory in the presence of steep gradients. It was shown that second-order time accuracy has a noticeable influence on unsteady simulations. The SIMPLEC method coupled with conjugate gradient methods seems to be an efficient Navier-Stokes solver. The results indicate that ELUD does provide high spatial accuracy. However, a grid independent solution is still difficult to achieve with complex problems such as engine flows. Higher order temporal accuracy makes a significant difference in resolving the time dependent features of complex flows such as those encountered in engine cylinders.