| In the past decades, the use of high order numerical methods is becoming more and more popular in computational fluid dynamics. Padé schemes and ENO schemes are two of those popular schemes and get a lot of applications. However, Padé schemes have been found to cause nonphysical oscillations if they are applied directly to discontinuous data and ENO schemes somehow show dissipative behavior in smoothly varied regions.; In the present dissertation study, a new, essentially non-oscillatory high order Padé-type (ENO-Padé) scheme has been developed by incorporating the ENO interpolation algorithm into the cell-centered Padé scheme. The ENO-Padé scheme is designed to eliminate the nonphysical oscillatory behavior of the conventional Padé scheme across discontinuities and to improve the numerical accuracy of the ENO scheme in smooth regions. The major steps of constructing the ENO-Padé scheme are illustrated by the solution of the scalar transportation equation and several important issues about its overall formal accuracy, resolution characteristic, and numerical stability are also discussed. Subsequently, the proposed ENO-Padé scheme is extended to the solution of compressible Euler equations. Two different methods to evaluate the interfacial fluxes are provided and the local characteristic decomposition is discussed. Lastly, the demonstration of how to apply the ENO-Padé scheme to simulation of incompressible flows is presented. In the present ENO-Padé approach for incompressible fluid flows, a curvilinear coordinate system is employed and the solution procedure is based on the famous SIMPLE algorithm and Rhie and Chow's treatment on a collocated, non-staggered grid system.; A number of benchmark test cases, including three scalar convection problems, five compressible flows, and three incompressible flows, are used to test and validate the proposed ENO-Padé scheme. In fact, the proposed ENO-Padé scheme exhibits great performance in capturing discontinuities in an essentially non-oscillatory fashion, accurately resolving the flow structures, and achieving smooth and stable solutions when small-scale structures can not be fully resolved on the given mesh, in the test cases. This validates that the proposed ENO-Padé scheme is an excellent compromise of the available schemes for resolving profiles over flow discontinuities while maintaining accurate flow structures in smooth regions. |
