Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics

Finite difference schemes used in the field of computational fluid dynamics is generally only of second-order accurate in representing the spatial derivatives. Numerical algorithms based on higher order finite difference schemes that can achieve fourth-order accuracy in space have been developed. Higher order schemes will enable a larger grid size with fewer grid points to sufficiently give fine results. A compact scheme, which preserves the smaller stencil size, is preferred due to its simplicity and computational efficiency, as opposed to the normal approach to expand the stencil to achieve higher accuracy. Two approaches are used to obtain the fourth-order accurate compact scheme. In Lax-Wendroff approach, the governing differential equations are used to approximate the leading truncation error in the second-order central difference of the governing equations. In Hermitian scheme, the fourth-order approximations to the derivatives are treated as unknowns. These unknowns are solved explicitly with Hermitian relations that relate the variables and its spatial derivatives. The numerical algorithms are first developed for viscous Burgers' equation on uniform and clustered grids. The fourth-order accuracy and convergence rate is demonstrated. The performance of the two different approaches are compared and found on par with each other. Second, the numerical algorithms are used to solve the quasi-one-dimensional subsonic-supersonic nozzle flow. The Hermitian scheme shows excellent agreement with the analytical result but the Lax-Wendroff approach failed to do so due to instability problem. Third, only the numerical algorithm based on Hermitian scheme is used to solve the flow past a backward-facing step. The reattachment lengths of the first separation bubble compare favourably with previously published results in the literature. The success of the fourth-order compact finite difference schemes in solving the viscous Burgers' equation is not repeated in the isentropic nozzle flow and the flow past a backward-facing step. Further efforts have to be made to improve the convergence rate of the numerical solution of the isentropic nozzle flow using the Hermitian scheme and to overcome the instability in the numerical solution of the same problem using the Lax-Wendroff approach.

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