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....
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| Format: | Thesis |
| Language: | English English |
| Published: |
2003
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/12188/1/FK_2003_38_.pdf |
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| Summary: | 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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