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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Main Author: Yap, Wen Jiun
Format: Thesis
Language:English
English
Published: 2003
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Online Access:http://psasir.upm.edu.my/id/eprint/12188/1/FK_2003_38_.pdf
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spelling my-upm-ir.121882024-07-04T02:57:24Z Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics 2003-02 Yap, Wen Jiun 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. Finite differences Fluid dynamics - Problems 2003-02 Thesis http://psasir.upm.edu.my/id/eprint/12188/ http://psasir.upm.edu.my/id/eprint/12188/1/FK_2003_38_.pdf text en public masters Universiti Putra Malaysia Finite differences Fluid dynamics - Problems Faculty of Engineering Asrar, Waqar English
institution Universiti Putra Malaysia
collection PSAS Institutional Repository
language English
English
advisor Asrar, Waqar
topic Finite differences
Fluid dynamics - Problems

spellingShingle Finite differences
Fluid dynamics - Problems

Yap, Wen Jiun
Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
description 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.
format Thesis
qualification_level Master's degree
author Yap, Wen Jiun
author_facet Yap, Wen Jiun
author_sort Yap, Wen Jiun
title Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
title_short Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
title_full Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
title_fullStr Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
title_full_unstemmed Application Of Higher Order Compact Finite Difference Methods To Problems In Fluid Dynamics
title_sort application of higher order compact finite difference methods to problems in fluid dynamics
granting_institution Universiti Putra Malaysia
granting_department Faculty of Engineering
publishDate 2003
url http://psasir.upm.edu.my/id/eprint/12188/1/FK_2003_38_.pdf
_version_ 1804888695386406912