Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes
Autor(a) principal: | |
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Data de Publicação: | 2011 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1678-58782011000400001 |
Resumo: | We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved. |
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Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemeslaminar boundary layersimilarity analysishigh-order-compact finite differencesWe present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM2011-12-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1678-58782011000400001Journal of the Brazilian Society of Mechanical Sciences and Engineering v.33 n.4 2011reponame:Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online)instname:Associação Brasileira de Engenharia e Ciências Mecânicas (ABCM)instacron:ABCM10.1590/S1678-58782011000400001info:eu-repo/semantics/openAccessDuque-Daza,CarlosLockerby,DuncanGaleano,Carloseng2012-01-26T00:00:00Zoai:scielo:S1678-58782011000400001Revistahttps://www.scielo.br/j/jbsmse/https://old.scielo.br/oai/scielo-oai.php||abcm@abcm.org.br1806-36911678-5878opendoar:2012-01-26T00:00Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) - Associação Brasileira de Engenharia e Ciências Mecânicas (ABCM)false |
dc.title.none.fl_str_mv |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
title |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
spellingShingle |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes Duque-Daza,Carlos laminar boundary layer similarity analysis high-order-compact finite differences |
title_short |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
title_full |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
title_fullStr |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
title_full_unstemmed |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
title_sort |
Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes |
author |
Duque-Daza,Carlos |
author_facet |
Duque-Daza,Carlos Lockerby,Duncan Galeano,Carlos |
author_role |
author |
author2 |
Lockerby,Duncan Galeano,Carlos |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Duque-Daza,Carlos Lockerby,Duncan Galeano,Carlos |
dc.subject.por.fl_str_mv |
laminar boundary layer similarity analysis high-order-compact finite differences |
topic |
laminar boundary layer similarity analysis high-order-compact finite differences |
description |
We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved. |
publishDate |
2011 |
dc.date.none.fl_str_mv |
2011-12-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1678-58782011000400001 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1678-58782011000400001 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S1678-58782011000400001 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM |
publisher.none.fl_str_mv |
Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM |
dc.source.none.fl_str_mv |
Journal of the Brazilian Society of Mechanical Sciences and Engineering v.33 n.4 2011 reponame:Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) instname:Associação Brasileira de Engenharia e Ciências Mecânicas (ABCM) instacron:ABCM |
instname_str |
Associação Brasileira de Engenharia e Ciências Mecânicas (ABCM) |
instacron_str |
ABCM |
institution |
ABCM |
reponame_str |
Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) |
collection |
Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) |
repository.name.fl_str_mv |
Journal of the Brazilian Society of Mechanical Sciences and Engineering (Online) - Associação Brasileira de Engenharia e Ciências Mecânicas (ABCM) |
repository.mail.fl_str_mv |
||abcm@abcm.org.br |
_version_ |
1754734681920634880 |