Unstable manifolds computation for the two-dimensional plane Poiseuille flow
In this work we study some aspects of the dynamics of the plane Poiseuille problem in dimension 2, in what refers to the connection among different configurations of the flow. The fluid is confined in a channel of plane parallel walls. The problem is modeled by the incompressible Navier-Stokes equat...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2003 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/899 |
| Acceso en línea: | https://hdl.handle.net/2117/899 |
| Access Level: | acceso abierto |
| Palabra clave: | Differentiable dynamical systems Fluid mechanics Poiseuille flow Unstable manifolds computation Sistemes dinàmics diferenciables Teoria ergòdica Fluids Vorticitat -- Teoria Classificació AMS::76 Fluid mechanics::76D Incompressible viscous fluids Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications |
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Unstable manifolds computation for the two-dimensional plane Poiseuille flowSánchez Casas, José PabloJorba, AngelDifferentiable dynamical systemsFluid mechanicsPoiseuille flowUnstable manifolds computationSistemes dinàmics diferenciablesTeoria ergòdicaFluidsVorticitat -- TeoriaClassificació AMS::76 Fluid mechanics::76D Incompressible viscous fluidsClassificació AMS::37 Dynamical systems and ergodic theory::37N ApplicationsIn this work we study some aspects of the dynamics of the plane Poiseuille problem in dimension 2, in what refers to the connection among different configurations of the flow. The fluid is confined in a channel of plane parallel walls. The problem is modeled by the incompressible Navier-Stokes equations ∂u ∂t + (u · ∇)u = −∇p + 1 Re ∆u, ∇ · u = 0, (1) being Re the Reynolds number, together with no-slip on the channel walls and L-periodic boundary conditions (L = 2π/α, being α the parameter wave number). We have considered two different formulations to drive the fluid through the channel: holding constant the total flux or the mean pressure gradient. For each of them we obtain a different definition of Re = hUc/ν, where h represents half of the channel length, Uc the velocity of the laminar flow in the centre of the channel, and ν the kinematic viscosity. To be precise ReQ = 3Q/ν, Rep = Gh3ρ/2µ2, corresponds to the Reynolds numbers when we keep Q as a constant flux or G as a constant mean pressure gradient respectively. The fluid is supposed of constant density ρ and viscosity µ. The numerical approximation is detailed in [1]. Roughly, it employs Fourier and Chebyshev spectral discretizations of velocities and pressure in the periodic and transversal directions respectively. The temporal variable is approximated by means of finite differences. In figure 1 we represent bifurcating curves of periodic and quasi-periodic flows obtained numerically in [1] using N = 8 and M = 70 spectral modes in the x and y spatial directions respectively. For the temporal discretizacion the time step has been prescribed to ∆t = 0.02. Each point of those curves corresponds to the amplitude A (distance to the laminar flow in L2-norm) of the flow (periodic or quasi-periodic) for the given value of Re. It is also marked on the curves the different stability regions, together with several Hopf bifurcations. In figure 1b at ReQ1, the Hopf bifurcation of periodic flows give rise to a family of quasi-periodic solutions, whose stability is also presented. The main load of the computations has been carried out in parallel in a Beowulf cluster of PCs.20032003-01-0120072007-05-07journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/899reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/8992026-05-27T15:37:01Z |
| dc.title.none.fl_str_mv |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| title |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| spellingShingle |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow Sánchez Casas, José Pablo Differentiable dynamical systems Fluid mechanics Poiseuille flow Unstable manifolds computation Sistemes dinàmics diferenciables Teoria ergòdica Fluids Vorticitat -- Teoria Classificació AMS::76 Fluid mechanics::76D Incompressible viscous fluids Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications |
| title_short |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| title_full |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| title_fullStr |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| title_full_unstemmed |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| title_sort |
Unstable manifolds computation for the two-dimensional plane Poiseuille flow |
| dc.creator.none.fl_str_mv |
Sánchez Casas, José Pablo Jorba, Angel |
| author |
Sánchez Casas, José Pablo |
| author_facet |
Sánchez Casas, José Pablo Jorba, Angel |
| author_role |
author |
| author2 |
Jorba, Angel |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
Differentiable dynamical systems Fluid mechanics Poiseuille flow Unstable manifolds computation Sistemes dinàmics diferenciables Teoria ergòdica Fluids Vorticitat -- Teoria Classificació AMS::76 Fluid mechanics::76D Incompressible viscous fluids Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications |
| topic |
Differentiable dynamical systems Fluid mechanics Poiseuille flow Unstable manifolds computation Sistemes dinàmics diferenciables Teoria ergòdica Fluids Vorticitat -- Teoria Classificació AMS::76 Fluid mechanics::76D Incompressible viscous fluids Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications |
| description |
In this work we study some aspects of the dynamics of the plane Poiseuille problem in dimension 2, in what refers to the connection among different configurations of the flow. The fluid is confined in a channel of plane parallel walls. The problem is modeled by the incompressible Navier-Stokes equations ∂u ∂t + (u · ∇)u = −∇p + 1 Re ∆u, ∇ · u = 0, (1) being Re the Reynolds number, together with no-slip on the channel walls and L-periodic boundary conditions (L = 2π/α, being α the parameter wave number). We have considered two different formulations to drive the fluid through the channel: holding constant the total flux or the mean pressure gradient. For each of them we obtain a different definition of Re = hUc/ν, where h represents half of the channel length, Uc the velocity of the laminar flow in the centre of the channel, and ν the kinematic viscosity. To be precise ReQ = 3Q/ν, Rep = Gh3ρ/2µ2, corresponds to the Reynolds numbers when we keep Q as a constant flux or G as a constant mean pressure gradient respectively. The fluid is supposed of constant density ρ and viscosity µ. The numerical approximation is detailed in [1]. Roughly, it employs Fourier and Chebyshev spectral discretizations of velocities and pressure in the periodic and transversal directions respectively. The temporal variable is approximated by means of finite differences. In figure 1 we represent bifurcating curves of periodic and quasi-periodic flows obtained numerically in [1] using N = 8 and M = 70 spectral modes in the x and y spatial directions respectively. For the temporal discretizacion the time step has been prescribed to ∆t = 0.02. Each point of those curves corresponds to the amplitude A (distance to the laminar flow in L2-norm) of the flow (periodic or quasi-periodic) for the given value of Re. It is also marked on the curves the different stability regions, together with several Hopf bifurcations. In figure 1b at ReQ1, the Hopf bifurcation of periodic flows give rise to a family of quasi-periodic solutions, whose stability is also presented. The main load of the computations has been carried out in parallel in a Beowulf cluster of PCs. |
| publishDate |
2003 |
| dc.date.none.fl_str_mv |
2003 2003-01-01 2007 2007-05-07 |
| dc.type.none.fl_str_mv |
journal article http://purl.org/coar/resource_type/c_6501 NA http://purl.org/coar/version/c_be7fb7dd8ff6fe43 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
| format |
article |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2117/899 |
| url |
https://hdl.handle.net/2117/899 |
| dc.language.none.fl_str_mv |
Inglés eng |
| language_invalid_str_mv |
Inglés |
| language |
eng |
| dc.rights.none.fl_str_mv |
open access http://purl.org/coar/access_right/c_abf2 Attribution-NonCommercial-NoDerivs 2.5 Spain http://creativecommons.org/licenses/by-nc-nd/2.5/es/ |
| dc.rights.openaire.fl_str_mv |
info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
open access http://purl.org/coar/access_right/c_abf2 Attribution-NonCommercial-NoDerivs 2.5 Spain http://creativecommons.org/licenses/by-nc-nd/2.5/es/ |
| eu_rights_str_mv |
openAccess |
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application/pdf |
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reponame:UPCommons. Portal del coneixement obert de la UPC instname:Universitat Politècnica de Catalunya (UPC) |
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