{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120197"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120197","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Nonlinear fluid dynamics in complex geometries","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","abstract_has_math":false,"creators":["Biagioli, Madeleine George"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Chemical Engineering","degree_department":null,"school":null,"contributors":["Higdon, Jonathan J.L.","Rao, Christopher V","Peters, Baron G","Kuzan, John D"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Fluid Dynamics","Hybridizable Discontinuous Galerkin Method","Hdg","High Performance Computation","Rosenbrock","Navier-stokes","Convection","Reynolds Number","Convective Diffusion","Complex Geometries","Extended Domains","Isoparametric Mapping","Nonlinear Flow","Nonlinear Differential Equations","Differential Algebraic Equations","Discontinuous Galerkin","Galerkin","Spectral Elements","High Order"],"languages":["en","eng"],"rights":["© 2023 Madeleine George Biagioli"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120197","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Higdon, Jonathan J.L.","Rao, Christopher V","Peters, Baron G","Kuzan, John D"]},{"key":"dc:creator","label":"Author","values":["Biagioli, Madeleine George"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-01-30"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Chemical Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fluid Dynamics","Hybridizable Discontinuous Galerkin Method","Hdg","High Performance Computation","Rosenbrock","Navier-stokes","Convection","Reynolds Number","Convective Diffusion","Complex Geometries","Extended Domains","Isoparametric Mapping","Nonlinear Flow","Nonlinear Differential Equations","Differential Algebraic Equations","Discontinuous Galerkin","Galerkin","Spectral Elements","High Order"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["© 2023 Madeleine George Biagioli"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120197"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","The student, Madeleine Biagioli, accepted the attached license on 2023-01-27 at 06:30.","The student, Madeleine Biagioli, submitted this Dissertation for approval on 2023-01-27 at 06:31.","This Dissertation was approved for publication on 2023-01-30 at 16:25.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18832 on 2023-09-01 at 17:07:15","A high-order spectral element method was developed to solve the 2D Navier-Stokes equations in convoluted geometries. The method is based on the hybridizable discontinuous Galerkin (HDG) method of Nguyen, et al., 2011 combined with spectral elements and isoparametric mapping for the geometry. The HDG method is further combined with a Rosenbrock time-stepping algorithm for unsteady problems, yielding high order spatial and temporal accuracy. The discretized incompressible Navier-Stokes equations, combined with the continuity equation, yield a system of differential algebraic equations (DAE) which are efficiently solved with the Rosenbrock algorithm given an appropriate set of stage coefficients attuned to the index of the DAE system. Computational performance data are presented demonstrating hp spectral convergence in spatial resolution and 3rd order convergence in time discretization. Scaling analysis and timing data are presented to select optimal parameters for computational efficiency with the three major components of CPU time including (1) a global sparse linear solve, (2) local elemental inverses, and (3) elemental back substitution for local variables. Nonlinear flows in highly convoluted geometries are relevant to applications in a wide array of chemical engineering problems; examples include flows in packed beds and other porous media, microfluidics, and meandering river dynamics. Applying isoparametric mapping to a spectral element mesh provides a high order of accuracy and stability without the need for lengthy re-gridding protocols as with standard finite element meshes. Furthermore, with the local solver based on the HDG method of Nguyen, the large coupled system is broken into global and local subproblems which allow local variables to be solved independently and efficiently on each element. Numerous examples are presented for both steady state and unsteady Navier-Stokes flows over a range of highly distorted flow channel geometries which exhibit many flow characteristics of interest. We observe separated flows and multiple steady states as well as oscillatory driving functions with possibly non-oscillatory system dynamics. We also observe departure from quasi-steady behavior with different flow configurations at given system states (Reynolds numbers) which differ from steady state results. This combined isoparametric mapped HDG-Rosenbrock platform is appropriate for extension to many areas of interest to chemical engineers, including microfluidics, empirical turbulent flow models, and non-Newtonian flows."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Nonlinear fluid dynamics in complex geometries"]}]}],"canonical_facts":{"dc:contributor":["Higdon, Jonathan J.L.","Rao, Christopher V","Peters, Baron G","Kuzan, John D"],"dc:creator":["Biagioli, Madeleine George"],"dc:date":["2023-05","2023-01-30"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms","The student, Madeleine Biagioli, accepted the attached license on 2023-01-27 at 06:30.","The student, Madeleine Biagioli, submitted this Dissertation for approval on 2023-01-27 at 06:31.","This Dissertation was approved for publication on 2023-01-30 at 16:25.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18832 on 2023-09-01 at 17:07:15","A high-order spectral element method was developed to solve the 2D Navier-Stokes equations in convoluted geometries. The method is based on the hybridizable discontinuous Galerkin (HDG) method of Nguyen, et al., 2011 combined with spectral elements and isoparametric mapping for the geometry. The HDG method is further combined with a Rosenbrock time-stepping algorithm for unsteady problems, yielding high order spatial and temporal accuracy. The discretized incompressible Navier-Stokes equations, combined with the continuity equation, yield a system of differential algebraic equations (DAE) which are efficiently solved with the Rosenbrock algorithm given an appropriate set of stage coefficients attuned to the index of the DAE system. Computational performance data are presented demonstrating hp spectral convergence in spatial resolution and 3rd order convergence in time discretization. Scaling analysis and timing data are presented to select optimal parameters for computational efficiency with the three major components of CPU time including (1) a global sparse linear solve, (2) local elemental inverses, and (3) elemental back substitution for local variables. Nonlinear flows in highly convoluted geometries are relevant to applications in a wide array of chemical engineering problems; examples include flows in packed beds and other porous media, microfluidics, and meandering river dynamics. Applying isoparametric mapping to a spectral element mesh provides a high order of accuracy and stability without the need for lengthy re-gridding protocols as with standard finite element meshes. Furthermore, with the local solver based on the HDG method of Nguyen, the large coupled system is broken into global and local subproblems which allow local variables to be solved independently and efficiently on each element. Numerous examples are presented for both steady state and unsteady Navier-Stokes flows over a range of highly distorted flow channel geometries which exhibit many flow characteristics of interest. We observe separated flows and multiple steady states as well as oscillatory driving functions with possibly non-oscillatory system dynamics. We also observe departure from quasi-steady behavior with different flow configurations at given system states (Reynolds numbers) which differ from steady state results. This combined isoparametric mapped HDG-Rosenbrock platform is appropriate for extension to many areas of interest to chemical engineers, including microfluidics, empirical turbulent flow models, and non-Newtonian flows."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120197"],"dc:language":["en","eng"],"dc:rights":["© 2023 Madeleine George Biagioli"],"dc:subject":["Fluid Dynamics","Hybridizable Discontinuous Galerkin Method","Hdg","High Performance Computation","Rosenbrock","Navier-stokes","Convection","Reynolds Number","Convective Diffusion","Complex Geometries","Extended Domains","Isoparametric Mapping","Nonlinear Flow","Nonlinear Differential Equations","Differential Algebraic Equations","Discontinuous Galerkin","Galerkin","Spectral Elements","High Order"],"dc:title":["Nonlinear fluid dynamics in complex geometries"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Chemical Engineering"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}