{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/106183"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/106183","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A high-order Legendre-WENO numerical scheme for the non-conservative kernel density equations modeling particle-laden flows","abstract":"A numerical scheme is developed to integrate the kernel density equations for the particle phase of disperse flows. The kernel density equations arise from least squares minimization of the approximation of the position-velocity phase-space particle distribution by a sum of kernel density functions. For concreteness in this work, Gaussian kernel density functions are used in approximating the particle distribution. The resulting kernel density equations form a non-conservative, hyperbolic system of PDE’s. The basic numerical scheme is a Roe method that has been generalized to non-conservative systems. The kernel density equations can become weakly hyperbolic; in this case an asymptotic expansion is introduced and singular terms are handled analytically. High-order spatial accuracy is achieved using a novel hybrid essentially non-oscillatory scheme with a Legendre basis. A standard WENO scheme is used by default, and a nonlinear mapping is introduced when needed to enforce physically relevant bounds in the solution, especially in areas of particle depletion. The strong stability preserving RK3 scheme is used to obtain high temporal accuracy. The new scheme has been applied to direct numerical simulation of particles in homogeneous turbulence.","abstract_html":"A numerical scheme is developed to integrate the kernel density equations for the particle phase of disperse flows. The kernel density equations arise from least squares minimization of the approximation of the position-velocity phase-space particle distribution by a sum of kernel density functions. For concreteness in this work, Gaussian kernel density functions are used in approximating the particle distribution. The resulting kernel density equations form a non-conservative, hyperbolic system of PDE’s. The basic numerical scheme is a Roe method that has been generalized to non-conservative systems. The kernel density equations can become weakly hyperbolic; in this case an asymptotic expansion is introduced and singular terms are handled analytically. High-order spatial accuracy is achieved using a novel hybrid essentially non-oscillatory scheme with a Legendre basis. A standard WENO scheme is used by default, and a nonlinear mapping is introduced when needed to enforce physically relevant bounds in the solution, especially in areas of particle depletion. The strong stability preserving RK3 scheme is used to obtain high temporal accuracy. The new scheme has been applied to direct numerical simulation of particles in homogeneous turbulence.","abstract_has_math":false,"creators":["Smith, Timothy Andrew"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Theoretical & Applied Mechans","degree_department":null,"school":null,"contributors":["Pantano, Carlos","Fischer, Paul","Ewoldt, Randy","Bodony, Daniel"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-03-02T21:58:08Z","date_published":"2020-03-02T21:58:08Z","updated_at":"2026-07-22T22:24:45Z","subjects":["dispersed flows","essentially non-oscillatory","finite volume"],"languages":["en"],"rights":["Copyright 2019 Timothy Andrew Smith"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/106183","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Pantano, Carlos","Fischer, Paul","Ewoldt, Randy","Bodony, Daniel"]},{"key":"dc:creator","label":"Author","values":["Smith, Timothy Andrew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-03-02T21:58:08Z","2019-11-22","2019-12"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Theoretical & Applied Mechans"]},{"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":["dispersed flows","essentially non-oscillatory","finite volume"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 Timothy Andrew Smith"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/106183"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A numerical scheme is developed to integrate the kernel density equations for the particle phase of disperse flows. The kernel density equations arise from least squares minimization of the approximation of the position-velocity phase-space particle distribution by a sum of kernel density functions. For concreteness in this work, Gaussian kernel density functions are used in approximating the particle distribution. The resulting kernel density equations form a non-conservative, hyperbolic system of PDE’s. The basic numerical scheme is a Roe method that has been generalized to non-conservative systems. The kernel density equations can become weakly hyperbolic; in this case an asymptotic expansion is introduced and singular terms are handled analytically. High-order spatial accuracy is achieved using a novel hybrid essentially non-oscillatory scheme with a Legendre basis. A standard WENO scheme is used by default, and a nonlinear mapping is introduced when needed to enforce physically relevant bounds in the solution, especially in areas of particle depletion. The strong stability preserving RK3 scheme is used to obtain high temporal accuracy. The new scheme has been applied to direct numerical simulation of particles in homogeneous turbulence.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-02-28 without embargo terms","The student, Timothy Smith, accepted the attached license on 2019-11-10 at 22:33.","The student, Timothy Smith, submitted this Dissertation for approval on 2019-11-10 at 22:46.","This Dissertation was approved for publication on 2019-11-22 at 17:32.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14539 on 2020-02-28 at 17:13:17","Made available in DSpace on 2020-03-02T21:58:08Z (GMT). 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For concreteness in this work, Gaussian kernel density functions are used in approximating the particle distribution. The resulting kernel density equations form a non-conservative, hyperbolic system of PDE’s. The basic numerical scheme is a Roe method that has been generalized to non-conservative systems. The kernel density equations can become weakly hyperbolic; in this case an asymptotic expansion is introduced and singular terms are handled analytically. High-order spatial accuracy is achieved using a novel hybrid essentially non-oscillatory scheme with a Legendre basis. A standard WENO scheme is used by default, and a nonlinear mapping is introduced when needed to enforce physically relevant bounds in the solution, especially in areas of particle depletion. The strong stability preserving RK3 scheme is used to obtain high temporal accuracy. The new scheme has been applied to direct numerical simulation of particles in homogeneous turbulence.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-02-28 without embargo terms","The student, Timothy Smith, accepted the attached license on 2019-11-10 at 22:33.","The student, Timothy Smith, submitted this Dissertation for approval on 2019-11-10 at 22:46.","This Dissertation was approved for publication on 2019-11-22 at 17:32.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14539 on 2020-02-28 at 17:13:17","Made available in DSpace on 2020-03-02T21:58:08Z (GMT). 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