{"id":{"repo_id":"ottawa-retro","oai_identifier":"oai:ruor.uottawa.ca:10393/45657"},"canonical_url":"https://search.dev.ndltd.org/etd/ottawa-retro/oai:ruor.uottawa.ca:10393/45657","repository":{"repo_id":"ottawa-retro","name":"University of Ottawa","base_url":"https://ruor.uottawa.ca/server/oai/request"},"display":{"title":"An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics","abstract":"The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [McDonald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost. Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution.","abstract_html":"The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [McDonald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost. Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution.","abstract_has_math":false,"creators":["Giroux, Fabien"],"institution":"Université d'Ottawa / University of Ottawa","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["McDonald, James Gerald"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-11-23T14:51:41Z","date_published":"2023-11-23T14:51:41Z","updated_at":"2026-07-24T03:39:40Z","subjects":["Moment Closure","Computational Fluid Dynamics","Boltzmann Equation","Maximum-Entropy"],"languages":["en"],"rights":["Attribution-NonCommercial-NoDerivatives 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["http://dx.doi.org/10.20381/ruor-29861"],"render_values":[{"text":"http://dx.doi.org/10.20381/ruor-29861","href":"http://dx.doi.org/10.20381/ruor-29861","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10393/45657","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McDonald, James Gerald"]},{"key":"dc:creator","label":"Author","values":["Giroux, Fabien"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-11-23T14:51:41Z","2023-11-23"]},{"key":"dc:publisher","label":"Institution","values":["Université d'Ottawa / University of Ottawa"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Moment Closure","Computational Fluid Dynamics","Boltzmann Equation","Maximum-Entropy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Attribution-NonCommercial-NoDerivatives 4.0 International","http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/10393/45657","http://dx.doi.org/10.20381/ruor-29861"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [McDonald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost. Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics"]}]}],"canonical_facts":{"dc:contributor":["McDonald, James Gerald"],"dc:creator":["Giroux, Fabien"],"dc:date":["2023-11-23T14:51:41Z","2023-11-23"],"dc:description":["The use of moment-closure methods to predict continuum and moderately rarefied flow offers many modelling and numerical advantages over traditional methods. The maximum-entropy family of moment closures offers models described by hyperbolic systems of balance laws. In particular, the twenty-one moment model of the maximum-entropy hierarchy offers a hyperbolic treatment of viscous flows exhibiting heat transfer. This twenty-one moment model has the ability to provide accurate solutions where the Navier-Stokes equations lose physical validity due to the solution being too far from local equilibrium. Furthermore, its first-order hyperbolic nature offers the potential for improved numerical accuracy as well as a decreased sensitivity to mesh quality. Unfortunately, higher-order maximum-entropy closures cannot be expressed in closed form. The only known affordable option is to propose approximations. Previous approximations to the fourteen-moment maximum-entropy model have been proposed [McDonald and Torrilhon, 2014]. Although this fourteen-moment model also predicts viscous flow with heat transfer, the necessary moments to close the system renders it more difficult to approximate accurately than the twenty-one moment model. The proposed approximation for the fourteen-moment model also has realizable states for which hyperbolicity is lost. Unfortunately, the velocity distribution function associated with the twenty-one moment model is an exponential of a fourth-order polynomial. Such a function cannot be integrated in closed form, resulting in closing fluxes that can only be obtained through complex numerical methods. The goal of this work is to present a new approximation to the closing fluxes that respect the maximum-entropy philosophy as closely as possible. Preliminary results show that a proposed approximation is able to provide shock predictions that are in good agreement with the Boltzmann equation and surpassing the prediction of the Navier-Stokes equations. Furthermore, Couette flow results as well as lid-driven cavity flows are computed using a novel approach to Knudsen layer boundary conditions. A dispersion analysis as well as an investigation of the hyperbolicity of the model is also shown. The Couette flow results are compared against Navier-Stokes and the free-molecular analytical solutions for a varying Knudsen number, for which the twenty-one moment model show good agreement over the domain. The shock-tube problem is also computed for different Knudsen numbers. The results are compared with the one obtained by directly solving the BGK equation. Finally, the lid-driven cavity flow computed with the twenty-one moment model shows good agreement with the direct simulation Monte-Carlo (DSMC) solution."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/10393/45657","http://dx.doi.org/10.20381/ruor-29861"],"dc:language":["en"],"dc:publisher":["Université d'Ottawa / University of Ottawa"],"dc:rights":["Attribution-NonCommercial-NoDerivatives 4.0 International","http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:subject":["Moment Closure","Computational Fluid Dynamics","Boltzmann Equation","Maximum-Entropy"],"dc:title":["An Approximation for the Twenty-One-Moment Maximum-Entropy Model of Rarefied Gas Dynamics"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T03:39:40Z"}