{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/94547"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/94547","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Solutions to three laminar viscous flow problems by an implicit finite-difference method","abstract":"This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method. The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body. The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method. In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method. The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed.","abstract_html":"This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method. The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body. The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method. In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method. The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed.","abstract_has_math":false,"creators":["Chyu, Wei Jao"],"institution":"Virginia Polytechnic Institute","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Engineering Mechanics","degree_department":"Engineering Mechanics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1965,"date_issued":"1965","date_published":"1965","updated_at":"2026-07-22T22:19:53Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/94547","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Engineering Mechanics"]},{"key":"dc:creator","label":"Author","values":["Chyu, Wei Jao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-10-10T19:28:02Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-10-10T19:28:02Z"]},{"key":"dc:date.issued","label":"Date","values":["1965"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering Mechanics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/94547"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method. The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body. The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method. In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method. The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Solutions to three laminar viscous flow problems by an implicit finite-difference method"]}]}],"canonical_facts":{"dc:contributor.department":["Engineering Mechanics"],"dc:creator":["Chyu, Wei Jao"],"dc:date.accessioned":["2019-10-10T19:28:02Z"],"dc:date.available":["2019-10-10T19:28:02Z"],"dc:date.issued":["1965"],"dc:description.abstract":["This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method. The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body. The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method. In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method. The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/94547"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Solutions to three laminar viscous flow problems by an implicit finite-difference method"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Engineering Mechanics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:19:53Z"}