\"Transient state solution of the Navier Stokes equation was obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry around the axis in the falling direction, the Navier Stokes equation and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. The governing partial differential equations were non-dimensionalized. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity closer to the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations were found to be at Reynolds number 20 for steady state flows and at Reynolds numbers 22.46 and 28.24 for transient state flows I with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions and drag coefficients were calculated for both steady and unsteady states. Good agreement was obtained by comparing with existing experimental data when steady state was reached\"--Abstract, pages iii-iv.
","abstract_html":"<p>"Transient state solution of the Navier Stokes equation was obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry around the axis in the falling direction, the Navier Stokes equation and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. The governing partial differential equations were non-dimensionalized. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity closer to the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations were found to be at Reynolds number 20 for steady state flows and at Reynolds numbers 22.46 and 28.24 for transient state flows I with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions and drag coefficients were calculated for both steady and unsteady states. Good agreement was obtained by comparing with existing experimental data when steady state was reached"--Abstract, pages iii-iv.</p>","abstract_has_math":false,"creators":["Lin, Ching-Liang"],"institution":"University of Missouri--Rolla","degree_name":"Ph. D. in Mechanical Engineering","degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-02-10T08:00:00Z","date_published":"2016-02-10T08:00:00Z","updated_at":"2026-07-24T03:19:30Z","subjects":["Mechanical Engineering"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarsmine.mst.edu/doctoral_dissertations/185","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lin, Ching-Liang"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2016-02-10T08:00:00Z"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation - Open Access"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D. in Mechanical Engineering"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Missouri--Rolla"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mechanical Engineering"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarsmine.mst.edu/doctoral_dissertations/185"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["\"Transient state solution of the Navier Stokes equation was obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry around the axis in the falling direction, the Navier Stokes equation and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. The governing partial differential equations were non-dimensionalized. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity closer to the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations were found to be at Reynolds number 20 for steady state flows and at Reynolds numbers 22.46 and 28.24 for transient state flows I with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions and drag coefficients were calculated for both steady and unsteady states. Good agreement was obtained by comparing with existing experimental data when steady state was reached\"--Abstract, pages iii-iv.
"]},{"key":"dc:title","label":"Title","values":["Transient state analysis of separated flow around a sphere"]}]}],"canonical_facts":{"dc:creator":["Lin, Ching-Liang"],"dc:date.available":["2016-02-10T08:00:00Z"],"dc:description.abstract":["\"Transient state solution of the Navier Stokes equation was obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry around the axis in the falling direction, the Navier Stokes equation and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. The governing partial differential equations were non-dimensionalized. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity closer to the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations were found to be at Reynolds number 20 for steady state flows and at Reynolds numbers 22.46 and 28.24 for transient state flows I with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions and drag coefficients were calculated for both steady and unsteady states. Good agreement was obtained by comparing with existing experimental data when steady state was reached\"--Abstract, pages iii-iv.
"],"dc:identifier":["https://scholarsmine.mst.edu/doctoral_dissertations/185"],"dc:subject":["Mechanical Engineering"],"dc:title":["Transient state analysis of separated flow around a sphere"],"dc:type":["Dissertation - Open Access"],"thesis:degree_name":["Ph. D. in Mechanical Engineering"],"thesis:institution_name":["University of Missouri--Rolla"]},"updated_at":"2026-07-24T03:19:30Z"}