On the geometry of developable surfaces in stratified incoming vortical flow over a cone
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| Natura: | Recurso digital |
| Lingua: | En |
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2026
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| _version_ | 1866901085608738816 |
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| author | Milyute E. Milyus A. |
| author_facet | Milyute E. Milyus A. |
| contents | <p>The paper considers a mathematical model of the formation of a developing surface in a stratified oncoming vortex flow when interacting with a conical obstacle. Similar structures arise in the hydrodynamics of swirling flows, atmospheric vortex systems, as well as in a number of technical applications related to turbomachines and aerodynamic devices. It is shown that the surface of the vortex layer descending from the cone can be described as a spiral ruled surface, the geometry of which is determined simultaneously by the shape of the obstacle and the structure of the oncoming flow. For the description, a system of equations of a stratified fluid in the Boussinesq approximation is used. In cylindrical coordinates, a parameterization of the surface is introduced through the radius function $r(\theta, z, t)$, depending on the angular coordinate, vertical coordinate and time. Expressions are obtained for the surface normal, coefficients of the first and second quadratic forms, Gaussian and mean curvature. The condition for the developability of the surface is derived and the criterion for the separation of the vortex layer from the surface of the cone is formulated. It is shown that the horizontal sections of the surface under consideration have the shape of an Archimedes spiral, and the pitch of the spiral is determined by the ratio of the layer descent velocity to the local angular velocity of the flow. The resulting model allows us to relate the geometry of the resulting surface with the parameters of the oncoming stratified vortex flow and the cone angle.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19141557 |
| institution | Zenodo |
| language | enc |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the geometry of developable surfaces in stratified incoming vortical flow over a cone Milyute E. Milyus A. oncoming stratified flow matter transfer vortex dynamics of substance circulation twist turbulence vortex layer separation criterion cone cavity developable surface Gaussian curvature vortex-sheet separation line torse <p>The paper considers a mathematical model of the formation of a developing surface in a stratified oncoming vortex flow when interacting with a conical obstacle. Similar structures arise in the hydrodynamics of swirling flows, atmospheric vortex systems, as well as in a number of technical applications related to turbomachines and aerodynamic devices. It is shown that the surface of the vortex layer descending from the cone can be described as a spiral ruled surface, the geometry of which is determined simultaneously by the shape of the obstacle and the structure of the oncoming flow. For the description, a system of equations of a stratified fluid in the Boussinesq approximation is used. In cylindrical coordinates, a parameterization of the surface is introduced through the radius function $r(\theta, z, t)$, depending on the angular coordinate, vertical coordinate and time. Expressions are obtained for the surface normal, coefficients of the first and second quadratic forms, Gaussian and mean curvature. The condition for the developability of the surface is derived and the criterion for the separation of the vortex layer from the surface of the cone is formulated. It is shown that the horizontal sections of the surface under consideration have the shape of an Archimedes spiral, and the pitch of the spiral is determined by the ratio of the layer descent velocity to the local angular velocity of the flow. The resulting model allows us to relate the geometry of the resulting surface with the parameters of the oncoming stratified vortex flow and the cone angle.</p> |
| title | On the geometry of developable surfaces in stratified incoming vortical flow over a cone |
| topic | oncoming stratified flow matter transfer vortex dynamics of substance circulation twist turbulence vortex layer separation criterion cone cavity developable surface Gaussian curvature vortex-sheet separation line torse |
| url | https://doi.org/10.5281/zenodo.19141557 |