Two cylindrical vortex sheets : evolution and singularity formation
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| Title: | Two cylindrical vortex sheets : evolution and singularity formation |
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| Authors: | Johnson, Jeremy David |
| Committee Members: | Nitsche, Monika; Monika Nitsche; Pedro Embid; Jens Lorenz |
| Summary: | Using Rosenhead's point-vortex approximation with correction terms, the evolution of two symmetrical, counter-rotating, initially cylindrical vortex sheets in an incompressible, potential fluid flow is studied. Simulations are performed in time up to the occurrence of branch-point curvature singularities in the vortex sheets' geometries. The numerical methods employed are discussed. Parameters pertaining to the asymptotics of the Fourier coefficients of the vortex sheets' positions are numerically fitted to gain insight into aspects of the singularity formation; these include the order of the branch-point singularities, and the times and locations of singularity formation. A smoothing over initial singularity formations is implemented by either the heat equation or through a local application of the vortex blob method in an attempt to gain details into further singularity formations. Lastly, the effects of the initially prescribed total circulation around the vortex sheets on their evolutions are studied, both up to the time of singularity formation, and with the implementation of the vortex blob method, past the times of singularity formation. |
| URL: | https://digitalrepository.unm.edu/math_etds/22 |
| Database: | OpenDissertations |
| Abstract: | Using Rosenhead's point-vortex approximation with correction terms, the evolution of two symmetrical, counter-rotating, initially cylindrical vortex sheets in an incompressible, potential fluid flow is studied. Simulations are performed in time up to the occurrence of branch-point curvature singularities in the vortex sheets' geometries. The numerical methods employed are discussed. Parameters pertaining to the asymptotics of the Fourier coefficients of the vortex sheets' positions are numerically fitted to gain insight into aspects of the singularity formation; these include the order of the branch-point singularities, and the times and locations of singularity formation. A smoothing over initial singularity formations is implemented by either the heat equation or through a local application of the vortex blob method in an attempt to gain details into further singularity formations. Lastly, the effects of the initially prescribed total circulation around the vortex sheets on their evolutions are studied, both up to the time of singularity formation, and with the implementation of the vortex blob method, past the times of singularity formation. |
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