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Singularities in the Complex Spatial Plane of a Vortex Sheet with Blob Regularization

Luo, Guo
http://rave.ohiolink.edu/etdc/view?acc_num=osu1250012314

Abstract Details

2009, Doctor of Philosophy, Ohio State University, Mathematics.
I study in this thesis a periodic version of Krasny's vortex blob equation (1987), whose solutions are known to converge to a classical weak solution of the Euler equation upon vanishing blob size (Liu and Xin 1995). The equation is analytically continued into the complex spatial plane of its parametrization variable, and for a specific initial condition used by Meiron et al. (1982), a simplified far-field equation is derived which is then studied using Taylor expansions in small time and small blob size. Like the unregularized vortex sheet studied by Cowley et al. (1999), the solutions of the far-field equation develop singularities immediately after the sheet starts to evolve. These singularities propagate in from infinity and local analysis suggests they are square-root branch points, possibly infinitely many in quantity. All analytic results are confirmed in a subsequent numerical study, and certain properties of the complex singularities, like their locations and strengths, are studied numerically in some detail. It appears that all singularities of the blob-regularized vortex sheet collapse to the 3/2-power singularity of the unregularized vortex sheet as the blob size vanishes, but the precise way in which this collapse happens is yet to be determined.
Gregory Baker, PhD (Advisor)
Saleh Tanveer, PhD (Committee Member)
Ed Overman, PhD (Committee Member)
179 p.
Mathematics
vortex sheet; vortex blob; complex singularity

Recommended Citations

Citations

  • Luo, G. (2009). Singularities in the Complex Spatial Plane of a Vortex Sheet with Blob Regularization [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1250012314

    APA Style (7th edition)

  • Luo, Guo. Singularities in the Complex Spatial Plane of a Vortex Sheet with Blob Regularization. 2009. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1250012314.

    MLA Style (8th edition)

  • Luo, Guo. "Singularities in the Complex Spatial Plane of a Vortex Sheet with Blob Regularization." Doctoral dissertation, Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=osu1250012314

    Chicago Manual of Style (17th edition)

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This open access ETD is published by The Ohio State University and OhioLINK.