Shape calculus for vortex patch equilibria and its application to lattice configurations

Uda, Tomoki

doi:10.1016/j.cam.2017.10.023

Cited by 1 publication

(1 citation statement)

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“…In this regard, it is useful to review some preceding results on the steady vortex structures on the doubly periodic flat torus. Stationary lattice structures of point vortices [17][18][19] and a vortex patch without changing its boundary shape [20] are known. Gurarie & Chow [21] investigated the solution to the sinh-Poisson equation, ∇ 2 ψ = σ sinh ψ, in a doubly periodic domain and showed the existence of an elongated 'cat-eye' pattern and a symmetric 'diagonal' configuration with smooth vorticity distributions.…”

Section: Introductionmentioning

confidence: 99%

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

Sakajo

2019

Proc. R. Soc. A.

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A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α = R / r . A comparison with the Stuart vortex on the flat torus is also made.

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Section: Introductionmentioning

confidence: 99%

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

Sakajo

2019

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

Shape calculus for vortex patch equilibria and its application to lattice configurations

Cited by 1 publication

References 18 publications

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

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