Abstract : Vortex dynamics is a classical but ever active topic in the study of fluid flows. Despite huge efforts to understand vortex phenomena, many aspects are still not properly understood. In this minisymposium, Elling and Jeong are presenting mathematical and rigorous results of self-similar vortices. Xu will describe computations of elliptical vortices. Kim and Krishnamurthy will discuss point vortex dynamics and generalized geostrophic models. Nitsche and Sohn speak on computational issues for interfacial flows and application to swimming. Krasny will present computations of plasma vortices in the Vlasov-Poisson equation.
Organizer(s) : Sun-Chul Kim, Robert Krasny, Sung-Ik Sohn
Abstract : We investigate the dynamics of logarithmic vortex for the two-dimensional incompressible Euler equations. More precisely, we consider vorticity which is invariant under the transformation $(r,\theta) \mapsto (\lambda r, \theta + c \ln(\lambda))$ for any $\lambda>0$ and some $c>0$. Within this class of vorticities, one can consider various types, including patches and sheets. We derive the equations of motion for logarithmic vortex and consider the limit problem where patches become sheets.
[00069] Dynamics of elliptical vortices
Format : Online Talk on Zoom
Author(s) :
Ling Xu (North Carolina Agricultural and Mechanical State University)
Robert Krasny (University of Michigan, Ann Arbor)
Abstract : We examine the dynamics of elliptical vortices in 2D ideal fluid using an adaptively refined and remeshed vortex method. Four cases are considered: the compact MMZ and POLY vortices, and noncompact Gaussian and smooth Kirchhoff vortices (SK). The vortices have the same maximum vorticity and 2:1 initial aspect ratio, but unlike the top-hat Kirchhoff vortex, they have continuous profiles with different regularity. In all cases the co-rotating phase portrait has two hyperbolic points. At early time two filaments emerge and form a halo around the core as vorticity is advected along the unstable manifold of each hyperbolic point. The Gaussian vortex rapidly axisymmetrizes, but later on the core begins to oscillate and two small lobes emerge adjacent to the core; this is attributed to a resonance. For the MMZ, POLY, and SK vortices, the core maintains its ellipticity for longer time and the filaments entrain fluid into two large lobes forming a non-axisymmetric tripole state; afterwards the lobes repeatedly detrain fluid into the halo; this is attributed to a heteroclinic tangle. While prior work suggested that elliptical vortices evolve to either an axisymmetric state or a non-axisymmetric tripole state, our results suggest that such vortices may oscillate between these states.
[00121] The N-vortex problem in doubly-periodic domains with background vorticity
Format : Talk at Waseda University
Author(s) :
Vikas Krishnamurthy (IIT Hyderabad)
Takashi Sakajo (Kyoto University)
Abstract : We study the N-vortex problem in a doubly periodic rectangular domain in the presence of a constant background vorticity field. Using a conformal mapping approach, we derive an explicit formula for the hydrodynamic Green's function. We show that the point vortices form a Hamiltonian system and that the two-vortex problem is integrable. Several fixed lattice configurations are obtained for general N, some of which consist of vortices with inhomogeneous strengths and lattice defects.
[00127] Swimming of a Fish-like Body by using a Vortex Shedding Model
Format : Talk at Waseda University
Author(s) :
SUNG-IK SOHN (Gangneung-Wonju National University)
Abstract : We consider the undulatory motion of a body translating through a quiescent fluid, which is motivated by the anguilliform swimming of aquatic animals, e.g., eels. We use an inviscid vortex shedding model to investigate the swimming dynamics. The model demonstrates the self-propulsion of the swimming body and yields pairs of anti-rotating vortices shed from the body. We examine the wake pattern and swimming efficiency which depends on the recoil motions of a body.
Sun-Chul Kim (Chung-Ang University, Seoul, Korea (Republic of))
Habin Yim (Chung-Ang University, Seoul, Korea (Republic of))
Sung-Ik Sohn (Gangneung-Wonju National University)
Abstract : We investigate the dynamics of geostrophic Bessel vortices focusing on the three-vortex case, where the possibility of self-similar motion and general dynamics for arbitrary strengths is studied. It is found that self-similar motions are limited to rigid rotations and self-similar triple collapse is impossible. For a general description, trilinear coordinates are adopted. The physical regions in the phase plane cannot be directly identified, but the boundary approaches the vertices of the triangle in trilinear coordinates in geostrophic vortices.
[00126] Self-similar vortical flows
Format : Talk at Waseda University
Author(s) :
Volker Wilhelm Elling (Academia Sinica, Taipei)
Abstract : Vortex spirals and vortex cusps are important features of self-similar vortical flows near stagnation points. Vortex sheets produced at triple points of Mach reflection have distinguished signs that determine whether interaction with walls or symmetry axes can be attached cusps or detached jets. Progress on analysis, modelling and numerics for such phenomena is discussed, along with applications to shock reflection or non-uniqueness of vortical flows.
[00150] Near-singular integrals in 3D interfacial Stokes and potential flows
Format : Talk at Waseda University
Author(s) :
Monika Nitsche (University of New Mexico)
Abstract : Boundary integral formulations yield efficient numerical methods to solve elliptic boundary value problems. They are the method of choice for interfacial fluid flow in either the inviscid vortex sheet limit, or the viscous Stokes limit. The fluid velocity at a target point is given by an integral over all interfaces. However, for target points near, but not on the interface, the integrals are near-singular and standard quadratures lose accuracy. While several accurate methods for near-singular integrals exist in planar geometries, they do not generally apply to the non-analytic case that arises in axisymmetric or 3D geometries. We present a method based on Taylor series expansions of the integrand about basepoints on the interface that accurately resolve a large class of integrals, and apply it to solve the near-interface problem in planar vortex sheet flow, axisymmetric Stokes flow, and Stokes flow in 3D. The application to multi-nested Stokes flow uses a novel representation of the fluid velocity.
[00147] The FARSIGHT Vlasov-Poisson code
Format : Talk at Waseda University
Author(s) :
Robert Krasny (University of Michigan)
Ryan Sandberg (Lawrence Berkeley National Laboratory)
Alexander Thomas (University of Michigan)
Abstract : We present electrostatic plasma simulations using a new semi-Lagrangian particle method for the Vlasov-Poisson equations called FARSIGHT. The electron density is represented on adaptively refined and remeshed panels in phase space, and the macroparticles are advected using a regularized electric field kernel and a GPU-accelerated barycentric Lagrange treecode. Results are presented for Landau damping, two-stream instability, and ion beam propagation. Work supported by AFOSR grant FA9550-19-1-0072.
