Bulletin of the American Physical Society
75th Annual Meeting of the Division of Fluid Dynamics
Volume 67, Number 19
Sunday–Tuesday, November 20–22, 2022; Indiana Convention Center, Indianapolis, Indiana.
Session G28: Vortex Dynamics: Theory |
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Chair: Mark Stremler, Virginia Tech Room: 236 |
Sunday, November 20, 2022 3:00PM - 3:13PM |
G28.00001: Variational Formulation of Vortex Dynamics Nabil M Khalifa, Haithem E Taha Vortex dynamics for a system of constant strength point vortices is described by the Kirchhoff-Routh (KR) function, where the resulting first-order differential equations have a special non-canonical Hamiltonian structure. Although this Hamiltonian is not derived from basic definitions in classical mechanics, it was widely adopted in Literature. In contrast, a new formulation of vortex dynamics is developed here using the principle of least action and formal variational principles for fluid mechanics. A system of non-deforming free vortex patches of constant strength is considered as a case study, where, for the first time, the resulting ODEs are second order in nature, defining acceleration. As a consequence, the dynamics is richer than that derived from the KR where vortices can attract and repel each other (i.e., there is a radial acceleration component along the line connecting each pair of vortices). Interestingly, the new formulation can recover the KR solution in the limit of vanishing core size (i.e., point vortices). Finally, the fact that the resulting model is derived from first principles (not devised to match certain dynamics) makes it applicable to arbitrary time-varying vortices. |
Sunday, November 20, 2022 3:13PM - 3:26PM |
G28.00002: Fluid Dynamics of a Periodic Array of Axisymmetric Thin-cored Vortex Rings Emad Masroor, Mark A Stremler It is well known that a spatially-periodic array of identical, axisymmetric, thin-cored vortex rings moves uniformly without change of shape or form in the direction of the central axis of symmetry, and is an equilibrium solution of Euler's equations. We revisit this classical result, due to Vasilev (1914), using the modern formalism of Borisov et. al (2013). In a frame of reference moving with the system of vortex rings, the motion of passive fluid particles is investigated as a function of the two nondimensional parameters that define this system: ε=a/R, the ratio of minor radius to major radius of the torus-shaped vortex rings, and λ=L/R, the separation of the vortex rings normalized by their radii. Two bifurcations in the streamline topology are found that depend significantly on ε and λ; these bifurcations delineate three distinct shapes of the "atmosphere" of fluid particles that move together with the vortex ring for all time. Analogous to the case of an isolated vortex ring, the atmospheres can be "thin-bodied" or "thick-bodied". Additionally, we find the occurrence of a "connected" system, in which the atmospheres of neighboring rings touch at an invariant circle of fluid particles that is stationary in a frame of reference moving with the vortex rings. |
Sunday, November 20, 2022 3:26PM - 3:39PM |
G28.00003: On the finite-time singularity for an inviscid vortex ring model Philip J Morrison, Yoshifumi Kimura In Moffat and Kimura [1], a low degree-of-freedom model for describing the approach to finite-time singularity of the incompressible Euler fluid equations was introduced. The model is obtained by assuming an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes and then approximating the subsequent evolution. In [2] we obtained the Hamiltonian structure of the inviscid limit of the model. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant were discovered. It is shown that the system is integrable with a solution that lies on the intersection of the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we showed the existence of finite-time singularity. This talk is a continuation of [3], with an emphasis on the physical interpretation of the invariants and subsequent evolution. |
Sunday, November 20, 2022 3:39PM - 3:52PM |
G28.00004: Role of Vortex Line Coiling in Vortex Breakdown Eric N Stout, Fazle Hussain We propose a conceptual foundation for the vortex breakdown mechanism within a column with axial flow using the dynamics of vortex line coiling to develop a breakdown criterion. The stagnation point at the onset of breakdown’s recirculation bubble can be found by setting the axial velocity of the column equal to the axial velocity induced by the coiled vortex lines. Deriving the coiled vorticity and associated axial flow yields a model for the breakdown criterion q (the ratio of inlet peak azimuthal to peak axial velocities) in terms of the core radius variation (δr) and axial length (zL) of the coiling. The model gives q2=[16πm(1+2µ+µ2)]/[7(2µ2+µ3)], where m=δr/zL and µ=δr/r1 (r1 is the upstream core radius, i.e. at the start of the core radius variation), as the breakdown threshold. Prior models result in particular q values, while ours adapts q to a variety of vortex spreading rates (which also controls the coiling rate). For a given sinuously varying vortex core, the model predicts a stagnation point (hence breakdown criterion) at q=1.2, which occurs numerically between q=1.15 and q=1.25; prior data is also compared to our model. We have thus derived a simple, intuitive model for breakdown via the generation of axial flow using only the core radius variation of a column. |
Sunday, November 20, 2022 3:52PM - 4:05PM |
G28.00005: Evolution of dissipative fluid flows with imposed helicity conservation Zhaoyuan Meng, Weiyu Shen, Yue Yang We propose the helicity-conserved Navier–Stokes (HCNS) equation by modifying the non-ideal force term in the Navier–Stokes (NS) equation. The corresponding HCNS flow has the strict helicity conservation, and retains major NS dynamics with finite dissipation. Using the helical wave decomposition, we show that the pentadic interaction of Fourier helical modes in the HCNS dynamics are more complex than the triadic interaction in the NS dynamics, and enhanced variations for left- and right-handed helicity components cancel each other in the HCNS flow to keep the invariant helicity. The comparative study of HCNS and NS flow evolutions with the direct numerical simulation elucidates the influence of the helicity conservation on flow structures and statistics in the vortex reconnection and isotropic turbulence. First, the HCNS flow evolves towards a Beltrami state with a −4 scaling law of the energy spectrum at high wavenumbers at long times. Second, large-scale flow structures are almost identical during the viscous reconnection of vortex tubes in the two flows, whereas much more small-scale helical structures are generated via the pentadic mode interaction in the HCNS flow than in the NS flow. Moreover, we demonstrate that the parity breaking at small scales can trigger a notable helicity variation in the NS flow. |
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