Bulletin of the American Physical Society
73rd Annual Meeting of the APS Division of Fluid Dynamics
Sunday–Tuesday, November 22–24, 2020; Virtual, CT (Chicago time)
Session P20: Vortex Dynamics and Vortex Flows: Theory (3:10pm  3:55pm CST)Interactive On Demand

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P20.00001: Selfsimilar motions of point vortices and related relative equilibrium states Takeshi Gotoda We talk about selfsimilar motions of point vortices, which are governed by the pointvortex (PV) system. The explicit formula for selfsimilar solutions of the PV system has been established for the three PV problem and specific examples of the four and five PV problems. We see that the families consisting of these selfsimilar solutions are expressed by oneparameter families, and their collapse time and Hamiltonian are functions of the same parameter. Then, the configurations at limits of the parameter are in relative equilibria. As for the $N$vortex problem, we consider the case that $N  1$ point vortices have a uniform vortex strength with the help of numerical computations. We show that families of selfsimilar collapsing solutions continuously depend on the Hamiltonian and relative equilibria appear in their limit states. In particular, we can prove the existence of relative equilibria for the four PV system. We also see some examples of four and seven point vortices with nonuniform vortex strengths. [Preview Abstract] 

P20.00002: Compressible axisymmetric swirling flow states in chambers with various geometries Yuxin Zhang, Zvi Rusak, Shixiao Wang The stability and breakdown of compressible swirling flows is an important problem for a variety of technological applications such as the aerodynamics of slender wings operating at high angles of incidence, flows in jet engine nozzles and in combustion chambers. We investigate the structure of inviscid, compressible, subsonic and axisymmetric swirling flow of a perfect gas in chambers with varying geometries. The inlet flow is described by profiles of circumferential and axial velocity and temperature together with a fixed azimuthal vorticity. The outlet flow is assumed to be a zero radial velocity state. The nonlinear interaction among flow compressibility, flow swirl and chamber geometry are investigated numerically. We solve the unsteady and axisymmetric Euler equations using the Steger and Warming flux vector splitting method. The solver provides the natural evolution of flows including the dynamics to states with lowspeed recirculation zones along the chamber centerline or attached to the wall. Results of the timeasymptotic states show agreement with theoretical predictions of steady compressible swirling flows in chambers. Results also shed light on the stability of various steady states and the nature of flow evolution. [Preview Abstract] 

P20.00003: Dynamics of two point vortices in the presence of a fixed vortex Sreethin Sreedharan Kallyadan, Priyanka Shukla The various trajectories exhibited by two mutually interacting point vortices of arbitrary nonzero circulations in the velocity field of a fixed point vortex is examined. The underlying dynamical system is categorized and studied based on the zero and nonzero values of the angular impulse. In each case, a phase plane analysis conducted on a transformed coordinate system to obtain a complete classification of the vortex motions and to characterize the behavior of the intervortex distance functions. Finally, The analytical predictions are verified numerically and illustrated for different sets of initial conditions and circulations. [Preview Abstract] 

P20.00004: Interaction of Streamwise Vortices with Surface Textures: Effects of Differential Flow Displacement and Acceleration Saikishan Suryanarayanan, David Goldstein, Edward White, Garry Brown Understanding the interaction of vortices with surface textures is essential for the optimization of control strips for roughness induced transition mitigation (Suryanarayanan et al. AIAA J. 58(7), 2951, 2020). Recently, a simple theoretical model that accounts for the effect of solid surfaces on streamwise vortex evolution by the diffusion of vortex sheets generated on the noslip surface has been proposed (Suryanarayanan et al. DFD 2019, AIAA 20203020). This model provides a bulk prediction of the effect of the strips on a streamwise vortex but requires the relative displacement of the vortex with respect to the strips as an input. This talk focuses on the subsequent development of the theoretical model  (1) prediction of the motion of the vortex axis as it goes over the strip that makes the model selfcontained and (2) incorporation of the effects of vortex stretching caused by the streamwise acceleration experienced by the vortex as it goes over the strip. These developments use a combination of existing theoretical ideas and analysis of simple DNS cases. The effects of streamwise evolution of vortex strength and vorticity scooped from the wall on the vertical motion of the vortex axis will also be discussed, along with the application to multiple vortices. [Preview Abstract] 

P20.00005: Vortex sound generation during vortex reconnection Yoshifumi Kimura, Keith Moffatt A formula is derived for the quadrupole vortex sound pressure generated by two tilted hyperbol{\ae} in a tentshaped configuration. This model, which was recently proposed by the authors as a model for vortex reconnection, exhibits nearly selfsimilar Leray scaling for the minimum separation of the hyperbol{\ae} and the maximum velocity and axial strainrate toward a finitetime singularity. The formula shows that if the lengthscale of the system follows exactly the selfsimilar development $\sim (t_ct)^{1/2}$, then the farfield quadrupole sound pressure vanishes. A nonvanishing quadrupole will therefore provide a sensitive indication of any departure from selfsimilarity. Also, if the vortices eventually reconnect through the action of viscosity, the timedependence of the lengthscale must differ significantly from that of the selfsimilar state, and any departure from selfsimilarity will then generate sound. Furthermore, as $t$ passes through $t_c$, there is an interchange of vortex pairs resulting in rotation of the quadrupole axis by $\pi/2$, an effect that could in principle be detected. This prediction agrees with the numerical result that the sound emission at superfluid vortex reconnections is in the form of an intense, localized and directional pulse. [Preview Abstract] 

P20.00006: Compressible point vortices Tianyi Chu, Stefan Llewellyn Smith, Zinan Hu The effect of weak compressibility on unsteady point vortices is studied for twodimesional adiabatic flow. The appropriately nondimensionalized Blokhintsev equation governs the motion, and is solved asymptotically in the region between vortices. This approximation fails near the vortex cores and in the far field. An appropriate asymptotic matching with the core region is carried out, coupled with the momentum balance over the cores, to obtain governing equations. The flow is expressed using a RayleighJansen expansion in $M^2$. To carry out the matching, it is necessary to consider the locations of vortices to vary over slow time scales of $O(M^2)$ and $O(M^2\log{M})$. Higherorder singlevalued complex potentials are obtained locally, and appropriate terms are added to remove unacceptable singularities. The equations of motion at different orders are calculated by implementing conservation of momentum. Matching with the far field is also considered to examine waves generated by the motion of the vortices. Results are presented for a cotraveling and a corotating dipole, and are compared to known steady solutions. [Preview Abstract] 

P20.00007: Investigation of finitetime singularity models of the NavierStokes equations Philip J. Morrison, Yoshifumi Kimura Recently proposed low degreeoffreedom models [1] for describing the approach to finitetime singularity of the incompressible NavierStokes equations are investigated. These models assume an initial finiteenergy configuration of two vortex rings placed symmetrically on two tilted planes. The noncanonical Hamiltonian structure [2] of the inviscid limit of the models will be presented and shown to elucidate the nature of the possible finitetime singularities in the model. \\ {\ } \\ \ [1] H. K.\ Moffatt and Y.\ Kimura, J.\ Fluid Mech.\ {\bf 861}, 930 (2019); {\bf 870\, R1}, (2019). \\ \ [2] P.\ J.\ Morrison, Rev.\ Mod.\ Phys.\ {\bf70}, 467 (1998). [Preview Abstract] 

P20.00008: ``Vortex Bursting'': its initiation and dynamics Eric Stout, Fazle Hussain Vortex bursting  an abrupt expansion of the core radius of a slender vortex column, but distinct from vortex breakdown  occurs when a vortex ring dipole forms within the column, advects radially outward, and erupts as a detached structure. The ring dipole forms when an initial perturbation to the column's core radius generates azimuthal vorticity, $\omega_{\theta }$, that then mutually interacts to increase the radius. Via DNS, we find that eruption of the dipole can be arrested due to the dipole's interaction with the column. The meridional velocity of each ring of the dipole generates opposite $\omega_{\theta }$ at the column's surface via tilting of the column's initially axial vortex lines; this tilting, hence coiling, forms a counterdipole at the column's surface, outside the first dipole; this counterdipole advects radially inward, arresting the erupting dipole's outward motion and decimating it via uncoiling of vortex lines. We model the generation of $\omega_{\theta }$ at the column's surface due to the meridional velocity of each ring of the erupting dipole and find quadratic growth, confirmed by DNS. Then, by assuming the $\omega_{\theta }$ in the dipole rings is generated at the initial core perturbation, we find that bursting occurs for a minimum axial variation of the core radius, $\Delta $r$_{\mathrm{0}}$/$\Delta $z, of 0.49  this occurs when the erupting dipole's $\omega_{\theta }$ is larger than the counterdipole's $\omega _{\theta }$. This criterion should enable seeding radius variations on a column to control or induce vortex bursting in various applications. [Preview Abstract] 
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