Session P20: Vortex Dynamics and Vortex Flows: Theory (3:10pm - 3:55pm CST)

Self-similar motions of point vortices and related relative equilibrium states

We talk about self-similar motions of point vortices, which are governed by the point-vortex (PV) system. The explicit formula for self-similar 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 self-similar solutions are expressed by one-parameter 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 self-similar 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 non-uniform vortex strengths.   

Compressible axisymmetric swirling flow states in chambers with various geometries

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 low-speed recirculation zones along the chamber centerline or attached to the wall. Results of the time-asymptotic 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.   

Dynamics of two point vortices in the presence of a fixed vortex

The various trajectories exhibited by two mutually interacting point vortices of arbitrary non-zero 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 non-zero 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 inter-vortex distance functions. Finally, The analytical predictions are verified numerically and illustrated for different sets of initial conditions and circulations.   

Interaction of Streamwise Vortices with Surface Textures: Effects of Differential Flow Displacement and Acceleration

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 no-slip surface has been proposed (Suryanarayanan et al. DFD 2019, AIAA 2020-3020). 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 self-contained 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.   

Vortex sound generation during vortex reconnection

A formula is derived for the quadrupole vortex sound pressure generated by two tilted hyperbol{\ae} in a tent-shaped configuration. This model, which was recently proposed by the authors as a model for vortex reconnection, exhibits nearly self-similar Leray scaling for the minimum separation of the hyperbol{\ae} and the maximum velocity and axial strain-rate toward a finite-time singularity. The formula shows that if the length-scale of the system follows exactly the self-similar development $\sim (t_c-t)^{1/2}$, then the far-field quadrupole sound pressure vanishes. A non-vanishing quadrupole will therefore provide a sensitive indication of any departure from self-similarity. Also, if the vortices eventually reconnect through the action of viscosity, the time-dependence of the length-scale must differ significantly from that of the self-similar state, and any departure from self-similarity 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.   

Compressible point vortices

The effect of weak compressibility on unsteady point vortices is studied for two-dimesional adiabatic flow. The appropriately non-dimensionalized 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 Rayleigh-Jansen 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 $O(M^2\log{M})$. Higher-order single-valued 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 co-traveling and a co-rotating dipole, and are compared to known steady solutions.   

Investigation of finite-time singularity models of the Navier-Stokes equations

Recently proposed low degree-of-freedom models [1] for describing the approach to finite-time singularity of the incompressible Navier-Stokes equations are investigated. These models assume an initial finite-energy 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 finite-time 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).   

``Vortex Bursting'': its initiation and dynamics

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 counter-dipole at the column's surface, outside the first dipole; this counter-dipole 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 counter-dipole's $\omega _{\theta }$. This criterion should enable seeding radius variations on a column to control or induce vortex bursting in various applications.