Session AU: Vortex Flows I

Experimental characterization of thermally driven superfluid flow

We characterize the flow of superfluid $^{4}$He by analyzing trajectories of solid hydrogen seed particles.~ A thermal counterflow is driven by heating the bottom of a cylindrical channel filled with superfluid $^{4}$He while also cooling the free surface. While all particles feel Stokes drag with the normal fluid, some subset are scattered or trapped by quantized vortices of the superfluid. Particle tracking is used, instead of particle image velocimetry cross-correlations, due to this lack of a single underlying velocity field. We find that upward moving particles show the expected normal fluid velocity for all measured temperatures and heat fluxes. Tracers affected by the quantized vortices move downward, opposing the motions driven by Stokes drag, and exhibit erratic trajectories.   

Bifurcation analysis of an infinite array of von Karman Streets

This research investigates the behavior of an infinite array of (inverse) von Karman streets.~Primary motivation is to model the wake dynamics in large fish schools. Ignoring the fish we focus on the dynamic interaction of multiple wakes.~In particular, we investigate~the problem of fluid transport between adjacent vortex streets for its relevance to understanding the transport of oxygen and nutrients to inner fish in large schools as well as understanding flow barriers to passive locomotion. We prove that the configuration of vortices is in relative equilibrium, meaning that the streamline pattern remains steady in the frame moving with vortices.~We~look at the topology of these streamline patterns plotted in the moving frame which lends insight to fluid transport through the mid-wake region.~ Fluid is advected along different paths depending on the distance separating two adjacent streets. When the streets are far apart, the dynamics is decoupled and fluid is transported globally between two adjacent streets. When the streets get closer to each other, the number of streets that enter into partnership in transporting fluid among themselves increases. This observation motivates a bifurcation analysis which links the distance between streets to the maximum number of streets transporting fluid among themselves.   

Characterization of the interaction of two unequal co-rotating vortices

The interaction of two co-rotating vortices in a viscous fluid is investigated. Two-dimensional simulations of initially equal sized vortices with varying relative strengths are performed. In the case of equal strength vortices (Brandt and Nomura, J. Fluid Mech., 2007), the mutually induced strain deforms and tilts the vortices which leads to a core erosion process. As the vortices are jointly entrained, they rapidly move towards each other and the flow eventually transforms into a single vortex. With unequal strengths, the disparity of the vortices alters the interaction and merger may not occur. The flow behavior is distinguished based on the relative onset of the core erosion process. Through scaling analysis and simulation results, a critical nondimensional strain rate characterizing the onset of erosion is determined. If the disparity of strengths is sufficiently large, the critical strain rate is not attained by the stronger vortex and the vortices do not merge.   

Wavelet analysis of vortex bursting

We study the quasi-periodic bursting of a three-dimensional vortex immersed in a laminar channel flow. We measure the velocity field by PIV and analyze the time evolution of the bursting process. We use the orthogonal wavelet transform to separate the flow into coherent and incoherent components and then the continuous wavelet transform to analyze the evolution of each component separately. We found that the coherent flow is intermittent, long-range correlated and sustain the turbulent cascade, while the incoherent flow is non intermittent, exhibits an enstrophy equipartition spectrum and leads to turbulent dissipation. In order to better understand the buildup of the turbulent cascade and to quantify the flow intermittency, we have designed new wavelet-based diagnostics which find out, when in time, where in space, and at which scale, the activity is dominant . We observe that the bursting process starts as an excitation of the small scales inside the vortex core, and then spreads in space and all over the inertial scales. We recover the Kolmogorov k$^{-5/3}$ scaling only for time averages, since the spectral slope of energy varies in time, between k$^{-1}$ during bursting, and k$^{-2}$ after vortex bursting.   

Determining the stability of steady inviscid flows through ``Imperfect Velocity-Impulse'' diagrams

More than a century ago, Lord Kelvin proposed a variational argument for determining the stability of steady inviscid flows; while the underpinnings of the method are well established, its application has been the subject of extensive debate. Considering, for example, a vortex configuration rotating at a rate $\Omega$ with impulse $J$ and energy $E$, Kelvin argued that an equilibrium corresponds to a stationary point of $H = E -\Omega J$. Since $H$ is conserved, the second variation $\delta^2 H$ constrains the dynamics and can be used to assess stability. Unfortunately, computation of $\delta^2 H$ is often impossible or impractical. Saffman \& Szeto (1980) suggested that extrema in a plot of $E$ vs $J$ could be used to identify changes in $\delta^2 H$. However, Dritschel (1985) later pointed out the lack of a firm link between $\delta^2 H$ and a plot of $E$ vs $J$. Furthermore, he stated that even if such link could be proven, changes of stability could also occur, at bifurcations, away from extrema in $E$ and $J$. We address both issues by proposing a new approach. We introduce a theorem from dynamical systems theory to prove that extrema in a plot of $J$ vs $\Omega$ (instead of $E$ vs $J$) are indeed related to the properties of $\delta^2 H$, while we use ideas from imperfection theory to ensure that bifurcations are detected by means of an ``imperfect velocity-impulse'' (IVI) diagram. By applying our approach to several classical flows, we obtain stability results in agreement with linear analysis, while additionally discovering new steady solutions.   

Elementary vortex systems and the generation of internal waves

As a first step toward improving our understanding of the behavior of a turbulent patch in a stratified fluid, we investigate the effects of stable stratification on different configurations of two-dimensional horizontally oriented vortices. The vortex systems considered, which consist of initially Lamb-Oseen vortices, include a single vortex, co-rotating vortex pair, counter-rotating vortex pair, two sets of co-rotating vortex pairs in a quadrapole configuration, and two sets of counter-rotating vortex pairs in a quadrapole configuration. Analytical and numerical methods are used to compute the linear and nonlinear interactions of the vortices and the generated internal waves, as well as the transport of vorticity and energy associated with each vortex configuration. These fundamental findings provide further insight on wave-vortex interactions and vortex structure decay.   

Chaotic scattering of two vortex pairs

Chaotic scattering of two vortex pairs with slightly different circulations was considered by Eckhardt \& Aref in 1988. A new numerical exploration suggests that the motion of two vortex pairs, with constituent vortices all of the same absolute circulation, also displays chaotic scattering regimes. The mechanisms leading to chaotic scattering are different from the ``slingshot effect'' identified by Price [{\it Phys. Fluids} A, {\bf 5}, 2479 (1993)] and occur in a different region of the four-vortex phase space. They may in many cases be understood by appealing to the solutions of the three-vortex problem obtained by merging two like-signed vortices into one of twice the strength, and by assuming that the four-vortex problem has unstable, periodic solutions similar to those seen in the thereby associated three-vortex problems. The integrals of motion, linear impulse and Hamiltonian, are recast in a form appropriate for vortex pair scattering interactions that provides constraints on the parameters characterizing the outgoing vortex pairs in terms of the initial conditions.   

Statistics of vortex tube properties in isotropic turbulence

The vortex tubes of isotropic turbulence are statistically analyzed by means of a feature-extraction algorithm applied to DNS data at several value of the Taylor Reynolds number. It is found that the main geometric parameters of the vortices (radius, induced velocity, core vorticity) exhibit log-normal distributions, and very nearly collapse in terms of Kolmogorov units. Consistent with pevious studies, we have found that vortex tubes are special instances of the vorticity field associated with intensity stronger than the mean and local alignment of the vorticity vector, which subjected to the r.m.s background strain. Contrary to previous findings, however, we find that the vortex core radius and the local strain are nearly statistically independent, thus raising doubts on the relevance of vortex models based on stretched axi-symmetric vortices. The analysis of the azimuthal velocity profiles indicate scaling of the induced velocity similar to recent experimental findings, but very different from Burgers vortex model.