Session D21: Vortex Dynamics II

A mathematical model of laminar wakes with four vortices per period

Laminar wakes behind vortex-shedding bluff bodies can exhibit a variety of patterns when the bodies oscillate or are in close proximity to one another. In many cases these patterns consist of regular groupings of four vortices. We present a two-dimensional potential flow model that assumes a spatially periodic arrangement of point vortices. This Hamiltonian system is made integrable by imposing an additional spatial symmetry that is motivated by the experimental wakes. The current model generalizes our previous results by allowing for unequal vortex strengths. The model shows a variety of dynamic modes that we classify using a bifurcation analysis of the phase space structure. Some initial conditions lead to relative equilibria in which the vortex configuration moves without change in size or shape. Scaled comparisons of the model with experiments show good results and allow for estimation of the experimental vortex strengths.   

Numerical Investigations of Reconnection of Quantized Vortices

Reconnection of quantized vortices in superfluid helium was conjectured by Feynman in 1955, and first observed experimentally by Bewley et al. (PNAS 105, 13708, 2007). The nature of this phenomenon is quantum mechanical, involving atomically thin vortex cores. At the same time, this phenomenon influences the large scale dynamics, since a tangle of vortices can change topology through reconnection and evolve in time. Numerically, the Gross-Pitaevskii (GP) equation allows detailed predictions of vortex reconnection as first shown by Koplik and Levine (1993). We have undertaken further calculations to characterize the dynamics of isolated reconnection events. Initial conditions have been analyzed carefully, different geometries have been considered and a new approach has been proposed. This approach consists in using the diffusion equation associated to the GP equation to set minimum energy initial vortex profiles. The underlying questions we wish to answer are the universality of vortex reconnection and its effect on energy dissipation to the phonon field.   

Cascade of vortex tube collisions at $Re_{\Gamma} = 10\,000$

We present simulations of the collision of two anti-parallel vortex tubes, with and without axial flow in a periodic box at $Re_{\Gamma} = 10\,000$ using a remeshed vortex method. In the non-axial flow case, after the first, well-known vortex reconnection of the tubes, a quiescent period is followed by a second vortex collision of the remaining structures. The characteristics of this second collision are an increase of energy in the small scales of the flow; remnant vorticity left behind in thread-like structures; a persistent $-7/3$ slope in the three-dimensional energy spectrum; and a significant increase in enstrophy and helicity in the flow. Characteristics of the secondary collision are also observed during the first reconnection of the vortex tubes with axial flow. The simulations indicate that vortical flows containing initially large-scale vortical structures can transfer energy from large scales to smaller scales through a cascade of vortex collisions.   

Merger of multiple vortices

We study the merger of three or more identical co-rotating vortices initially arranged on the vertices of a regular polygon, and compare it to the merger of two like-signed vortices. The latter is a well studied problem, with the merger process there consisting of four stages. In the multiple (three or more) vortex case, we find a new stage in the merger process, where an annular vortical structure is formed and is long-lived. We find that merger on the whole is slowed down significantly as the number of vortices goes up, and the formation of the annular structure is primarily responsible for the delaying of the merger. Vortices initially elongate radially, and then reorient their long axis closer to the azimuthal direction, and then diffuse out to form an annulus. The inviscid case is similar at short times, but at longer times, rather pronounced filaments are visible, which are practically absent in the viscous case. The formation of an annular structure is further impeded since the azimuthal alignment is reduced. The annular stage will be contrasted with the ``second diffusive stage'' in two-vortex merger.   

Formation number of particle-laden starting jets

The dynamics of a starting jet is studied under conditions where the injected fluid is laden with small spherical particles. The pinch-off process and its associated time scale, the formation number, are studied via a series of two-way coupled particle-laden, large eddy simulations with Lagrangian tracking of the order of 10$^{5}$ particles. The particles are small enough for the point-particle approximation to be valid, and inter-particle forces are neglected since the particle to injected fluid volume fraction is smaller than 10$^{-3}$. Forces acting on the particle include drag force and gravitational force. The numerical code is validated by reproducing formation numbers for pure as well as positively and negatively buoyant starting jets. A systematic study is carried out to investigate the effect of particle size and density on the formation number. Results at Re=5000 indicate that the presence of particles 10 times heavier than the fluid results in significantly enhanced total and head vortex circulations, leading to formation numbers distinct from the pure starting jet case. A particle-laden injection is thus shown to behave similar to an injection of a fluid heavier than the ambient.   

LBM Simulations for the Head-on Collisions of Vortex Pairs and Vortices-Wall Interactions in Two Dimensions

Vortex dynamics has been studied in two dimensions with lattice Botzmann method for the basic interactions of vortices. With different off-center distances the head-on collisions of two equal pairs of vortices(two dipoles) different regimes have been found: exchanging, merging and diffracting. Meanwhile the interactions of a vortex dipole with solid wall have been also investigated and comparison with existing experiments has been shown certain agreement. More complicated cases for three dimensions are left for future works.   

Investigation of the critical strain rate for co-rotating vortex pairs

Two-dimensional interactions of a co-rotating vortex pair in a viscous fluid are investigated. With symmetric vortex pairs, i.e. those with two vortices of equal size and strength (Brandt \& Nomura, \itshape{J. Fluid Mech.}\normalfont, vol. 592, 2007), the mutually-induced strain deforms and tilts the vortices, which leads to a core detrainment process. The weakened vortices are mutually entrained and rapidly move towards each other as they intertwine and destruct, thereby developing into a single compound vortex. With asymmetric pairs, the interactions are more complex and may result in different outcomes. These can be classified according to the relative timing of core detrainment and core destruction of the vortices. A critical strain rate parameter which characterizes the establishment of core detrainment has been identified and determined. In rudimentary studies, this critical value was found to be consistent for a range of vortex strength ratios (Brandt \& Nomura, \itshape{J. Fluid Mech.}\normalfont, vol. 646, 2010). In the present study, further analysis and numerical simulations are performed to ascertain the degree of universality of the critical strain rate value. A larger range of flow parameters is considered. Results clarify the relationship between the parameter and the behavior of these vortex flows.   

Stirring, Stretching and Transport Generated by a Vortex Pair

We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. The rotational and co-rotational motion of the vortices stir the fluid and subdivide the domain into inner core, inner recirculation, outer recirculation regions and outer flow. We quantify stirring using stretching of the interface and the mix-norm. Our numerical simulations show that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mixnorm is dominated by the laminar transport induced within the outer recirculation regions. Stirring is sensitive to the geometry of the initial concentration field. We consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation and show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms. Funding provided by NSERC, contract RGPIN217169.   

Aeroacoustics of viscous vortex reconnection

Reconnection of two anti-parallel vortex tubes is studied by direct numerical simulations and large-eddy simulations of the incompressible Navier-Stokes equations over a wide range (2000-50,000) of the vortex Reynolds number (Re). A detailed investigation of the flow dynamics is performed and at high Re, multiple reconnections are observed as the newly formed ``bridges'' interact by self and mutual induction. To investigate acoustics produced by the recoil action of the vortex threads, M\"{o}hring's theory of vortex sound is applied to the flow field and evaluated at varying far-field locations. The acoustic solver is verified against calculations of laminar vortex ring collision. For anti-parallel vortex reconnection, the resulting far-field spectra are shown to be grid converged at low-to-mid frequencies. To assess the relevance to fully turbulent jet noise, the dependence of reconnection upon Reynolds number is investigated.   

Formation regimes of vortex rings in negatively buoyant starting jets

The formation of vortex rings (VR) in negatively buoyant starting jets has been studied numerically for different values of the Richardson number covering the range of weak to moderate buoyancy effects ($0 \le \mathrm{Ri} < 0.20$). We have identified two different regimes in the vortex formation whose transition takes place at about $\mathrm{Ri} \approx 0.03$. The vorticity distribution inside the VR after pinching-off as well as the total amount of circulation it encloses (characterized by the formation number, $\mathrm{F}$) show different behaviors with $\mathrm{Ri}$ in both regimes. Nevertheless, the physical mechanism limiting the circulation of the vortex, or equivalently the formation number, is the same in both cases. Thus, the formation number of a negatively buoyant VR, whose propagating velocity is slower than that of a neutrally buoyant one due to gravity effects, can be determined considering that it is nearly the same as that of a neutrally buoyant VR moving with the velocity corresponding to the negatively buoyant vortex. Based on this simple idea, a phenomenological model is presented to quantitatively describe the evolution of the formation number with the Richardson number, $\mathrm{F}(\mathrm{Ri})$, obtained numerically. We also discuss the limitations of different vortex identification to evaluate the vortex properties in buoyant flows. Supported by the Spanish Ministry of Science grant DPI2008-06369.