Session EI: Vortex Dynamics and 3D Vortex Flows II

Adaptive Quad-tree Surface Representation for 3-D Vortex Sheet Motion

We consider a new local, adaptive, higher order, tree-based quadrature and point insertion method to describe 3-D vortex sheet motion. This method for the first time enabled us to consider long time behavior of unstable vortex rings and vortex rings collision (oblique and head-on). The method is not limited to vortex ring motion and can be applied to other vortex sheets such as jets and wakes.   

Low Dimensional Modeling of Transient Flow Fields Using Double Proper Orthogonal Decomposition

Proper Orthogonal Decomposition (POD) has been demonstrated in the past as a powerful tool for low dimensional modeling of periodic flow fields. When applied to transient flows, though, the method leads to spatial modes that have to span the different flow states at the beginning and end of the transient. Thus, the spatial modes tend to loose their association with physical phenomena in the flow. Here, we present an extension to POD that remedies this problem, by using POD based expansion of the physical modes present in each cycle of the transient flow behavior. This is done using a double POD (DPOD) decomposition: First, all individual shedding cycles within a transient flow field are modeled using POD. This leads to an ensemble of similar, albeit---due to the transient effects---slightly varying sets of POD modes. A second POD decomposition is then used to represent this set of modes and their changes in time, completing the DPOD procedure. We demonstrate the performance of DPOD by applying it to transient wake flow data.   

Relative Equilibria of Identical Point Vortices

The problem of finding relative equilibria of identical point vortices is classical and was considered by Kelvin and J. J. Thomson almost immediately after the model had been introduced by Helmholtz in 1858. At the time relative equilibria of vortices were proposed as models of atoms. Apart from the intrinsic interest of the problem, and its mathematical challenge, such equilibria have been used as models for stationary states of distributed vortices, and have been observed in rotating superfluids, most recently in spectacular images of BECs. Simple equilibria such as regular polygons (both open and centered) were found and analyzed in the 19th century. Double rings and more recently triple rings have been found analytically. However, the numerically known relative equilibria continue to greatly outnumber those that are analytically known. A major numerical exploration was undertaken by Campell \& Ziff in 1978 resulting in what is known as the {\it Los Alamos Catalog}. We will explore the results in this catalog and what we have learned since then, and present details on the quest for an analytical understanding of these intriguing states.   

Influence of axial flow and aspect ratio on the stability of vortex pairs

A planar vortex pair with axial flow is simulated using DNS. The evolution of the flow is stopped when the aspect ratio of the dipole has reached a desired value. Values between 0.2 and 0.4 are investigated. A tail of vorticity forms downstream the dipole as it evolves toward higher aspect ratios. Using three dimensional stability analyses, we study the sensibility of the Widnall modes of the dipole about the swirl number of the base flow and the aspect ratio. Those short-wave modes are recovered for two-dimensional vortex pairs. New modes appear as we progressively increase the axial flow. The increase in aspect ratio seems to widen the bands of instability. Long time instability modes are then compared to short time modes that are found by a code of optimal perturbation based on an adjoint method. We show that the perturbation that gives the highest increase in energy at short time is localized in the vicinity the centre of each vortex whereas the perturbation that gives the highest increase in energy at long time is localized at the periphery of the vortices.   

Mixing characteristics of laminar two-dimensional co-rotating vortex merging

The mixing characteristics of laminar, two-dimensional, co-rotating vortex merging were studied. The unsteady, incompressible two-dimensional Navier--Stokes equations were solved with fourth-order Runge--Kutta temporal discretization and fourth-order compact schemes for spatial discretization. Previous experimental research has uncovered four meaningful physical stages in the merging of co-rotating vortices. In this study, passive Lagrangian tracers were used to explore the transport of fluid elements during these stages and to provide a detailed view of the merge. The dynamics of Lagrangian tracers improved the characterization of the core dynamics and identified the fluid entrainment regions in the final vortex core. Chaotic advection was studied in the merging process by examining patterns of spatial variation in finite-time Lyapunov exponents. These patterns characterized the chaotic mixing regions and pinpointed transport barriers throughout the evolution.   

Transport induced by vortex formation

This presentation will address the transport structure of vortex ring formation and the induced stirring and mixing properties. Vortex rings are empirically generated in water by a piston- cylinder apparatus. Digital Particle Image Velocimetry (DPIV) is used to obtain cross-section measurements of the velocity field within and around the cylinder exit. Dynamical systems methods are used to compute Lagrangian coherent structures (LCS) from the DPIV data during the vortex formation. The LCS clearly indicate the geometric structure of how fluid is entrained to form the vortex ring. This topological picture is compared with traditional mechanisms of vortex formation. Additionally, the LCS are shown to offer a convenient tool for predicting the stirring/mixing of the surrounding fluid. The information provided from the LCS is compared with traditional stirring/mixing metrics and paradigms. This study provides valuable insight on how vortex rings form, and how vortical flows can be used to produce efficient stirring/mixing of fluids.   

The dispersal, decay and instability of multiple trailing-line vortices

We consider a number of aspects concerning the downstream dispersal, decay and instability of systems of trailing-line vortices. Three basic upstream configurations are considered, namely: (i) two counter-rotating vortices; (ii) a four-vortex (symmetrical) configuration; (iii) a symmetrical four co-linear vortex configuration. First the downstream flow evolution is studied, based on a high Reynolds number (long boundary-layer) methodology. This leads to predictions for the translation and decay of the vortex systems, including the asymptotic far-downstream behavior of these types of flow. Second, in cases of sufficiently large adverse freestream pressure gradients, it is shown that a novel three-dimensional breakdown of the vortex system is possible at a finite downstream location, linked to the onset of streamwise flow reversal, and this is fully analysed. Third, the stability of the developing flow is considered, using short-wavelength stability analysis. The asymptotic analysis is self consistent throughout.   

Simulations of temporal evolution of isolated trailing vortices

The temporal evolution of turbulent trailing vortices is studied using pseudo-spectral direct numerical simulations of the vorticity transport equations. A range of initial conditions are used, covering a wide spectrum of swirl numbers, different base flow profiles and varied perturbations. In the inviscidly-unstable swirl number range, the flow is characterized by the dynamics of saturating helical instabilities, as has been observed previously. In the low swirl number range, the evolution is dominated by structures in the outer part of the vortical core, resulting in enhanced angular momentum transport and a significant circulation overshoot. Though markedly different from the high swirl number cases in the transient regime, low swirl number Batchelor vortices ultimately reach an equilibrium self-similar state. The long time evolution of each of these cases is investigated. The mechanisms of turbulent production and angular momentum transport are detailed.