Session GT: Vortex Dynamics and Vortex Flows IV

Trajectory of an Elliptic Vortex Ring

An elliptic vortex ring is known to be unstable due to its oscillatory deformation while propagating. It oscillates periodically at low aspect ratios, but deforms and breaks up into two smaller rings at high aspect ratios. Although studies on elliptic vortex rings have been conducted before, certain aspects of the vortex ring behavior remain unclear; in particular the influence of Reynolds number on their trajectories. Moreover, most of the earlier experimental studies were conducted using flow visualization techniques, which provide only qualitative description on the motion of elliptic vortex rings and not their velocity and vorticity fields. In the present investigation, we focus our attention on the vorticity field during various stages of the vortex ring deformation; particularly the effect of Reynolds number and aspect ratio on the vortex ring trajectory. Experiments are conducted in a water tank using elliptic nozzles of aspect ratios 1, 2 and 3. Obviously, the nozzle aspect ratio of 1 represents a circular nozzle, and the results are included here for comparison. Preliminary results show that the trajectory of elliptic vortex ring of aspect ratio 2 is insensitive to changes in the Reynolds numbers, but this is not the case with the aspect ratio 3, where noticeable deviation of the trajectory at lower Reynolds numbers is observed. The cause of this deviation and its implication will be discussed.   

Vortex ring impacting on wall

Three dimensional vortex ring impacting a wall at different angles of incidence has been numerically investigated using the lattice Boltzmann model. The detailed flow behavior, vortex evolution, and pressure distribution on wall have been studied systematically with the Reynolds number of 100 $<$ Re $<$ 1000, and the impact angle of the range of 0\r{ } $< \quad \theta \quad <$ 60\r{ }. Our results show that only when Re $>$ 100, the interaction of the vortex ring with the wall can generates the secondary vortex rings. The evolution of vortex structure is also strongly influenced by $\theta $. The increase of $\theta $ will cause a wrap process of the secondary vortex ring and an obvious suppression of the tertiary vortex ring generation. Further more, the study identified new features of vortex structure and its interaction with wall, in particular, for the oblique impact scenarios. A simple model is adopted to describe the basic characteristics of pressure distribution on the wall along the symmetry vortex ring plane at low Reynolds number.   

Comparison of DNS Determination of the Dynamics of Vortex Rings in Viscous Fluids and Experiment

We have been studying vortex rings in water for some time [1] and recently became aware of an important paper studying vortex rings by direct numerical simulation (DNS) from Coleman's group at Southampton [2]. There is clearly much to be learned from a comparison of the results in [1] and [2]. The first insight is a comparison of slowing vortex rings, where we find quite similar decay rates at comparable Reynolds numbers. A second insight is gained by noting that they find a time t* needs to elapse before the core adjusts to its vorticity distribution. We find photographically that the ring needs to propagate at least one gun diameter before it adjusts its vorticity. A third insight is that the rings in Fig. 5(b) of [2] do not change much in radius, consistent with the results in Table 2 of our paper [1]. The talk will cover more recent comparisons of the two works including observations of the growth of vortex waves.\\[4pt] [1] I. S. Sullivan, J. J. Niemela, R. Hershberger, D. Bolster and R. J. Donnelly, J. Fluid Mech. 609 319 (2008).\\[0pt] [2] P. J. Archer, T. G. Thomas and G. N. Coleman, J. Fluid Mech.598 201 (2008).   

Interaction of the 2D vortex patch with the wall. Eruption of the boundary layer phenomenon.

The boundary layer eruption phenomenon caused by a 2D patch of vorticity moving above a wall was investigated. It was shown that eruption phenomenon depends on the viscosity (or Reynolds number, Re) of the fluid. There exists a threshold value of Re above which the eruption takes place. The initiation of the eruption goes through the creation of a small recirculation zone near the solid wall. For small Re numbers it disappears but for larger it is strongly stretched in the direction perpendicular to the wall. The terminal state is appearance of a saddle point on streamlines inside the recirculation zone. Next this zone is torn off and portion of the fluid particles from the near wall region are abruptly ejected into the other flow. Further increase of the Reynolds number causes more complex flow. One can observe that eruption is regenerative and that the vortex patch can produce a cascade of secondary vortices. The vortex-in-cell method was employed to investigate the eruption phenomenon.   

Local Stability Analysis of Fat Vortex Rings

The stability of fat vortex rings is studied by the geometrical optics method. It is found that Hill's vortex is subject not only to the elliptical instability but also to the curvature instability, which is due to the curvature of vortex tubes and first found for the vortex ring with thin core. A new type of instability is also found; it is a coupled mode of the elliptical and curvature instabilities. The strongest instability is the elliptical instability for a wide area of the vortex, while the coupled instability surpasses the elliptical instability near the surface. The effects of swirl on the instability are investigated. The maximal growth rate becomes small as the magnitude of swirl becomes large.   

DPIV Measurements of Vortex Ring Interaction with Multiple Permeable Screens

Flow visualization of the interaction of a vortex ring impinging on several parallel, transparent permeable screens was made previously for screens with 84{\%} open area ratio. The results indicated the vortex ring split into smaller vortical structures after its interaction with the first screen and exhibited a continuous break down into increasingly irregular flow after interaction with subsequent screens. The flow did not reorganize into a transmitted vortex ring as was observed with vortex rings impinging on a single permeable screen. The present work seeks to provide a more quantitative assessment of the flow through screens using DPIV. DPIV measurements were made using an aqueous solution that was refractive index matched to the transparent screens. Measurements were made for vortex rings interacting with screens with variable spacing and open area ratios of 58{\%}-84{\%}. The vortex rings were generated with a piston-cylinder vortex ring generator using piston stroke-to-diameter ratios of 2-4 and jet Reynolds numbers of 1000-2000. Preliminary results show splitting and decay of the flow vorticity in agreement with the flow visualization.   

Stability of relative equilibria of three vortices

Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear. While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved. The only comprehensive analysis available in the literature, by Tavantzis \& Ting [{\it Phys. Fluids}, {\bf 31}, 1392 (1988)], is not easy to follow nor is it very physically intuitive. The symmetry between the three vortices is lost in this analysis. A different analysis is given based on explicit formulae for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium. A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used. This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding. The main results are summarized in a diagram that gives both the stability or instability of the various collinear relative equilibria and their sense of rotation.   

Remarks on Continuation of Inviscid Vortex Flows in the Presence of the Kutta Condition

Our investigation concerns solutions of the steady--state Euler equations in two dimensions featuring finite--area regions with constant vorticity embedded in a potential flow. Using elementary methods of the functional analysis we derive precise conditions under which such solutions can be uniquely continued with respect to their parameters, valid also in the presence of the Kutta condition concerning a fixed separation point. Our approach is based on the Implicit Function Theorem and perturbation equations derived using shape--differentiation methods. These theoretical results are illustrated with careful numerical computations carried out using the Steklov--Poincar\'e method which show the existence of a global manifold of solutions connecting the point vortex and the Prandtl--Batchelor solution, each of which satisfies the Kutta condition.   

Reynolds number effects on the dynamics of the turbulent horseshoe vortex: High resolution experiments and numerical simulations

Turbulent flows past wall-mounted obstacles are dominated by dynamically rich, slowly evolving coherent structures producing most of the turbulence in the junction region. Numerical simulations [Paik et al., \textit{Phys. of Fluids} 2007] elucidated the large-scale instabilities but important questions still remain unexplored. One such question is with regard to the effect of the Reynolds number on the dynamics of the turbulent horseshoe vortex (THV). We carry out high-resolution laboratory experiments for the flow past a wall mounted cylinder in a laboratory water tunnel for Re$_{D}$= 26000, 48000 and 117000. We employ the Time-Resolved Particle Image Velocimetry technique to resolve the dynamics of the flow at the symmetry plane of the cylinder and analyze the instantaneous velocity fields using the Proper Orthogonal Decomposition technique. The experimental study is integrated with coherent-structure-resolving numerical simulations providing the first comprehensive investigation of Reynolds number effects on the dynamics of the THV.