In this dissertation, the early development of turbulent vortex rings at two Reynolds numbers is studied using two-dimensional and Stereoscopic Particle Image Velocimetry. In the late 1980s? a similarity theory of turbulent vortex rings was proposed and tested primarily using a twochannel tracking Laser Doppler Velocimeter. However, due to the limitations of the experimental technique the tests were inconclusive and important assumptions could not be checked. Since single-point measurements were used, turbulent vortex ring structures could only be inferred using a complex signal-analysis technique. In the present study, two-dimensional and stereoscopic Particle Image Velocimetry techniques provide spatial and temporal resolved measurements of the full field of the cross-section of turbulent vortex rings, from which a more rigorous investigation of the similarity theory is possible. Since the region over which the similarity theory appears to hold starts at about 2.5 orifice diameters downstream, this study focusses on the early development region from this point to ten diameters downstream. Finally, the ensembleaveraged turbulent ring velocity contours, vorticity contours, pressure field contours, as well as Reynolds stresses and turbulence production contours, are presented. The effects of the turbulent vortex ring position dispersion and tilting angle variation on the measurement results are also studied and quantified. An effort is also made to reconstruct a three-dimensional turbulent vortex ring velocity field by adopting Taylor?s hypothesis. Some important features are successfully captured. An azimuthal-averaging method is also developed in an attempt to estimate the turbulence quantities in cylindrical coordinates. However, because of various limitations, the three-dimensional reconstruction method is not perfect, and room for future improvement is discussed.

This thesis presents an investigation into the dynamics of vortex rings impacting a sharply-stratified density interface. This problem has a long history and is important for understanding how individual eddies in stratified turbulence mix the density field. We tackle this problem using a combination of experimental, numerical and modelling techniques to understand the flow instability and subsequent mixing induced by the impinging vortex ring. Our findings demonstrate that there exists a critical Richardson number, corresponding to a mixing transition, beyond which the mixing efficiency is constant. Using a novel Stereo Particle Image Velocimetry (Stereo-PIV) technique, we analyze a series of vortex ring experiments. By amalgamating an ensemble of these experiments, we measure the full, time-resolved, three-dimensional velocity field of the vortex-ring interaction. These measurements capture the instability that is produced on the baroclinically generated vorticity field. This instability is identified as a Crow-like instability. At low Richardson numbers, the timescale of the interface rebound is faster than that of the instability. As a result, there exists a critical Richardson number below which the Crow-like instability will not have sufficient time to grow to large amplitude. By generating a large number of vortex-ring interactions, we measure the incremental change to the stratification. After an initialization period, there is strong evidence to suggest that the mixing due to each vortex ring becomes constant over a moderate range of Richardson numbers. We suggest that the mixing efficiency of the vortex rings does drop at low Richardson numbers (below unity) in agreement with the analysis of the Stereo-PIV measurements. A model of the system accurately predicts the dependence of the mixing rate on the Richardson number. Based upon our study of the vortex-ring system, we construct a one-dimensional turbulence model that includes the energy advection from the vortex rings. This model is validated with both physical experiments and numerical simulations of repeated vortex-ring generations. The constant mixing efficiency regime is recovered in all three methodologies. Through examining the detailed dynamics of the flow, this work suggests that there exists a critical Richardson number corresponding to a transition between mixing regimes, and that this critical Richardson number is a result of the growth of a Crow-like instability. We have highlighted how to improve current mixing-models to capture this physics. New avenues of future research are currently underway to study the mixing produced by a stratified mixing-box experiment in light of these new developments.

Experimental studies have been conducted on the topic of the interaction of compressiblevortex rings on stationary surfaces. Throughout the campaign experimentswere carried out at pressure ratios of ! 4, 8, and 12. In the classical set up of airas both the driver and driven gas, these corresponded to theoretical incident Machnumbers Ms of 1.34, 1.54, and 1.61.Experiments were conducted on vortex rings impinging on a stationary surfacelocated at three (increasing) distances (1.66, 3.33, and 5.00 inner diameters) fromthe shock tube exit and on a stationary surface at a set distance but at three anglesinclinations (75, 60, and 45deg at 3.33 inner diameters). Results of the impingementof a vortex ring on a stationary solid surface perpendicular to the flow showed asymmetrical impingement process. A boundary layer is generated over the surfacewith an associated increase in pressure. An increase in velocity due to the radialexpansion causes the pressure over the surface to decrease. This expansion leads tothe development of azimuthal wave instabilities along the core. Pressure was seen toincrease with an increase in incident Mach number value. The variation in distanceresulted in an increase in pressure with an increase in distance. This counter-intuitiveresult can be explained by the higher translational velocity at impingement, alongwith the absence of the initial radial expansion of the counter-rotating vortex rings. The variation in surface angle inclination introduced several degrees of asymmetry. One core of the vortex ring impinges first on the surface due to its closerproximity to it, while the other core is still free to propagate. This process generatesan asymmetric boundary layer over the surface, and a higher rate of stretching ofthe lower core, resulting in its dissipation. At higher incident Mach numbers, theembedded rearward facing shock is reflected and propagates perpendicularly to thesurface. At the inclination angles of 60 and 45deg, the counter-rotating vortex ringsare fully deflected upwards and orbit around the main vortex. This phenomenonresult in a significant difference in pressure distribution between the upper and lowersections of the surface.

The objective of this research was to investigate parameters affecting vortex ring formation and propagation and their application to mixing of fluids. To this end both empirical and numerical simulation experiments were conducted. The empirical experiments involved observations and measurement of the volume, displacement and velocity of vortex rings generated from a 5 cm diameter tube. The results revealed that there is an optimal range of generation injection velocity for various mixing requirements. The numerical simulations were done using a commercial package, FLUENT. Both tube type and plate orifice type vortex ring generators were investigated. Also the affects of a central shaft and various projections on the control of the motion of a vortex ring. All models considered a polar model cylindrical tank with a diameter to height ratio of 3:10. The average injection velocity was in the range of 0.7 m/s to 3 m/s. When simulating the tube type generator various injection velocity profiles and value were investigated, which resulted in a fitted correlations of nondimension displacement versus non-dimension time as a function of infection profile. In order to control the forward motion of vortex rings some obstructions were considered. It was found that the trajectory and energy of a vortex ring can be controlled with shape and geometries of baffles. In the simulations for the orifice plate type generator, a moving mesh technique was used. As expected a pair of vortex rings were produced per half cycle of the plate oscillation, but they did not travel as fast as expected. Recommendations have been made to improve the simulation accuracy. It has also been found that the FLUENT package will not properly simulate turbulent vortex rings. However, this may be because a vortex ring is not truly homogeneously turbulent. The use of a laminar model appears to give quite good agreement with empirical data for tube type vortex ring generator. The results of this research are expected to be useful for the optimization of the design of vortex ring mixing systems. / Thesis / Master of Engineering (ME)

Left ventricular diastole (filling) is a complex process with many features and coupled compensatory mechanisms which coordinate to maintain optimal filling and ejection of the left ventricle. Diastolic filling is controlled by the left ventricular recoil, relaxation, and compliance as well as atrial and ventricular pressures making left ventricular diastolic dysfunction very difficult to understand and diagnose. An improved understanding of these unique flows is important to both the fundamental mechanics of the cardiac diastolic filling as well as the development of novel and accurate diagnostic techniques. This work includes studies of in-vivo and in-vitro vortex rings. Vortex rings created in the left ventricle past the mitral valve during diastole are produced in a confined domain and are influenced by the left ventricular walls. Therefore, an in-vitro analysis of the formation and decay of vortex rings within confined cylindrical domains using particle image velocimetry was conducted. Varying mechanisms of vortex ring breakdown were observed over a wide range of Reynolds numbers, and an analytical model for vortex ring circulation decay of laminar vortex rings was developed. Also, in this work a novel method for analyzing color M-mode echocardiography data using a newly developed automated algorithm is introduced which examines the pressure gradients and velocities within the left ventricle. From this analysis, a new diagnostic filling parameter is introduced which displays a greater probability of detection of diastolic dysfunction over the conventionally used diagnostic parameter. / Ph. D.

Previous research has indicated aperiodicity in the positions of tip vortices emitted from a helicopter rotor blade in hover. The objective of the current study is to develop an analysis of the tip vortex aperiodicity in hover and to validate it with measurements on a reduced-scale, 1m diameter, four-bladed rotor. A “vortex ring emitter model” (VREM) was developed to study the statistics of the tip vortices emitted from a rotor blade during hover. In order to better model the rotor wake, a number of independent vortex blobs were used to describe a vortex ring. An empirical model for viscosity was also considered which helped model the core radius growth of the vortex ring with vortex age. A parametric analysis was then performed to obtain a comprehensive qualitative and quantitative convergence study of the time step, viscosity parameter, initial core size, number of rings shed, number of blobs and overlap factor. It was observed that the solution converged rapidly for all the parameters used. The locations of tip vortex cores for vortex ages ranging from 0◦ to 260◦ were measured on the reduced-scale rotor using a stereo PIV system. The blade loading for the reduced scaled rotor was Ct /σ = 0.044 and the blade rotational speed was 1520 RPM, which corresponds to a tip Reynolds number of 248,000. The 95 % confidence region for the position of tip vortex cores exhibited an anisotropic, aperiodic pattern, approximating an ellipse. It was seen that the principal axis of this ellipse appeared to be aligned perpendicular to the slipstream boundary. The analytical model showed good correlation with experimental data in terms of the orientation and extent of the anisotropy. Moreover, an estimate of the total thrust produced and spanwise loading along the rotor blade was also obtained and compared with Blade Element Momentum Theory (BEMT). It was seen that by using more blobs to represent a vortex ring, the solution converged to the BEMT estimate. / text

One of the most prominent open problems in mathematical physics is determining whether solutions to the incompressible three-dimensional (3D) Navier-Stokes system, corresponding to arbitrarily large smooth initial data, remain regular for arbitrarily long times. A promising approach to this problem relies on the fact that both the smoothness of classical solutions and the uniqueness of weak solutions in 3D flows are ultimately controlled by the growth properties of the $H^1$ seminorm of the velocity field U, also known as the enstrophy. In this context, the sharpness of analytic estimates for the instantaneous rate of growth of the $H^2$ seminorm of U in two-dimensional (2D) flows, also known as palinstrophy, and for the instantaneous rate of growth of enstrophy in 3D flows, is assessed by numerically solving suitable constrained optimization problems. It is found that the instantaneous estimates for both 2D and 3D flows are saturated by highly localized vortex structures. Moreover, finite-time estimates for the total growth of palinstrophy in 2D and enstrophy in 3D are obtained from the corresponding instantaneous estimates and, by using the (instantaneously) optimal vortex structures as initial conditions in the Navier-Stokes system and numerically computing their time evolution, the finite-time estimates are found to be uniformly sharp for 2D flows, and sharp over increasingly short time intervals for 3D flows. Although computational in essence, these results indicate a possible route for finding an extreme initial condition for the Navier-Stokes system that could lead to the formation of a singularity in finite time. / Thesis / Doctor of Philosophy (PhD)

In this thesis, the effect of the following three aspects related to the generation and efficient transport of vortex rings were studied. They included: 1) the initial boundary condition where the vortex ring is generated. A comparison between previous results and the data obtained in this study showed that vortex rings generated at a tube orifice were both slower in velocity and larger in size than vortex rings generated at a plate orifice under similar conditions. 2) the presence of a stratified layer of fluid in a vessel. Flow visualization experiments showed that after a vortex ring penetrated through the interface of the stratified layer, it was able to mix the fluid inside the ring with the surrounding fluid. The amount of mixing depended on the depth of penetration of the ring into the stratified layer. An empirical relationship was obtained to predict the maximum penetration depth of a vortex ring into a stratified layer. It is: Xp/Rm - -29.7 log10Ri - 22.7 3) the proximity of a wall or another vortex ring to the path of the primary ring. Through numerical simulation, it was shown that the primary ring would slow down and turn away from its original path. Eventually, this ring would either crash into the wall or collide with another ring. In order to prevent this turning of a vortex ring from happening, the centre of the generation orifice should be 7.5 times the radius of the injection orifice (Rm) from the wall or (Rm) between two generation orifices. These results can be used to optimize the design and positioning of vortex ring mixers for various mixing vessel geometries and mixing processes. / Thesis / Master of Engineering (ME)

Vortex breakdown is an important phenomenon observed in swirling flows involving the development of a stagnation point on the axis of the vortex followed by a region of recirculation when the swirl increases beyond a particular level. It has been studied extensively over past 50 years and various theories have been proposed to explain its various aspects. However, a single model explaining all the aspects together is yet to emerge. Numerical simulations of breakdown have been performed using a variety of grid-based as well as vortex methods. Vortex methods are a Lagrangian alternative to grid-based methods wherein the motion of the vorticity is determined by the local fluid velocity convection, with models for viscous effects when considered. The fluid velocity is obtained from the vorticity field. Only the rotational regions of the flow need to be considered leading to significant economy of computational effort for simulations of vorticity dominated flows, such as vortex breakdown. The inviscid vortex filament method has been used to simulate several aspects of the vortex breakdown phenomenon. The vortex filament method however, cannot easily simulate viscous effects. To simulate the viscous effects the viscous vortex particle method needs to be used. This work was intended to be a first step towards this end by initially evaluating the effectiveness of the inviscid version of the vortex particle method in simulating the breakdown phenomenon. The inviscid vortex particle method was found to satisfactorily simulate most qualitative aspects involved in the formation of vortex breakdown such as the retardation of axial velocity along centerline, radial swelling of the vortex core, formation of stagnation points, creation of azimuthal vorticity gradient from axial vorticity gradient and the turning of vortex lines along with the formation of a bubble-like structure with recirculating flow within. The effect of a wall placed adjacent to the vortex core was simulated by using image vortices. The wall was not found to influence the location of breakdown. However, the initiation of the spiral mode was found to occur earlier when a wall was present. For a quantitative assessment, a simulation of the experimental results of Faler and Leibovich (1978) was attempted. The simulation managed to predict the location of the breakdown and the extent of the bubble. The shape and height of the bubble obtained however were not in accord with the experimental observations. A single vortical cell was obtained in the interior of the bubble.

We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics. Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size. The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.