Session HT: Vortex Dynamics and Vortex Flows V

Global Description of Bifurcation Branches and Nonlinear Dynamics of Vortex Flow in a Pipe

Details of the global dynamics of the transition to vortex breakdown in the high Reynolds number, high swirl, axisymmetric vortex flow in a finite pipe, are investigated. A first global map of fixed point branches that bifurcate from the columnar flow is revealed, along with a detailed characterization of the nonlinear, local dynamics near fixed points. The description includes the fixed point states, linear and nonlinear stability analysis modes that dominate the fluctuations near fixed points, secondary bifurcations along branches, and dynamic interconnections between them. A central role in the investigation is played by low and least order, mean-field Galerkin models of the local dynamics near bifurcation branches. Indeed, as this is a first case study of mean field Galerkin models in a flow configuration with multiple coexisting attractors, the technical and conceptual aspects of these models and of their identification, are of independent interest. The unveiled picture is that of an inertial manifold that is well approximated with only few dominant coherent structures, or modes that are well characterized by temporal and spatial frequencies, and that continuously deform with changes in the operating conditions.   

Vortex breakdown of a swirling light gas jet discharging into a heavier ambient gas

The effect of co-flow in the structure of laminar gas swirling jets with very small jet-to-ambient density ratios has been investigated by solving the high Reynolds number parabolic equations. The study is of interest for the design of Hydrogen swirl combustors. We find that the critical swirl number for vortex breakdown decreases for increasing co-flow ratios, as in a swirling jet discharging into the same ambient gas, but the critical swirl for breakdown in a light gas jet can be significantly larger. As the co-flow increases, the difference between both critical swirl numbers decreases, are equal for co-flow ratio unity, and, for co-flow ratios larger than unity, the critical swirl for a light gas jet becomes smaller than that for a homogeneous jet. These behaviors are explained by the differences in the pressure distributions generated by the swirl when the jet is much lighter than the ambient gas, in relation to a homogeneous jet. The situation becomes more complex when compressibility effects are taken into account, owing to the interplay between temperature, density and pressure differences generated by the swirl. We characterize the critical swirl for vortex breakdown as a function of the density ratio, the co-flow ratio, and the Mach number.   

Interaction of multiple solid objects and vortices in a two-dimensional potential flow

The generalized Lagally theorem states that the unsteady hydrodynamic force acting on a rigid body exhibiting arbitrary motion in an inviscid and incompressible fluid can be expressed explicitly in terms of singularities such as sources and dipoles which generate the flow field. This work extends the theorem by considering the presence of point vortices in the flow, which results in additional force components when formulating the equation of motion for the solid bodies. The present model has been applied to describe the motion of a circular cylinder that interacts with a vortex. The obtained result is in exact agreement with the result derived by other researchers considering the conservation of the sum of the solid momentum and the fluid impulse. The current model is further applied to study a moving cylinder towards a fixed one, while the former motion is coupled with two tail vortices with designated strength and initial configurations, which are prescribed by the solution to the F\"{o}ppl's problem. It is found that the configuration and the velocity of the tail vortices change due to the fixed cylinder, which in turns generates a hydrodynamic force affecting the approach of the moving cylinder.   

The dynamics of a solid-body rotation flow in a finite-length pipe

The dynamics of a perturbed, incompressible, inviscid and axisymmetric solid-body rotation flow in a finite-length, straight, circular pipe is studied through a theoretical analysis of the steady-state equations and direct numerical simulations. The flow is subjected to non-periodic inlet and outlet conditions where the outlet flow is columnar. The computed bifurcation diagrams from both approaches agree as the pipe length increases. The results show the natural evolution of the flow at incoming swirl levels above critical into either a state with a breakdown (stagnation) zone, centred around the pipe axis, or into a state with a wall separation zone (where there is no axial and radial velocity). Each solution is a result of a different domain of attraction of initial perturbations. The work emphasizes that the solid-body rotation dynamics is dominated by the linear evolution of perturbations.   

The dynamics of axisymmetric swirling flows in a diverging or contracting circular pipe

This paper describes a study of the effect of pipe divergence and contraction on the stability and breakdown of axisymmetric swirling flows in a long, finite-length, circular pipe. The work extends the theory of Wang \& Rusak (1997). The approach is based on a rigorous analysis of the axisymmetric, steady and inviscid flow equations with non-periodic boundary conditions. The analysis firmly establishes the global bifurcation of flow states in the pipe (solutions of the Squire-Long PDE) by relating it to the bifurcation of solutions of the columnar flow problem (solutions of the resulting ODE) and using a new flow force relationship between the inlet and outlet states. This technique provides a simple, yet exact, method of analyzing the complex flow behavior including transitions from near-columnar vortex states to flow fields with large separation (stagnation) zones along the pipe centerline (breakdown states) or along the pipe wall (swirl induced wall separation). Bifurcation diagrams for base vortex models including the solid- body rotation and the Burgers vortex are presented. The stability characteristics of the various branches of solutions and the flow dynamics in the pipe under various perturbations are discussed. Results show that pipe divergence or contraction significantly modify the global flow behavior in a straight pipe and shed light on the effect of pipe geometry on the mechanism of vortex breakdown.   

Active Flow Enhancement for Heat Ejection in a Diverging Channel

Small-scale motions that are induced within the core flow of a heated, high-aspect ratio diverging channel are investigated experimentally. The small scale motions are effected by the time-periodic vortex shedding from streamwise-embedded cantilevered reeds driven in resonance by integrated piezoelectric actuators and span the entire channel height. The induced vortices are advected with the core flow, disrupt the thermal boundary layers and result in a significant enhancement of the local heat transfer coefficient along the channel. Deliberate interactions between the reeds and a given core flow along secondary channels that are formed downstream of a center partition lead to controlled flow oscillations over a range of vibration frequencies. These interactions and the flow characteristics between the tip of the vibrating reed and the partition are investigated using high resolution particle image velocimetry (PIV). Of particular interest are the effects of variation in the reed motion and its distance to the partition on the induced small-scale motions and mixing and thus, on local and global heat transfer across the channel. Supported by DARPA and UTRC.   

Flow asymmetry and vortical structures behind a rotating tire

The wake behind both stationary and rotating tires is dominated by two strong counterrotating vortices. Experimental observations and numerical simulations have shown that one vortex may become stronger thus altering the wake dynamics and stability. The objective of this study is to investigate the main causes of near-wake and far-wake asymmetry. Factors such as tire camber angle, hub cavities, spoke geometry and the flow through the brake system, are all investigated as possible sources of asymmetry; it is shown that the most critical factor in flow asymmetry is the flow through the hub of the tire. When this is primarily in one direction (inboard to outboard side), the inboard vortex increases in size and intensity, eventually overwhelming the outboard vortex. We also study the transient features and timescales associated with the counterrotating vortex pair. Transient feature extraction is typically not possible with traditional techniques like PIV and RANS computer simulations. As a result, an LES simulation is used to extract eddy turnover timescales and vortex core trajectories for a rotating tire at Re=500,000.   

On point vortex equilibria in a bounded circular domain

We develop a general and systematic approach for solving point vortex equilibria in a bounded domain. The motivation for this work is that most, if not all, vortex equilibria in the physical world occur within a confined container filled with a finite amount of fluid. However, most of the existing models for vortex equilibria are in the unbounded plane. Furthermore, the few manuscripts that analyze vortex equilibria in the presence of a solid boundary present results for very simple configurations. We use the Hamiltonian point vortex model and the circle theorem for the equations of motion. We formulate the problem as one in linear algebra with the positions of the vortices given, and we use singular value decomposition to determine the vortex strengths necessary for relative equilibrium. We illustrate the feasibility of this technique with a few examples that are physically motivated from experiments, and we comment on linear stability.   

Swirling Strength Vortex Study in Confined Rectangular Jet

Vortex behavior in confined rectangular jet (Re = 20K, Re = 50K) were examined by using vortex swirling strength as a defining characteristic. Instantaneous velocity fields were collected for by using Particle Image Velocimetry(PIV). Swirling strength fields were calculated from velocity fields, and then filtered with a universal threshold of 1.5 times of swirling strength RMS value. By identifying clusters in filtered swirling strength fields, vortex structures were defined. Both instantaneous swirling strength field data and vortex population calculation indicate that the positively (counterclockwise) rotating vortices are dominant on the left side of the jet, and negatively (clockwise) rotating vortices are dominant on the right side. As flow develops further downstream, vortex population decreases and the flow approach channel flow. At the locations of the left peak of turbulent kinetic energy, two point spatial cross-correlation of swirling strength with velocity fluctuation were calculated. Linear stochastic estimation was also used to interpret the spatial correlation results and to determine conditional flow structures. High speed PIV data were also analyzed by using swirling strength technique to trace development of vortices. Vortex trajectories were found by tracing individual swirling strength clusters. The speed and strength of individual vortex were also studied by using this method.   

Numerical simulation of impurity propagation in sea channels

Building the dike (2003) in Kerch channel (between Black and Azov seas) from Taman peninsula is an example of technological influence on the fluid flow and hydrological conditions in the channel. Increasing velocity flow by two times in a fairway region results in the appearance dangerous tendencies in hydrology of Kerch channel. A flow near the coastal edges generates large scale vortices, which move along the channel. A shipwreck (November 11, 2007) of tanker ``Volganeft-139'' in Kerch channel resulted in an ecological catastrophe in the indicated region. More than 1300 tons of petroleum appeared on the sea surface. Intensive vortices formed here involve part of the impurity region in own motion. Boundary of the impurity region is deformed, stretched and cover the center part of the channel. The adapted vortex singularity method for the impurity propagation in Kerch channel and analyze of the pollution propagation are the main goal of the report.