Session BT: Vortex Dynamics and Vortex Flows II

Deriving Kelvin's argument through the principle of ``virtual vortex work''

In 1875, Lord Kelvin proposed an energy argument to define equilibrium and stability in fluid flow. Kelvin stated that steady flows would realize a stationary point of the energy, for given vorticity and impulse. Intriguingly, Kelvin proposed this idea without proof; analytical confirmation was presented a century later by Benjamin (1975). Unfortunately, to this date, we have no indication as to how Kelvin's argument may be derived from a fundamental physical principle. Indeed, the path that led Kelvin to his statement remains unknown. A derivation based on a fundamental principle may enable generalizations to novel applications, which cannot be investigated by the current formulation of Kelvin's argument. In this presentation, we employ the fundamental principle of virtual work, and show that the requirement for a system to be stationary in a {\it moving} frame leads naturally to a constraint on the impulse. In the context of fluid flow, formulating the principle leads us to introduce the concept of {\it virtual vortex work}. We show this to be equivalent to Kelvin's argument. We exploit our derivation to devise generalizations of Kelvin's argument for a variety of fluid flows, including vortical flows, gravity waves and compressible flows. This, in turn, allows us to instantly deduce stability properties in 2D and 3D through an ``imperfect-velocity-impulse'' approach (which takes the form of ``IVI'' diagrams) recently proposed by the authors.   

Time resolved measurements of vortex-induced-vibration of a tethered sphere

Time resolved, high-speed PIV measurements were performed to study the spatio-temporal dynamics of a tethered, stainless steel sphere ($D$ = 5/16'', $m^{\ast }$=7.86, $L^{\ast }$=2.90), mounted in a water channel and exposed to uniform, free stream velocities up to 0.64 m/s. To study the coupled interaction between vortex and sphere dynamics, we performed measurements in a horizontal plane, intersecting the sphere's center. Below the threshold velocity for which vortex induced vibration occurs, wake dynamics are those of a stationary sphere. As the free stream velocity increases, modes of periodic and non-stationary intermittent sphere dynamics are observed in the plane transverse to the flow. We simultaneously track the sphere and vortex centers, the latter through maximum values of the swirling strength. Vortex tracking starts at the sphere interface through the separation point and away from the sphere. Also, time dependent turbulent stresses, terms of the TKE production are presented. This information together with the sphere's motion itself reveals clues to the intricate, coupled flow-structure interaction.   

A mathematical model of a ``2P mode'' vortex wake

The standard von Karman vortex street, also known as the ``2S'' mode, is the most common vortex configuration to appear in the wake of a bluff body. The next most common configuration is the ``2P'' mode, in which two pairs of vortices are shed per cycle. ~We consider a simple model of the ``2P'' mode consisting of a singly-periodic Hamiltonian system of four point vortices with identical strength magnitudes and zero net strength. An imposed spatial symmetry results in integrable dynamics that depend only on the relative vortex positions. Comparison of our model with a recent experimental result (Schnipper, Andersen, and Bohr, JFM 2009) suggests that this model approach can be used to characterize the experimental vortex motion and estimate the experimental vortex strengths.   

Vorticity Measurements in the Wake of an Inclined Prolate Spheroid

The generation and evolution of axial vorticity in the wake of an inclined 6:1 prolate spheroid is studied experimentally, with comparison to Computational Fluid Dynamics (CFD) results. 2D Particle Image Velocimetry (PIV) measurements were obtained in planes normal to the flow at several stations along the body and at downstream distances up to one body length, at angles of attack of 5, 10, and 20 degrees and body Reynolds numbers (Re$_{L}$=UL/$\nu )$ of {\{}13.7, 27.3, 45.6{\}} x 10$^{4}$. As an extension of previous numerical and experimental studies on the vortex roll-up on the body of a 6:1 Prolate Spheroid [for example, Fu \textit{et al} (1994), Tsai and Whitney (1999)], this study is focused on characterizing the downstream vorticity distribution as a function of the angle of attack and body Reynolds number. Long time average measurements of the circulation, core size, and core location are presented as a function of the angle of attack and the free stream velocity. In addition, measurements of turbulence characteristics of the wake are presented. Vortex migration velocities are found to be less than those estimated from inviscid vortex dipole theory. Experimental results for the 10-degree case are compared. Reynolds Average Navier-Stokes (RANS) CFD calculations show significant differences in the vorticity distribution near the stern, but with good agreement at one body length downstream.   

Organization of Cylinder Wake Using a Splitter-Plate Active Flow Control

It is well-known that a splitter plate in the wake of a circular cylinder prohibits the formation of the classic von Karman vortex street. Here we present an experimental study which shows the near wake can be manipulated using flow control to restore the vortex shedding in the presence of a splitter plate. Three splitter plate locations along with three cylinder diameters were analyzed using spectral analysis and proper orthogonal decomposition of time resolved digital particle image velocimetry (TRDPIV) data. As an example, in one case the splitter plate was placed 1.9 diameters downstream of the cylinder and spectral measurements of the TRDPIV results indicated its presence decreased the Strouhal number from 0.19 to 0.12 as anticipated. When activated the flow control restored the wake to a Strouhal number of 0.19 and a 2P vortex shedding mode was clearly visible. The data suggests that the jet excited the circular cylinder shear layers causing instability, roll up, and subsequent vortex shedding.   

Experimental Study of Synchronization and Phase Dynamics in Flapping Wing Propulsion

Experiments are conducted on a two dimensional heaving airfoil to determine whether or not natural flight can be modeled as a limit cycle process as well as the degree to which the wing motion, any vortices shed from upstream bodies and the fluid force response act as dynamically coupled oscillators. The heaving airfoil mechanism is constructed to permit the experimental simulation of a freely flying system in which the forward speed of the system is determined by the Strouhal number and reduced frequency of the motion. Also examined is how the structure of the flow behind a freely flying system differs from that of a strongly forced system, where the position of a flapping airfoil mechanism is fixed and the forward velocity is imposed.   

Vortex rings behind an oscillating sphere

The talk presents the results of an investigation of the formation, stability and control of localized vortex ring structures due to a periodic motion of a sphere in an inviscid incompressible fluid. The low order model based upon Dyson's vortex rings is employed. The intensity of generated vortex rings is estimated by the total vorticity, which takes place in a viscous boundary layer on the sphere. Numerical results of simulations of the transport processes in the near-wall zone, the determination of regions of low and high domains of passive admixtures are discussed. Finally, we compare numerical and analytical solutions with the results of analogue laboratory experiments for helium II.   

Dynamics of collision of a vortex ring and a planar surface

The dynamics of the impact between a vortex ring and a planar surface orientated perpendicular to the direction of travel are presented. High Reynolds number vortex rings are injected into a quiescent tank of water using a piston-cylinder generator before colliding with a target at a long distance. Both the pressure at the stagnation point on the surface and the force imparted to the target by the ring impact are measured directly. The changes in both are related to the ring motion and deformation captured by high speed digital video, and DPIV measurements. These relations are used to develop a scaling law relation between impact force and vortex ring circulation, speed, and size.   

Strain dynamics for vortex ring mixing process

Simultaneous PIV-PLIF measurements were carried out to investigate the mixing occurring in a laminar vortex ring flow during the formation stage (Re=357-1072). In the first part of the work a control volume analysis was used to determine the variation in time of the scalar concentration mean, variance, and probability density function. In the second part the advection-diffusion differential equations of the scalar, $\xi$, and of its energy, 0.5 $\xi^{2}$, were studied in depth to gain insight into the effect of the strain rate tensor, \textbf{S}, on the local scalar concentration for increasing \textit{Re}. The measurements were obtained with a high spatial resolution (12 $\mu$m for the PLIF) in order to resolve the scalar dissipative scales. Reliable estimates of the scalar dissipation rate ($\nabla \xi \cdot \nabla \xi$), and of the symmetric contraction term ($\nabla \xi \cdot \textbf{S} \cdot \nabla \xi$), shown in equation 1, were obtained. $\nabla \xi \cdot \textbf{S} \cdot \nabla \xi$ accounts for the reduction of scalar dissipation due to the straining component directed as the local scalar gradient (see Southerland et al.\footnote{Southerland K B., Porter III J. R., Dahm, W. J. A., Buch K. A., An experimental study of the molecular mixing process in an axisymmetric laminar vortex ring, Phys. Fluids A 3 (5), May 1991}) Equation 1: $\left( \frac{\partial }{\partial t}+\vec {u}.\nabla +\frac{1}{ReSc}{\nabla ^2} \right)\frac{1}{2}\left( {\nabla \xi .\nabla \xi } \right)=-\left( {\nabla \xi .S.\nabla \xi } \right)-\frac{1}{ReSc}\nabla (\nabla \xi ):\nabla (\nabla \xi$)   

Experimental Investigation of Vortex Ring Interaction with Inclined Surfaces

A number of experimental and numerical studies have described the collision of a laminar vortex ring with an inclined surface and noted similarities with hairpin vortices found in turbulent boundary layers. However, the dependence of the observed flow on the vortex ring properties and angles of collision have largely been neglected. In the present investigation, vortex ring interactions with an inclined plate were studied experimentally to determine the effects of plate angle on the flow evolution and draw comparisons with coherent structures in turbulent boundary layers. Vortex rings were generated using a mechanical piston-cylinder vortex ring generator at jet Reynolds numbers ranging from 1000 to 3000 and stroke length-to-piston diameter ratios from 0.5 to 2. The plate angle relative to the initial axis of the vortex ring ranged from 3 to 70 degrees. Flow observations were made using planar laser induced fluorescence, 2D digital particle image velocimetry (DPIV), and 3D defocusing DPIV (DDPIV). Results show deformation and stretching of the vortex ring into a loop-like vortex and the generation of secondary vorticity at the surface of the plate.