A vortex sheet model is used to numerically study vortex ring dynamics. A Lagrangian numerical method and an adaptive tree-code is used to simulate the smoothed vortex sheet. Results are presented for azimuthal perturbations of an initially axisymmetric vortex ring and for the off-axis collision of two vortex rings.
[Ci.02] Structure and Dynamics of Vortex Rings Exiting Inclined Orifices
D.R. Webster, E.K. Longmire (University of Minnesota)
Impulsive flows exiting non-axisymmetric orifices are examined at two Reynolds numbers (2800 and 23,000). The flow is initiated with a round piston/cylinder apparatus and the cylinder exits are inclined with lengths of 0, D/4, D/2 and D. Instantaneous velocity fields were measured in a number of azimuthal planes with particle image velocimetry (PIV). Due to the inclined exit, the velocity field exiting the cylinder was spatially non-uniform leading to non-axisymmetric entrainment and the formation of a complex vortex structure. The structure consisted of a primary vortex ring with non-uniform circulation around the circumference and branched vortex tubes. The structure migrated off the centerline toward the short cylinder side. With increasing incline angle, the vortex structure migrated further away from the centerline. The flow also became increasingly disorganized resulting in reduced propagation speed and penetration distance. The general flow features were the same for each Re and for varying piston geometry.
[Ci.03] Vortex Sheet Evolution in Three Dimensions
Mark Brady, Anthony Leonard, Dale Pullin (California Institute of Technology)
A computational method is presented to track the evolution of three dimensional vortex sheets through an inviscid, constant-density fluid. The sheet surface is represented by a triangulated mesh with interpolating functions locally defined inside each triangle. C^1 continuity is provided between triangles via combinations of cubic Bézier triangular interpolants. The self-induced sheet motion generally results in a highly deformed surface which is adaptively refined as needed to capture regions of increasing curvature. Mesh refinement is performed automatically by an advancing front technique. Sheet motion is regularized by adding a length scale cut-off to the Biot-Savart kernel, a 3D extension of the Rosenhead--Krasny desingularization. At this point, toroidal and periodic-cylinder vortex sheets are simulated, modeling vortex rings and vortex/jet combinations respectively. Comparisons with previous 2D results are favorable and 3D results are presented.
[Ci.04] The interaction of two slender vortex rings and their generated sound
K. Ishii (Dept. Comp. Sci. Eng., Nagoya Univ.,), S. Adachi (RIKEN,Wako,351-01)
A vortex ring is characterized by two nondimensional parameters, i.e. the slenderness ratio of core radius to ring radius and Reynolds number. When the slenderness ratio is small and the Reynolds number is large, there are various different time scales in the motion of a vortex ring, which are lifetime of vortex ring, moving time scale and revolution time scale of fluid particle and so on. The phenomena of vortex dynamics ( i.e. stretching, vanishing, the cut-and-connect of vortex lines ) in the interaction of two vortex rings are also characterized by these time scales. In addition, the estimation of far-field acoustic pressure excited by a time-dependent localized vorticity distribution will strongly be related to these time scales. In order to study these relationship, the numerical simulations of oblique collisions of two same vortex rings with various angles have been done. Two slenderness ratios( 0.1 and 0.2) are chosen. Time evolution of the vorticity field is obtained by solving a viscous incompressible vorticity equation with a vorticity-potential method. The rings with different slenderness ratio and/or with different collision angles experience the cut-and-connect or vanishing of vortex lines in the different time stages of collision of rings. As the results, the different time histories of strength of multipole sound modes are obtained , while they have peaks at the mostly same time. Using these data, we discuss the relationship between the vortical phenomena and the emitted vortex sound.
[Ci.05] The Vortex Ring Structure of a Bursting Bubble
James H.J. Buchholz, Lorenz W. Sigurdson (Dept. of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada)
For the first time, the evolution of the vortex ring structure created by a bursting bubble has been photographically studied and modeled. It has similarities to the structure created by an above-ground atomic test and by a water drop impacting a pool of still fluid. The bubble is situated on a thin, stationary layer of fluid on the top of a cylinder. Two upward-moving vortex rings with collinear axes are formed shortly after bubble rupture. The two rings leapfrog and then form a structure consisting of a vortex ring which sheds "lobes" of fluid which are sometimes similar in appearance to hairpin vortex loops. A two-dimensional point vortex computational model was created which produced images with resemblance to the physical structure. Both the experimental and computational results suggest that the lobes are rotational in at least some cases. The kinematics of the bubble film were also studied since the initial conditions of the vorticity are created when the bubble bursts. It was discovered that the edge of the film traces a spherical surface as it retracts, and that its velocity is not constant but greatest in the initial stages of retraction. This work was supported by NSERC Grant No. OGP004174 and the Central Research Fund of the University of Alberta.
[Ci.06] A Dynamic Lagrangian Large Eddy Simulation Scheme for the Vorticity Transport Equation
John Mansfield, Omar Knio, Charles Meneveau (Department of Mechanical Engineering, Johns Hopkins University)
An LES scheme is developed for simulating turbulent flows with concentrated large-scale vorticity. The scheme combines an adaptive, viscous, three-dimensional vortex element method with a dynamic eddy viscosity model of the sub-filter scale stresses. Dynamic implementation of sub-filter scale model relies on determining model coefficients through test-filtering of Lagrangian particle representation of the filtered vorticity field.
The vorticity-based dynamic model is examined using a priori tests based on direct numerical simulations of forced homogeneous, isotropic turbulence. These tests show a fair correlation of the turbulence model with the sub-filter scale convection of vorticity. Moreover, the computed value of the dynamic model coefficient is in agreement with predictions based on enstrophy balances. The Lagrangian LES scheme is then applied to the simulation of the evolution of a single vortex ring in free space, and of the co-axial collision of two vortex rings.
[Ci.07] Core dynamics instability of a vortex in shear: a prototypical cascade mechanism
D. S. Pradeep, Fazle Hussain, Wade Schoppa (Univ. of Houston)
Using DNS, we study 3D instability and transition of a vortex column with nonuniform core size in a uniform plane shear normal to its axis. Internal (core) dynamics leads to propagation of vorticity wavepackets along the column which can extract energy from shear, and amplifies oscillations of core size and vortex line twisting. The exponential 3D energy growth rate increases with increasing shear strength. Collision of oppositely propagating wavepackets forms a low-enstrophy pocket in the core surrounded by a thin sheath of intense vorticity which rolls up by a local Kelvin-Helmholtz type instability into an array of fine-scale vortices along the periphery. Reconnection of folded vortex lines within the core leads to concurrent fine-scale generation. These constitute physical-space scenarios of cascade mechanisms, quite likely to be prototypical in turbulent flows.
[Ci.08] Dynamics of Co-rotating Vortex Pairs in the Wakes of Flapped Airfoils
A. L. Chen (University of California at Berkeley), J. D. Jacob (University of Kentucky), O. Savas (University of California at Berkeley)
The behavior of a pair of co-rotating vortices in the wake of a flapped airfoil is experimentally studied in a water towing tank. Three rectangular airfoils with an aspect ratio of 6 and flap spans of 0%, 30%, and 67% are used in the experiments. Reynolds numbers based on total circulation of the vortices range from 1.8\cdot 10^4 to 6.5\cdot 10^4. Velocity vector fields and their gradients are derived from PIV images using an adaptive Lagrangian parcel tracking algorithm. The wakes behave inviscidly at these Reynolds numbers. Merger of the co-rotating vortex pair is observed at all Re_\Gamma. A key observation is that the merger occurs within the time required for one vortex to rotate around the other. The merger appears inviscid and three-dimensional. First order statistics of the flow field remain invariant during the merger. The higher order moments of the vorticity distribution show strong time dependence which implies three dimensionality of the flow resulting from vortex stretching. The strengths of the individual vortices before merger are constant, and the total circulation after merger remains constant within the range of observations. The trajectory of the center of vorticity remains unaffected by the merger process. Kinetic energy and angular momentum of the flow are conserved during the merger. The merger is preceded by a splitting of the weaker vortex into filaments. Depending on the relative strengths of the vortices, the splitting can be in the radial direction, the axial direction, or somewhere in between.
[Ci.09] Nonlinear stability, energy exchange and solitons on vortex cores
Kamran Eftekhari, Ernst W. Mayer (\urllinkDept. of Mechanical and Aerospace Engineering http://k2.scl.cwru.edu/cse/emae, \urllinkCase Western Reserve Universityhttp://www.cwru.edu)
Weakly nonlinear, nonaxisymmetric wave propagation on concentrated vortex
cores is considered under the assumption of inviscid, incompressible flow.
A formulation of the type first used by Leibovich et al.~is used, in which
the linear-theory perturbation is modulated by a slowly varying complex
amplitude function having finite amplitude. The only place where the full
details of the above authors' analysis are presented is an internal memorandum
by Ma~(H.-Y. Ma, Rept.~FDA-82-10, Sibley School of Engineering, Cornell U.)
which contains several inconsistencies, the major one being the absence of
a quadratic amplitude term at second order. This type of term is present
in the later papers of Leibovich, Yang, Brown and Patel, but in a different
form than used by us. The solvability condition at third order yields a
cubically nonlinear Schrödinger equation (NLS) in one (slow) space dimension
for the envelope function, which means that solitary waves are possible for
certain wavenumber ``windows.'' We examine these in detail for the families
of linear disturbances which exist on the well-known Q-vortex model flow.
Most interestingly, we find a set of singularities in the coefficient of
the nonlinear term in NLS, which occur at discrete values of the axial
wavenumber. These may serve physically as a wavenumber selection mechanism
in nonlinear phenomena such as those leading to vortex breakdown.
[Ci.10] Long's Vortex Revisited
Sih-Tsan Lee, Ernst W. Mayer (\urllinkDept. of Mechanical and Aerospace Engineering http://k2.scl.cwru.edu/cse/emae, \urllinkCase Western Reserve Universityhttp://www.cwru.edu)
The conically self-similar vortex of Long (1962) is reconsidered, with a view toward understanding what, if any, relationship exists between it and the recent similarity solutions of Mayer and Powell (1992), which are a rotational-flow analogue of the Falkner-Skan boundary layer flows. It is found that, when minor differences in the formulations are accounted for, the Long and Mayer/Powell (MP) flows in fact satisfy the same system of coupled ODEs, subject to different boundary conditions.
Long's equations thus can be seen as a special case of the MP equations corresponding to conical vortex growth, which can occur only if the outer axial velocity decelerates in a z^-1 fashion, implying a severe adverse pressure gradient. For pressure gradients this adverse MP were not able to find solutions of the similarity equations which satisfied a physicality criterion based on monotonicity of the total-pressure profile of the flow. It is shown that Long's solutions also violate this criterion in an extreme fashion, hence are nonphysical. However, the fact that the Long's flows fit into a more general similarity framework means that nonconical analogues of these flows should also exist. The generalized Long's flows which do satisfy the physicality criterion are described.