Previously we studied the two-dimensional problem of planing of a flat plate on the free surface of an incompressible inviscid fluid of infinite depth. We assumed that the Froude number U /(gL)(1/2) was large, with U the horizontal velocity of the plate, L the length of the plate, and g the acceleration of gravity. We solved this problem by the method of matched asymptotic expansions, with gLU(-2) as the small parameter. The solution contained a jet flowing up along the forward-facing surface of the plate. When the jet reached the upper edge of the plate, it became a free jet that followed a parabolic path and fell back onto the free surface. Now we consider the case where the jet falls off the plate before it reaches the upper edge, and we determine where it leaves the plate.

Grobli (1877) laid the foundation for the analysis of the motion of three point vortices in a plane by deriving governing equations for triangular configurations of the vortices. Synge (1949) took this formulation one step further to that of a similar triangle of unit perimeter, via trilinear coordinates. The final reduced problem is governed by an integrable two-dimensional system of differential equations with solutions represented as planar trajectories. Another key to Synge's analysis was his classification of the problem into three distinct cases: elliptic, hyperbolic and parabolic corresponding, respectively, to the sum of products of pairs of vortex strengths being positive, negative or zero. The reduction of the vortex configuration, a curve in space to a planar curve is one-to-one, except along a critical planar curve C in the parabolic case. Each point on C represents a triangle of unit perimeter corresponding to a family of similar vortex configurations, expanding or contracting. The latter would lead to coelescence of the three vortices. Tavantzis and Ting (1988) filled most of the gaps left by Synge regarding the dynamics of the problem, and showed in particular that points on C corresponding to similar expanding families of vortex configurations are stable while those corresponding to similar contracting families are unstable. Their investigations yielded an exhaustive description of the motion and stability of three vortices in a plane except for the global behavior of the vortex configurations in a narrow strip containing C. The main contribution of this paper is a complete description of the global dynamics in such a strip, which emphaticallly demonstrates that three distinct vortices almost never coalesce.

AbstractThe motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half‐plane, are also Hamiltonian, but not completely integrable. Yet these perturbed systems may still have periodic trajectories and invariant tori close to those for the unperturbed dynamics.Extending recent work by the authors [4], invariant 2‐tori approximating those for the unperturbed system are located and analyzed using a combination of classical analysis, asymptotics, and Hamiltonian methods. It is shown that the results and approximation methods used are applicable to several perturbations of three vortex dynamics such as three vortices in a half‐plane, the restricted four vortex problem in the plane, and three coaxial vortex rings in 3‐space. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

AbstractWe study an N ‐vortex problem having J of them forming a cluster, which means the distances between the vortices in the cluster is much smaller by O (ε) than the distances, O (ℓ), to the N – J vortices outside of the cluster. With the strengths of N vortices being of the same order, the velocity and time scales for the motion of the J vortices relative to those of the N – J vortices are O (ε–1) and O (ε2) respectively. We show that this two‐time and two‐length scale problem can be converted to a standard two‐time scale problem and then the leading order solution of the N ‐vortex problem can be uncoupled to two problems, one for the motion of J vortices in the cluster relative to the center of the cluster and one for the motion of the N – J vortex plus the center of the cluster. For N = 3 and J = 2, the 3‐vortex problem is uncoupled to two binary vortices problems in the length scales ℓ and ℓε respectively. When perturbed in the scale ℓ, say by a fourth vortex even of finite strength, the binary problem becomes a 3‐vortex problem, admitting periodic solutions. Since 3‐vortex problems are solvable, the uncoupling enables us to solve 3‐cluster problems having at most three vortices in each cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Vortex-Dominated Flows and General Theory.- Motion and Decay of Vortex Filaments - Matched Asymptotics.- Nonlinear Dynamics of Nearly Straight Vortex Filaments.- Numerical Simulation of Slender Vortex Filaments.- Numerical Simulations of the Merging of Vortices or Filaments.- Flow Generated.- Sound Generated Flow.- Epilogue.

It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated with completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half plane, three coaxial slender vortex rings in three space, and “restricted” four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser- and Poincaré-Birkhoff-type arguments and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.

Each convex planar set K has a perimeter C, a minimum width E, an area A, and a diameter D. The set of points (E,C, A1/2, D) corresponding toall such sets is shown to occupy a cone in the non-negative orthant of R4with its vertex at the origin. Its three-dimensional cross section S in theplane D = 1 is investigated. S lies in a rectangular parallelepiped in R3. Results of Lebesgue, Kubota, Fukasawa, Sholander, and Hemmi are used todetermine some of the boundary surfaces of S, and new results are given for the other boundary surfaces. From knowledge of S, all inequalities amongE, C ,A, and D can be found.

The asymptotic theory of incompressible slender vortex filaments is extended to account for the compressibility effect of high swirling flow in the vortical core. We derive the equations of motion of a filament and equations for the evolution of the core structure, the large swirling and axial velocity and the density profiles. We highlight how the equations and solutions for the compressible flows differ from those for the incompressible flows.

A weak shock incident upon an obstacle produces weak reflected and diffracted shocks.The resulting flow is analyzed near a point P where either the incident shock or a reflected shock meets a diffracted shock.The path of P is called a singular ray, and it is analogous to a ray on a shadow boundary.For self-similar problems, the flow near P is shown to be the solution of a nonlinear elliptic free boundary problem.This flow is regular, in contrast to the singular flow given by linear theory.An analogous problem is found for the interaction of a weak rarefaction wave with a weak shock.Both problems are also applicable to steady self-similar supersonic flows past bodies.Hunter's analysis (SIAM J. Appl.Math.48, (1988), pp.1-37.)implies that these problems are canonical, i.e. that they apply to general hyperbolic systems in any number of dimensions.Simplified forms of both problems are solved numerically.

We give a brief review of the asymptotic theory of slender vortex filaments with emphases on the choices of scalings characterizing the physical problems and the corresponding assumptions and/or restrictions introduced in the formation of the asymptotic theory of Callegari and Ting (1978) and its extension by Klein and Ting (1992). In particular, the slender filaments considered are assumed to be forming loops or tori. Because of this restriction, the theory is not applicable to the trailing vortex system of a rotorcraft. We describe the multiple length scales characterizing the vortex system, formulate the expansion scheme, derive the governing equations and then identify the assumptions or restrictions inherent in the multi-scale analysis and needed for the validity of the asymptotic theory of the trailing vortex system.

We give a brief review of the asymptotic theory of slender vortex filaments to emphasize i) the choices of scalings, small parameters and the distinguished limit, ii) the consistency conditions, iii) the optimum similar and non-similar viscous vortical core structures and iv) their applications to complement experimental investigations. We present highlights of several extensions of the asymptotic theory: the analyses for core structures with axial variation, for the interaction of filaments with a solid body and sound generation and for a filament in a background rotational flow. We then outline the vortical flow problems currently under investigation.

Interactions between a slender vortex filament and a stationary rigid sphere are analyzed using a vortex element scheme which tracks the motion of the filament centerline. The filament velocity is expressed as the sum of a self-induced velocity and potential velocity due to the presence of the sphere. The self-induced velocity is estimated numerically using a line Biot–Savart integral which is carefully desingularized so as to reflect the correct asymptotic behavior of the core vorticity distribution under the influence of stretching and viscous diffusion. Meanwhile, the potential velocity is evaluated from a recently derived formula, which expresses it as a line integral along the image of the filament centerline in the sphere with regular weight functions. From the far-field behavior of an unsteady vortical flow outside a stationary sphere, formulas for the acoustic far field are obtained. It is shown that the interaction between the slender vortex filament and the sphere generates dipoles and quadrupoles in addition to the quadrupoles generated by the filament alone in space. The strengths and orientations of the dipoles and quadrupoles are completely determined by the time evolution of the weighted first and second moments of vorticity. The formulas are applied to compute the far-field sound generated by the passage of a slender vortex ring over the sphere. Both coaxial and noncoaxial passage events are analyzed in the computations, as well as the effects of initial core size and asymmetric perturbations.

Interactions between a slender vortex filament and a stationary rigid sphere are analyzed using a vortex element scheme which tracks the motion of the filament centerline. The filament velocity is expressed as the sum of a self-induced velocity and potential velocity due to the presence of the sphere. The self-induced velocity is estimated numerically using a line Biot–Savart integral which is carefully desingularized so as to reflect the correct asymptotic behavior of the core vorticity distribution under the influence of stretching and viscous diffusion. Meanwhile, the potential velocity is evaluated from a recently derived formula, which expresses it as a line integral along the image of the filament centerline in the sphere with regular weight functions. From the far-field behavior of an unsteady vortical flow outside a stationary sphere, formulas for the acoustic far field are obtained. It is shown that the interaction between the slender vortex filament and the sphere generates dipoles and quadrupoles in addition to the quadrupoles generated by the filament alone in space. The strengths and orientations of the dipoles and quadrupoles are completely determined by the time evolution of the weighted first and second moments of vorticity. The formulas are applied to compute the far-field sound generated by the passage of a slender vortex ring over the sphere. Both coaxial and noncoaxial passage events are analyzed in the computations, as well as the effects of initial core size and asymmetric perturbations.

Interactions of a slender vortex ring with and a stationary rigid sphere are analyzed using a 3D, Lagrangian vortex element scheme which discretizes and tracks the filament centerline using smoothed vortex elements. The filament self-induced velocity is obtained from a desingularized Biot-Savart law that reflects the correct asymptotic behavior of the core vorticity distribution. The effect of the sphere is represented in terms of a potential velocity field that is expressed as a line integral along the image of the filament centerline with regular weight functions. It is shown that the acoustic emission due to the interaction between the filament and the sphere essentially consists of dipoles and quadrupoles whose strengths and orientations are determined by the time-evolution of the weighted first and second moments of vorticity, respectively. The scheme is used to analyze the sound generated during the passage of slender vortex rings near a solid sphere.

Formulas are presented for an incompressible inviscid velocity field V with a vorticity field ${\bf \Omega}$ outside of a rigid sphere and for the far-field sound generation. The velocity V is expressed as the sum of an image velocity v* and a known velocity v in $\Re^3$, which is induced by the same vorticity field ${\bf \Omega}$ outside the sphere and the extension ${\bf \Omega}=0$ inside. We derive formulas for the image velocity v* and the corresponding image potential $\Phi^*$, which in turn yields the far-field sound. These formulas are applied to define the velocity of a slender vortex filament in the presence of a rigid sphere and the associated far-field sound.

The interaction of an acoustic wave and a clamped elastic panel is considered. Here we present numerical results comparing the exact solution found by solving the coupled acoustic wave — elastic panel problem to two decoupling approximations using an on surface boundary condition. These on surface conditions are derived in the limit where the ratio of the acoustic sound speed in the fluid to the surface wave speed in the panel is small.

: Perturbation methods and numerical methods were employed to study four problem areas in fluid dynamics. The areas and the progress were: (1) Viscous vortical flows - We showed how to combine the asymptotic theory and experiments to study slender vortex filaments and how to specify the numerical parameters needed for the vortex element method to predict correctly the motion of slender filament(s) in space. We presented formulas relating a rotational flow outside a sphere to the rotational flow in space and formulas defining the far-field sound. These formulas were used to study the interaction of a vortex filament with a sphere. (2) Shock wave interactions - A canonical nonlinear elliptic problem was formulated and used to correct the defect of linear theory near a singular ray where a weak shock interacts with a diffracted wave. A similar canonical problem was formulated to solve the interaction of a weak expansion wave with a diffracted wave. (3) Wave propagation - Rules were formulated for determining the multiplicity of acoustic signals and the retarded times for media moving with unsteady speed ranging from subsonic to supersonic. In the analysis of structural/acoustic interactions, the solution for the panel oscillation was uncoupled from the acoustic field by the formulation of the on surface conditions taking into account the acoustic effect. (10) Free boundary problems - We obtained solutions for the formation of a drop after the breaking up of symmetric slender jets or thin sheets. (AN)