Takeshi Miyazaki

Three-Dimensional Distortions of a Vortex Filament with Axial Velocity

Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrodinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N -soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Eulers elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.

Three-dimensional instability of strained vortices in a stably stratified fluid

The linear stability of unbounded strained vortices in a stably stratified fluid is investigated theoretically. The problem is reduced to a Floquet problem which is solved numerically. The three‐dimensional elliptical instability of Pierrehumbert type [Phys. Rev. Lett. 57, 2157 (1986)] is shown to be suppressed by the stable stratification and it disappears when the Brunt–Vaisala frequency exceeds unity. On the other hand, two classes of new instability mode are found to occur. One appears only when the Brunt–Vaisala frequency is less than 2, whereas the other persists for all values of the Brunt–Vaisala frequency. The former mode is related to a parametric resonance of internal gravity waves, and the latter modes are related to superharmonic parametric instability.

Time‐dependent thermal convection in a stably stratified fluid layer heated from below

Time‐dependent thermal convection resulting when a stably stratified fluid layer is heated from below is analyzed as an initial‐value problem of a linear model. The effect of stratification on its development and structure is investigated. The convection is found to develop in three stages. Initially the given disturbance induces oscillatory motion with the corresponding Brunt–Vaisala frequency. In the second stage the stratification works to form secondary cells above the primary cell. This multiple cell pattern, however, changes slowly but periodically with time. Finally, the convection grows with an exponential growth rate, retaining a similar cell pattern. The vertical scale of the primary cell decreases with the stratification level, while the horizontal scale remains constant. The onset time of the manifest convection is predicted to be delayed as the stratification shifts from neutral to stable conditions. Attaining a maximum, it then becomes slightly earlier. Experimental results supporting the mo...

The motion of a small sphere in fluid near a circular hole in a plane wall

The Stokes flow due to the motion of a small particle in arbitrary directions is investigated in the presence of a circular hole in an infinite thin plane wall separating a quiescent viscous fluid. The solutions of the boundary-value problem are obtained in closed forms to the point-force approximation in toroidal coordinates, by the use of the Green and Neumann functions supplemented by the edge function to remove the singularity at the rim of the hole. The volume flux through the hole and the force and torque experienced by the small spherical particle are determined on the basis of this solution. The case of linear motion parallel to the plane of the wall is discussed in detail.

Axisymmetric problem of vortex sound with solid surfaces

A general formulation of the axisymmetric vortex sound is given, and three typical problems are considered in detail, in which the acoustic wave is generated by a vortex ring interacting with either a sphere, a circular disk, or a circular aperture of a plane wall in an axisymmetric manner. The velocity potentials induced by the vortex in the presence of the body are determined by the method of dual integral equations or the image vortex. From the vortex trajectories, the acoustic wave fields are determined. The method of matched asymptotic expansions yields the result that the force exerted on the body is related to the profile of the dipole component. It is also found that the wave amplitude is expressed in terms of the value of a streamfunction at the vortex position, which represents a hypothetical potential flow around the body and is defined appropriately in each problem.

AXISYMMETRIC WAVES ON A VERTICAL VORTEX IN A STRATIFIED FLUID

Axisymmetric wave propagation along a vertical vortex core in a stably stratified fluid is considered theoretically. The fluid is assumed to be inviscid, incompressible, nondiffusive, and exponentially stratified. A linear analysis under the Boussinesq approximation shows that discrete inertial modes (bounded modes) are allowed in addition to continuous internal gravity waves (unbounded modes), when the stratification is not too strong. These inertial modes, whose eigenfunctions are confined to the vorticity region, disappear if the Brunt–Vaisala frequency N2 exceeds the maximum value of the Rayleigh function. Concrete results are given for the Burgers vortex. A weakly nonlinear analysis indicates that inertial modes (if permitted) are generated through the resonant interactions between two internal gravity waves. The time evolution of its amplitude is described by a cubic nonlinear Schrodinger equation, which admits envelope soliton solutions for shorter carrier waves only, viz., the soliton window has a...

On thermophoresis of relatively large aerosol particles suspended near a plate

Abstract Thermal creep flow of a gas around a spherical particle of an arbitrary thermal conductivity suspended near a plate is analyzed, neglecting shear velocity slip. The thermal force acting on the particle at an arbitrary distance from the plate is given as a function of ratio of the thermal conductivity of the particle to that of the gas. Effects of the collector plate on the drift velocity of a precipitating particle are clarified.

Diffusion equation coupled to Burgers' equation

Scalar diffusion in one-dimensional Burgers flow is considered. When the Prandtl number is unity, the diffusion equation with convective term is reduced to a simple diffusion equation by a generalized Cole-Hopf transformation. An exact solution of an initial value problem is obtained in a closed form. When the Prandtl number is arbitrary, a similar analytical treatment is possible for limited classes of Burgers flow (expansion wave and single shock). The statistics of scalar field are discussed briefly.

Three-dimensional distortions of a vortex filament: Exact solutions of the localized induction equation

The three-dimensional motion of a thin vortex filament in an inviscid incompressible fluid is investigated theoretically, on the basis of the localized induction equation(LIE). It is shown that the N-soliton solution, obtained through Hirotas bilinear method, does not exhibit clear phase-advance during head-on collisions as observed in the experiment by Maxworthy et al. In order to resolve this discrepancy an effect of axial flow within the vortex core is incorporated into the LIE and a new integrable equation is derived. The bilinear procedure as well as the soliton surface approach gives the N-soliton solution which is identical to that of the LIE except for the dispersion relation. Besides, this equation predicts that a certain class of helicoidal vortices with axial flow is neutrally stable against any small perturbations.