We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman ( Phil. Trans. R. Soc. Lond. A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.
Axisymmetric contour dynamics for buoyant vortex rings
The present work uses a reducedorder model to study the motion of a buoyant vortex ring with nonnegligible core size. Buoyancy is considered in both nonBoussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integrodifferential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shocklike profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.
 Award ID(s):
 1706934
 Publication Date:
 NSFPAR ID:
 10189440
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 887
 ISSN:
 00221120
 Sponsoring Org:
 National Science Foundation
More Like this


This paper investigates the steady axisymmetric structure of the cold boundarylayer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The inviscid swirling motion found outside the boundary layer, driven by the entrainment of the buoyant turbulent plume of hot combustion products that develops above the fire, is described by an irrotational solution, obtained by combining Taylor's selfsimilar solution for the motion in the axial plane with the azimuthal motion induced by a line vortex of circulation $2 {\rm \pi}\Gamma$ . The development of the boundary layer from a prescribed radial location is determined by numerical integration for different swirl levels, measured by the value of the radialtoazimuthal velocity ratio $\sigma$ at the initial radial location. As in the case $\sigma =0$ , treated in the seminal boundarylayer analysis of Burggraf et al. ( Phys. Fluids , vol. 14, 1971, pp. 1821–1833), the pressure gradient associated with the centripetal acceleration of the inviscid flow is seen to generate a pronounced radial inflow. Specific attention is given to the terminal shape of the boundarylayer velocity near the axis, which displays a threelayered structure that is described by matched asymptotic expansions. The resulting composite expansion, dependent onmore »

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thinlayer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The threecurve equations of contour dynamics that fully describe the doublelayer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initialvalue evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the singlelayer equation of Moore ( Stud. Appl. Math. , vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the firstorder equations to smallamplitude perturbations shows Kelvin–Helmholtz instability when the farfield fluid velocities on either side of the double layer are unequal. Equal velocities define a circulationfree double vorticity layer,more »

In this work, an exact inviscid solution is developed for the incompressible Euler equations in the context of a bidirectional, cyclonic flowfield in a rightcylindrical chamber with a hollow core. The presence of a hollow core confines the flow domain to an annular swirling region that extends into a toroid in threedimensional space. The procedure that we follow is based on the Bragg–Hawthorne framework and a judicious assortment of boundary conditions that correspond to a wallbounded cyclonic motion with a cylindrical core. At the outset, a selfsimilar stream function is obtained directly from the Bragg–Hawthorne equation under the premises of steady, axisymmetric, and inviscid conditions. The resulting formulation enables us to describe the bidirectional evolution of the socalled inner and outer vortex motions, including their fundamental properties, such as the interfacial layer known as the mantle; it also unravels compact analytical expressions for the velocity, pressure, and vorticity fields, with particular attention being devoted to their peak values and spatial excursions that accompany successive expansions of the core radius. By way of confirmation, it is shown that removal of the hollow core restores the wellestablished solution for a fully flowing cylindrical cyclone. Immediate applications of cyclonic flows include liquid andmore »

Downburst events initialized at various hours during the evening transition (ET) period are simulated to determine the effects of ambient stability on the outflow of downburst winds. The simulations are performed using a pseudospectral large eddy simulation model at high resolution to capture both the largescale flow and turbulence characteristics of downburst winds. First, a simulation of the ET is performed to generate realistic initial and boundary conditions for the subsequent downburst simulations. At each hour in the ET, an ensemble of downburst simulations is initialized separately from the ET simulation in which an elevated cooling source within the model domain generates negatively buoyant air to mimic downburst formation.
The simulations show that while the stability regime changes, the ensemble mean of the peak wind speed remains fairly constant (between 35 and 38 m s^{−1}) and occurs at the lowest model level for each simulation. However, there is a slight increase in intensity and decrease in the spread of the maximum outflow winds as stability increases as well as an increase in the duration over which these strongest winds persist. This appears to be due to the enhanced maintenance of the ring vortex that results from the lowlevel temperature inversion, increased ambientmore »