The motion of a two-dimensional buoyant vortex patch, i.e. a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength by applying Euler's equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength is obtained as an integral equation. The equations are also examined in the limit of a patch of vanishing size or a buoyant point vortex.
Previous studies have found that the flickering of buoyant diffusion flames is associated with the periodic shedding of a toroidal vortex, which is formed under gravity-induced shearing at the flames. Moreover, numerous experimental investigations have reported the scaling relation, $f \propto D^{-1/2}$, where $f$ is the flickering frequency and $D$ is the diameter of the fuel inlet. However, the connection between the toroidal vortex mechanism and the scaling relation has not been clearly understood. The current study theoretically revisits this problem from the perspective of vortex dynamics…
A general formulation is presented for studying the motion of buoyant vortices in a homogeneous ambient fluid. It extends the well-known Hamiltonian framework for interacting homogeneous point vortices to include buoyancy effects acting on the vortices. This is then used to systematically examine the buoyant 1-, 2-, and 3-vortex problems. In doing so we find that 2 buoyant vortices may either evolve as a pair in bounded circular orbits, or as two independent unbounded vortices that drift apart, and a criteria is found to distinguish these cases. Special attention is given to the buoyant vortex…
In this fluid dynamics video we study the dynamics of miscible vortex rings falling in ambient strongly (near two-layer) stratified fluid. Experiments and direct numerical simulations using the variable density Navier-Stokes (VARDEN) solver are presented and compared. Critical phenomena are identified depending upon the key parameters of the experiment (fluid and ring densities, upper layer vortex travel distance, etc) in which the descending dense vortex ring may experience complete trapping, partial trapping, or fissioning into a cascade of smaller vortices. The interaction of the vortex rin…
Anti-phase and in-phase flickering modes of dual buoyant diffusion flames were numerically investigated and theoretically analyzed in this study. Inspired by the flickering mechanism of a single buoyant diffusion flame, for which the deformation, stretching, or even pinch-off of the flame surface result from the formation and evolution of the toroidal vortices, we attempted to understand the anti-phase and in-phase flickering of dual buoyant diffusion flames from the perspective of vortex dynamics. The interaction between the inner-side shear layers of the two flames was identified to be respo…
We present experiments on the instability and fragmentation of volumes of heavier liquid released into lighter immiscible liquids. We focus on the regime defined by small Ohnesorge numbers, density ratios of order one, and variable Weber numbers. The observed stages in the fragmentation process include deformation of the released fluid by either Rayleigh-Taylor instability or vortex ring roll-up and destabilization, formation of filamentary structures, capillary instability, and drop formation. At low and intermediate Weber numbers, a wide variety of fragmentation regimes is identified. Those …
Small-scale flickering buoyant diffusion flames in externally swirling flows were computationally investigated with a particular interest in identifying and characterizing various distinct dynamical behaviors of the flame oscillators under different swirling flow conditions. By varying the external swirl, six distinct flame dynamical modes, such as the flickering flame, the oscillating flame, the steady flame, the lifted flame, the spiral flame, and the flame with a vortex bubble, were computationally identified in both physical and phase spaces and analyzed from the perspective of vortex dyna…
By introducing an equivalence between magnetostatics and the equations governing buoyant motion, we derive analytical expressions for the acceleration of isolated density anomalies, a.k.a. thermals. In particular, we investigate buoyant acceleration, defined as the sum of the Archimedean buoyancy $B$ and an associated perturbation pressure gradient. For the case of a uniform spherical thermal, the anomaly fluid accelerates at $2B/3$, extending the textbook result for the induced mass of a solid sphere to the case of a fluid sphere. For a more general ellipsoidal thermal, we show that the buoya…
Buoyant plumes, evolving free of boundary constraints, may develop well-defined mushroom shaped heads. In normal plumes, overturning flow in the head entrains less buoyant fluid from the surroundings as the head rises, robbing the plume of its driving force. We consider here a new type of plume in which the source of buoyancy is an autocatalytic chemical reaction. The reaction occurs at a sharp front which separates reactants from less dense products. In this type of plume, entrainment assists the reaction, producing new buoyancy which fuels an accelerating plume head. When the head has grown …
The deformation of two-dimensional vortex patches in the vicinity of fluid boundaries is investigated. The presence of a boundary causes an initially circular patch of uniform vorticity to deform. Sufficiently far away from the boundary, the deformed shape is well approximated by an ellipse. This leading order elliptical deformation is investigated via the elliptic moment model of Melander, Zabusky & Styczek [M. V. Melander, N. J. Zabusky & A. S. Styczek, J. Fluid. Mech., 167, 95 (1986)]. When the boundary is straight, the centre of the elliptic patch remains at a constant distance from the bo…
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.
We perform direct numerical simulations of rotating Rayleigh--Bénard convection of fluids with low ($Pr=0.1$) and high ($Pr=5$) Prandtl numbers in a horizontally periodic layer with no-slip top and bottom boundaries. At both Prandtl numbers, we demonstrate the presence of an upscale transfer of kinetic energy that leads to the development of domain-filling vortical structures. Sufficiently strong buoyant forcing and rotation foster the quasi-two-dimensional turbulent state of the flow, despite the formation of plume-like vertical disturbances promoted by so-called Ekman pumping from the viscou…
We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash-Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case, some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations adv…
Buoyant, finite-size or inertial particle motion is fundamentally unlike neutrally buoyant, infinitesimally small or Lagrangian particle motion. The de-jure fluid mechanics framework for the description of inertial particle dynamics is provided by the Maxey-Riley equation. Derived from first principles - a result of over a century of research since the pioneering work by Sir George Stokes - the Maxey-Riley equation is a Newton-type-law with several forces including (mainly) flow, added mass, shear-induced lift, and drag forces. In this paper we present an overview of recent efforts to port the…
In this paper we examine the role of weak magnetic fields in breaking Kelvin's circulation theorem and in vortex breakup in two-dimensional magnetohydrodynamics for the physically important case of a low magnetic Prandtl number (low $Pm$) fluid. We consider three canonical inviscid solutions for the purely hydrodynamical problem, namely a Gaussian vortex, a circular vortex patch and an elliptical vortex patch. We examine how magnetic fields lead to an initial loss of circulation $Γ$ and attempt to derive scaling laws for the loss of circulation as a function of field strength and diffusion as …
We show that a vortex matter, that is a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with sign-like eddies, exhibits (i) a boundary layer of vorticity (vorticity layer), and (ii) a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if the vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation exhibiting solitons with a quantized total vorticit…
The study of vortex dynamics using a variational formulation has an extensive history and a rich literature. The standard Hamiltonian function that describes the dynamics of interacting point vortices of constant strength is the Kirchhoff-Routh (KR) function. This function was not obtained from basic definitions of classical mechanics (i.e., in terms of kinetic and potential energies), but it was rather devised to match the already known differential equations of motion for constant-strength point vortices given by the Bio-Savart law. Instead, we develop a new variational formulation for vorte…
In the spirit of Prandtl's conjecture of 1926, for turbulence at high Reynolds number we present an analogy with the kinetic theory of gases, with dipoles made of quasi-rigid and 'dressed' vortex tubes as frictionless, incompressible but deformable quasi-particles. Their movements are governed by Helmholtz' elementary vortex rules applied locally. A contact interaction or 'collision' leads either to random scatter of a trajectory or to the formation of two likewise rotating, fundamentally unstable whirls forming a dissipative patch slowly rotating around its center of mass which is almost at r…
In this paper, we highlight the importance of the boundary effects on the construction of quasi-periodic vortex patches solutions close to Rankine vortices and whose existence is not known in the whole space due to the resonances of the linear frequencies. Availing of the lack of invariance by radial dilation of Euler equations in the unit disc and using a Nash-Moser implicit function iterative scheme we show the existence of such structures when the radius of the Rankine vortex belongs to a suitable massive Cantor-like set with almost full Lebesgue measure.
