A popular model for a generic fat-cored vortex ring or eddy is Hill's spherical vortex (Phil. Trans. Roy. Soc. A vol. 185, 1894, p. 213). Here we find an exact solution for such a spherical vortex steadily propagating along the axis of a rotating ideal fluid. The spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines form in this outer fluid region and hence fluid is carried along with the spherical vortex. As the rotation rate is further increased, further concentric "sibling" vortex rings are formed, counter rotating with respect to the original spherical vortex.
In 1894 M.J.M. Hill published an article describing a spherical vortex moving through a stationary fluid. Using cylindrical coordinates and assuming the azimuthal velocity component zero, Hill found a simple solution that described this flow. A similar modern problem in the MHD framework was put forth in 1987 by A. A. Bobnev and in 1995 by R. Kaiser and D. Lortz who applied the setup to model a ball lighting. We present a much simpler derivation of Hill's spherical vortex using the Bragg-Hawthorne equation. In particular, by using the moving frame of reference, the Euler equations reduce to eq…
The present study performs a numerical simulation of the interference and heat transfer between hairpin vortices formed in wakes behind staggered hills in a laminar boundary layer. Hairpin vortices are periodically shed in the wake of a row of hills, causing interference between the hairpin vortices. As the spanwise distance between the hills decreases, interference increases and the hairpin vortices become strong. At that time, because the interference between the legs of the hairpin vortex and the Q2 ejection becomes strong, the head of each hairpin vortex rises sharply. When the hill spacin…
The Hill vortex is a three-dimensional vortex structure solution of the Euler equations. For small amplitude axisymmetric disturbances on the external surface from the linear stability analysis by \citet{moff78} emerged the formation of a tail. Successive studies confirmed this fact, and in addition observed a different shape of the tail with azimuthal small amplitude disturbances. In this paper the Navier-Stokes equations are solved at high values of the Reynolds number imposing large amplitude axisymmetric and three-dimensional disturbances on the surface of the vortex. The conclusion is tha…
We consider the linear stability of Hill's vortex with respect to axisymmetric perturbations. Given that Hill's vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a 3D axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretize…
The generation of initial or inflow synthetic turbulent velocity or scalar fields reproducing statistical characteristics of realistic turbulence is still a challenge. The synthetic eddy method, previously introduced in the context of inflow conditions for large eddy simulations, is based on the assumption that turbulence can be regarded as a superposition of coherent structures. In this paper, a new type of synthetic eddy method is proposed, where the fundamental eddy is constructed by superposing three Hill's vortices, with their axes orthogonal to each other. A distribution of Hill's vortic…
Until now it was not a success to identify a universal triggering mechanism for the formation of the observed cyclon-anticyclon vortex asymmetry phenomenon or the corresponding breaking of chiral vortex symmetry in the rotating medium. In this paper the new linear universal instability mechanism of breaking chiral vortex symmetry due to the super threshold medium rotation frequence in the presence of linear Eckman friction is established. A new non stationary solution to the problem of the disc oscillations in fluid in the connection with experiments on the rotating superfluid helium-II has be…
The paper presents a numerical investigation of the leading-edge vortices generated by rotating triangular wings at Reynolds number $Re=250$. A series of three-dimensional numerical simulations have been carried out using a Fourier pseudo-spectral method with volume penalization. The transition from stable attachment of the leading-edge vortex to periodic vortex shedding is explored, as a function of the wing aspect ratio and the angle of attack. It is found that, in a stable configuration, the spanwise flow in the recirculation bubble past the wing is due to the centrifugal force, incompressi…
We deal with the Hill's spherical vortex, which is an exact solution to the Euler equation, and manage the solution to satisfy the incompressible Navier-Stokes(INS) equations with a viscous term. Once we get a viscous solution to the INS equations, we will be able to analyze the flows with discontinuities in vorticity. In the same procedure, we also present a time developing exact solution to the INS equations, which has a rotation on the axis besides the Hill's vortex.
In this paper, we study a new type of large-scale instability in obliquely rotating stratified fluids with small scale non-helical turbulence. The small-scale turbulence is generated by the external force with zero helicity and low Reynolds number. The theory uses the method of multiscale asymptotic developments. The nonlinear equations for large scale motions are obtained in the third order of the perturbation theory. In this paper, we consider the linear instability and the stationary nonlinear modes. We obtain solutions in the form of nonlinear Beltrami waves and localized vortex structures…
In rotating Rayleigh-Bénard convection, columnar vortices advect horizontally in a stochastic manner. When the centrifugal buoyancy is present the vortices exhibit radial motions that can be explained through a Langevin-type stochastic model. Surprisingly, anomalous outward motion of cyclones is observed in a centrifugation-dominant flow regime, which is contrary to the well-known centrifugal effect. We interpret this phenomenon as a symmetry-breaking of both the population and vorticity magnitude of the vortices brought about by the centrifugal buoyancy. Consequently, the cyclones submit to t…
We present spatial-resolved measurements of the columnar vortex structures in rotating Rayleigh-Bénard convection. The scaled radial profiles of the azimuthal velocity $u_ϕ(r)$ and vertical vorticity $ω(r)$ of the vortices are analyzed and compared with the predictions of the asymptotic theory. The results reveal that the asymptotic theory predicts accurately $u_ϕ(r)$ and $ω(r)$ in the geostrophic convection regime, but extension of the theory in the weak rotation regime is needed to interpret the rotation-dependence of the experimental data. Our measurements of the mean velocity, vorticity of…
Using direct numerical simulations of rotating Rayleigh-Bénard convection, we explore the transitions between turbulent states from a 3D flow state towards a quasi-2D condensate known as the large-scale vortex (LSV). We vary the Rayleigh number $Ra$ as control parameter and study the system response (strength of the LSV) in terms of order parameters assessing the energetic content in the flow and the upscale energy flux. By sensitively probing the boundaries of the region of existence of the LSV in parameter space, we find discontinuous transitions and we identify the presence of a hysteresis …
Existence of a stationary mode for a Hamiltonian dynamic system of two point vortexes with different signs on different latitudes of a uniform rotating sphere complying with observed data is stated. It is shown that such mode realization is possible only in the case when the more intensive cyclonic vortex has greater latitude than that of the anticyclonic vortex. A criterion of exponential instability of the stationary vortex mode taken into account impact of the polar vortexes is obtained. Compliance of the theory to observed data and reanalysis for coupled quasi-stationary systems of cycloni…
We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is d…
Pattern forming with externally imposed symmetry is ubiquitous in nature but lightly studied.We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. We observe symmetric convection patterns in form of regular vortex lattice near the instability onset, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which can be well described by a Ginzburg-Landau-like model. With increasing buoyancy strength the eff…
Several recent studies have demonstrated how large-scale vortices may arise spontaneously in rotating planar convection. Here we examine the dynamo properties of such flows in rotating Boussinesq convection. For moderate values of the magnetic Reynolds number ($100 \lesssim Rm \lesssim 550$, with $Rm$ based on the box depth and the convective velocity), a large-scale (i.e. system-size) magnetic field is generated. The amplitude of the magnetic energy oscillates in time, nearly out of phase with the oscillating amplitude of the large-scale vortex. The large-scale vortex is disrupted once the ma…
