Toroidal Vortices of Energy in Tightly Focused Second-Order Cylindrical Vector Beams

In this paper, we simulate the focusing of a cylindrical vector beam (CVB) of second order, using the Richards–Wolf formula. Many papers have been published on focusing CVB, but they did not report on forming of the toroidal vortices of energy (TVE) near the focus. TVE are fluxes of light energy in longitudinal planes along closed paths around some critical points at which the flux of energy is zero. In the 3D case, such longitudinal energy fluxes form a toroidal surface, and the critical points around which the energy rotates form a circle lying in the transverse plane. TVE are formed in pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). We show that when light with a wavelength of 532 nm is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 μm (0.85 λ) from the axis (with a period along the z-axis of 0.8 μm (1.5 λ)). The vortices arise in pairs: the vortex nearest to the focal plane is twisted clockwise, and the next vortex is twisted counterclockwise. These vortices are accompanied by saddle points. At higher distances from the z-axis, this pattern of toroidal vortices is repeated, and at a distance of about 0.7 μm (1.3 λ), a region in which toroidal vortices are repeated along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, these toroidal vortices are not observed.

Compressible Two-Phase Viscous Flow Investigations of Cavitation Dynamics for the ITTC Standard Cavitator

In this paper, the ITTC Standard Cavitator is numerically investigated in a cavitation tunnel. Simulations at different cavitation numbers are compared against experiments conducted in the cavitation tunnel of SVA Potsdam. The focus is placed on the numerical prediction of sheet-cavitation dynamics and the analysis of transient phenomena. A compressible two-phase flow model is used for the flow solution, and two turbulence closures are employed: a two-equation unsteady RANS model, and a hybrid RANS/LES model. A homogeneous mixture model is used for the two phases. Detailed analysis of the cavitation shedding mechanism confirms that the dynamics of the sheet cavitation are dictated by the re-entrant jet. The break-off cycle is relatively periodic in both investigated cases with approximately constant shedding frequency. The CFD predicted sheet-cavitation shedding frequencies can be observed also in the acoustic measurements. The Strouhal numbers lie within the usual ranges reported in the literature for sheet-cavitation shedding. We furthermore demonstrate that the vortical flow structures can in certain cases develop striking cavitating toroidal vortices, as well as pressure wave fronts associated with a cavity cloud collapse event. To our knowledge, our numerical analyses are the first reported for the ITTC standard cavitator.

Toroidal polarization vortices in tightly focused beams with singularity

In this paper, we numerically investigated tight focusing of cylindrical vector beams of the second order using Richards-Wolf formulae. It was shown that intensity rings where the Poynting vector was equal to zero appeared not only in the focal plane but also in nearby planes. For example, a lens with numerical aperture NA=0.95 was shown to generate periodical toroidal vortices with a 0.8-m period along the z-axis at a distance of about 0.45 m from the axis. The vortices were generated pairwise, with the closest-to-focus vortex having clockwise helicity and the subsequent being anticlockwise. The vortices were also characterized by saddle points. When focusing an optical beam passed through a narrow annular aperture, no toroidal vortices were observed.

Quasi-Stationary Turbulent Flow in Cylinder Channel with Irregular Walls

A simple asymptotic model of a stationary rotationally symmetric flow in an infinite cylindrical channel with irregular walls is constructed and investigated. We assume that kinematic viscosity is depends on the radial and axial coordinates. The kinematic condition corresponding to the impermeability of the boundary for the liquid is chosen. It is assumed that the classical non-slip condition is completed only in straight segments of the boundary. For parts of the boundary that are not straight, any additional terms, in addition to the kinematic conditions, not required. It is shown that at any fluid flow rate in domains where the curvature of the cylinder boundary is negative, there are toroidal vortices in the stationary flow. The solution of the problem is constructed in analytical form. This solution is valid in any domain with sufficiently smooth cylinder boundaries. As an example, we consider the case of turbulent kinematic viscosity vanishing at the cylinder boundary. To prevent the velocity singularities the roughness of the wall is introduced. The proposed model can simulate the blood flow through the vessels and, in particular, be used to study the quasi-stationary motion of impurities, for example, erythrocytes, in the flow with the known structure.

E. coli Inactivation Kinetics Modeling in a Taylor-Couette UV Disinfection Reactor

A simple model was developed to predict the survival behavior of E. coli subjected to UV disinfection in a Taylor-Couette reactor. The model includes the CFD evaluation of the counterrotating toroidal vortices developed within the annular space of two coaxial cylinders. The UV lamp was located within the diameter of the internal rotating cylinder. The residence time of the bacteria near the UV lamp is, therefore, a function of both the size of the vortex and its angular velocity. The effect of angular velocity on the formation of counterrotating toroidal vortices and their impact on the kinetics of UV microbial inactivation was experimentally evaluated. The kinetics of microbial inactivation follow an apparent first-order kinetic equation between 300 and 2000 revolutions per minute. Therefore, in this range of angular velocities, a set of k values (indirectly taking into account the hydrodynamic pattern and UV irradiance) was obtained for a given concentration of bacteria. Then, the set of k values was correlated with the range of angular velocities applied using the polynomial equation. A k value can be obtained for an unknown angular velocity through the polynomial equation. Therefore, a simulation curve of microbial inactivation can be obtained from the first-order kinetic equation. The efficiency of bacteria removal improves depending on the angular velocity applied. A good agreement is observed between the simulation of the survival behavior of the microorganisms subjected to UV disinfection with the experimental data.

The stability and nonlinear evolution of quasi-geostrophic toroidal vortices

We investigate the linear stability and nonlinear evolution of a three-dimensional toroidal vortex of uniform potential vorticity under the quasi-geostrophic approximation. The torus can undergo a primary instability leading to the formation of a circular array of vortices, whose radius is approximately the same as the major radius of the torus. This occurs for azimuthal instability mode numbers $m\geqslant 3$, on sufficiently thin tori. The number of vortices corresponds to the azimuthal mode number of the most unstable mode growing on the torus. This value of $m$ depends on the ratio of the torus’ major radius to its minor radius, with thin tori favouring high mode $m$ values. The resulting array is stable when $m=4$ and $m=5$ and unstable when $m=3$ and $m\geqslant 6$. When $m=3$ the array has barely formed before it collapses towards its centre with the ejection of filamentary debris. When $m=6$ the vortices exhibit oscillatory staggering, and when $m\geqslant 7$ they exhibit irregular staggering followed by substantial vortex migration, e.g. of one vortex to the centre when $m=7$. We also investigate the effect of an additional vortex located at the centre of the torus. This vortex alters the stability properties of the torus as well as the stability properties of the circular vortex array formed from the primary toroidal instability. We show that a like-signed central vortex may stabilise a circular $m$-vortex array with $m\geqslant 6$.

A vortex-dynamical scaling theory for flickering buoyant diffusion flames

The flickering of buoyant diffusion flames is associated with the periodic shedding of toroidal vortices that are formed under gravity-induced shearing at the flame surface. Numerous experimental investigations have confirmed the scaling,$f\propto D^{-1/2}$, where$f$is the flickering frequency and$D$is the diameter of the fuel inlet. However, the connection between the toroidal vortex dynamics and the scaling has not been clearly understood. By incorporating the finding of Gharibet al.(J. Fluid Mech., vol. 360, 1998, pp. 121–140) that the detachment of a continuously growing vortex ring is inevitable and can be dictated by a universal constant that is essentially a non-dimensional circulation of the vortex, we theoretically established the connection between the periodicity of the toroidal vortices and the flickering of a buoyant diffusion flame with small Froude number. The scaling theory for flickering frequency was validated by the existing experimental data of pool flames and jet diffusion flames.