Vortex breakdown phenomena in the axial vortices is an important feature which occurs frequently in geophysical flows (tornadoes and hurricanes) and in engineering flows (flow past delta wings, Von-Kerman vortex dynamo) We analyze helicity for axisymmetric vortex breakdown and propose a simplified formulation For such cases, negative helicity is shown to conform to the vortex breakdown A model problem has been analyzed to verify the results The topology of the vortex breakdown is governed entirely by helicity density in the vertical plane Our proposed methodology may be regarded as the prototype for identifying and characterize the breakdowns/eye in more complicated large-scale flows such as tornadoes/hurricanes

Helicity in axisymmetric vortex breakdown

Bubble-type vortex breakdown in axial vortices is investigated numerically through a model problem of flow inside a cylinder with a top rotating lid, referred to as ‘Vogel–Escuider flow’. The parameters of the flow are Reynolds number (. We show that a correlation exists between planar helicity and the vortex breakdown bubble. We also show that the topology of the breakdown bubbles is described by the planar helicity. Using only this planar helicity, the entire breakdown bubble is reconstructed for axisymmetric as well as non-axisymmetric flows.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/632874B7C31893DF2C65797D1F48015A/S0022112020000439a.pdf/div-class-title-on-the-correlation-between-vortex-breakdown-bubble-and-planar-helicity-in-vogel-escudier-flow-div.pdf

On the correlation between vortex breakdown bubble and planar helicity in Vogel─Escudier flow

Axial vortices are ubiquitous in nature and in engineering, and the breakdown of these vortices can have far-reaching consequences. In the presence of a temperature gradient, topological behavior dramatically changes depending on the various flow parameters. Quantities such as heat transport, scalar mixing, and turbulent correlations are influenced immensely by the interplay between rotation and convection.

Heat transfer scaling in natural convection with shear due to rotation

This work describes an inviscid solution for a cyclonic flowfield evolving in a hemispherical chamber configuration. In this context, a vigorously swirling motion is triggered by a gaseous stream that is introduced tangentially to the inner circumference of the chamber's equatorial plane. The resulting updraft spirals around while sweeping the chamber wall, reverses direction while approaching the headend, and then tunnels itself out through the inner core portion of the chamber. Our analysis proceeds from the Bragg–Hawthorne formulation, which proves effective in the treatment of steady, incompressible, and axisymmetric motions. Then using appropriate boundary conditions, we are able to obtain a closed-form expression for the Stokes streamfunction in both spherical and cylindrical coordinates. Other properties of interest are subsequently deduced, and these include the principal velocities and pressure distributions, vorticity, swirl intensity, helicity density, crossflow velocity, and the location of the axial mantle; the latter separates the outer annular updraft from the inner, centralized downdraft. Due in large part to the overarching spherical curvature, the cyclonic motion also exhibits a polar mantle across which the flow becomes entirely radial inward. Along this lower and shorter spherical interface, the polar velocity vanishes while switching angular direction. Interestingly, both polar and axial mantles coincide in the exit plane where the ideal outlet size, prescribed by the mantle position, is found to be approximately 70.7% of the chamber radius. We thus recover the same mantle fraction of the cyclonic flow analog in a right-cylindrical chamber where an essentially complex-lamellar motion is established.

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A quasi complex-lamellar solution for a hemispherically bounded cyclonic flowfield

The effect of the mixed convection with anisotropic thermal diffusion on the bubble breakdown inside a cylindrical cavity with a rotating top and the stationary bottom is investigated in this article. The lattice Boltzmann multiple relaxation time axisymmetric method with the D2Q9 model is used. The three distribution functions, one for axial and radial components of the velocity field, second for an azimuthal component of the velocity field, and third for the temperature field, are used. The code is validated for the fluid flow inside a lid-driven cylindrical cavity and for hot rotating-top-lid cylindrical cavity. The results are compared with the benchmark data. The effect of mixed convection on the Bodewadt boundary layer thickness and the temperature boundary layer thickness is investigated. The simulations are perfermed for various Reynolds number (Re) from of 990 and 2494, the Richardson number (Ri in 0.01 and 1, and Rayleigh number (Ra from 9801 and 6.22 × 10 6). This study concludes that the Bodewadt boundary layer thickness ( δ B ) is ∝ to Ri and the temperature boundary layer thickness ( δ T c ) is ∝ to Ri for 0.01 ≤ Ri ≤ 0.1. Further, the δ B ≈ δ T c for isotropic thermal diffusion. For an anisotropic thermal diffusion, the δB remains constant and δ T c increases at Ri = 0.01 with an increase in the ratio of thermal diffusivity values (n). The bubble breakdown vanishes with an increase in Ri at a constant Re. The main application of this study is in the mixing/blending processes with convection inside cylindrical shaped reactors.

Axisymmetric lattice Boltzmann formulation for mixed convection with anisotropic thermal diffusion and associated bubble breakdown

In this paper, we investigate mixing in flows dominated by bubble-type vortex breakdown. Three-dimensional Navier–Stokes equations in cylindrical polar coordinates are solved to simulate flow in a cylindrical container. We have found that in steady regime of the flow, the vortex breakdown bubble is axisymmetric and apparent non-axisymmetric features observed in experiments are artifacts of imperfections in experimental set-ups. We also find that the heteroclinic manifold joining hyperbolic points of the vortex breakdown bubble is stable in the absence of any perturbation and no chaotic advection was found within vortex breakdown bubble. This makes the vortex breakdown bubble impermeable to outer fluid and hence, the vortex breakdown bubble inhibits mixing. We conclude that symmetry is a barrier to mixing.

https://iopscience.iop.org/article/10.1088/1402-4896/ab0097/pdf

Axisymmetric vortex breakdown: a barrier to mixing

The topology and the dynamics of Vogel–Escudier flow, which is the flow inside a circular cylinder with a top rotating lid, are presented in this paper. A three-dimensional direct numerical simulation of the Navier–Stokes equations in cylindrical coordinates is used to investigate the flow. Various combinations of Reynolds number and aspect ratio are studied and classified based on the flow topology. The flow is found to exhibit steady axisymmetric, unsteady axisymmetric, rotating azimuthal waves, and weak turbulence regimes. The perturbations found in the system are characteristically different for various flow regimes and are used for the classification of flow. The presence of several modes at high Reynolds number suggests a weak turbulence state, and a Taylor–Gortler type instability wave is found in the sidewall boundary layer.

Manjul Sharma

Papers

On the correlation between vortex breakdown bubble and planar helicity in Vogel─Escudier flow

Axisymmetric vortex breakdown: a barrier to mixing

Synopsis of Vogel–Escudier flow