The evolution of a viscous vortex ring from thin to thick-cored form is considered using an improved asymptotic solution, which is obtained after impressing a spatially uniform drift on the first-order solution of the Navier–Stokes equations. The obtained class of rings can be considered as the viscous analog solution to the Norbury vortices and classified in terms of the ratio of their initial outer radius to the core radius. The model agrees with the reported theoretical and experimental results referring to the post-formation and the formation stages. By using the matching procedure suggested earlier and the obtained properties of the viscous vortex ring, it is found that when the length-to-diameter aspect ratio L∕D reaches the limiting value 4.0 (“formation number”), the appropriate values of the normalized energy and circulation become around 0.3 and 2.0, respectively. An approach that enables to predict the “formation number” is proposed.

It is known that the cross section of the vortex ring core takes an approximately elliptical shape with increasing Reynolds number. In order to model this feature, the functional form of a vortex ring solution of the Stokes equations is modified so as to be able to model higher Reynolds number rings. The model introduces two nondimensional parameters that govern the shape of the vortex core:λ ≥ 1 and β ≥ 1. Based on this modification, new expressions for the translation velocity, energy, circulation, and streamfunction are derived for a wide range of section ellipticity that are specific to such vortices. To validate the model, the data adapted from the numerical study of vortex ring at Reynolds number Re = 1400 performed by Danaila and Helie [Phys. Fluids 20, 073602 (2008)], is used. In this case, the appropriate values of λ and β are calculated by equating the normalized energy E d and circulation d of the theoretical vortex to the corresponding values obtained from the numerical data. The model provides a good prediction of the ring velocity evolution at high Reynolds numbers. C 2012 American Institute of Physics. [http://dx.

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A three-dimensional model of particulate flows using the Reynolds Averaged NavierStokes method is presented. The governing equations of the gas-solids flow are supplemented with appropriate closure equations to take into account all the relevant forces exerted on the solid particles, such as particle-turbulence interactions, turbulence modulation, particle-particle interactions, particle-wall interactions, as well as gravitational, viscous drag, and lift forces. A finite volume numerical technique was implemented for the numerical solution of the problem. The method has been validated by comparing its results with the limited number of available experimental data for the velocity and turbulence intensity of the gas-particle flow. The results show that the presence of particles in the flow has a significant effect on all the flow variables. Most notably, the distribution of all the parameters becomes asymmetric, because of the gravitational effect on the particles and particle sedimentation.