A model for confined vortex rings with elliptical-core vorticity distribution

Abstract

We present a new model for an axisymmetric vortex ring confined in a tube. The model takes into account the elliptical (elongated) shape of the vortex ring core and thus extends our previous model [Danaila, Kaplanski and Sazhin, J. of Fluid Mechanics, 774, 2015] derived for vortex rings with quasi-circular cores. The new model offers a more accurate description of the deformation of the vortex ring core, induced by the lateral wall, and a better approximation of the translational velocity of the vortex ring, compared with the previous model. The main ingredients of the model are the following: the description of the vorticity distribution in the vortex ring is based on the previous model of unconfined elliptical-core vortex rings [Kaplanski, Fukumoto and Rudi, Physics of Fluids, 24, 2012]; Brasseur's approach [Brasseur, PhD Thesis, 1979] is then applied to derive a wall-induced correction for the Stokes stream function of the confined vortex ring flow. We derive closed formulae for the flow stream function and vorticity distribution. An asymptotic expression for the long time evolution of the drift velocity of the vortex ring as a function of the ellipticity parameter is also derived. The predictions of the model are shown to be in agreement with direct numerical simulations of confined vortex rings generated by a piston-cylinder mechanism. The predictions of the model support the recently suggested heuristic relation [Krieg and Mohseni, J. Fluid Engineering, 135, 2013] between the energy and circulation of vortex rings with converging radial velocity. A new procedure for fitting experimental and numerical data with the predictions of the model is described. This opens the way for applying the model to realistic confined vortex rings in various applications including those in internal combustion engines.

Formation number of confined vortex rings

“…Threshold values have ranged from 2% and 5% in numerical studies [3,9,38,44], to approximately 10% in experiments [45]. In our previous studies [29][30][31], the value ω cut = 5% proved optimal for identifying the vortex ring in the post-formation phase. In the present study, we defined the vorticity patch Ω 0.05 = {(r, x) | ω(r, x)/ω max ≥ 0.05} and followed its evolution during both formation and post-formation phases.…”

“…In the present work, we use the model recently suggested by Danaila et al [30] to describe confined vortex rings with elliptically shaped vortex cores. We anticipate that taking into account the ellipticity of the core of the vortex ring will lead to a more accurate prediction of the formation number.…”

“…To further improve the VRC model for the confined vortex ring, Danaila et al [30] took into account the elliptical (elongated) shape of the vortex ring core. This model is hereafter referred to as the VRE (for vortex rings with elliptical core).…”

“…Note that the vorticity distribution (18) was derived by imposing weak perturbations on the first-order solution (13), in order to find an approximation of the non-linear solution to the Navier-Stokes equations for high Reynolds numbers. Using the vorticity function (18) within Brasseur's approach to derive the stream function correction due to the tube wall involved some technical analytical developments, presented in detail in [30]. Closed formulae were obtained for the two terms in (9).…”

“…For example, the vorticity distribution in the vortex core is assumed to be linear, i. e. proportional to the distance to the axis of symmetry (as in Hill's spherical vortex model), which is far from the quasi-Gaussian distribution observed for unconfined [27,28] or confined vortex rings [20,29,30]. The model recently suggested by Danaila et al [30] to describe confined and viscous vortex rings with an elliptical vortex core is used in this study. It will be shown that taking into account the ellipticity of the vortex ring core and the influence of the lateral wall leads to a more accurate prediction of the formation number.…”