Quadratic formula for determining the drop size in pressure-atomized sprays with and without swirl

Abstract

We use a theoretical framework based on the integral form of the conservation equations, along with a heuristic model of the viscous dissipation, to find a closed-form solution to the liquid atomization problem. The energy balance for the spray renders to a quadratic formula for the drop size as a function, primarily of the liquid velocity. The Sauter mean diameter found using the quadratic formula shows good agreements and physical trends, when compared with experimental observations. This approach is shown to be applicable toward specifying initial drop size in computational fluid dynamics of spray flows.

The Reynolds stress in turbulence from a Lagrangian perspective

“…However, in the Lagrangian perspective, u′v′ is simply a forward transport of v′ momentum, and as such subject to the same Newton's second law. A shift in the perspective, in this instance and in [18], leads to a great simplification, due to cancellation of the mean velocity with that of the control volume (figure 2). Implications of this work are that the Reynolds stress can be written explicitly in terms of basic turbulence parameters so that it furnishes the u′v′ term in the Reynolds-averaged Navier-Stokes (RANS) equation.…”

“…Thus, a conversion of the perspective from Eulerian to Lagrangian transforms the turbulence momentum transport to a simple, intuitive form. At times, a change in perspective (Eulearian to Lagrangian in this case, and from differential to integral analysis in [18]) can lead to unexpected solutions to a seemingly complex fluid mechanics problem [18].…”

Determination of the Drop Size During Atomization of Liquid Jets in Cross Flows

“…When combined with the conservation of mass, this method leads to a cubic formula for the drop size as a function of the injection parameters. This framework has worked quite well in predicting the drop size in pressure-atomized sprays with and without swirl, as a function of all the relevant injection parameters and liquid/gas properties [18,[20][21][22]. Some of the resulting expressions could also be cast in terms of Reynolds and Weber numbers, after some proper normalization [22].…”

“…Recently, we developed a theoretical framework using integral form of the conservation equations which led to a closed-form solution for the drop size in pressure-atomized sprays with and without swirl [18]. As will be described below, this method does not involve any ad-hoc assumptions or non-physical descriptions of the atomization process, and thus represents a new theoretical formulation for the spray atomization in cross flows.…”

“…Therefore, as the title of that work ("Energy considerations in twin-fluid atomization [19]") indicates, it is a zero-dimensional energy consideration, not a full energy balance in a control volume setting. The current formulation is a complete set of mass, energy (and momentum), where the momentum balance has also been used in some of our previous work for a complete solution of the spray atomization problem [18,[20][21][22]. Current approach by-passes the complex physics of the atomization process by enveloping the spray in a control volume, where the incoming and outgoing energy terms are balanced according to the conservation of energy.…”