Journal of Fluids

Volume 2015 (2015), Article ID 405696, 15 pages

http://dx.doi.org/10.1155/2015/405696

## Separation Criteria for Off-Axis Binary Drop Collisions

^{1}United Technologies Research Center, 411 Silver Lane, MS 129-19, East Hartford, CT 06108, USA^{2}Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada M5S 3G8

Received 15 December 2014; Accepted 14 April 2015

Academic Editor: Robert Spall

Copyright © 2015 Mary D. Saroka and Nasser Ashgriz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Off-axis collisions of two equal size droplets are investigated numerically. Various governing processes in such collisions are discussed. Several commonly used theoretical models that predict the onset of separation after collision are evaluated based on the processes observed numerically. A separation criterion based on droplet deformation is found. The numerical results are used to assess the validity of some commonly used phenomenological models for drop separation after collision. Also, a critical Weber number for the droplet separation after grazing collision is reported. The effect of Reynolds number is investigated and regions of permanent coalescence and separation are plotted in a Weber-Reynolds number plane for high impact parameter collisions.

#### 1. Introduction

Many engineering applications and natural process involve drop collisions in which the final outcome has a direct impact on the eventual success or failure of the application or process. In sprays, for example, the behavior is characterized by many small liquid drops with a particular size distribution and with each individual drop moving along a particular trajectory with a prescribed velocity. By singling out and focusing on just two drops within the spray, one can readily see that the interaction between these drops plays a part in the overall evolutional characteristics of the spray. One can further imagine that the likelihood of two drops colliding along the same axis of trajectory is less likely to occur than the situation in which the two drops collide along different trajectories. It is the latter case, typically referred to as off-axis collisions, that is the subject of this investigation.

Over the years, many researchers have studied the collision dynamics of the binary drops. The early studies on drop collision were motivated by understanding of the physical processes that occur during rain fall [1–12]. Rain drops may collide with each other and break into smaller drops or coalesce and generate larger drops. For instance, Park [5] did experiments on water drops in humid environment to determine conditions for the drop separation after collision. Several phenomenological models for the drop separation after collision are provided, for instance, the rotational energy model [7] and the variation principal model for the minimum potential energy for the stretching separation [12]. The later studies were motivated by drop collision in various spray systems [13–28]. Collisions of many other liquids, such as hydrocarbons [13–18], heavy oils [19, 20], and mercury [21], in addition to water, have also been studied. The bulk of quantitative information on the drop collision is obtained through experiments [21–28], while numerical studies are used to provide detailed understanding of this complex process. The front tracking techniques [29–32] and a combination of level set and VOF methods are used to solve the dynamics of the free surfaces [33–40]. A detailed review of the droplet collision process is provided by Brenn [41]. Review of this body of research indicates that, depending upon the initial energy of the two drops and the angle at which they collide, different types of outcomes will occur. These outcomes can be categorized as bouncing, partial coalescence, permanent coalescence, separation, or shattering. There are also several subcategories for each outcome. Of these five, permanent coalescence and separation are the most frequently observed outcomes, while the other three, namely, bouncing, partial coalescence, and shattering, represent special cases in which the drops have either very small velocities, large size differences, or very high velocities.

The present study is on the off-axis collision of two equal size droplets. This study is a continuation of a previous study by the authors on the head-on collision of two drops [40]. Droplet collisions in dynamically inert surroundings are considered. Therefore, bouncing collisions, which are due to air entrapment between the two approaching droplets, are not observed. In the next sections, first the problem statement and the numerical methods are described, followed by the results and analysis.

#### 2. Problem Statement

A general schematic depicting the positions and velocity vectors of two drops just prior to contact is shown in Figure 1. Suppose that one of the drops has a velocity defined as , while the other has a velocity of . These individual velocities are used to define a relative velocity, , of the interacting drop. This is done by combining the drop velocities and their impact angle, , through the following relation: . Another important parameter necessary to define a particular collision involves the distance between the drops’ centers. This distance is referred to as the impact parameter, noted as . As the impact parameter increases the amount of “interaction” is reduced, while the converse is true. Using this information, it is convenient to define various dimensionless numbers to represent the phenomena under consideration. Three dimensionless parameters, namely, a Weber number, a nondimensional impact parameter, and a drop diameter ratio, are defined based on the drop diameter, liquid density, , and the coefficient of surface tension, : whereIn addition, Reynolds and Ohnesorge numbers are used to correlate effect of viscosity, , and its relationship with surface tension:The Ohnesorge number is unique in the fact that it is only a function of fluid properties and the drop size, while both the Weber and Reynolds numbers are functions of the fluid properties, drop geometry, and impact velocity. In the present study, We, Re, and Oh numbers are calculated based on the liquid properties.