An experimental and computational study of bouncing and deformation in droplet collision

An experimental and computational investigation of the collision of equal-sized liquid droplets was conducted. The Navier-Stokes equations for the fluid motion both inside and outside the droplets were solved using the numerical method of "front tracking". The calculated history of the droplet collision process was found to agree well with the experimental results. The configuration of the gas gap between the droplet surfaces was analyzed in order to understand the mechanism which controls droplet bouncing. The minimum gap thickness was found to exhibit a non-monotonic dependence on the droplet kinetic energy, thereby explaining the non-monotonic transition between the collision regimes of coalescence and bouncing. Both computational and experimental results further showed that the droplet collision time is close to its natural oscillation period. Recognizing the large deformation of the droplet surface during collision, large amplitude droplet oscillation with viscous dissipation was also numerically studied. Introduction The dynamics of binary droplet collision is of interest to a variety of natural and technological problems including, for example, raindrop formation, nuclear fusion, as well as various spraying processes such as the spray painting, insecticide spraying, and spray combustion within liquidfueled combustors. A comprehensive understanding of the global phenomenology of droplet collision has emerged based on studies using water droplets (see, for example, * Graduate Student ** Professor *** Robert H. Goddard Professor, Fellow AIAA Copyright @ 1996 by C. K. Law. Published by the American Institute of Aeronautics and Astronautics, Inc, with permission. Adam & Linblad 1968; Park 1970; Brazier , Jennings & Latham 1972; Ashgriz & Poo 1990) and more recent studies using hydrocarbon droplets (Jiang, Umemura & Law 1992; Qian & Law 1996). Specifically, the collision outcome for a given liquid-gas system can be classified into five distinct regimes (figure 1) according to the collision Weber number (We) and collision impact parameter (B), which are respectively defined for identical droplets as 2Rp,U/o and X/(2R), with R being the droplet radius, U the relative velocity, x trie projection of the separation distance between the droplet centers in the direction normal to that of U, and p, and a the density and the surface tension of the liquid respectively. Thus We is the initial kinetic energy normalized by the droplet surface tension energy, while B=0 and 1 respectively designate head-on and grazing collisions. The five regimes of collision outcome are respectively characterized by: (I) coalescence after minor droplet deformation, (II) bouncing, (HI) coalescence after substantial droplet deformation, (IV) coalescence followed by separation for near head-on collisions, and (V) coalescence followed by separation for off-center collisions. Time-resolved photographic images of the collision sequence in these five regimes are shown in Jiang, Umemura & Law (1992) and Qian & Law (1996). A particularly interesting aspect of the collision outcome (Qian & Law 1996) is the observed sensitive dependence of the boundary between bouncing and coalescence on the density of the gaseous medium. Increasing density by increasing either the pressure or the molecular weight of the gas medium promotes bouncing and widens Regime II, while the opposite holds for reduced density. In particular, by progressively reducing the environment pressure, Regime II first vanishes for B=0 and eventually for all values of B. As such, while Regime II does not exist for the head-on collision of water droplets at one atmosphere pressure, it emerges as the system pressure is gradually increased to 2.7 atmospheres. It was suggested that the propensity for bouncing or merging is a consequence of the readiness with which the gaseous mass in the inter-droplet gap can be squeezed out of the gap by the colliding masses in order for the droplet surfaces to make contact. Since in most instances the collision event involves large deformation, theoretical studies of the collision response have been mainly phenomenological in nature, for example through algebraic balances of force and energy. Advances, however, have been recently made on the numerical simulation of droplet collision. Specifically, adopting the front tracking method of Unverdi (1990) and Nobari & Tryggvason (1994), which treats the multi-phase flow field as a single domain while delineating the different phases by tracking the interface movement, Nobari, Jan & Tryggvason (1995) simulated the head-on collision event and gave a detailed description of the evolution of the droplet surface geometry as well as the pressure and velocity distribution in the entire flow field. Recognizing the advantages offered by the numerical approach, especially those associated with large-scale deformation and the extremely small dimension of the inter-droplet spacing, which respectively are not readily amenable to analytical and experimental investigations, the present study was initiated with the overall objective of coupling the experimental and numerical results to gain further understanding of the droplet collision process. Three specific issues are addressed herein. First, we have performed a quantitative comparison between the experimentally and computationally determined evolution of the droplet collision configurations. Since the experimental images can be determined quite precisely, they can be used as benchmark data for the validation of the computational code. Second, we have simulated the evolution of the inter-droplet configuration and related the results to the propensity of bouncing. Third, the droplet collision time has been simulated and experimentally determined, hence providing useful information on the characteristic time scale governing collision and its dependence on the impact inertia, surface tension, and viscous loss, for the situation of large deformation which is of particular interest here. The numerical and experimental methods are specified in the next section, which is followed by presentation and discussion of the results. Formulation The front tracking numerical method adopted herein was developed by Unverdi (1990) and discussed in Unverdi & Tryggvason (1992). The actual code is an axisymmetric version of the method, described in Jan & Tryggvason (1995). The physical problem and the computational domain is sketched in Fig. 2. The domain is axisymmetric, with the droplets initially placed near each end and the origin located at the mid-point of the line connecting the two droplet centers. The Navier-Stokes equations are valid for the fluid constituting the droplet as well as the ambient gas. A single set of momentum equations can be written for entire domain: 2R d(pV) UT dt 2ti -<W + W + A-(pW) = -Vl 8 We' KnS(r rf )da where V, p and p are the velocity, density and pressure, respectively normalized by half of the relative velocity U/2, liquid density p, and the dynamic pressure p(U/4. Furthermore, t is time normalized by the droplet oscillation time T=2rt(p| RVSa)", which is analytically derived based on inviscid droplet oscillation with small amplitude (Lamb 1932). In Eq. (1) surface tension is added as a delta function in the integration, resulting in a force distribution that is smooth and continuous over the droplet surface. Here K is twice the mean curvature, n the unit normal vector of the droplet surface, r the space vector, and the subscript f designates the interface. Finally g is a body force that is turned off at a prescribed droplet separation distance before collision, and is used to give the droplets an initial unity velocity toward each other. We further note that p is unity in the liquid phase but is given by Y=p,/Pi in the gas phase. Similarly, the effective viscosity coefficient, 2(i/p,RU, in equation (1) is 4/Re and 4X/Re in the liquid and gas phases respectively, where X is the viscosity ratio between gas and liquid, and the Reynolds number is defined based on the liquid density and viscosity, as 2p,RU/|i. Therefore, the collision outcome depends on four nondimensional parameters: We, Re, 7 and X. To solve the Navier-Stokes equation, we used a fixed, regular, staggered grid, a conservative, second order centered difference scheme for the spatial variables, and explicit first order time integration. In the computation mass is always conserved within a fraction of a percent. The interface is represented by separate computational points that are moved by interpolating their velocity from the grid. These points are connected to form a front which is used to keep the density and viscosity stratification sharp and to calculate the surface tension force. The method and the code have been tested in various ways (Nobari 1993; Nobari, Jan & Tryggvason 1995), for example by extensive grid refinement and comparison with published works (e.g. rising bubble, droplet oscillation). Figure 3 shows the droplet interface contour h(r) from a test of grid refinement for water droplet collision at t=0.63, We=2.25, and under 14.6 atm of argon. The deformed droplet surface shape and the gas gap configuration are compared for grids of 90 by 90 and 180 by 180 mesh points. It is seen that the two grids lead to essentially the same configuration, indicating that the code yields converged numerical result. Since the present numerical simulation is able to yield fine resolution of the geometry of the thin gap, which is of the order of 0.1 |j.m, all of the following numerical results were computed on a grid of 90 by 90. The time required for each run ranged from 10 to 20 hours on a DEC 3000 workstation, depending on the governing parameters. Generally the method is very robust for moderate density ratios (about 100 or less), while at large density ratios difficulties in obtaining convergence in the pressure solver were sometimes encounted. In the present code we used a relatively simple SO