15.11.1 Introduction

When your simulation includes tracking of droplets, ANSYS FLUENT provides an option for estimating the number of droplet collisions and their outcomes in a computationally efficient manner. The difficulty in any collision calculation is that for $N$ droplets, each droplet has $N-1$ possible collision partners. Thus, the number of possible collision pairs is approximately $\frac{1}{2}N^2$. (The factor of $\frac{1}{2}$ appears because droplet A colliding with droplet B is identical to droplet B colliding with droplet A. This symmetry reduces the number of possible collision events by half.)

An important consideration is that the collision algorithm must calculate $\frac{1}{2}N^2$ possible collision events at every time step. Since a spray can consist of several million droplets, the computational cost of a collision calculation from first principles is prohibitive. This motivates the concept of parcels. Parcels are statistical representations of a number of individual droplets. For example, if ANSYS FLUENT tracks a set of parcels, each of which represents 1000 droplets, the cost of the collision calculation is reduced by a factor of 10 $^6$. Because the cost of the collision calculation still scales with the square of $N$, the reduction of cost is significant; however, the effort to calculate the possible intersection of so many parcel trajectories would still be prohibitively expensive.

The algorithm of O'Rourke [ 255] efficiently reduces the computational cost of the spray calculation. Rather than using geometry to see if parcel paths intersect, O'Rourke's method is a stochastic estimate of collisions. O'Rourke also makes the assumption that two parcels may collide only if they are located in the same continuous-phase cell. These two assumptions are valid only when the continuous-phase cell size is small compared to the size of the spray. For these conditions, the method of O'Rourke is second-order accurate at estimating the chance of collisions. The concept of parcels together with the algorithm of O'Rourke makes the calculation of collision possible for practical spray problems.

Once it is decided that two parcels of droplets collide, the algorithm further determines the type of collision. Only coalescence and bouncing outcomes are considered. The probability of each outcome is calculated from the collisional Weber number ( $We_c$) and a fit to experimental observations. Here,

where $U_{rel}$ is the relative velocity between two parcels and $\overline{D}$ is the arithmetic mean diameter of the two parcels. The state of the two colliding parcels is modified based on the outcome of the collision.