Numerical Issues in Droplet Collision Modeling

A faster, more accurate replacement for existing collision algorithms has been developed. The method, called the NTC algorithm, is not grid dependent, and is much faster than older algorithms. Calculations with sixty thousand parcels required only a few CPU minutes. However, there is a significant need to develop mesh-independent momentum coupling between the gas and spray, so that the collision algorithm’s full accuracy can be fully realized. Introduction Droplet collisions have proven very difficult to calculate. In addition to the complex physics, droplet collision has presented several numerical difficulties. These difficulties include high computational cost and large numerical errors. In light of the large role that droplet collisions can have in determining drop size, this situation is most unfortunate. In response to the need for faster and more accurate collision calculations, a new algorithm has been developed for calculating the incidence of droplet collision. The most widely used approach currently is O'Rourke's algorithm, as implemented in Kiva (Amsden, l989). O'Rourke's algorithm has had remarkable success and is based on sound reasoning. O'Rourke's algorithm is consistent with the stochastic nature of spray simulations, where only a sub-sample of droplets is tracked. The tracked drops represent parcels of varying numbers of drops. The size, number, and velocities of parcels determine the probability of colliding with other parcels. Collision partners are chosen stochastically. Only parcels within the same gas phase cell are permitted to collide. Schmidt and Rutland (2000) showed that this approach is second order accurate in space, but fails dramatically with typical mesh resolution. The problems of mesh dependency are particularly severe when using a Cartesian mesh, as shown in Fig. 1. The grid dependency can clearly be seen in the "clover leaf" pattern that forms from what should be an axisymmetric calculation. The grid dependency also causes large quantitative fluctuations in the predicted drop size. Figure 1. Simulation of a hollow-cone spray using the standard Kiva3V Release 2 code with collision (O'Rourke's) turned on and break-up and turbulence turned off. Simulation is done with the injector on a vertex of a Cartesian mesh. Injection is directed towards the viewer, and the drops are colored by diameter. * Author to whom correspondence should be addressed: [email protected] The new algorithm, first reported in Schmidt and Rutland, (2000) is based on the No Time Counter (NTC) method used in gas dynamics for Direct Simulation Monte Carlo (DSMC) calculations. However, Schmidt and Rutland rederived the algorithm from first principles so that it could be applied to sprays, where the number of droplets per parcel is variable. Like O'Rourke's algorithm, the NTC algorithm is first order accurate in time and second order accurate in space. The algorithm was tested against analytical solutions and found to converge to the exact answer. There are two significant improvements with the NTC algorithm. The first improvement is speed. The NTC algorithm achieves its results without consuming significant amounts of CPU time. For example, the calculation of collision between sixty thousand parcels requires less than three minutes of CPU time for twenty milliseconds of simulated injection duration. The second improvement is the use of a fully automatically generated collision mesh. There is no direct connection between the gas flow field and the incidence of droplet collisions. Consequently, it is possible to generate a collision mesh that is optimized for accuracy. Like O'Rourke's algorithm, The collision mesh achieves very high spatial resolution without incurring significant CPU cost. The mesh is cylindrical, oriented around the injection axis, and sized just large enough to contain all of the parcels. The mesh resolution is then set as fine as possible while maintaining a significant sample size of drops in each collision cell. This approach allows the code to achieve millimeter-level resolution of the spray while only costing the user a few CPU minutes. Because the algorithm is both convergent and consistent, the user can be assured of getting very accurate results.