Finite-volume droplet trajectories for icing simulation

During a Lagrangian icing simulation, it is necessary to calculate a large number of droplet trajectories to determine the water catch, and as a result it is important that this procedure is as rapid as possible. In order to arrive at a method with minimum complexity, a finite volume representation of streamlines is extended to incorporate the equations of motion for a droplet. Using a finite volume representation means that the accuracy of the droplet motion is consistent with the underlying flow simulation, and that any flow cell may be crossed in a single timestep as the velocity is constant and the trajectory is therefore a sequence of straight line segments. However, since cells vary greatly in size, the method must be implicit to avoid a stability restriction which would otherwise degrade performance. Therefore, an implicit method is implemented by carrying out a handful of coupling iterations for every cell for each timestep, so that the droplet motion is tightly coupled to the underlying flow. By crossing every cell in a single step, and by using the mesh connectivity to track the droplet motion between cells, any need for costly searches is eliminated and the resulting method is very efficient. The final method is able to find 100000 trajectories on a mesh of 460000 cells in only 2-3 minutes, using standard hardware and unoptimised code and carrying an I/O overhead. I. Introduction Calculation of droplet or solid particle trajectories is a problem that arises across a wide range of disciplines, ranging from icing simulation to combustion modelling and chemical engineering. Modern techniques apply CFD to calculate the flow field surrounding droplets, and then integrate Newton’s second law to determine the trajectory. However, this means using the flow field information (which usually consists of just the velocity, but it might also include temperature or other flow parameters), which is Eulerian in nature, within a Lagrangian calculation, and hence requires a method of tracking. The question of tracking a massless particle (to give a streamline) through a finite-volume mesh is straightforward, but does pose the question of finding the most efficient way to perform this tracking. Many hundreds of thousands of droplets may ultimately need to be tracked, so it is important to carry out this part of the calculation as quickly as possible. The level of complexity is substantially increased by introducing the droplet equations of motion, as these can imply stability restrictions, and for efficiency it is necessary to resolve these numerical limitations with the method of tracking. This work starts by describing the simplest possible way to represent streamlines on a finite-volume mesh, before moving on to incorporate the equations of motion and thereby give a trajectory. The final result is a method for finding droplet trajectories that is fast, adaptive to the volume mesh resolution and without timestep restrictions. The basic method is first order accurate in space and time, but these may be increased at the expense of reliability or complexity if desired.