# Simulation of multiphase flows and droplets

Institute for Space Systems

Particle methods offer an efficient way to visualize non-equilibrium effects in multiphase flows.

Both in nature and in technical applications, the interaction of droplets with their adjacent gaseous phase is characterized by a variety of mechanisms. Many of these processes are far from equilibrium, such as the injection of liquid fuel into the combustion chambers of rocket engines or the interaction of droplets in the exhaust gases of space engines. In the case of non-equilibrium interactions, the correct simulation of microscopic phenomena such as evaporation and nucleation is of particular interest.

Simulation programs based on the Navier-Stokes equations or similar equations are often used to simulate such processes. However, these equations are not able to model the behavior of atoms or molecules at the interface between liquid and vapor, which is necessary for an accurate prediction of the evaporation and condensation processes. This is because these equations assume an only slightly perturbed Maxwell-Boltzmann velocity distribution of particle velocities. This may not be the case under such non-equilibrium conditions where the particles experience high accelerations due to surface tension. Therefore, these evaporation and condensation processes must be integrated into these continuum models as analytical or empirical models. Kinetic solvers can help to verify these models, such as a Direct Simulation Monte Carlo (DSMC) solver that solves the Enskog-Vlasov equation. This equation is a generalization of the Boltzmann equation for dense gases and liquids and describes the forces acting on the particles, divided into attraction and collisions. The formation of liquid and vapor phases and their interface region is part of the solution space of this equation. Therefore, no additional assumptions or models are required in the interface area, which enables an undisturbed investigation of this area.

As part of the international research training group "Droplet Interaction Technologies" (GRK 2160/2: DROPIT), the sub-project SP-A3 is dedicated to the gas-kinetic simulation of micro-droplet-gas interactions and is being worked on by extending the gas-kinetic simulation framework PICLas at the Institute of Space Systems. For this purpose, a module for the numerical solution of the Enskog-Vlasov extension of the Boltzmann equation was implemented, in which a DSMC method is used to solve the Enskog-Vlasov equation, which takes into account the attraction of the particles as well as their diameters during the collision process and can thus simulate multiphase flows as an intrinsic solution without further assumptions at the phase boundary. The implemented Enskog-Vlasov solver utilizes the natural symmetries of droplets to reduce the original 3D problem to an axisymmetric 2D problem or even to a spherically symmetric 1D problem, depending on the effects under investigation, by preserving the overall structure of the spherical or oblate droplet itself. This approach can significantly reduce computation times compared to full 3D computations.

In addition to the Vlasov-Enskog solver, a second module based on a Fokker-Planck approach for dense gases was implemented to overcome the numerical challenges arising from the Enskog-Vlasov equation, such as a higher stiffness of the collision operator for higher densities, since the mean collision time and the mean free path have to be resolved. In the implemented dense Fokker-Planck scheme, the Enskog collision operator is approximated by an equivalent drift and diffusion process using the Fokker-Planck model. By using a drift and diffusion term to describe the jump process of the Enskog collision operator, the resulting paths become continuous, so that the discretization constraints are relaxed and the computational cost can be decoupled from the mean collision time.

### Contact

### Claudia Marianowski

**M.Sc.**

Research Associate

### Raphael Tietz

**M.Sc.**

Research Associate

### Marcel Pfeiffer

**Dr.-Ing.**