4.11. Fokker-Planck Collision Operator

The implementation of the FP-based collision operator is based on the publications by [57] and [58]. It is a method, which allows the simulation of gas flows in the continuum and transitional regime, where the DSMC method is computationally too expensive. The collision integral is hereby approximated by a drift and diffusion process

\[ \left.\frac{\partial f}{\partial t}\right|_{\mathrm{coll}}\approx-\sum_{i=1}^3 {\frac{\partial }{\partial v_i}(A_i f)+\frac{1}{2}\sum_{i=1}^3 \sum_{j=1}^3\frac{\partial ^2 }{\partial v_i\partial v_j}(D_{ij}f)}, \]

where \(\mathbf{A}\) is the drift vector and \(\mathcal{D}\) the diffusion matrix.

The current implementation supports:

  • 2 different methods: Cubic and Ellipsoidal Statistical (ES)

  • Single species, monatomic and polyatomic gases

  • Thermal non-equilibrium with rotational and vibrational excitation (continuous or quantized treatment)

  • 2D/Axisymmetric simulations

  • Variable time step (adaption of the distribution according to the maximal relaxation factor and linear scaling)

Relevant publications of the developers:

  • Implementation of the cubic Fokker-Planck in PICLas [58]

  • Comparison of the cubic and ellipsoidal statistical Fokker-Planck [59]

  • Simulation of a nozzle expansion (including the pressure chamber) with ESBGK, ESFP and coupled ESBGK-DSMC, comparison to experimental measurements [60]

To enable the simulation with the FP module, the respective compiler setting has to be activated:

PICLAS_TIMEDISCMETHOD = FP-Flow

A parameter file and species initialization file is required, analogous to the DSMC setup. It is recommended to utilize a previous DSMC parameter file to ensure a complete simulation setup. To enable the simulation with the FP methods, select the Fokker-Planck method, cubic (=1) and ES (=2):

Particles-FP-CollModel = 2

The vibrational excitation can be controlled with the following flags, including the choice between continuous and quantized vibrational energy:

Particles-FP-DoVibRelaxation = T
Particles-FP-UseQuantVibEn   = T

An octree cell refinement until the given number of particles is reached can be utilized, which corresponds to an equal refinement in all three directions (x,y,z):

Particles-FP-DoCellAdaptation = T
Particles-FP-MinPartsPerCell  = 10

A coupled FP-DSMC simulation can be enabled, where the FP method will be utilized if the number density \([\text{m}^{-3}]\) is above a certain value:

Particles-CoupledFPDSMC       = T
Particles-FP-DSMC-SwitchDens  = 1E22

The flag Particles-DSMC-CalcQualityFactors controls the output of quality factors such as mean/maximal relaxation factor (mean: average over a cell, max: maximal value within the octree), max rotational relaxation factor, which are defined as

\[ \frac{\Delta t}{\tau} < 1,\]

where \(\Delta t\) is the chosen time step and \(1/\tau\) the relaxation frequency. The time step should be chosen as such that the relaxation factors are below unity. The FP_DSMC_Ratio gives the percentage of the sampled time during which the FP model was utilized. In a couple FP-DSMC simulation this variable indicates the boundary between FP and DSMC. However, a value below 1 can occur for pure FP simulations due to low particle numbers, when an element is skipped. Additionally, the Prandtl number utilized by the ESFP model is given.