Extension of a Chemical Reaction Model in the Fokker-Planck Framework and its Application to Supersonic Flows

Abstract

Numerical investigations of space flight and atmospheric re-entry requires the modeling of thermochemical effects for shock related dissociation and recombination. These phenomena also play an important role during post-combustion in plumes and nozzle flows. Modeling of the aforementioned processes often requires simulating flows in the rarefied and transitional regimes, where standard continuum assumptions are no longer valid and the usual Navier–Stokes equations can not be applied. For the rarefied regime, the Direct Simulation Monte Carlo method (DSMC) has proven to be a viable method capable of simulating complex multi-scale problems with thermochemical non-equilibrium effects [Bird 1994]. However, in the transitional regime (corresponding to Knudsen numbers of 0.1 < Kn < 10), it becomes computationally very expensive. Therefore, some method is required to bridge the computational gap between the continuum and rarefied flow regimes and provide a computationally efficient modeling capability for simulating transitional flows.

The particle Fokker–Planck approach (FP) is one such method [Jenny et al. 2010]. It is based on rigorous kinetic theory formalism and provides a way to simulate flows across the whole range of rarefaction with no change in computational cost. Similar to DSMC, the FP method relies on simulated particles to transport mass, momentum and energy through the flow domain, but does not resolve individual collisions, which makes it more efficient as the collision frequency increases with lower Kn numbers.

Moreover, its particle-based nature provides a natural way for its coupling with DSMC for multi-scale simulations [Gorji et al. 2014]. At DLR a FP method has been implemented as an extension to the DSMC code SPARTA [Plimpton et al. 2019] developed at Sandia National Labs. Recent extensions to the method include the multi-species formulation [Hepp et al. 2020], the incorporation of internal degrees of freedom for diatomic [Gorji et al. 2013, Hepp et al. 2020, Kim et al. 2024], polyatomic gases [Mathiaud et al. 2016, Basov et al. 2022, Nagel et al. 2023], and possibility of modeling chemical reactions [Basov et al. 2023, Basov et al. 2025]. However, not all the physical phenomena relevant to aerospace applications can currently be simulated by the FP approach.

In the present work, we present for the first time ever, an extension of the framework for simulating chemical reactions in the FP method to correct modeling of forward and backward reaction rates ensuring detailed balance and correct chemical equilibrium in simulations. The reaction scheme is verified against systems of master equations [Nagnibeda et al. 2009]. Additionally, the model is extended to include three-body reactions, such as recombination reactions.

The developed particle Fokker–Planck framework with thermochemical non-equilibrium modeling is then applied to the simulation of several model 2-dimensional hypersonic air flows for various rarefaction regimes. Comparisons are carried out with the DLR TAU [Mack et al. 2002] solver in the continuum regime, and with the SPARTADSMC code in the rarefied regime, in order to verify the new approach. The impact of the degree of rarefaction on the flow is investigated, and the performance of the new model is compared to DSMC simulations.