Resonant VV Exchange in the DSMC Method with the Larsen-Borgnakke Model

Abstract

The Direct Simulation Monte Carlo (DSMC) method is currently the principal numerical technique for simulation of rarefied nonequilibrium flows at the kinetic level. In the DSMC method, the gas flow is represented by ∼ 10 − 10 model molecules. The process of continuous motion and collisions of molecules at a time step ∆t is divided into two consecutive stages: free-molecular transfer and collisional relaxation. The DSMC method may be treated as a stochastic numerical approach to the solution of the Boltzmann equation. One of the important areas of application of the DSMC method is high-altitude aerothermodynamics of space vehicles. The formation of a high-temperature region behind the bow shock wave is associated with the translational-rotational nonequilibrium and the excitation of vibrational molecular modes, which have significant impact on spacecraft aerothermodynamics at altitudes below 90 km. This necessitates modeling of energy exchange between translational, rotational, and vibrational modes of molecules, and raises requirements to accuracy and efficiency of energy transfer models used in the DSMC method. A simple Larsen-Borgnakke model has been successfully used for a long time to model the rotationtranslation (RT) and vibration-translation (VT) energy exchange in DSMC. In this model, post-collisional energies are sampled according to the local Boltzmann distribution. The Larsen-Borgnakke model is constructed in such a way that it allows one to exactly reproduce the solutions of relaxation equations for the mean rotational and vibrational energies in the homogeneous case. 4 The temperature dependences of rotational and vibrational relaxation times are well known for many diatomic molecules, which allows one to determine temperature-dependent probabilities of the corresponding collisional processes. The main advantages of this model over more complicated and accurate models are the possibility of using it for multispecies gas mixtures consisting of molecules of different type (including polyatomic molecules), simplicity of implementation, and numerical efficiency. Exchange of vibrational energy between the colliding molecules (VV exchange) is a process that plays an important role in aerothermodynamics of space vehicles, gas-dynamic lasers, gas discharge, etc. This process has been extensively studied theoretically for several decades (see, e.g., Refs. 5-9). There are several papers where VV transitions for diatomic molecules were modeled by the DSMC method. The model is worth noting: both VT exchange and VV exchange were modeled, and multiquantum VT and VV transitions were considered on the basis of the theoretical models. 6 One should also note the paper where VT and VV transitions for SHO were considered, the paper where the near-resonant VV exchange for anharmonic oscillators was studied (it was also noted there that the model does not satisfy the detailed balance principle), and the simple statistical model with single-quantum VT transitions and VV