Bayesian estimation-based information exchange techniques for coupled Navier-Stokes equation and DSMC algorithms

In the calculation process of coupled Navier-Stokes (N-S) equation and direct simulation Monte Carlo (DSMC) algorithms, the DSMC method suffers from a limited number of statistical samples. This leads to significant statistical fluctuations in its results, which adversely affect the solution of the N-S equations. To address this issue, a Bayesian estimation approach is introduced into the two-way information exchange process. By leveraging the theory of conjugate prior distributions, a preliminary Bayesian estimation-based information exchange framework for N-S/DSMC coupling is established. Specifically, conjugate prior distributions of the normal type are adopted for density, velocity, and translational temperature. For rotational and vibrational temperatures, conjugate priors of the gamma type are employed. In this method, the computational results obtained from the N-S solver at the previous step are taken as prior information. These priors are then combined with the statistical samples provided by the DSMC method to yield the posterior distributions of the various flow parameters. The Bayesian estimates of these parameters are subsequently used as the DSMC results for information exchange with the N-S solver. To validate the proposed method, numerical simulations are performed for two test cases: a hypersonic thermochemical nonequilibrium flow over a cylinder, and a high-speed flow in the transitional regime over a flight vehicle under experimental conditions. The results demonstrate that, even in the presence of large variances, with appropriately selected hyperparameters, the proposed Bayesian approach yields results that agree well with those obtained using the conventional under-relaxation information exchange technique. The maximum discrepancy in predicted aerodynamic characteristics between the two methods is 3.2%. When compared against wind tunnel experimental data, the maximum error is approximately 5%. These findings confirm the effectiveness of the Bayesian method in suppressing statistical fluctuations caused by small sample sizes in DSMC computations.