# Statistical methods for the analysis of DSMC simulations of hypersonic shocks

## Abstract

In this work, statistical techniques were employed to study the modeling of a hypersonic shock with the Direct Simulation Monte Carlo (DSMC) method, and to gain insight into how the model interacts with a set of physical parameters. Direct Simulation Monte Carlo (DSMC) is a particle based method which is useful for simulating gas dynamics in rarefied and/or highly non-equilibrium flowfields. A DSMC code was written and optimized for use in this research. The code was developed with shock tube simulations in mind, and it includes a number of improvements which allow for the efficient simulation of 1D, hypersonic shocks. Most importantly, a moving sampling region is used to obtain an accurate steady shock profile from an unsteady, moving shock wave. The code is MPI parallel and an adaptive load balancing scheme ensures that the workload is distributed properly between processors over the course of a simulation. Global, Monte Carlo based sensitivity analyses were performed in order to determine which of the parameters examined in this work most strongly affect the simulation results for two scenarios: a 0D relaxation from an initial high temperature state and a hypersonic shock. The 0D relaxation scenario was included in order to examine whether, with appropriate initial conditions, it can be viewed in some regards as a substitute for the 1D shock in a statistical sensitivity analysis. In both analyses sensitivities were calculated based on both the square of the Pearson correlation coefficient and the mutual information. The quantity of interest (QoI) chosen for these analyses was the NO density profile. This vector QoI was broken into a set of scalar QoIs, each representing the density of NO at a specific point in time (for the relaxation) or a specific streamwise location (for the shock), and sensitivities were calculated for each scalar QoI based on both measures of sensitivity. The sensitivities were then integrated over the set of scalar QoIs to determine an overall sensitivity for each parameter. A weighting function was used in the integration in order to emphasize sensitivities in the region of greatest thermal and chemical non-equilibrium. The six parameters which most strongly affect the NO density profile were found to be the same for both scenarios, which provides justification for the claim that a 0D relaxation can in some situations be used as a substitute model for a hypersonic shock. These six parameters are the pre-exponential constants in the Arrhenius rate equations for the N2 dissociation reaction N2 + N ⇄ 3N, the O2 dissociation reaction O2 + O ⇄ 3O, the NO dissociation reactions NO + N ⇄ 2N + O and NO + O ⇄ N + 2O, and the exchange reactions N2 + O ⇄ NO + N and NO + O ⇄ O2 + N. After identification of the most sensitive parameters, a synthetic data calibration was performed to demonstrate that the statistical inverse problem could be solved for the 0D relaxation scenario. The calibration was performed using the QUESO code, developed at the PECOS center at UT Austin, which employs the Delayed Rejection Adaptive Metropolis (DRAM) algorithm. The six parameters identified by the sensitivity analysis were calibrated successfully with respect to a group of synthetic datasets.