Vibrational Modeling with an Anharmonic Oscillator Model in Direct Simulation Monte Carlo

Abstract

Vehicles undergoing hypersonic speed experience extreme aerothermodynamic conditions. Real gas effects cannot be neglected, and thus internal degrees of freedom of molecules being partially/fully excited must be carefully predicted in order to accurately capture the physics of the flowfield. Within direct simulation Monte Carlo solvers, a harmonic oscillator (HO) model, where the quantum levels are evenly spaced, is typically used for vibrational energy. A more realistic model is an anharmonic oscillator (aHO), in which the energy between quantum levels is not evenly spaced. In this work, the Morse-aHO model is compared against HO. The Morse-aHO model is implemented in the dsmcFoam+ solver, and the numerical results are in excellent agreement with analytical and potential energy surface solutions for the partition function, mean vibrational energy, and degrees of freedom. A method for measuring the vibrational temperature of the gas when using the anharmonic model in a direct simulation Monte Carlo solver is presented, which is essential for returning macroscopic fields. For important thermophysical properties of molecular oxygen, such as the specific heat capacity, it is shown that the aHO and HO models begin to diverge at temperatures above 1000 K, making the use of HO questionable for all but low-enthalpy flows. For the same gas, including the electronic energy mode significantly improves the accuracy of the specific heat prediction, compared to experimental data, for temperatures above 2000 K. For relaxation from a state of thermal nonequilibrium, it is shown that the aHO model results in a slightly lower equilibrium temperature. When applied to hypersonic flow over a cylinder, the aHO model results in a smaller shock standoff distance and lower peak temperatures.