Modeled Boltzmann Equation and Its Application to Direct Aeroacoustic Simulation

Abstract

The Bhatnagar, Gross, and Krook modeled Boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success. However, its application to direct aeroacoustic computation is less successful. One possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound. The present study reports on the development of an improved modeled Boltzmann equation for aeroacoustics simulation. The approach is to modify the Maxwellian distribution normally assumed for the equilibrium particle distribution function. Constraints imposed are the exact recovery of the state equation for a diatomic gas and the Euler equations without invoking the small Mach number assumption. Thus formulated, a distribution function consisting of the Maxwellian distribution plus three other terms that attempt to account for particle-particle collisions is obtained. A velocity lattice method is used to solve the improved modeled Boltzmann equation using an equivalent lattice equilibrium distribution function. The simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the Euler equations. The results of the improved modeled Boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain, thus verifying the viability and correctness of the modified equilibrium distribution function.