Investigation of Amplitude Dependence on Nonlinear Acoustics Using the Direct Simulation Monte Carlo Method

Traditionally, acoustics is concerned with the treatment of the fluid as a continuum using macroscopic quantities such as velocity and pressure as dependent variables. However, the continuum model breaks down for Knudsen numbers (Kn) greater than roughly 0.05, where Kn is defined as the ratio of mean free path to wavelength. Particle methods are necessary for, but not limited to, problems with Kn > 0.05. The Knudsen number is large for sound propagation in very dilute gases, or at high frequencies and thus requires a particle method (or Boltzmann equation) solution. In our studies we have used a particle method, the direct simulation Monte Carlo (DSMC) method, for the direct physical modeling of particle motions and intermolecular collisions in nonlinear acoustics problems. Using DSMC to study nonlinear acoustics allows us to explore real gas effects for all values of Kn with a molecular model that continuum methods cannot offer. DSMC results for the absorption of sound have shown that absorption depends heavily on Kn and amplitude and deviates significantly from the continuum classical assumption for large Kn. In our current DSMC calculations we have explored the relationship between absorption and nonlinearity as a function of amplitude for a range of Knudsen numbers. INTRODUCTION There is a hierarchy of mathematical models available to solve fluid dynamics problems. These models have varying degrees of approximation but can be categorized into two groups: continuum and non-continuum methods. Continuum methods, which are popular for acoustic problems, model the fluid as a continuous medium. This model describes the state of the fluid with macroscopic level using quantities such as density, velocity, and temperature. The continuum approximation is valid when the characteristic length of the problem is much larger than the molecular spacing between fluid particles. This assumption is satisfied for many engineering problems, and thus fluid evolution can be described using continuum equations such as the Navier-Stokes, Euler or wave equations. However, the continuum model has its limitations. The macroscopic model assumes deviations from thermal equilibrium are small, and it is the failure of the closure of the Navier-Stokes equations that limit the applications of this approach. The Knudsen number (Kn) which is defined to be the mean free path divided by a characteristic length scale is a measure of the nonequilibrium or viscous effects of the gas. The Knudsen number is also used to distinguish the regimes where different governing equations of fluid dynamics are applicable. The NavierStokes equations are valid for Kn < 0.05, and the Navier-Stokes equations reduce to the Euler equations as Kn approaches 0. The Boltzmann equation is the mathematical model for noncontinuum methods and is valid for all Kn, although most efficient for high Kn. Therefore, noncontinuum (or particle methods) are necessary for, but not limited to, problems where the Knudsen number is greater than 0.05. Non-continuum methods are based on molecular models that realize the particle nature of the gas and describe the state of the gas at the microscopic level. Despite the fact that the Boltzmann equation was derived using a microscopic approach, it can be shown that the