Date of Award


Degree Type


Degree Name

Master of Science in Mechanical Engineering (MSME)


Mechanical Engineering

Committee Chair

Carlo Scalo

Committee Member 1

Jingwei Hu

Committee Member 2

Fabio Semperlotti


A numerical investigation of the effects of gas rarefaction on the steepening of planar nonlinear acoustic waves has been presented here. To this end, a problem setup consisting of a piston-tube assembly with adiabatic walls and the piston oscillating at the fundamental harmonic frequency of the tube was considered. The Knudsen number, which is a metric of the degree of rarefaction of the gas, considered for the assembly ranges from 0.1 to 10−5 with decrements of an order of magnitude between successive cases. The computational setup consists of numerical schemes based on the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation as well as fully compressible 1-dimensional Navier Stokes equations. The BGK equation was used to solve for cases up to a Knudsen number of 0.01 and the Navier Stokes equations were used for cases with Knudsen number less than 0.01. Based on a comparison of the acoustic wave at the asymptotic finite amplitude solution for each of the cases, it was found that for the same piston Reynolds number, Rep, a decrease in Knudsen number leads to steepening of the planar acoustic wave. The amplitude of the resonant wave relative to the piston oscillation amplitude was found to increase as the gas becomes denser.The temporal and spatial spectra of pressure perturbation and velocity were also evaluated which illustrate the broadening of the range of harmonics excited, concurrent with steepening of the acoustic wave. Scaling parameters were introduced to collapse the spectra for all the cases considered and estimate the smallest scales associated with the system. The shock thickness in the domain is proportional to these length scales which represent the point beyond which thermoviscous dissipation dominates.