Piston-generated dynamic compression and expansion of an inert gas in a cylinder

The behavior of an inert gas confined between a variable speed piston and a fixed cylinder endwall is investigated theoretically using unsteady Euler equations. The characteristic piston Mach number is {dollar}Msb{lcub}p{rcub}{dollar} = 0.05. Acoustic waves generated by piston motion and/or initial disturbances reflect back and forth in the cylinder on the acoustic time scale {dollar}tsbsp{lcub}a{rcub}{lcub}*{rcub}{dollar}, while the piston traverses the cylinder on the longer piston time scale {dollar}tsbsp{lcub}p{rcub}{lcub}*{rcub}{dollar}.; Perturbation methods, based on small piston Mach number {dollar}Msb{lcub}p{rcub}{dollar} {dollar}sim tsbsp{lcub}a{rcub}{lcub}*{rcub}/tsbsp{lcub}p{rcub}{lcub}*{rcub}{dollar}, are employed to find the velocity and thermodynamic properties of the gas. Solutions are constructed in terms of infinite Fourier series. The application of multiple-time scaling methods allows the instantaneous acoustic field to be separated from the accumulated bulk response of the gas to piston motion. The latter is found to be identical to the classical quasistatic results derived from equilibrium thermodynamics. The evolution of the {dollar}{lcub}cal O{rcub}(Msb{lcub}p{rcub}{dollar}) acoustic phenomena, including nonlinear wave deformation and weak shock formation during the piston time period, are studied in detail. In the nonlinear regime the Fourier series technique is equivalent to a spectral-type of numerical method in which the time dependent Fourier coefficients are computed from a truncated system of coupled nonlinear ordinary differential equations. This method is shown to be capable of describing wave deformation processes and shock formation.; In this study three types of acoustic wave generation arising from piston motion are considered. A fast, smooth piston acceleration occuring on the acoustic time scale produces an acoustic wave of {dollar}{lcub}cal O{rcub}(Msb{lcub}p{rcub}{dollar}), which is ultimately transformed into a weak shock during the piston time period. In contrast, a slower, smooth piston acceleration that occurs on a piston time scale induces an {dollar}{lcub}cal O{rcub}(Msbsp{lcub}p{rcub}{lcub}2{rcub}){dollar} acoustic field, which remains linear always. Finally the shock wave generated by an impulsively started piston motion is investigated. The shock strength is related to the initial piston Mach number and the subsequent continuously varying piston velocity by a first order nonlinear ordinary differential equation. It is demonstrated that cyclic piston motion damps the shock wave field.