Computational Verification of Acoustic Damping in High-Enthalpy Environments

Abstract

T HE motivation for studying acoustic propagation in highenthalpy environments lies partially in the role that acoustics play in promoting laminar–turbulent boundary-layer transition. At high speeds, boundary-layer transition on slender-body vehicles is primarily due to Mack’s second-mode instability [1]. These secondmode disturbances are associated with acoustic waves that become trapped in the boundary layer flowing over the vehicle. Thus, the interaction of an acoustic wave with the boundary layer can either promote or inhibit the growth of second-mode disturbances [2–5]. If the growth of disturbances is sufficiently reduced, then laminar– turbulent boundary-layer transition could be delayed or prevented. In many cases, maintaining laminar flow over a high-speed vehicle is beneficial because a laminar boundary layer exerts significantly less heating and shear forces on the vehicle than a turbulent boundary layer.Accurately capturing the physics of transitionwould allow for a more precise prediction of the transition location. Less uncertainty in the transition location would then allow for optimization of thermal protection systems and design of high-speed vehicles. Thus, it is important to understand the interaction of an acoustic wave with a high-enthalpy flow environment. It has long been postulated that internal molecular relaxation processes can damp acoustic waves. Griffith [6] stated that Jeans [7] first proposed the idea of a lagging internal energy mode due to changes in the gas state. The first experimental evidence of acoustic damping was shown by Pierce [8], who studied the speed of highfrequency sound through various gases. The theoretical modeling of the acoustic damping and dispersion process due to molecular vibration was first formulated a few years later by Herzfeld and Rice [9]. Lighthill [10] performed an analysis similar to Herzfeld and Rice [9], showing the dependency of acoustic damping on the bulk viscosity response of a gas. Vincenti and Kruger [11] also analyzed several aspects of a single internal energy mode on acoustic waves propagating through a gas in equilibrium and found the damping to be greatest when the frequency of sound is near the relaxation rate of the internal mode. Clarke and McChesney [12] performed a similar analysis of the acoustic response to a single chemical reaction. Fujii and Hornung [2,13,14] were able to improve on the modeling of acoustic damping by including several relaxation modes. In doing this, they were able tomodel realisticmixtures of gases in equilibrium. Fujii and Hornung found that various gases had similar damping properties at different equilibrium states and frequencies, meaning that all of thesegases tested couldbe used in acoustic damping applications. A sample plot of Fujii and Hornung’s work is included in Fig. 1a and shows the variation in damping properties based on the temperature range and frequency. Fujii and Hornung showed that the results of the boundary-layer stability calculations by Johnson et al. [15] were due to carbon dioxide’s ability to damp the acoustic frequencies associated with second-mode transition. As demonstrated in Fig. 1b, carbon dioxide proves to be very effective at the temperatures and pressures considered in the experiment and computation. For the current study, we seek to verify our computational solver against an extension of the acoustic damping theory given by Fujii and Hornung [2,13,14] as well as investigate the various features of acoustic waves propagating through high-temperature gases. First, we highlight the generic formula to calculate the optimum damping frequency for an internal molecular process. Next, we introduce the dispersion relation for an acousticwave traveling through a gaswith a mean flow velocity. Based on this dispersion relation, we determine the variation of the optimum damping frequency with the mean flow Mach number. We then use a computational fluid dynamics (CFD) solver to simulate the propagation of slow and fast acoustic waves through carbon dioxide. We compare the acoustic damping rate as calculated from the simulation to the damping rate based on theory. We end with a brief investigation of the physical differences between the disturbance quantities of fast acoustic waves and those of slow acoustic waves.