Time-Resolved Planar Particle Image Velocimetry of the 3-D Multi-Mode Richtmyer Meshkov Instability

Abstract

An experimental investigation of the Richtmyer-Meshkov instability (RMI) is carried out using a single driver vertical shock tube. A diffuse, stably stratified membrane-less interface is formed between air and sulfur hexafluoride (SF6) gases (Atwood number, $ A = \frac{\rho_1 - \rho_2}{\rho_1+\rho_2} \approx0.67$) via counterflow, where the light gas (air) enters the tube from the top of the driven section, and the heavy gas (SF$_6$) enters from the bottom of the test section. A perturbation is imposed at the interface using voice coil drivers that cause a vertical oscillation of the column of gases. This oscillation results in the Rayleigh-Taylor unstable growth of random modes present at the interface, and gives rise to Faraday waves which invert with half the frequency of the oscillation. The interface is initially accelerated by a Mach 1.17 (in air) shock wave, and the development of the ensuing mixing layer is investigated. The shock wave is then reflected from the bottom of the apparatus, where it interacts with the mixing layer a second time (reshock). The experiment is initialized with two distinct perturbations - high amplitude experiments where the shock wave arrives at the maximum excursion of the perturbation, and low amplitude experiments where it arrives near its minimum. Time resolved Particle Image Velocimetry (PIV) is used as the primary flow diagnostic, yielding instantaneous velocity field estimates at a rate of 2 kHz. Measurements of the growth exponent $\theta$, where the mixing layer width $h$ is assumed to grow following $h(t) \approx t^\theta$, yield a value of $\theta\approx 0.51$ for high amplitude experiments and $\theta\approx0.45$ for low amplitude experiments following the incident shock wave when estimated using the width of the mixing layer approximated by the width of the turbulent kinetic energy containing region. Following interaction with the reflected shock wave, $\theta \approx 0.33$ for high amplitude experiments, and $\theta \approx 0.50$ for low amplitude experiments. It is observed that the low amplitude experiments grow faster than the high amplitude experiments following reshock, likely owing to the presence of steeper density gradients present in the relatively less developed mixing layer. $\theta$ is also estimated using the decay of turbulent kinetic energy for experiments where dissipation is significant. Theta estimates using both methods are found to be in good agreement for the high amplitude case following the incident shock, with $\theta\approx0.51$. $\theta \approx 0.46$ is found following reshock, which is larger than the value found when fitting $\theta$ to width data. Low amplitude experiments do not exhibit significant dissipation, and a value of $\theta \approx 0.68$ is found for low amplitude experiments following the incident shock, and $\theta \approx 0.62$ following reshock. Persistent anisotropy is a commonly observed phenomenon in the RMI mixing layer, owing to the stronger velocity perturbation components in the streamwise direction following the passage of a shock wave. High amplitude experiments are observed to reach a constant anisotropy ratio (defined as the ratio of streamwise to spanwise turbulent kinetic energy, or TKX/TKY), an indication of self-similarity, shortly following the passage of the incident shock wave with value of $\approx 1.8$. Low amplitude experiments do not reach a constant value during the experimental observation window, suggesting that the flow is still evolving even after a second shock interaction. Examination of the spanwise average anisotropy tensor reveals asymmetry in the anisotropy for low amplitude experiments, with the heavy gas exhibiting a slightly larger degree of anisotropy. The high amplitude experiments exhibit transitional outer Reynolds numbers ($Re\equiv\frac{h\Dot{h}}{\nu} > 10^4$) using the criterion proposed by Dimotakis shortly following the passage of the initial shock wave, while the low amplitude experiments largely remain below this threshold. Following reshock, both sets of experiments are elevated to $Re \approx 10^5$, which is a strong indication that mixing transition should occur and an inertial range will form. However, extended length scale analysis proposed by Zhou that accounts for the temporal evolution of scales which are a prerequisite for the formation of an inertial range indicates that neither high or low amplitude experiments have entered a transitional regime even following reshock. Furthermore, the $\theta \approx 0.5$ growth of the outer length scale in these experiments suggests that transition will not occur even if longer observation windows were possible. The lack of an inertial range is evident in spectral analysis of the mixing region.