Measurement of traveling pressure waves inside a droplet

Abstract

Shock wave-droplet interactions have been receiving increasing attention due to their relevance in aviation fuel combustion and minimally invasive medical treatments, yet quantifying them experimentally remains a challenge. In this study, we propose a background-oriented schlieren (BOS) technique for quantitative spatiotemporal measurements of shock wave-droplet interaction, employing a novel ray-tracing correction, a synchronization system, and a projected background. Underwater shock waves propagating both inside and outside a millimetric perfluorohexane droplet immersed in water are experimentally measured. The quantified density-gradient and pressure fields are compared with numerical simulations, and the BOS measurements-including sound speeds, the shock-focusing location, and the maximum pressure-are found to be in close agreement with the numerical results. Notably, the technique successfully captures the phase shift before and after shock focusing that had previously only been hypothesized.

Figures (15)

• Figure 1: Schematic of the experimental setup for the proposed BOS technique for quantitative measurements inside a droplet. A perfluorohexane (PFH) droplet is placed on an agarose gel substrate in water to stabilize its position. The origin of the ($x,y,z$) coordinate system is set at the center of the droplet. A shock wave propagates along the $x$-axis from left to right. A background pattern is projected and positioned at $z = l_b$ along the optical axis, and the camera is focused on the background.
• Figure 2: A schematic of the BOS theory.
• Figure 3: Schematic illustrating the optical path distortion considered in the coordinate correction. A light ray passing through point $P_0$ is refracted by the droplet on the $y$–$z$ plane, as shown by the green line. The dashed green line indicates the corresponding ray imaged onto the camera sensor, which is displaced due to refraction at the droplet interface. The red line represents the projected background.
• Figure 4: The geometry of light rays refracted by the droplet on the $y$–$z$ cross-section. The refraction angles in the surrounding fluid and within the droplet are denoted as $\theta_1$ and $\theta_2$, respectively. $R$ represents the droplet radius. $l_b$ denotes the distance from the droplet center to the projected background. The yellow and green lines represent the light rays passing through the droplet in the presence and absence of the measurement target, respectively.
• Figure 5: (a) Experimental setup of the BOS technique. Top view: The center of the droplet is defined as the origin $O$. A light source, a background with a checker pattern, two lenses, the droplet, and a camera are aligned along the $z$-axis. The background pattern is projected into the droplet through the two lenses. An enlarged view of the droplet region is shown within the dashed rectangle. The droplet is placed on an agarose gel substrate immersed in water. A vapor bubble and a spherically propagating shock wave are generated by a laser focused along the $x$-axis. Side view: A hydrophone is positioned at $R_{\rm hyd}$, which was a safe distance from the shock source to record the shock wave. (b) The black line indicates shock pressure at $R_{\rm bos}$ measured by the BOS in the absence of both the droplet and the agarose gel substrate. The dashed red line represents the pressure profile estimated using the Friedlander function (Eq. \ref{['eq:friedlanderEquation']}), based on the maximum pressure value of the black line. The gray dashed line with a single dot indicates 0 MPa.