Instability and Atomization of Liquid Cylinders after Shock Wave’s Impacting

Abstract

This paper describes an experimental study on the instability and atomization of liquid cylinders after the impact of shock waves. Single row water column, in-line double rows water columns and alongside triple rows water columns were evaluated in a horizontal shock tube. The diameter of water column and the Mach number in the experiments were 2.0–4.14 mm and 1.10–1.25, respectively. The global instability along the axial direction of water cylinders was focused. Using a high-speed camera, the developments of spike height, bubble depth and turbulent mixing zone, width were measured. Some comparison was also made between the present experimental results and the existing theoretical model.

1. Introduction

The interaction between a shock wave and a liquid cylinder can be seen in atomization and combustion in Scramjet engines [1,2,3]. Research on the instability model of the liquid is key to evaluate efficiency in fuel mixing and combustion. On the other hand, results of the Richtmyer–Meshkov (RM) instability at a gas/liquid interface can be applied in design of inertial confined fusion (ICF) because it provides a case of large Atwood number [4,5,6].

In fact, people have been familiar with the phenomenon of shock wave interaction with a liquid droplet. In 1959, Gordon [7] established a theoretical model of predicting the breakup time of a droplet in highspeed gas stream after shock wave’s impact. Later, Wierzba and Takayama [8] and Hisang and Faeth [9,10] experimentally investigated different types of droplet breakup induced by shock wave. In 2001, Igra and Takayama [11] simulated shock wave interaction with a water cylinder using CIP format. In 2008, Chen and Liang [12] conducted a similar numerical work using a compressible multiphase solver with a five-equation model. In 2016, Wang et al. [13] simulated deformation and disintegration of a two-dimensional water drop, in which their numerical results are in agreement with the experimental results. They found the effect of liquid viscosity on the vortex structure behind the drop and investigated the influence of Weber number and Ohnesorge number on breakup of the drop. Recently, based on unsteady Reynolds-averaged Navier–Stokes approach and VOF method, Rossano [14,15,16] effectively reproduced the aerobreakup process under the shear-induced entrainment regime. Wherein the results presented by the hybrid VOF–Lagrangian method are very commendable. Moreover, the adaptive numerical mesh approach can well capture the phase interface, which provides a good example for the numerical simulation method to study RM instability.

Research on wave dynamics has also been widely concerned by scholars [17,18,19]. For example, Guan et al. [18] used numerical simulations to focus on the internal flow pattern of a droplet and observed that a high pressure develops in the leeward region of the droplet due to the collision of the transmitted and diffracted shock wave, which flattens the entire droplet. These studies have clarified in detail that the complex wave structure generated by the interaction between shock wave and droplets, is fully capable of deforming the droplet at the early stage of the impact.

Perhaps the earliest experiment on shock wave interaction with a liquid cylinder was conducted at Tohoku University, Japan, using a thin gel layer to represent a water cylinder [20]. In 2016, the Royal Institute of Technology, Sweden conducted an experiment of shock wave’s impact on a 5 mm height water cylinder aiming to reveal the wave propagation in the cylinder [21]. Further, Xu and Wang [22] investigated the interaction between the plane shock and two concentric/eccentric cylinders, which are formed by the improved soap film technique.

However, none of the experiments have been concerned with the global deformation and disintegration along the axial direction of the cylinder. This paper will report an experimental work on the instability and atomization of relatively long water cylinders produced by gravity.

2. Experimental Devices

Figure 1 shows the horizontal shock tube applied in our experiment, whose total length is about 12 m. The high-pressure gas tank 1 (nitrogen N2 gas), round cross low pressure section 5 and square cross low-pressure section 6 composes the shock wave generator. The experimental section 9 is connected to the square cross low-pressure section and vacuum chamber 14 by flanges and two pressure transducers 7 for measuring shock Mach number are set on the top of the square cross low-pressure section 6. Figure 2 is the enlarged schematic of the experimental section 8. The interior sectional dimension of the 500 mm long experimental section is 120 mm × 120 mm. A highspeed camera (Photon Fastcam SA5) is set at one side of the experimental section 8 which is windowed by two transparent PMMA plates. If shadowgraph or schlieren optics is applied, the shock waves can also be visualized. On the upper and lower walls of the experimental section 8, square holes are machined respectively to arrange the water cylinder adapter 16, testing plate 17 and lower drainage board 18. The water cylinder adapter 16 is fixed on the testing plate 17 through screw structure. The outlet of the adapter 16 is machined carefully to ensure that a long smooth water cylinder can be obtained [23]. In Figure 2, it is also shown that the in-line double rows water columns can be obtained with 50 mm spacing distance. Using the shock tube, we already successfully conducted experiments on shock wave interaction with water droplet (see [13,24]).

After the shock wave’s impact, the water cylinder (column) will experience processes of compression, RM instability, disintegration, atomization, etc. The RM instability will produce the so-called spike and bubble structures. Therefore, measurements of the spike height and bubble depth $h_b$ are important to describe how the instability develops, which are shown in Figure 3. In Figure 3a, the two successive photos are overlapped each other so that the increment of the spike height $\Delta h_s$ and increment of bubble depth $\Delta h_b$ can be obtained. Then, the next time the spike height or bubble depth becomes $h_2=h_1+\Delta h$. From Figure 3b,c, the displacement of the vertical axial base line $\Delta s$ equals to that taken from high-speed photographs (e.g., Figure 4 and Figure 5). By definition, if we consider the gas/liquid interface as a sinusoidal curve, the distance from the base line to the crest is $h_s$; the distance from the base line to the trough is $h_b$; then the turbulent mixing zone is that $h=h_s+h_b$. Averaging the results along the axial direction of the cylinder, we have the global parameter:

$$h_i=\frac{1}{m}{\displaystyle \sum _{n=1}^m{h}_{im}(i=s,b; m=1,2,3\dots )}$$

(1)

3. Results and Discussion

Figure 4 shows high-speed photographs of $d_0=4.14$ mm diameter single row water column after interaction with a $Ma=1.10$ shock wave. It is seen that from Figure 4(1–5), the cylinder is compressed and moves to the right after shock wave’s impact. The compression duration is longer than that of $d_0=2.76$ mm water cylinder [25]. It should be noted that compression of the liquid column, as mentioned in the introduction, is due to the high-pressure zone formed by the interaction between the diffracted shock wave and the transmitted shock wave [17,18,19]. From Figure 4(6), the liquid is stripped off and many thin ligament tails are formed on the leeward side of the column. From Figure 4(8) to Figure 4(12), the windward side of the column starts to deform and bend but the surface is still smooth. At the moment, the instability has not been developed obviously. From Figure 4(13–16), the spike and bubble structures appear on the windward side of the column. Then, they develop gradually and merge together.

Figure 5 gives high-speed photographs of RM instability after a $Ma=1.10$ shock wave impacts on $d_0=3.38$ mm diameter in-line double rows water columns. We mark the first one as the left column (L) and the second one as the right column (R). As shown in Figure 5(1–6), after shock wave’s impact, the two columns are compressed and move to the right direction. The motion of the right column lags behind the left one. From Figure 5(7), many thin ligament tails appear on the leeward side of the left column. Meanwhile, in Figure 5(7), the streamwise size of the right column is compressed to minimum. From Figure 5(8–11), as disintegration and atomization of the liquid develop further, the two columns expand gradually and the wakes on the leeward sides integrate with the surrounding air. The spike and bubble structures appear on the windward side of the left column from Figure 5(12) and similar phenomenon occurs on the right column from Figure 5(13). From Figure 5(14–24), the spike and bubble structures develop further and merge with each other; at the later stage, the two columns interact mutually and mix together to form a complicated turbulent cloud. Noticeably, the left column takes precedence to develop because it firstly meets the shock wave. In comparison with the single row water cylinder, the in-line double rows water columns deform and break up more slowly to some degree. This can be explained due to retardation of the left column to the shock wave.

The measured spike height $h_s$, bubble depth $h_b$ and turbulent mixing zone $h$ are presented in Figure 6, Figure 7 and Figure 8. It is clear that at $Ma=1.10$, the parameters are all linear with time in spite of the single row water column or in-line double rows water cylinders (see Figure 6 and Figure 8). However, as the increase of Mach number to $Ma=1.25$, $h_s$, $h_b$ and $h$ all become nonlinear with time (see Figure 7).

The measured spike height $h_s$, bubble depth $h_b$ and turbulent mixing zone $h$ are presented in Figure 6, Figure 7 and Figure 8. Next, we introduce the results of the shock wave interaction with alongside triple rows water columns with 2 mm space of $d_0=2.0$ mm diameter at Mach number $Ma=1.20$. The shadowgraphs and numerical schlieren pictures of cross-sectional view are given in Figure 9. It is known that the reflection and diffraction of shock wave from the neighbour cylinders cause a very complex wave system, so that this may bring a nonlinear RM instability. The measurement results of the spike height and bubble depth confirm this postulation (see Figure 10). The results of three dimensional numerical simulations based on Fluent software and VOF method [26,27] give a similar trend. Additionally, a comparison with the single row water column experiment of $d_0=2.0$ mm diameter at Mach number $Ma=1.20$ at identical conditions demonstrates that the RM instability in case of triple columns has obvious nonlinear characteristics [26,27].

As already known, Richtmyer [28] proposed his famous acoustic linear impact model for the disturbance on a fluid interface as following:

$$h=2\alpha tA\Delta \nu$$

(2)

$$A=\frac{{\rho }_2-{\rho }_1}{{\rho }_2+{\rho }_1}$$

(3)

In the Atwood number $A$, ${\rho }_1$ and ${\rho }_2$ are densities of light fluid and heavy fluid respectively, $\alpha$ is an experimental coefficient. Bases on single mode of disturbance and two bubbles model, Alon et al. [29] obtained a theory for RT and RM instabilities, that is, at any Atwood number $h_b\propto t^{0.4}$, and when $A\to 1$, $h_s\propto t$. Nevertheless, according to the experimental results in this paper ($A\to 1$), at $Ma=1.10$, $h_s \& h_b\propto t$; at $Ma=1.25$, $h_s \& h_b\propto t^2$. The reasons for nonlinear growth may be complex. It is also related to R-T instability and K-H instability in addition to the complex wave structure effects as just described. In addition, the growth trend of R-T instability is exponential. Therefore, we give the following formula for a better explanation:

$$h(t)=h_{\mathrm{RM}}(t)+h_{\mathrm{RT}}(t)+h_{\mathrm{KH}}(t)+\dots$$

(4)

4. Conclusions

In this paper, we reported the first study on global Richtmyer–Meshkov instability along the axial direction of liquid cylinders. The experimental results reveal that the spike height, bubble depth and turbulent mixing zone can grow linearly with time when the Mach number of shock wave is not great. As the increase in Mach number, the RM instability can become nonlinear. For multiple liquid cylinders, the reflection and diffraction of shock wave from the neighbour cylinders may bring a nonlinear effect into RM instability. The experimental data of this paper are measured at a relatively later stage of the instability. A direct measurement on the very early stage of the instability on a gas/liquid interface is expected in future work.