Numerical simulations of the early stages of high-speed droplet breakup

Abstract

Experiments reported in the literature are reproduced using numerical simulations to investigate the early stages of the breakup of water cylinders in the flow behind normal shocks. Qualitative features of breakup observed in the numerical results, such as the initial streamwise flattening of the cylinder and the formation of tips at its periphery, support previous experimental observations of stripping breakup. Additionally, the presence of a transitory recirculation region at the cylinder’s equator and a persistent upstream jet in the wake is noted and discussed. Within the uncertainties inherent to the different methods used to extract measurements from experimental and numerical results, comparisons with experimental data of various cylinder deformation metrics show good agreement. To study the effects of the transition between subsonic and supersonic post-shock flow, we extend the range of incident shock Mach numbers beyond those investigated by the experiments. Supersonic post-shock flow velocities are not observed to significantly alter the cylinder’s behavior, i.e., we are able to effectively collapse the drift, acceleration, and drag curves for all simulated shock Mach numbers. Using a new method that minimizes noise errors, the cylinder’s acceleration is calculated; acceleration curves for all shock Mach numbers are subsequently collapsed by scaling with the pressure ratio across the incident shock. Furthermore, we find that accounting for the cylinder’s deformed diameter in the calculation of its unsteady drag coefficient allows the drag coefficient to be approximated as a constant over the initial breakup period.

Appendices

Appendix A: Grid resolution

As part of the grid resolution study, we simulated the Mach 1.47 shock wave case at three different resolutions. A time history of these simulations is shown in Fig. 19. Though the wake structure becomes more detailed as the grid is refined, the overall qualitative features of the breakup process remain similar. Features characteristic of stripping breakup, such as the initial flattening of the cylinder and the formation of tips at the cylinder’s periphery, are present at all three grid resolutions. The recirculation regions at the cylinder’s equator and the presence of an upstream jet in the wake are also observable at all levels of grid refinement. Quantitative measurements of cylinder deformation and center-of-mass properties, used in the comparison with experimental data, do not show significant differences between the original and doubled resolution. As an example, Fig. 20 plots the center-of-mass drift calculated at the three different grid resolutions, and shows negligible differences between the two finest grid sizes.

Fig. 19

Numerical schlieren images (top) and filled pressure contours (bottom) of the breakup of a 4.8 mm cylinder at $t^*=$ a 0.017 b 0.171 c 0.262 d 0.444 e 0.626 f 0.808 g 1.036 behind a Mach 1.47 shock. Isocontours are shown for $\alpha _l \ge 0.9$. Grid resolutions correspond to 600x300 (left), $1,\!200 \times 600$ (middle), and $2,\!400 \times 1,\!200$ (right) cells

Fig. 20

Center-of-mass drift measurements at three grid resolutions

Appendix B: Experimental visualization comparison

The holographic interferograms of [19] used in Fig. 6 were originally stated to be at $t = 23\,\upmu$s and $t = 43\,\upmu$s after “the interaction between the incident shock wave and the water column” [19]. We interpret this to mean the time after the shock reaches the leading edge of the water cylinder. A comparison of the experimental interferograms and the numerical schlieren images from our simulations at these times is shown in Fig. 21. It is clear from the figure that any comparison is difficult to make since the images appear to be taken at different times. In an attempt to reconcile the discrepancy, digital measurements of the distance traversed by the incident shock were taken from the interferograms. Our measurements indicate that the times should perhaps be closer to 16 and 32 $\upmu$s, respectively. Numerical schlieren images at these modified times are compared to the experimental interferograms in Fig. 6, and are seen to match the incident and reflected shock locations. There is an inherent uncertainty in the exact location of the boundary of the water cylinder, owing to the thick ring on the holographic interferograms. Measurements to obtain times of 16 and 32 $\upmu$s were taken by assuming the boundary to be located in the middle of the thick ring. Measurements taken from the edge of the ring resulted in alternate times of approximately 22 and 42 $\upmu$s, respectively, which are closer to the reported times in [19]. It is unclear whether this discrepancy in time is a result of a reporting error in the original work, or a misinterpretation, on our part, of what is meant by “the interaction between the incident shock wave and the water column” [19].

Fig. 21

Holographic interferograms (left) [19] and numerical schlieren images (right) at $t=$ a 23$\,\upmu$s b 43$\,\upmu$s. Reprinted from [19] with permission from Springer