^{1,a)}, John H. J. Niederhaus

^{1}, Jason G. Oakley

^{1}, Mark H. Anderson

^{1}, Riccardo Bonazza

^{1,b)}and Jeffrey A. Greenough

^{2}

### Abstract

The interaction of a planar shock wave with a spherical bubble in divergent shock-refraction geometry is studied here using shock tube experiments and numerical simulations. The particular case of a helium bubble in ambient air or nitrogen is considered, for . Experimental planar laser diagnostics and three-dimensional multifluid Eulerian simulations clearly resolve features arising as a consequence of divergent shock refraction, including the formation of a long-lived primary vortex ring, as well as counter-rotating secondary and tertiary upstream vortex rings that appear at late times for . Remarkable correspondence between experimental and numerical results is observed, which improves with increasing , and three-dimensional effects are found to be relatively insignificant. Shocked-bubble velocities, length scales, and circulations extracted from simulations and experiments are used successfully to evaluate the usefulness of various analytical models, and characteristic dimensionless time scales are developed that collapse temporal trends in these quantities. Those linked directly to baroclinicity tend to follow time scales based on shock wave speeds, while those linked to interface deformation and vortex- or shear-induced motion tend to follow a time scale based on the postshock flow speed, though no single time scale is found to be universally successful.

This work was partially supported by the U.S. Department of Energy Grant No. DE-FG52-06NA26196. This work was also partially performed under the auspices of the U.S. Department of Energy at the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.

I. INTRODUCTION

II. SHOCK-BUBBLE INTERACTIONS IN DIVERGENT GEOMETRY: OVERVIEW

III. EXPERIMENTAL DESIGN

IV. COMPUTATIONAL SCHEME AND SETUP

V. FLOW VISUALIZATION

A. Helium bubble in air,

B. Helium bubble in air,

C. Helium bubble in air,

D. Helium bubble in air at late postshock times

VI. RESULTS: VORTEX RING VELOCITY

VII. RESULTS: LENGTH SCALES

VIII. RESULTS: CIRCULATION

IX. CONCLUSIONS

### Key Topics

- Shock waves
- 91.0
- Vortex rings
- 87.0
- Bubble dynamics
- 58.0
- Mach numbers
- 37.0
- Vortex dynamics
- 35.0

## Figures

Schematic view of a shock-bubble interaction in divergent refraction geometry : (a) compression phase, just before shock passage; (b) formation of primary vortex ring; (c) seeding of secondary vortex ring; (d) late-time separation of primary and secondary vortex rings.

Schematic view of a shock-bubble interaction in divergent refraction geometry : (a) compression phase, just before shock passage; (b) formation of primary vortex ring; (c) seeding of secondary vortex ring; (d) late-time separation of primary and secondary vortex rings.

Schematic view of experimental setup (not drawn to scale), showing the configuration (1) before and (2) after retraction of the bubble injector.

Schematic view of experimental setup (not drawn to scale), showing the configuration (1) before and (2) after retraction of the bubble injector.

Sauter mean diameter of spherical droplets produced in shock-wave atomization of a spherical drop having the same mass as the experimental film layer, plotted against the Mach number of the incident shock wave.

Sauter mean diameter of spherical droplets produced in shock-wave atomization of a spherical drop having the same mass as the experimental film layer, plotted against the Mach number of the incident shock wave.

Schematic view of three-dimensional Cartesian mesh used for the present simulations (not drawn to scale). “S” indicates symmetry boundary conditions; “O” indicates outflow boundary conditions.

Schematic view of three-dimensional Cartesian mesh used for the present simulations (not drawn to scale). “S” indicates symmetry boundary conditions; “O” indicates outflow boundary conditions.

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 2.4, (b) 2.5, (c) 11.5, (d) 2.4, (e) 4.8, (f) 7.7, (g) 21.6, (h) 38.7, (i) 57.9, (j) 21.8, (k) 38.9, and (l) 57.9. The width of the field of view in each image is .

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 2.4, (b) 2.5, (c) 11.5, (d) 2.4, (e) 4.8, (f) 7.7, (g) 21.6, (h) 38.7, (i) 57.9, (j) 21.8, (k) 38.9, and (l) 57.9. The width of the field of view in each image is .

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 2.0, (b) 6.6, (c) 41.7, (d) 2.2, (e) 5.8, and (f) 37.9. The width of the field of view in each image is .

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 2.0, (b) 6.6, (c) 41.7, (d) 2.2, (e) 5.8, and (f) 37.9. The width of the field of view in each image is .

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 1.3, (b) 4.0, (c) 7.7, (d) 1.3, (e) 4.1, (f) 7.7, (g) 11.4, (h) 11.6, (i) 23.8, (j) 11.5, (k) 11.8, and (l) 23.6. The width of the field of view in each image is .

Flowfield evolution for . Numerical images show (on the left) the total density with the isosurface of plotted in red, and (on the right) vorticity magnitude. Dimensionless times are (a) 1.3, (b) 4.0, (c) 7.7, (d) 1.3, (e) 4.1, (f) 7.7, (g) 11.4, (h) 11.6, (i) 23.8, (j) 11.5, (k) 11.8, and (l) 23.6. The width of the field of view in each image is .

Late-time flowfield visualizations. Numerical images show (on the left) the helium volume fraction on a logarithmic gray scale peaked at , and (on the right) vorticity magnitude . The left column [(a), (d)] is , the center column [(b), (e)] is , and the right column [(c), (f)] is . Dimensionless times are (a) 105.8, (b) 63.4, (c) 69.5, (d) 105.8, (e) 63.5, and (f) 63.5. The width of the field of view in each image is . White dashed lines denote cropping locations for experimental images.

Late-time flowfield visualizations. Numerical images show (on the left) the helium volume fraction on a logarithmic gray scale peaked at , and (on the right) vorticity magnitude . The left column [(a), (d)] is , the center column [(b), (e)] is , and the right column [(c), (f)] is . Dimensionless times are (a) 105.8, (b) 63.4, (c) 69.5, (d) 105.8, (e) 63.5, and (f) 63.5. The width of the field of view in each image is . White dashed lines denote cropping locations for experimental images.

Normalized translation speed of primary vortex ring in simulations, plotted on a dimensionless time scale based on the ambient shocked flow speed . Horizontal lines on the right margin indicate normalized translation speeds predicted by the Picone–Boris model.

Normalized translation speed of primary vortex ring in simulations, plotted on a dimensionless time scale based on the ambient shocked flow speed . Horizontal lines on the right margin indicate normalized translation speeds predicted by the Picone–Boris model.

Streamwise dimension of the shocked bubble, plotted on a dimensionless time scale based on (a) the incident shocked ambient flow speed , (b) the incident shock wave speed , and (c) the transmitted shock wave speed . Symbols represent experimental data; lines represent simulation results.

Streamwise dimension of the shocked bubble, plotted on a dimensionless time scale based on (a) the incident shocked ambient flow speed , (b) the incident shock wave speed , and (c) the transmitted shock wave speed . Symbols represent experimental data; lines represent simulation results.

Lateral dimension of the shocked bubble, plotted on a dimensionless time scale based on (a) the incident shocked ambient flow speed , (b) the incident shock wave speed , and (c) the transmitted shock wave speed . Symbols represent experimental data; lines represent simulation results.

Lateral dimension of the shocked bubble, plotted on a dimensionless time scale based on (a) the incident shocked ambient flow speed , (b) the incident shock wave speed , and (c) the transmitted shock wave speed . Symbols represent experimental data; lines represent simulation results.

Schematic diagram of shock-bubble interaction at the instant of shock passage, showing integration path and flow variables used in the one-dimensional-gasdynamics reconstruction model for circulation, given in Eqs. (24) and (25).

Schematic diagram of shock-bubble interaction at the instant of shock passage, showing integration path and flow variables used in the one-dimensional-gasdynamics reconstruction model for circulation, given in Eqs. (24) and (25).

Decomposed circulation for the scenario, plotted on a dimensionless time scale based on . (Dark lines) Numerical results obtained using Eq. (19). (Open symbols) Numerical results obtained using Eq. (21) (Kelvin’s model). (Closed symbols) Experimental results obtained using Eq. (21). (Light lines) Analytical models.

Decomposed circulation for the scenario, plotted on a dimensionless time scale based on . (Dark lines) Numerical results obtained using Eq. (19). (Open symbols) Numerical results obtained using Eq. (21) (Kelvin’s model). (Closed symbols) Experimental results obtained using Eq. (21). (Light lines) Analytical models.

Decomposed circulation for the scenario, plotted on a dimensionless time scale based on . (Dark lines) Numerical results obtained using Eq. (19). (Open symbols) Numerical results obtained using Eq. (21) (Kelvin’s model). (Closed symbols) Experimental results obtained using Eq. (21). (Light lines) Analytical models.

Normalized circulation: ratio of the positive component of circulation obtained from simulations to the circulation predicted by the one-dimensional gasdynamics reconstruction (R) model.

Normalized circulation: ratio of the positive component of circulation obtained from simulations to the circulation predicted by the one-dimensional gasdynamics reconstruction (R) model.

## Tables

Initial properties of gases present in the calculations. The initial pressure and temperature in the system are and , respectively.

Initial properties of gases present in the calculations. The initial pressure and temperature in the system are and , respectively.

Overview of flow parameters for three simulations, including the incident shock Mach number , the Atwood number at the unshocked interface, and lab-frame speeds , , and of the incident shock wave, shocked ambient gas, and transmitted shock wave, respectively.

Overview of flow parameters for three simulations, including the incident shock Mach number , the Atwood number at the unshocked interface, and lab-frame speeds , , and of the incident shock wave, shocked ambient gas, and transmitted shock wave, respectively.

Normalized primary vortex ring translation speed, computed from simulations, measured in experiments, and predicted by the Picone–Boris (PB) and Rudinger–Somers (RS) models. In experiments and simulations, the value is computed as a mean of velocities measured for .

Normalized primary vortex ring translation speed, computed from simulations, measured in experiments, and predicted by the Picone–Boris (PB) and Rudinger–Somers (RS) models. In experiments and simulations, the value is computed as a mean of velocities measured for .

Mean circulation extracted from simulations and experiments for , and predicted by analytical models. Circulation values are given in units of .

Mean circulation extracted from simulations and experiments for , and predicted by analytical models. Circulation values are given in units of .

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