Dynamics of jets produced by bursting bubbles

Sea spray is the main source of aerosols above the ocean. One of the pathways for sea spray production is through bubble bursting, which ejects myriads of droplets. We present a detailed description of the velocity of jets formed by bubble bursting, obtained through extensive comparison between experimental results and numerical simulations for a wide range of physical parameters. We discuss the importance of the shape of the cavity on the jet velocity and the regime of parameters for which drop ejection is observed. We present a phenomenological formula that predicts the jet velocity for the full range of parameters and compare it to a theoretical prediction based on curvature reversal. The results are then discussed in light of their fundamental applications in the understanding of the phenomena.

Figure 1

Direct numerical simulations of the axisymmetric Navier-Stokes equations, using Gerris, of a bursting bubble and subsequent jet formation. Top: Bond number $\text{Bo}=0.1172$, and Laplace number $\text{La}=6.7\times {10}^4$, or Morton number $\text{Mo}=2.63\times {10}^{-11}$ corresponding to a bubble radius $R_0=934\mu \text{m}$ in water. Several fast capillaries propagate toward the center of the cavity, with significant vorticity being generated, and leading to the creation of a jet, that then pinches off to release droplets. Times of the snapshots are $t/t_{\text{ic}}=0,0.16,0.32,0.40,0.48,0.54,$ and 0.60. Bottom: bursting close to the optimal Laplace number, with bond number $\text{Bo}=0.01$, and Laplace number $\text{La}={10}^3$. During the cavity collapse, a single capillary wave is seen propagating toward the center with relatively low vorticity being generated before focusing. The collapse can be described by a self-similar dynamics and gets close to the finite-time singularity. At focusing, a very strong vorticity field is observed, associated with a very thin and fast jet, small drops being quickly ejected. Times of the snapshots are $t/t_{\text{ic}}=0,0.16,0.32,0.43,0.45,0.46,$ and 0.48. The color field represents vorticity $\mathrm{\Gamma }$ in the gas and liquid (with maximum vorticity in red, $\mathrm{\Gamma }{t}_{\text{ic}}=190$, and minimum in blue $\mathrm{\Gamma }{t}_{\text{ic}}=-190$). Maximum refinement corresponds to a grid equivalent to ${\left(2^{13}\right)}^2$, corresponding to 1638 grid points per bubble diameter.

Figure 2

Jet dynamics of a bursting bubble in water, bubble radius is $R_0=934\mu \text{m}$. Bond number $\text{Bo}=0.1172$, Laplace number $\text{La}=6.7\times {10}^4$, or Morton number $\text{Mo}=2.63\times {10}^{-11}$. Left: Experimental time sequences superposed with numerical profiles. The top four panels show the event below the free surface, while the bottom four panels display the bubble above the free surface. The top four panels take place before the first image of the bottom panels. Excellent agreement between the numerical and experimental data is observed for the cavity collapse dynamics, as well as for the jet formation and the first ejected droplet. Differences appear for the dynamics of the jet after the first drop is ejected and for the following satellite drops. Right: Time evolution of the vertical velocity of the apex of the cavity, becoming the tip of the jet, $\text{Ca}\left(t\right)=\left[\rho \nu V_{\text{tip}}\left(t\right)\right]/\gamma$. Time is shifted by $t_0$, the time at which the jet crosses the $z=0$ reference point and normalized by $t_{\text{ic}}$. Black circles are laboratory data from Ref. [7], while color lines are DNS with increasing resolutions (from $2^{11}$ to $2^{13}$ equivalent grid size). Excellent agreement is observed between the laboratory and numerical results, as well as mesh convergence. The velocity of the jet is still decreasing when it crosses the $z=0$ horizontal line, and it reaches a constant value before the formation of the first drop, which we define as $V_{\mathrm{tip}}$. The first drop detachment is indicated with an arrow. Inset: same data for the vertical position of the central point of the cavity, becoming the tip of the jet.

Figure 3

Experimental time sequences superposed with numerical profiles of a typical jetting event following a bubble bursting at a free surface of silicon oil. The top four panels show the event as seen from above the free surface, while the bottom four panels display the bubble under the free surface. The bottom three right panels take place between the first two images of the top panels. Bubble radius is $R_0=904\mu \text{m}$. Bond number $\text{Bo}=0.14$, Laplace number $\text{La}=1697$, or Morton number $\text{Mo}=4.86\times {10}^{-8}$. A single capillary wave propagates towards the center of the cavity, creating a sharp corner and leading to a very thin jet, corresponding to the optimal Laplace number and a dynamic close to the finite-time singularity. Very good agreement between experimental and numerical profiles is observed up to the formation of the first drop. The jet velocity $V_{\mathrm{tip}}$ is defined as the velocity before drop ejection.

Figure 4

Left panels (a, c, e) Time evolution of the cavity profile, showing the collapse and jet formation for three Laplace numbers, at $\text{Bo}=0.01$. Time is color coded and increases from blue to red, with $\mathrm{\Delta }t=0.03t_{\text{ic}}$ between each profile. (a) $\text{La}={10}^4$: several capillary waves are seen propagating toward the center, creating a jet that eventually pinches off and forms drops. No air bubble entrainment is observed. (c) $\text{La}=500$: a single capillary wave is visible propagating toward the center of the cavity, leading to a thinner jet, which does not pinch off. Air entrainment in the liquid is observed when the jet is formed. (e) $\text{La}=1000$, the self-similar collapse comes very close to a finite time singularity, with a single capillary wave focusing to the center of the cavity, leading to a very fast and thin jet, which pinches off quickly and forms drops. Air entrainment is observed when the jet is formed. Right panels (b, d, f) Time evolution of the vertical velocity of the apex of the cavity, becoming the tip of the jet $\text{Ca}\left(t\right)=\left[\rho \nu V_{\text{tip}}\left(t\right)\right]/\gamma$, for three different Laplace numbers and three Bond numbers, representative of the dynamics at high Laplace number ($\text{La}={10}^4$), low Laplace number with no drop formation ($\text{La}=500$), and Laplace number close to the capillary singularity ($\text{La}=1000$). Time is normalized, $(t-t_0)/t_{\text{ic}}$. For each case, the velocity for three Bond numbers is shown ($\text{Bo}=0.1, \text{Bo}=0.01, \text{Bo}=0.001$), with the velocity decreasing when increasing the Bond number. (f) $\text{La}=1000$, the dynamic is close to the finite time singularity for the smallest Bond numbers, while the velocity of the jet is much smaller in the $\text{Bo}=0.1$ case, due to gravity effects on the shape of the cavity. Arrows indicate the time just before drop formation.

Figure 5

Color symbols are the runs at a constant $\text{Mo}$ (and by definition $\text{La}={(\text{Bo}/\text{Mo})}^{1/2}$). Each color is a different liquid with increasing Morton number: blue, $\text{Mo}=2.63\times {10}^{-11}$ (water), light blue $\text{Mo}=5.99\times {10}^{-10}$, green $\text{Mo}=3.2\times {10}^{-9}$, light red $\text{Mo}=2.3\times {10}^{-8}$, and dark red $\text{Mo}=4.8\times {10}^{-8}$. DNS are open symbols (diamond and circles are $2^{12}$ and $2^{11}$ mesh size, respectively) and solid dots are experimental data. Mesh convergence is observed as well as good agreement between DNS and laboratory data. (a) Parameter space $(\text{La},\text{Bo})$ where drop formation is observed in the numerical simulations and experiments, with at least one drop ejected by the ascending jet. Black symbols are from the $(\text{Bo},\text{La})$ sweep. The black dashed line indicates a critical Laplace number below which no drop ejection is observed, ${\text{La}}_c=500$. The solid black line is the boundary between the regimes with and without drop ejection, $\text{Bo}\propto {\text{La}}^{3/2}$. (b) Jet capillary number $\text{Ca}=V_{\text{tip}}\mu /\gamma$ as a function of the Bond number. Dashed lines are $\text{Ca}\propto {\text{Bo}}^{-0.6}$ relationships for each working liquid, observed at high $\text{Bo}$ while a change of regime, or saturation of the capillary velocity is observed at lower $\text{Bo}$. (c) Jet capillary number $\text{Ca}$ as a function of $\text{La}$. The singularity is observed around $\text{La}\approx 1000$. At a given Laplace number, the jet capillary number decreases when increasing the Morton number, due to Bond number effects. Very good agreement is found between numerical and experimental data except near the singularity $\text{La}\approx 1000$, where some differences are visible due to the very high velocity of the jet. Solid lines are $\text{Ca}\propto {\text{La}}^{-1}$ relationships for each working liquid, observed at high $\text{La}$. (d) $\text{We}$ as a function of $\text{Bo}$. Solid lines are $\text{We}\propto {\text{Bo}}^{-0.6}$ relationships for each working liquid at high Bond number and dashed line is $\text{We}\propto {\text{Bo}}^{-0.33}$ at low Bond number for water. The value of the $\text{We}$ seems to saturate, for a given liquid, to a constant value and a $\text{We}\propto {\text{Bo}}^0$.

Figure 6

(a) Jet capillary number $\text{Ca}$ as a function of the Laplace number, for four different Bond numbers ($\text{Bo}=0.001$ in dark blue, $\text{Bo}=0.01$ in light blue, $\text{Bo}=0.1$ in yellow, and $\text{Bo}=0.5$ in red), the Bond number being color-coded. For $\text{La}>2000$, at a given Laplace number, the velocity increases with decreasing Bond number. The optimal Laplace number for which the highest velocity is observed increases with the Bond number, the highest value of the capillary jet velocity being obtained for small Bond numbers ($\text{Bo}\ll 0.1$). Asymptotically, at high Laplace numbers, the capillary jet velocity scales like $\text{Ca}\propto {\text{La}}^{-3/4}$, with a prefactor that depends on the Bond number. Dashed lines are the asymptotic solutions at high Laplace number $\text{Ca}\propto {\text{La}}^{-3/4}$. The solid lines are $\text{Ca}=k_v\left(\text{Bo}\right){\text{La}}^{-3/4}{({\text{La}}_{*}^{-1/2}-{\text{La}}^{-1/2})}^{-3/4}$ [Eq. (5)]. The critical Laplace number is ${\text{La}}_{*}={\text{La}}_c=500,$ below which no drop ejection is observed. The prefactor $k_v\left(\text{Bo}\right)$ is fitted to the data and we find $k_v(\text{Bo}=0)=k_v(\text{Bo}\ll 0.1)=19$, while at finite Bond number, $k_v(\text{Bo}=0.1)=14$ and $k_v(\text{Bo}=0.5)=10$. Inset shows $k_v\left(\text{Bo}\right)$, with the fitted curve in black $k_v\left(\text{Bo}\right)/k_v(\text{Bo}=0)={(1+\alpha \text{Bo})}^{-3/4}$, with $\alpha =2.2$ fitted to the data. (b) Rescaling with a Bond number correction, $\text{Ca}$ as a function of $\text{La}(1+\alpha \text{Bo})$. All DNS data are presented, including the $(\text{Bo},\text{Mo})$ sweep. The solid line describes the regime for $\text{La}$ larger than the singularity value, with $\text{Ca}=k_v\left(\text{Bo}\right){\text{La}}^{-3/4}{({\text{La}}_{*}^{-1/2}-{\text{La}}^{-1/2})}^{-3/4}$, and $k_v\left(\text{Bo}\right)=k_v(\text{Bo}=0){\text{La}}_{*}^{3/8}{(1+\alpha \text{Bo})}^{-3/4}$. Diamond and triangles are $2^{12}$ mesh size, while squares and circles are $2^{11}$ mesh size.

Figure 7

Influence of the initial cavity shape on the jet velocity. $\text{Bo}=0.01$ in air-water ($\text{Mo}=2.63\times {10}^{-11}, \text{La}=1.96\times {10}^4$). (a) Initial shapes of the tested cavity: (i) the static bubble shape, (ii) the spherical cavity with exponential reconnection to the mean level, (iii) the spherical cavity cut at the mean water level, with large hole, (iv) the spherical cavity cut at the mean water level, with small hole. (b) Same figure with a zoom on the cap and hole to see the differences in hole size. (c) Jet capillary number as a function of the normalized time $(t-t_0)/t_{\text{ic}}$, for the four cavity shapes. Differences in the final tip velocity as large as 20% can be observed due to the change in the initial conditions. (d) Profiles of two cavities (static bubble shape and large hole) at different times, showing that the generated capillaries have different wavelengths when the hole size changes. Shorter capillary waves are selected for a smaller hole (static case), leading to faster waves, leading to a faster jet.

Figure 8

Influence of the initial cavity shape on the jet velocity. $\text{Bo}=0.1172$ in air-water ($\text{Mo}=2.63\times {10}^{-11}, \text{La}=7\times {10}^4$). (a) Initial shapes of the tested cavity: (i) the static bubble shape, (ii) the spherical cavity with exponential reconnection to the mean level, (iii) the spherical cavity cut at the mean water level, with large hole, (iv) the spherical cavity cut at the mean water level, with small hole. (b) Same figure with a zoom on the cap and hole to see the differences in hole size. (c) Jet capillary number $\text{Ca}$ as a function of the normalized time, $(t-t_0)/t_{\text{ic}}$, for the four cavity shapes. Differences in the final tip velocity as large as 30% can be observed due to the change in the initial conditions. (d) Profiles of two cavities (static bubble shape, and large hole) at different times, showing that the generated capillaries have different wavelength when the hole size is changed. Smaller capillaries wavelengths are selected for a smaller hole (static case), leading to faster waves, leading to a faster ejected jet.