Bursting bubbles and the formation of gas jets and vortex rings

Abstract

Bubble bursting is important in air–sea interactions, food science, and industry, but the process of the pressurized gas escaping from inside the bursting bubble is not well understood. The fluid dynamics of gas jets and vortex rings produced by the bursting of 440 µm to 4 cm diameter smoke-filled bubbles resting at an air–water interface is investigated using high-speed stereophotogrammetry. The initial speed of the gas jet released from the bubbles increases with parent bubble size until the Bond number reaches unity and subsequently increases more slowly. The slow, low Reynolds number jets characteristic of small bubbles are attributed to high film retraction speeds which produce relatively large holes in the bubble cap, and these jets roll up into spherical, slow-growing vortex rings which travel short distances. However, the low film retraction speeds characteristic of larger bubbles produce high speed, high Reynolds number jets emitted through relatively small holes which roll up into highly oblate, fast-growing, far-traveling vortex rings. The tiniest bubbles eject only a thin stem-like jet which does not form a vortex ring. Finally, a simple scaling relationship relating the gas jet Reynolds number to the square root of the parent bubble Bond number is proposed.

Graphic abstract

Appendix

1.1 Modeling

We use a quasi-one-dimensional nozzle model to theoretically predict the velocity of the gas jet generated by the bubble bursting. It is assumed that the shape of the bubble is spherical for small bubbles (2R < 10 mm) and hemispherical for large bubbles, and that the temperature of the gas inside of the bubble is the same as that of the surrounding air. We also assume the volume of the bubble is constant in the time interval of interest. If there is no choking in the flow, the mass flow rate of the gas, $\dot{m}$, in steady isentropic compressible flow through a quasi-one-dimensional nozzle is given by Anderson (2003) as

$$\dot{m} = C_{\text{d}} \frac{{AP_{\text{b}} }}{{\sqrt {\Re T_{\text{b}} } }}\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{P}{{P_{\text{b}} }}} \right)^{{\frac{2}{\gamma }}} \left[ {1 - \left( {\frac{P}{{P_{\text{b}} }}} \right)^{{\frac{\gamma - 1}{\gamma }}} } \right]} ,$$

(4)

where $P$ is the pressure, $T$ is the temperature, $\Re$ is the specific gas constant, $\gamma$ is the specific heat ratio, $A$ is the nozzle exit area (i.e., the area of the hole in the bubble), and $C_{\text{d}}$ is the discharge coefficient. Here the subscript ${\text{b}}$ represents the properties of the air inside the bubble. In addition, the mass flow rate through the opening hole in the bubble film satisfies the following relation:

$$\dot{m} = \rho v_{\text{h}} A_{\text{h}} ,$$

(5)

where $v_{\text{h}}$ and $A_{\text{h}}$ are the velocity through the hole and the area of the hole, respectively, and ρ is the density of air. The initial pressure inside the bubble $P_{{{\text{b}}0}}$ can be given from the Young–Laplace equation as

$$\Delta P = P_{{{\text{b}}0}} - P = \frac{4\sigma }{R},$$

(6)

where P = 101325 Pa is the atmospheric pressure. When the gas is released through the hole in the bubble cap, choking may occur if

$$\frac{{P_{\text{b}} }}{P} \ge \left( {\frac{2}{\gamma + 1}} \right)^{{\frac{\gamma }{1 - \gamma }}} .$$

(7)

For air $(\gamma = 1.4),$ the condition is $\frac{{P_{\text{b}} }}{P} \ge 1.893.$ For the smallest bubble considered in this model (2R = 4.9), the pressure ratio $\frac{{P_{\text{b}} }}{P} = 1.002$ is much smaller than the choking condition. Therefore, choking cannot occur for any of the bubbles we consider in this model.

If we assume that both the bubble volume V_b and the pressure inside the bubble remain constant over the time period of interest during which the gas is initially released ($P_{\text{b}} = P_{{{\text{b}}0}}$ and V_b = V_b0), then using Eqs. (4) and (5) with the assumption $T \approx T_{\text{b}}$ we can obtain a “constant P and V” model for the velocity at the hole:

$$\left. {v_{\text{h}} } \right|_{{P_{\text{const}} V_{\text{const}} }} = C_{\text{d}} \sqrt {\Re T_{\text{b}} } \left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{{{\text{b}}0}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$

(8)

In reality, both the pressure inside the bubble and the bubble volume can be expected to decrease as the gas is released. We now consider an alternate “constant V” model in which the pressure in the bubble is allowed to decrease but the bubble volume remains constant over the time of interest. The mass reduction rate of the gas in the bubble then can be written using the equation of state of an ideal gas $P_{\text{b}} = \rho_{\text{b}} \Re T_{\text{b}}$ and the assumption $T \approx T_{\text{b}}$ as

$$\frac{{{\text{d}}m_{\text{b}} }}{{{\text{d}}t}} = \frac{{{\text{d}}\left( {\rho_{\text{b}} V_{\text{b}} } \right)}}{{{\text{d}}t}} \approx \frac{{V_{\text{b}} }}{\Re T}\frac{{{\text{d}}P_{\text{b}} }}{{{\text{d}}t}}.$$

(9)

From Eqs. (4) and (9), we can obtain the following equation for the pressure ratio $\frac{{P_{\text{b}} }}{P}$:

$$\frac{{{\text{d}}\left( {P_{\text{b}} /P} \right)}}{{{\text{d}}t}} = - C_{\text{d}} \sqrt {\Re T_{\text{b}} } \frac{{A_{\text{h}} }}{{V_{\text{b}} }}\left( {\frac{{P_{\text{b}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{\text{b}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{\text{b}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$

(10)

The above equation can be numerically integrated using an ordinary differential equation solver such as the fourth-order Runge–Kutta scheme. The equation of the velocity at the hole for this “constant V” model is then given as

$$\left. {v_{\text{h}} } \right|_{{V_{\text{const}} }} = C_{\text{d}} \sqrt {\Re T_{\text{b}} } \left( {\frac{{P_{\text{b}} }}{P}} \right)\sqrt {\left( {\frac{2\gamma }{\gamma - 1}} \right)\left( {\frac{{P_{\text{b}} }}{P}} \right)^{{ - \frac{2}{\gamma }}} \left[ {1 - \left( {\frac{{P_{\text{b}} }}{P}} \right)^{{\frac{1 - \gamma }{\gamma }}} } \right]} .$$

(11)

However, the jet speed at the hole could not be measured from experiments whereas the speed of the jet front could be measured. Thus, to compare model and experimental results, the velocity at the jet front $V_{\text{jet}}$ can be derived from the total momentum conservation relation once the velocity at the hole is obtained. Thus, consider the gas jet at $t = t_{\text{jet}}$ of Fig. 2c. The total momentum of this gas jet is

$${\mathbf{M}}_{\text{jet}} = \mathop \int \limits_{0}^{{L_{\text{jet}} }} v\rho A{\text{d}}l,$$

(12)

where $L_{\text{jet}}$ is the length of the gas jet from the bubble film to the jet front. Since the gas jet can be considered as incompressible flow, using the mass conservation relation $\rho vA = \rho V_{\text{jet}} A_{\text{jet}} ,$ Eq. (12) can be rewritten as

$${\mathbf{M}}_{\text{jet}} = \rho V_{\text{jet}} A_{\text{jet}} L_{\text{jet}} .$$

(13)

Here, as shown in Fig. 2c, A_jet is the cross sectional area of the jet front. Because ${\mathbf{M}}_{\text{jet}}$ is equal to the total amount of the momentum incoming through the hole from $t = 0$ to $t_{\text{jet}} ,$

$${\mathbf{M}}_{\text{in}} = \mathop \int \limits_{0}^{{t_{\text{jet}} }} \dot{m}v_{\text{h}} {\text{d}}t = \rho \mathop \int \limits_{0}^{{t_{\text{jet}} }} A_{\text{h}} v_{\text{h}}^{2} {\text{d}}t,$$

(14)

the velocity at the jet front can be given as

$$V_{\text{jet}} = \frac{1}{{A_{\text{jet}} L_{\text{jet}} }}\mathop \int \limits_{0}^{{t_{\text{jet}} }} A_{\text{h}} v_{\text{h}}^{2} {\text{d}}t.$$

(15)

To compute values of V_jet from both the “constant P and V” and “constant V” versions of the model, values of L_jet, A_jet, A_h, and C_d are required. The jet length L_jet, jet frontal diameter D_jet, and the opening hole diameter D_h were measured for bubble diameters of 2R = 4.90, 6.22, 11.2, 15.1, 16.8, 21.1, 22.9, 26.5, 28.2, and 41.0 mm. Smaller bubbles were not modeled because the bubble cap had completely disintegrated by this time point. The parameters A_h and A_jet were then calculated assuming a circular cross section of the expanding hole in the bubble film and of the jet, respectively.

To integrate these equations over time from t = 0 ms to t_jet = 0.4 ms, it was necessary to determine A_h as a function of time. Thus, for bubbles with 2R > 10 mm, the diameter of the hole in the bubble cap film $D_{\text{h}}$ is interpolated with a quadratic curve as a function of time, using a diameter of zero at t = 0 ms and measured values of $D_{\text{h}}$ at t = 0.2 and 0.4 ms. For bubbles with 2R < 10 mm, D_h was linearly interpolated between t = 0 ms and t = 0.2 ms and again between t = 0.2 ms and t = 0.4 ms because a quadratic fit resulted in negative values of D_h. Then, the hole area is readily obtained as $A_{\text{h}} = \frac{\pi }{4}D_{\text{h}}^{2}$. Finally, the value of the discharge coefficient is obtained by fitting the data of Hollingshead (2011) for a Venturi meter (smooth beta in the range of 100 < Re < 1000: $C_{\text{d}} = - 0.1413(\log Re_{\text{jet}} )^{2} + 0.9085\log Re_{\text{jet}} - 0.542).$ Values of C_d ranged from C_d = 0.75051 for a 2R = 4.9 mm bubble up to C_d = 0.89278 for a 2R = 41.0 mm bubble. The model was solved using MATLAB software.

1.2 Model results

Figure 14 shows the comparison of V_jet from the “constant P and V” and “constant V” theoretical models with the experimental results at t = 0.4 ms. Measured values of the model parameters L_jet, D_jet, and D_h and calculated values of Re_jet at t = 0.4 ms are shown in the inset of Fig. 14. In general, the “constant P and V” model overestimates the experimental data. The model most closely matches the experimental data for the largest bubble size (2R = 41 mm), where the percent difference between experimental and model values is 12.7%. However, with decreasing bubble size, the model and experimental values substantially diverge, with the model overestimating the experimental value by approximately six fold for the bubble with 2R = 4.9 mm. In addition, V_jet decreases with increasing bubble diameter in the model whereas V_jet shows the opposite trend in the experimental data. This “constant P and V” model unrealistically assumes that the bubble volume and the pressure in the bubble do not change as the gas inside is released in a jet. The constant volume assumption is most realistic for the larger bubble sizes. As seen in Fig. 3 for a large bubble (2R = 41 mm), the bubble volume changes negligibly from time of bursting to t = 0.4 ms. In contrast, the bubble cap films of small bubbles are largely retracted or even destroyed by t = 0.4 ms (e.g., Fig. 5), leading to large reductions in bubble volume. In addition, the pressure decreases substantially as the gas is released, but this decrease in pressure likely depends on bubble size. If the bubble is small, the pressure reduction in the bubble is relatively large because the amount of the released gas is large compared to the initial bubble volume. However, this ratio of the released gas to the initial bubble volume decreases as the bubble size increases. The pressure reduction in the bubble thus decreases accordingly for larger bubbles. The “constant P and V” model thus matches the experimental data most closely for the largest bubble size with the smallest hole opening speed for which this condition most closely holds (e.g., Fig. 3) and for which the bubble volume does not change much. In contrast, the “constant P and V” model fails for small bubbles with the largest hole opening speeds (e.g., Fig. 5) and large changes in bubble volume.

Fig. 14

Comparison of measured gas jet front speeds and theoretical gas jet speeds from “constant pressure and volume” and “constant volume” models as a function of bubble equivalent diameter 2R at t = 0.4 ms. The inset shows measured values of the hole diameter D_h, jet length L_jet, and jet frontal diameter D_jet and calculated values of jet Reynolds number Re_jet at t = 0.4 ms as a function of bubble equivalent diameter 2R

In contrast to the “constant P and V” model, the “constant V” model consistently underestimates V_jet, generating similar values (of 0.6–1.2 m/s) regardless of the bubble size. It is believed that the main reason for this underestimation is that the pressure drop in the bubble in the model occurs much faster than in reality. For example, the “constant P and V” model predicts that the pressure inside the bubble with 2R = 21.1 mm decreases to the atmospheric pressure after about t = 0.2 ms and the jet velocity at the hole becomes zero accordingly. However, Fig. 4 shows that, in a similarly sized bubble, the gas continues to be released well after that time. The faster pressure drop in the model thus results in jet speeds slower than those measured in the experiments. One possible reason for this discrepancy is that, while the gas jet is being released, a bubble shrinks slightly, which can lead to an increase in pressure inside the bubble or at least slow the decrease in pressure over time. For example, according to the model proposed by Lhuissier and Villermaux (2011), the pressure inside the bubble increases with decreasing bubble size as it “deflates”. However, this assumption leads to jet speeds which increase with time, which contradicts the jet speeds measured here. Two other reasons may contribute to the mismatch between experiments and theory. First, the gas jet is actually a highly unsteady flow in contrast to the steady flow assumption. Second, it is assumed that the pressure inside the bubble has a constant pressure of $P_{\text{b}}$ spatially, but in reality, the pressure inside the bubble varies depending on the location. It is noted that the computed jet velocities of both models are inversely proportional to the jet area and the jet length as shown in Eq. (15). The hole area linearly increases with the jet velocity for the “constant P and V” model while it has little effect on the jet velocity for the “constant V” model due to a fast depressurization. Finally, the two models can be considered as the upper and the lower bounds of the experimental results because it is expected that the actual pressure drop inside the bubble is between those of the models.