Interactions of multiple spark-generated bubbles with phase differences

Abstract

This paper aims to study the complex interaction between multiple bubbles, and to provide a summary and physical explanation of the phenomena observed during the interaction of two bubbles. High-speed photography is utilized to observe the experiments involving multiple spark-generated bubbles. Numerical simulations corresponding to the experiments are performed using the Boundary Element Method (BEM). The bubbles are typically between 3 and 5 mm in radius and are generated either in-phase (at the same time) or with phase differences. Complex phenomena are observed such as bubble splitting, and high-speed jetting inside a bubble caused by another collapsing bubble nearby (termed the ‘catapult’ effect). The two-bubble interactions are broadly classified in a graph according to two parameters: the relative inter-bubble distance and the phase difference (a new parameter introduced). The BEM simulations provide insight into the physics, such as bubble shape changes in detail, and jet velocities. Also presented in this paper are the experimental results of three bubble interactions. The interesting and complex observations of multiple bubble interaction are important for a better understanding of real life applications in medical ultrasonic treatment and ultrasonic cleaning. Many of the three bubble interactions can be explained by isolating bubble pairs and classifying their interaction according to the graph for the two bubble case. This graph can be a useful tool to predict the behavior of multiple bubble interactions.

Appendix: The boundary element method

Numerical simulations serve to enhance the understanding of complex bubble interactions by providing details of the bubble shape changes (when experimental spatial and temporal resolutions are not enough), and by giving other valuable information such as the jet velocity or pressure plots, which are often not accessible from experimental data. Bubble dynamics phenomena are known to be largely governed by inertial effects. It is therefore possible to introduce a velocity potential in the fluid (see Blake et al. 1986; Plesset and Prosperetti 1977; Klaseboer et al. 2005b). The Boundary Element Method (BEM) establishes a relationship between this potential, ϕ, and the normal velocity $v_{n} = {\mathbf{n}} \cdot \nabla \phi = \partial \phi /\partial n$ on the boundaries ‘S’ of all bubbles. The unit normal vector pointing out of the fluid domain is denoted as n. This relationship can be written based on Green’s second identity (Wrobel 2002) as

$$ c({\mathbf{x}})\phi ({\mathbf{x}}) + \int\limits_{s} {\left[ {\phi ({\mathbf{y}})\frac{{\partial G({\mathbf{x}},{\mathbf{y}})}}{{\partial n({\mathbf{y}})}} - v_{n} ({\mathbf{y}})\,G({\mathbf{x}},{\mathbf{y}})} \right]{\text{dS}}({\mathbf{y}}) = 0} , $$

(3)

where x is a vector pointing to a location situated on the surface S of any of the bubbles; y is a vector pointing to an integration point; G(x, y) = 1/|x − y| is the free space Green’s function and c(x) the solid angle. Thus, BEM only needs a mesh on the surfaces of the bubbles and not in the whole fluid domain, which considerably reduces the computing effort and storage space required. The bubble’s surfaces are divided into triangular linear elements. The potential on the bubble surface is updated for each time step using the unsteady Bernoulli equation

$$ \rho \frac{D\phi }{Dt} = p_{\text{ATM}} - p_{v} + \frac{1}{2}\rho \left| {\nabla \phi } \right|^{2}\,-\,p_{b} , $$

(4)

with D/Dt as the material or convective derivative, ρ the density of the surrounding fluid (water: 1,000 kg m⁻³), p _ATM the atmospheric pressure and p _v the vapor pressure of the bubble contents. The gas pressure inside the bubble is assumed to behave adiabatically, such that p _b  = p _b0(V ₀/V)^γ, where p _b0 and V ₀ are the initial pressure and volume, respectively. As usually employed in underwater explosion bubbles, the value of the isentropic coefficient is γ = 1.25 (see Cole 1948 or Klaseboer et al. 2005a, b). If (3) is applied for each node, then a matrix equation can be found relating all potentials and normal velocities of all nodes for all bubbles, which can be solved for the unknown normal velocities. The positions of the nodes are updated using the velocity vector and the ‘elastic mesh’ principle of Wang et al. (2003). With this framework the motion and deformation of the bubbles can be simulated as a function of time. Full details on the 3D implementation of BEM have been documented in previous works and will not be repeated here. Interested readers can refer to: Wang (1998), Zhang et al. 2001 or Klaseboer et al. (2005a, b)).

The initial conditions to be used in the simulations, p _b0 and V ₀ are not known for each bubble. It is assumed that there is some analogy with explosion bubbles, where several studies have been performed on these initial conditions (Cole 1948). For example, if similar bubbles were to be generated using explosives, the value of p _b0 would be 362.4 bar with a corresponding initial bubble radius R _0,i  = 0.0940 × R _max,i , where R _0,i corresponds to the initial volume (V ₀) and R _max,i to the maximum radius of bubble i (obtained from experiment). It is further assumed that each bubble is initially spherical. These initial conditions have been used for all simulations in this article. On a side note, it was observed that the vapor pressure, p _v, inside these spark bubbles might not always be negligible, as shown in Buogo and Cannelli (2002), who found p _v = 0.3 bar or Lew et al. (2006), who found p _v = 0.4 bar. Similar values are found in the current study.

The hydrostatic pressure is negligible in the current setup. Even though the bubbles observed in this article are millimeter in size, they oscillate fast enough to justify neglecting the influence of gravity in the numerical method, as opposed to non-oscillating bubbles, where buoyancy forces will eventually cause the bubble to rise. On the other hand, the bubbles are sufficiently large for surface tension effects to be negligible as well. Both effects can be easily accounted for in the numerical model if necessary (Fong et al. 2008).