NON-SPHERICAL COLLAPSE OF AN AIR BUBBLE SUBJECTED TO A LlTHOTRIPTER PULSE

Seattle, Washington 98105 [email protected] In order to better understand the contribution of bubble collapse to stone comminution in shockwave lithotripsy, the shockinduced and Rayleigh collapse of a spherical air bubble is investigated using numerical simulations, and the free-field collapse of a cavitation bubble is studied experimentally. In shock-induced collapse near a wall, it is found that the presence of the bubble greatly amplifies the pressure recorded at the stone surface; the functional dependence of the wall pressure on the initial standoff distance and the amplitude are presented. In Rayleigh collapse near a solid surface, the proximity of the wall retards the flow and leads to a more prominent jet. Experiments show that re-entrant jets form in the collapse of cavitation bubbles excited by lithotripter shockwaves in a fashion comparable to previous studies of collapse near a solid surface. NOMENCLATURE Ro Initial bubble radius H Initial stand-off from the wall * Address all correspondence to this author. 285 Tim Colonius Division of Engineering and Applied Science California Institute of Technology Pasadena, California 91125 [email protected] Michael R. Bailey Center for Industrial and Medical Ultrasound Applied Physics Laboratory University of WaShington Seattle, Washington 98105 [email protected] P Pulse amplitude M Mach number cr Pulse width p Density u Velocity vector p Pressure E Total energy y Ratio of specific heats (for gases) P= Stiffness constant c Sound speed Vj Maximum jet speed Va Maximum speed of distal side INTRODUCTION Shockwave lithotripsy (SWL) is the most common treatment for kidney stones [1]. The main advantage of this procedure consists in avoiding surgery altogether by generating shockwaves extracorporeally and focusing them onto kidney stones [2]. As a result, the stones are broken into fragments small enough that they can be passed naturally by the human body. Copyright © 2007 by ASME Because of the complex physics governing the process, the stone comminution mechanism is not yet fully understood. Lithotripter pulses consist of a steep compressive front followed by a long expansion tail with a tensile component. Since kidney stones are typically immersed in urine, this tension enhances the formation and growth of bubbles near the stone. Consequently, two mechanisms bear important roles in stone comminution: spallation due to stress waves reflected within the stone [3,4], and cavitation erosion due to violent bubble collapse along the stone surface [S,6]. It has been proposed that the combined effect of the internal stress waves and of cavitation erosion pulverizes the stones [4]. More recently, it has been observed that the interference of shear waves generated by the passage of the pulse along the stone may play an important role [7]. The impact of bubble dynamics on stone comminution is still unclear. After the passage of a lithotripter pulse, bubbles generally gather in clusters that grow and collapse near the stone surface, thereby generating shockwaves [8,9]. In order to provide an estimate for cavitation damage, the collapse of a single bubble near a wall is considered. In SWL, single bubble collapse near the stone takes place on two time scales: a short one, where the collapse is induced by the passage of the pulse (shockinduced collapse'" l,us), and a long one, where the bubble grows to a large size due to cavitation induced by the tensile part of the pulse and subsequently collapses (Rayleigh collapse'" 200,us). When a bubble collapses near a solid surface, a re-entrantjet directed towards the surface forms and penetrates the bubble [10]. The impact of the jet onto the stone leads to pitting and has been regarded as the primary cause of cavitation erosion in SWL [S,6]. Re-entrant jets also form when bubbles are impacted by shockwaves, even when far from solid surfaces [11-13]. In the present study, a high-order accurate, quasiconservative, shockand interface-capturing scheme [14] is used to simulate both shock-induced and Rayleigh collapse near a solid surface. A detailed presentation of the problem is provided in Section I, while the numerical model is briefly described in Section 2. Preliminary results from shock-induced and Rayleigh collapse are presented and discussed in Section 3, along with experiments on the collapse of a cavitation bubble. Finally, the findings are summarized in Section 4. 1 COMPUTATIONAL SET-UP A spherical air bubble of radius, Ro , in water is in equilibrium with its surroundings and located at some distance, H, from a rigid wall, which represents the kidney stone. The computational domain consists of a cylinder, along whose centerline the bubble is located; azimuthal symmetry is assumed, so that the computation is axisymmetric. A slice through the center is depicted in Figure I, where the dashed line denotes the domain. A lithotripter pulse of amplitude, P, and width, cr, starts on the right of the bubble and propagates at normal incidence to286 wards the walL Reflecting and symmetry boundary conditions are used along the centerline and the wall, while incoming and non-reflecting boundary conditions are specified along the right and about the circumference of the domain. The wall along the left side is removed for Rayleigh collapse in a free field. ········1········ ..........•.........•....