Focusing of Compression Waves in a Pipe Containing a Cylindrical Bubbly Cluster

The dynamics of two-dimensional waves in a water-filler pipe containing cluster of bubbles has been investigated. The waves are generated by a step-like increase in the pressure at one end of the pipe. The liquid with bubbles is described by a system of macroscopic equations of conservation of mass, bubble number, and momentum, as well as an equation of the pressure in the bubbles. A comparison with the case of the wave impact on a bubbly layer near the the pipe wall is also conducted. The dependence of the maximum pressure amplitude in the pipe on the bubbly cluster size and the initial volumetric gas content in the cluster is studied. A map of zones with maximum pressure in the pipe is constructed. The results show that besides the zone where the volumetric content is a determining factor, there is a zone where the size of the bubbly region play a crucial role. There is an optimal combination of the bubbly region size and the initial volumetric bubble content that ensures maximum pressure growth in the pipe.