Comparison and Validation of Two Mathemetical Model of Underwater Spark Simulation Using Cylindrical and Elleptical Coordinates

Abstract

The mathematical simulation of an underwater discharge generated in a long gap attracts a particular interest due to an increased resistance of a spark channel. Loosely speaking, with the increase of interelectrodes distance, the value of the spark resistance approaches to an impedance of pulse power source. As a consequence, a discharge regime becomes close to transient and a greater efficiency of shock wave generation is achieved. In this case, a cylindrical coordinates system is used for an analytical model of the underwater spark expansion [1]. However, the experiment observations show the long spark channel finally develops into a spherical bubble. Therefore, the real dynamic of an underwater spark cannot be described as an expanding cylinder. Using of elliptical coordinates allows one to describe the underwater spark channel as an expanding ellipsoid with only one coordinate depending on time [2]. In this study, we compare the numerical results received by using the cylindrical and elliptical coordinates for incompressible approach with the experimental data. Also, we define the region of applicability for both models. An elliptical model provides a close approximation of the spark expansion dynamic observed in the experiment. The model expressed in the cylindrical coordinates gives only approximate estimation of the temporal thickness of the spark channel. After reaching some radius value (usually half of interelectrodes distance) one have to convert a model from cylindrical coordinates to spherical coordinates [3]. The spark channel expressed in the elliptical coordinates gradually transforms from the prolate ellipsoid to spheroid shape. Nevertheless, due to complexity of a shock wave pressure simulation using elliptical coordinates, the cylindrical model remains a main tool for the shock wave calculation for a while.