Hydrodynamic modelling of transient cavities in fluids generated by high voltage spark discharges

Application of a voltage pulse having a rise time of tens of nanoseconds to electrodes immersed in water results in streamer development and the formation of a highly conductive plasma channel between the electrodes. The electrical resistance of such channels decreases rapidly from a few ohms to a few tens of milliohms due to Joule heating resulting from the high current which flows through the plasma. The dynamics of the plasma resistance depend on the parameters of the discharge circuit and the medium in which the discharge takes place. The resistance of the channel reaches a minimum value approximately at the moment of the peak current for under-damped current oscillations. During the resistance collapse, the pressure inside the channel rises to several GPa, causing a rapid expansion of the channel which forms a cavity in the liquid resulting in a high power ultrasound pulse. The cavity expands to a maximum size which is dependent on the circuit driving the discharge and the properties of the plasma discharge channel. The cavity then collapses producing a second acoustic pulse. In this paper the dynamic resistance of the spark channel is described using a phenomenological model based on the plasma channel energy balance equation used by Braginskii. The model which links the hydrodynamic characteristics of the channel and the resulting cavity with the parameters of the electric driving circuit allows the development of the plasma channel and cavity to be predicted. The peak high-power ultrasound (HPU) pressures calculated using this approach are compared with the pressure values estimated by an analytical model which uses a constant value of the spark channel resistance derived from experimental data. Comparisons are also made with direct measurements of HPU output made using a Pinducer sensor. Although the model is based on a phenomenological description of the plasma channel dynamics and its resistance and requires the value of the spark constant, the results obtained using this approach provide a reasonable agreement with experimental measurements and could therefore be used for the estimation of HPU pulse characteristics in practical applications of spark discharges in water.