An engineering model for temperature and pressure evolution during and after nanosecond discharges

Abstract A nanosecond voltage pulse applied between two pin electrodes creates an elongated plasma kernel at high temperature and high pressure. The kernel relaxes through a series of flow dynamics and thermal effects, which affect the temperature and pressure in the inter-electrode region. This work presents an engineering model that predicts the temperature and pressure changes throughout (i) the pulse and (ii) the interpulse period. The pulse model returns the temperature and pressure after the pulse, depending on the discharge regime and the governing parameters (pulse energy, nature of the gas, gap length, and discharge radius). The interpulse model is a sequence of four simple submodels that account for the three dominant flow mechanisms identified in CFD simulations and for vibrational relaxation. The first phase involves the rapid expansion of a blast wave. At the end of this phase, the temperature of the core has been about halved, the pressure has relaxed to the ambient pressure, and the gas has acquired vorticity in a narrow region close to the electrodes. However, this vorticity has not yet had time to disturb the core of the kernel. The second phase corresponds to the vibrational relaxation of the gas towards thermal equilibrium in the case of non-thermal sparks in molecular gases. The third phase is the cooling of the kernel by radial heat diffusion. The fourth phase corresponds to the development of a large recirculation zone from the localized vorticity seeds deposited during the first phase. Two vortices entrain fresh gases into the inter-electrode region. The interpulse model is successfully validated by comparison with a large database of post-discharge CFD simulations with varying inter-electrode gaps and kernel temperatures. In addition, the model shows good agreement with the cumulative heating shown by the temporal temperature profiles measured in literature experiments. It is computationally efficient (a few seconds vs 10 3 –10 6 CPU hours) and can be incorporated into zero-dimensional kinetic solvers to study the chemistry of the post-discharge. The model can also be used to get criteria on the discharge configuration (temperature, gap, frequency) that ensures full cooling between successive pulses.