The cylindrical pressure wave resulting from instantaneous energy release along a line in a quiescent atmosphere has been studied by numerical integration of the equations of gas dynamics. Atmospheres obeying both the ideal gas law, and a realistic equation of state for air at high temperatures, were employed. The effects of varying the initial distribution of mass and energy in space were also investigated. The computations were carried well into the weak shock region, and agree well with asymptotic solutions for very strong and very weak shock waves. The effects of deviations from the initial assumptions of the strong shock asymptotic solutions are discussed. An approximate equation for the radial dependence of shock strength, applicable to most of the numerical solutions, is presented. Experimental measurements of shock strengths from detonation of long high explosive charges are shown to be in good agreement with the numerical solutions.
The numerical model of a spark discharge in air, described in the preceding paper [Phys. Fluids 14, 2111 (1971)], is used to compute the properties of the discharge channel of a lightning return stroke and its acoustic wave. Initial conditions and electric current waveforms were varied over a wide range, to correspond with the observed variability of natural lightning events. Computed temperatures, pressures, and electron densities in the discharge channel agree well with available observational data. The model predicts temperatures near the upper limit of the range of spectroscopically determined temperatures. Discharge channel radii agree well with similarity solutions for the growth of a spark channel. The energy dissipated per unit length of the discharge channel and the strength of the pressure wave lie well below most literature estimates. The computed energy inputs fall between 10 and 60 J/cm, and depend primarily on the maximum values of the discharge current.
A numerical model is presented which describes the evolution with time of a short segment of a spark channel in air and its associated acoustic wave. The model assumes a straight, cylindrical conducting column in which local thermodynamic equilibrium exists at every point. The electrical energy input to the column is determined by a prescribed electrical current waveform, coupled with a computation of the plasma conductivity. The evolution with time of the conducting column and its surrounding flow field is then found by numerical integration of the equations of gas dynamics. The model employs a realistic equation of state for air at high temperatures, and incorporates kinetic and radiative energy transport processes. It is shown that a satisfactory description of the properties of a spark channel cannot be achieved when radiative transport processes are neglected. The model agrees well with experimental measurements of spark channel radii, temperatures, pressures, and electron densities, and predicts the resultant shock wave strengths closely. The voltage gradients along the spark channel predicted by the model, and the total energy input to the channel, are not as uniformly in agreement with experiment. Possible reasons for these discrepancies are discussed.
A theoretical model of the shock wave from a lightning discharge ranging from the strong blast wave region out to the acoustic limit is given for the first time. The trajectory and overpressure of the strong shock wave are described by the well‐known equations for cylindrical blast waves. In the intermediate shock strength region (1.1 < M < 3.3), the shock trajectory is given by the ‘correct limit’ equation of Vlases and Jones. We derive an additional ‘correct limit’ equation for overpressure that is valid out to the acoustic limit. The correct limit equations predict a much slower decay of the intermediate shock wave; thus, the shock wave is much stronger at large distances from the discharge than was previously believed. Consequently, the range of action of the lightning discharge via its shock wave, as it affects the shattering and freezing of supercooled hydrometeors, may be large.
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