Finite-difference computation of the wave motion generated in a gas by a rapid increase in the bounding temperature

Abstract

Rapid heating of a solid surface bounding a region of gas generates a slightly supersonic wave with a positive amplitude in pressure, temperature, density, and mass velocity. Computed results are presented for both a discrete step in wall temperature and a bounded exponential rate of increase. The numerical predictions are in good qualitative agreement with prior experimental measurements of the shape, amplitude, and rate of decay of the wave in terms of pressure. Quantitative comparisons are not possible because the greatest times and distances that were feasible computationally are less than the least times and distances that were feasible experimentally. A value for the time constant for the exponential heating was identified, below which the behavior differs negligibly from that for a discrete step in temperature. The computations encompass a range of gases, imposed temperature differences, initial pressures, and initial temperatures. The absence of spurious oscillations and the absence of a time of induction for the onset of the wave were found to be satisfactory criteria for convergence with respect to grid size and time step. All prior finite-difference computations, with one exception, were found to be in significant error owing to the use of insufficient discretization by factors of about 10 4 in both distance and time. The false prediction of an infinite rate of heat transfer by the Fourier equation is shown to be due to the postulate of incompressibility. The hyperbolic equation for conduction avoids the prediction of an infinite rate of heat transfer but is in error fundamentally and in all of its predictions because of the incompatible postulates of incompressibility and of a wave motion. On the other hand, the asymptotic, analytical solution of Trilling is found to provide reasonable predictions for most but not all aspects of the wave motion.