The internal structure of shock waves

The nonlinear Boltzmann equation has been solved for shock waves in a gas of elastic spheres. The solutions were made possible by the use of Nordsieck's Monte Carlo method of evaluation of the collision integral in the equation. Accurate solutions were obtained by the same numerical procedure for eight values of the upstream Mach numbers M1 ranging from 1.1 to 10, even though the corresponding degree of departure from equilibrium varies by a factor greater than 100. Many more characteristics of the internal structure of the shock waves have been calculated from the solutions than have hitherto been available. Each solution of the Boltzmann equation requires about 108 multiplications to obtain statistical errors of 3% in values of the velocity distribution function and collision integral and much smaller errors in the moments of these functions. The reciprocal shock thickness is in agreement with that of the Mott-Smith shock (u2 moment) from M1 = 2.5-8. The density profile is asymmetric with an upstream relaxation rate (measured as density change per mean free path) approximately twice as large as the downstream value for weak shocks and equal to the downstream value for strong shocks. The temperature density relation is in agreement with that of the Navier-Stokes shocks for Mach numbers in the range 1·1-1-56. The Boltzmann reciprocal shock thickness is smaller than the Navier-Stokes value in this range of Mach number because the viscosity-temperature relation computed is not constant as predicted by the linearized theory.The velocity moments of the distribution function are, like the Mott-Smith shock, approximately linear with respect to the number density; however, the deviations from linearity are statistically significant. Four functionals of the distribution function that are discussed show maxima within the shock. The entropy is a good approximation to the Boltzmann function for all M1. The solutions obtained satisfy the Boltzmann theorem for all Mach numbers. The ratio of total heat flux q to qx (that associated with the longitudinal degree of freedom) correlates well with local Mach number for all Ml in accordance with a relation derived by Baganoff & Nathenson (1970). The Chapman-Enskog linearized theory predicts that this ratio is constant. The (effective) transport coefficients are larger than the Chapman-Enskog equivalents by as much as a factor of three at the mid-shock position.At M1 = 4, and for 40% of the velocity bins, the distribution function is different from the corresponding Mott-Smith value by more than three times the 90% confidence limit. The r.m.8. value of the percentage difference in distribution functions is 15% for this Mach number. At MI = 1-59, the half width and several other characteristics of the function \[ \int f\,dv_y dv_z \] differ from that of the Chapman-Enskog first iterate, and many of the deviations are in agreement with an experiment by Muntz & Harnett (1970).