The internal structure of shock waves

Abstract

The non-linear 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 method for the whole range of upstream Mach numbers M^ from 1.1 to 10 even though the corresponding degree of departure from equilibrium varies by a factor greater than 1000. Many characteristics of the internal structure of the shock waves have been calcu lated from the solutions and compared with Navier-Stokes, Mott-Smith and Krook descriptions which, except for low Mach numbers, are not based upon the Boltz mann equation itself. Among our conclusions are the following: 1. The reciprocal shock thickness is in agreement with that of the 2 Mott-Smith shock (u-moment) from M^ of 2.5 to 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. 2. The temperature density relation is in agreement with that of the Navier-Stokes shocks for the lower Mach numbers in the range of 1"1 to 1.56. The Boltzmann reciprocal shock thickness is smaller than the Navier-Stokes value at this range of Mach number because the viscosity-temperature relation computed is not constant as predicted by the linearized theory. 2 3. 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. The four functionals of the distribution function discussed show maxima within the shock. 4. The entropy is a good approximation to the Boltzmann function for all M^. The solutions obtained satisfy the Boltzmann theorem for all Mach numbers. The increase in total temperature within the shock is small, but the increase is significantly different from zero. 5. The ratio of total heat flux q to (associated with the longi tudinal degree of freedom) correlates well with local Mach number for all M^ in accord with a relation derived by Baganoff and Nathenson. The Chapman-Enskog linearized theory predicts that the 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. 6. At M 1=4, and for 40% of velocity bins, the distribution function is different from the corresponding Mott-Smith value by more than three times the 90% confidence limit. The rms value of the percent difference, in distribu tion function is 15% for this Mach number. The halfwidth and several other characteristics of the function Jfdw^dvz differ from that of the Chapman-Enskog first iterate, and many of the deviations are in agreement with an experiment by Muntz and Harnett. 7. The ratio of the collision integral (found from our solution of the Boltzmann equation) to that calculated from Mott-Smit...

New relations between macroparameters in shock wave

“…The calculations have shown that this relation is not observed only in the area close to U 2 . The parabolic dependence up xx against velocity is identical to the proportional dependence of the transverse stress against the gas density in the shock wave described in [2,3], The results analogous to (2) were obtained also for the gas consisting of hard sphere molecules (k = oo) and in the case when k = 10. When k = oo the main difference is that coefficient 40/33 is replaced by coefficient 40/32 = 1.25 in relation (2).…”

“…Fig.l shows dependence T(u) for maxwellian molecules at M = 11. The solid line shows the DSMC results, the dashed line -the values obtained by the momentum conservation equation and approximation (2). It should be pointed out that the parabolic dependence of the temperature on velocity establishes the relation of their profiles in the shock wave.…”

“…Fig.lashows dependence(2) presented by a symbol for the shock waves at M = 3 and M =11. The line presents the DSMC results.…”

Atomistic phenomena in dense fluid shock waves

“…is less than unity whereas it is found to be greater than unity in dilute gases for comparable Mach numbers [28,29]. Not surprisingly, the Navier-Stokes equations do not even reproduce the shock thickness well for M s > 2 [30].…”

Non-equilibrium Shock Structure Calculation with High-order Modified Navier–Stokes Equations

“…In typical rarefied gas regimes, the available techniques can be classified into numerical and analysis methods [1]. Particle simulation methods such as direct simulation Monte Carlo (DSMC) method [2,3] and direct numerical simulation method of Boltzmann equations [4] are classified as numerical methods. Another important approach is called the analysis method, which includes linear Boltzmann equation [5,6], moment [7], and model equation methods [8].…”