Shock-wave structure and intermolecular collision laws

Abstract

The nonlinear Boltzmann equation has been solved for shock waves in a Max-wellian gas for eight upstream Mach numbers M1 ranging from 1·1 to 10. The numerical solutions were obtained by using Nordsieck's method, which was revised for use with the differential cross-section corresponding to an intermolecular force potential following an inverse fifth-power law. The accuracy of the calculations of microscopic and macroscopic properties for this collision law is comparable with that for elastic spheres published earlier (Hicks, Yen & Reilly 1972).We have made comparisons of the detailed characteristics of the internal shock structure in a Maxwellian gas with those in a gas of elastic spheres. The purpose of this comparative study is to find the shock properties that are sensitive as well as those which are insensitive to the change in collision law and to find effective ways to study them.The variation of thermodynamic and transport properties of interest with respect to density and to each other was found to depend only weakly on the change in collision law. The principal effect on the macroscopic shock structure due to the change in intermolecular potential is in the spatial variation of the macroscopic properties. The spatial variation of macroscopic properties may be determined accurately from the corresponding moments of the collision integral, especially in the upstream and downstream wings of the shock wave. The results for the velocity distribution function exhibit the microscopic shock characteristics influenced by a difference in intermolecular collisions, in particular the departure from equilibrium in the upstream wing of the shock and the relaxation towards equilibrium in the downstream wing. The departure of several characteristics of weak shock waves from those of the Chapman-Enskog linearized theory and the Navier-Stokes shock is also insensitive to the change in collision law. The deviation of the half-width of the function ∫fdvyduz from the Chapman-Enskog first iterate at M1 = 1·59 is in agreement with an experiment (Muntz & Harnett 1969).

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).…”

Higher moments of the velocity distribution function in dense-gas shocks

“…We do not expect that this is a dense gas effect. Experimental [24][25][26] and numerical [27][28][29][30][31][32][33][34][35] data for dilute gases, from which the higher moments can be extracted, is available in the literature, but to the authors' knowledge, the effect has not been reported previously. The location where the higher out-of plane moments first deviate from zero does not depend on he order of the moment, i.e., the trend for the lower moments that the temperature (second moment) changes upstream of the flow velocity (first moment) and the density (zeroth moment) is not continued or it approaches a limit asymptotically.…”

Asymptotic Approximation for the Distribution of Pairs of Molecules in a Shock Wave

“…Заметим, что эффект перехлёста продольной температуры является фундаментальным научным фактом. Данный эффект был обнаружен экспериментально [7] и получен строго теоретически на основе законов сохранения потоков массы и импульса в ударной волне [8]. Непосредственным следствием работы [8] является то, что любая новая теория, связанная с исследованием структуры фронта ударной волны, обязательно должна быть проверена на предмет совпадения с эталонной кривой для продольной температуры T x .…”