A formulation for the shock-wave structure is devised by the approximation of Boltzmann’s equation by a simpler kinetic model. Initially, the distribution function in Boltzmann’s collision integral is expressed in terms of a function of deviation from local equilibrium, the magnitude of which is unrestricted, and the analysis is specialized to hard sphere molecules. A model is then derived by assigning to the deviation function the first-order term of Chapman–Enskog’s sequence which leads to Navier–Stokes equations. The model equation is shown to possess a description of a gas in a non-equilibrium state and to imply a Prandtl number value of 2/3, the formulation also containing the Bhatnagar–Gross–Krook model as a special case. In applying the kinetic model to the shock problem, the collision frequency of the loss term is replaced by a set of mean frequencies (independent of the molecular velocity) each of which characterizes a specific macroscopic quantity. The shock equations are evaluated numerically for argon employing an interation scheme that is initiated by the Navier–Stokes solution. One iteration only to the flow variables is performed. For weak shocks the iteration proves to be in very close agreement with the Navier-Stokes solution for a Prandtl number of 2/3; at higher Mach numbers, the iteration predicts a progressively larger deviation, especially in the temperature profile. In addition, the density and velocity profiles exhibit a ‘kink’ at Mach numbers 5 and 10. Unlike the Navier–Stokes predictions, the results also show that for high Mach numbers the total enthalpy within the shock no longer remains sensibly constant.
Reinterpreting shock wave structure predictions using the Navier–Stokes equations