Improvement of the Grad 13 moment method for strong shock waves

Abstract

The classical problem of the mathematical limitation of the 13 moment theory for strong shock waves has been reexamined by introducing the Mott-Smith bimodal function as a reference function in the Grad moment method. The shock wave of monatomic Maxwell molecules has been formulated in time-dependent equations by using the Boltzmann equation. In the steady state, the proposed theory improves the original Grad theory. The obtained 13 moment equation gives a solution curve which connects a critical saddle to a critical node in the phase space when M<4.14, while the original Grad theory gives the solution curve only when M<1.65. Numerical calculations show that the present theory also gives a solution curve when M⩾4.14 by connecting two critical saddles. The saddle-saddle connection in the phase space is possible due to the existence of the regular singular point located between the saddles. The predicted shock profile for monatomic Maxwell molecules shows a reasonable agreement between the theory and the Monte Carlo direct simulation. Also, the theory predicts a small overshoot of the kinetic temperature profile at the downstream wing when M>3.3.