Prediction of shock structure using the bimodal distribution function

A modification of Mott-Smith method for predicting the one-dimensional shock wave solution is presented. Mott-Smith distribution function is used to construct the system of moment equations to study the steady-state structure of shock wave in a gas of Maxwell molecules and in argon. The predicted shock solutions using the newly proposed formalism are compared to the experimental data, direct-simulation Monte Carlo (DSMC) solution, and the solutions predicted by other existing theories for Mach numbers $M<11$. The density, temperature, heat flux profiles, and shock thickness calculated at different Mach numbers have been shown to have good agreement with the experimental and DSMC solutions. In addition, the predicted shock thickness is in good agreement with the DSMC simulation result at low Mach numbers.

Figure 1

Comparison of the computed inverse density thicknesses, which are plotted against the Mach number. The currently predicted results are compared to those based on the theories of Navier-Stokes, Mott-Smith, and DSMC.

Figure 2

Plot of the computed values of temperature-density separation which is plotted against the Mach number. Notation—see Fig. 1.

Figure 3

Density profile plotted as the function of distance. Comparison of the currently predicted density profile to the DSMC, Navier-Stokes, and Mott-Smith simulation results against $x$ at $M=1.7$.

Figure 4

Comparison of the heat flux profiles which are plotted as the function of distance. Notation—see Fig. 3.

Figure 5

Predicted density profiles plotted as the function of distance at $M=4.0$. Notation—see Fig. 3.

Figure 6

Predicted heat flux profiles plotted as the function of distance at $M=4.0$. Notation—see Fig. 3.

Figure 7

Predicted density profiles plotted as the function of distance at $M=8.0$. Comparison of the currently computed results to the results of Nanbu and Watanabe [40] and Bird [16] DSMC, Navier-Stokes, and Mott-Smith simulations at $M=8.0$

Figure 8

Predicted heat flux profiles plotted as the function of distance. Comparison of the currently computed results to the results of Nanbu and Watanabe [40] and Bird [16] DSMC, Navier-Stokes, and Mott-Smith simulations at $M=8.0$

Figure 9

Comparison of the predicted density profiles at $M=10$. Notation—see Fig. 3.

Figure 10

Comparison of the predicted heat flux profiles at $M=10$. Notation—see Fig. 3.

Figure 11

Comparison of the predicted heat flux profiles at $M=2$. Notation—see Fig. 3

Figure 12

Comparison of the predicted temperature profiles at $M=10$. Notation—see Fig. 3. Note that the predicted temperature shows an overshot.

Figure 13

Comparison of the predicted values of temperature-density separation which is plotted against the Mach number. Diamonds—DSMC results of Pham-Van-Diep [4]; circles-DSMC results of Nanbu and Watanabe [40].

Figure 14

Comparison of the predicted inverse density thicknesses to the experimental data—circles [2, 3].

Figure 15

Density profile plotted as the function of distance. Comparison of the currently predicted density profile to the DSMC, Navier-Stokes, and Mott-Smith simulation results against $x$ at $M=11$; $s=0.72$.

Figure 16

Temperature profile plotted as the function of distance. Comparison of the currently predicted density profile to the DSMC, Navier-Stokes, and Mott-Smith simulation results against $x$ at $M=11$; $s=0.72$.

Figure 17

Plots of the higher-order moment of the distribution function ${\overline{q}}_x$ as the function of distance at different Mach numbers.