Shock-wave structure according to a linear irreversible thermodynamic model

In this work we present a phenomenological model to look for a better understanding of the shock-wave structure in dilute monatomic gases. The model is based on the principles of linear irreversible thermodynamics, where we have been aware of the flow anisotropy caused by the shock-wave propagation. Then a new coupling appears between the stress tensor and the heat flux. The comparisons with the experimental data available for argon as well as the direct simulation Monte Carlo method calculations are done and shown to support our proposal.

Figure 1

Normalized density and temperature profiles for argon at $M=1.55$. (a) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Solid circles: Experiments by Alsmeyer [20]; open circles: Bird's DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] at $M=1.5$. (b) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Solid circles: Experiments by Alsmeyer [20]; open circles: Bird's DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; asterisks: Bird's DSMC for $\sigma =1.2$ and $\alpha =1$; diamonds: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] at $M=1.5$. (c) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Legend as in (b). (d) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Boxes: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] at $M=1.5$. (e) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Open circles: Bird's DSMC for $\sigma =1.2$; asterisks: Bird's DSMC for $\sigma =1.2$ and $\alpha =1$; diamonds: Bird's DSMC for for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] at $M=1.5$. (f) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Legend as in (e).

Figure 2

Normalized density and temperature profiles for argon at $M=9$. (a) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Solid circles: Experiments by Alsmeyer [20]; open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; solid line: DSMC by Sharipov and Dias [55] for $M=10$. (b) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Solid circles: Experiments by Alsmeyer [20]; open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; asterisks: Bird's DSMC for $\sigma =0.72$ and $\alpha =1$; diamonds: Bird's DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] for $M=10$. (c) Normalized density profiles vs Almeyer's reduced distance, ${\rho }_n$ vs $s_{{}_A}$. Legend as in (b). (d) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; solid line: DSMC by Sharipov and Dias [55] for $M=10$. (e) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; asterisks: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=1.0$; diamonds: Bird's DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$; solid line: DSMC by Sharipov and Dias [55] for $M=10$. (f) Normalized temperature profiles vs Almeyer's reduced distance, ${\tau }_n$ vs $s_{{}_A}$. Legend as in (e).

Figure 3

Reduced temperature vs reduced velocity; $\tau$ vs $v$ for argon for different Mach numbers. (a) $\tau$ vs $v$ for argon at $M=1.55$. open circles: Bird's DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; diamonds: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; solid line: NSF hydrodynamic model [1]. (b) $\tau$ vs $v$ for argon at $M=9$. open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; diamonds: Bird's DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$; solid line: NSF hydrodynamic model [1]. (c) $\tau$ vs $v$ for argon at $M=1.55$. open circles: Bird's DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; diamonds: Bird's DSMC for $\sigma =1.0$ and ${\alpha }^{-1}=0.6525$; Solid circle: Down-flow (hot part of the shock wave); solid line: NSF hydrodynamic model [1]. (d) $\tau$ vs $v$ for argon at $M=9$. Open circles: Bird's DSMC for $\sigma =0.72$ and ${\alpha }^{-1}=0.6015$; diamonds: Bird's DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6525$; boxes: Bird's DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$; solid line: NSF hydrodynamic model [1].

Figure 4

Normalized density profiles, ${\rho }_n$ vs $s_A$, for $M=1.55$ in two different regions [(a) and (b)]. Solid circles: Experiments [20]; open circles: DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where the values for $a_2$ and $\sigma$ are given in Table 1.

Figure 5

Normalized density profiles, ${\rho }_n$ vs $s_A$, for $M=3.38$ in two different regions [(a) and (b)]. Solid circles: Experiments [20]; open circles: DSMC for $\sigma =0.75$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 6

Normalized density profiles, ${\rho }_n$ vs $s_A$, for $M=8$ in two different regions [(a) and (b)]. Solid circles: Experiments [20]; open circles: DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6015$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 7

Orbits, $\tau$ vs $v$, for different Mach numbers. (a) $M=1.55$. Circles: DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Diamonds: DSMC for $\sigma =0.75$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Circles: DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6015$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 8

Normalized temperature profiles, ${\tau }_n$ vs $s_A$, and different Mach numbers. (a) $M=1.55$. Circles: DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Diamonds: DSMC for $\sigma =0.75$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Circles: DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6015$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 9

Reduced longitudinal temperature profiles, ${\tau }_{xx}$ vs $s_A$, for different Mach numbres. (a) $M=1.55$. Circles: DSMC for $\sigma =1.2$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Diamonds: DSMC for $\sigma =0.75$ and ${\alpha }^{-1}=0.6525$; solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Circles: DSMC for $\sigma =0.68$ and ${\alpha }^{-1}=0.6015$; solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 10

Reduced viscous pressure tensor profiles, $\mathrm{\Pi }$ vs $s_A$, for different Mach numbers. (a) $M=1.55$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 11

Reduced heat flux profiles, $q$ vs $s_A$, for different Mach numbers. (a) $M=1.55$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 12

Reduced entropy production profiles, $\mathrm{\Sigma }$ vs $s_A$. (a) $M=1.55$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 13

Reduced entropy change profiles, $\mathrm{\Delta }S$ vs $s_A$, for different Mach numbers. (a) $M=1.55$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (b) $M=3.38$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1. (c) $M=8$. Solid line: NSF; dashed line: LIT2; dotted line: LIT1, where $a_2,\sigma$ values are given in Table 1.

Figure 14

Normalized density profiles vs Bird's reduced distance, ${\rho }_n$ vs $s_{{}_B}$, for different times. Crosses: DSMC calculation for one time step (whose value is 0.75 $\times {10}^{-6}$ s); circles: DSMC calculations for 100 time steps; boxes: DSMC calculations for 1000 time steps; solid line: DSMC calculations for $10000$ time steps.