Shock-wave structure for a polyatomic gas with large bulk viscosity

The structure of a standing plane shock wave in a polyatomic gas is investigated on the basis of kinetic theory, with special interest in gases with large bulk viscosities, such as ${\mathrm{CO}}_2$ gas. The ellipsoidal statistical model for a polyatomic gas is employed. First, the shock structure is computed numerically for various upstream Mach numbers and for various (large) values of the ratio of the bulk viscosity to the shear viscosity, and different types of profiles, such as the double-layer structure consisting of a thin upstream layer with a steep change and a much thicker downstream layer with a mild change, are obtained. Then, an asymptotic analysis for large values of the ratio is carried out, and an analytical solution that describes the different types of profiles obtained by the numerical analysis, such as the double-layer structure, correctly is obtained.

Figure 1

Schematic figure of the profiles of types A, B, and C. The figure is a reproduction of Fig. 1 in Ref. [14] (courtesy of S. Taniguchi).

Figure 2

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=5$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-200\le x_1\le 3600$; (b) profiles for $-20\le x_1\le 100$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for ${\mu }_b/\mu =100$, the dashed lines for ${\mu }_b/\mu =200$, the dot-dashed lines for ${\mu }_b/\mu =500$, the dot-dot-dashed lines for ${\mu }_b/\mu =1000$, and the dotted lines for ${\mu }_b/\mu =2000$. In panel (b), the black dotted lines indicate the profiles of $\check{\rho },\check{v}$, and $\check{T}$ for ${\mu }_b/\mu =\infty$.

Figure 3

Profiles of ${\check{T}}_{\mathrm{tr}},{\check{T}}_{\mathrm{int}}$, and $\check{T}$ at $M_{-}=5$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-200\le x_1\le 3600$, (b) profiles for $-20\le x_1\le 100$. The red curves indicate ${\check{T}}_{\mathrm{tr}}$, the green curves ${\check{T}}_{\mathrm{int}}$, and the blue curves $\check{T}$. See the caption of Fig. 2 about the types of lines. In panel (b), the black dotted lines indicate the profiles of ${\check{T}}_{\mathrm{tr}},{\check{T}}_{\mathrm{int}}$, and $\check{T}$ for ${\mu }_b/\mu =\infty$.

Figure 4

Profiles of ${\widehat{p}}_{11}-\widehat{p},{\widehat{p}}_{22}-\widehat{p}$, and $-{\widehat{q}}_1$ at $M_{-}=5$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-200\le x_1\le 3600$; (b) profiles for $-20\le x_1\le 100$. The red curves indicate ${\widehat{p}}_{11}-\widehat{p}$, the green curves ${\widehat{p}}_{22}-\widehat{p}$, and the blue curves $-{\widehat{q}}_1$. See the caption of Fig. 2 about the types of lines. In panel (b), the black dotted lines indicate the profiles of ${\widehat{p}}_{11}-\widehat{p},{\widehat{p}}_{22}-\widehat{p}$, and $-{\widehat{q}}_1$ for ${\mu }_b/\mu =\infty$.

Figure 5

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=5$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000 in the new coordinate $y_1$. (a) Profiles for $-0.2\le y_1\le 1$; (b) profiles for $-0.06\le y_1\le 0.06$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$ and the blue curves $\check{T}$. See the caption of Fig. 2 about the types of lines.

Figure 6

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=5$ for $\mathrm{Pr}=0.761,A_c=\mathrm{const},{\mu }_b/\mu =1000$, and $\delta =3$, 4, 5, and 8. (a) Profiles for $-100\le x_1\le 1800$; (b) profiles for $-14\le x_1\le 20$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for $\delta =3$, the dashed lines for $\delta =4$, the dot-dashed lines for $\delta =5$, and the dot-dot-dashed lines for $\delta =8$.

Figure 7

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.2$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-1000\le x_1\le 24000$; (b) profiles for $-100\le x_1\le 2000$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for ${\mu }_b/\mu =100$, the dashed lines for ${\mu }_b/\mu =200$, the dot-dashed lines for ${\mu }_b/\mu =500$, the dot-dot-dashed lines for ${\mu }_b/\mu =1000$, and the dotted lines for ${\mu }_b/\mu =2000$. In panel (b), the black dotted lines indicate the profiles of $\check{\rho },\check{v}$, and $\check{T}$ for ${\mu }_b/\mu =\infty$.

Figure 8

Profiles of ${\check{T}}_{\mathrm{tr}},{\check{T}}_{\mathrm{int}}$, and $\check{T}$ at $M_{-}=1.2$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-1000\le x_1\le 24000$, (b) profiles for $-100\le x_1\le 2000$. The red curves indicate ${\check{T}}_{\mathrm{tr}}$, the green curves ${\check{T}}_{\mathrm{int}}$, and the blue curves $\check{T}$. See the caption of Fig. 7 about the types of lines. In panel (b), the black dotted lines indicate the profiles of ${\check{T}}_{\mathrm{tr}},{\check{T}}_{\mathrm{int}}$, and $\check{T}$ for ${\mu }_b/\mu =\infty$.

Figure 9

Profiles of ${\widehat{p}}_{11}-\widehat{p},{\widehat{p}}_{22}-\widehat{p}$, and $-{\widehat{q}}_1$ at $M_{-}=1.2$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-1000\le x_1\le 24000$; (b) profiles for $-100\le x_1\le 2000$. The red curves indicate ${\widehat{p}}_{11}-\widehat{p}$, the green curves ${\widehat{p}}_{22}-\widehat{p}$, and the blue curves $-{\widehat{q}}_1$. See the caption of Fig. 7 about the types of lines. In panel (b), the black dotted lines indicate the profiles of ${\widehat{p}}_{11}-\widehat{p},{\widehat{p}}_{22}-\widehat{p}$, and $-{\widehat{q}}_1$ for ${\mu }_b/\mu =\infty$.

Figure 10

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.2$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000 in the new coordinate $y_1$. (a) Profiles for $-0.5\le y_1\le 8$; (b) profiles for $-0.12\le y_1\le 0.2$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. See the caption of Fig. 7 for the types of lines.

Figure 11

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.2$ for $\mathrm{Pr}=0.761,A_c=\mathrm{const},{\mu }_b/\mu =1000$, and $\delta =3$, 4, and 5. (a) Profiles for $-1000\le x_1\le 12000$; (b) profiles for $-60\le x_1\le 240$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for $\delta =3$, the dashed lines for $\delta =4$, and the dot-dashed lines for $\delta =5$.

Figure 12

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.138...\left({\tilde{M}}_{-}=1\right)$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-2000\le x_1\le 28000$; (b) profiles for $-1000\le x_1\le 4000$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for ${\mu }_b/\mu =100$, the dashed lines for ${\mu }_b/\mu =200$, the dot-dashed lines for ${\mu }_b/\mu =500$, the dot-dot-dashed lines for ${\mu }_b/\mu =1000$, and the dotted lines for ${\mu }_b/\mu =2000$.

Figure 13

Profiles of ${\check{T}}_{\mathrm{tr}}$ and ${\check{T}}_{\mathrm{int}}$ at $M_{-}=1.138...\left({\tilde{M}}_{-}=1\right)$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-2000\le x_1\le 28000$; (b) profiles for $-1000\le x_1\le 4000$. The red curves indicate ${\check{T}}_{\mathrm{tr}}$, and the green curves ${\check{T}}_{\mathrm{int}}$. See the caption of Fig. 12 about the types of lines.

Figure 14

Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.05$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$, 200, 500, 1000, and 2000. (a) Profiles for $-40000\le x_1\le 60000$; (b) profiles for $-10000\le x_1\le 10000$. The red curves indicate $\check{\rho }$, the green curves $\check{v}$, and the blue curves $\check{T}$. The solid lines indicate the profiles for ${\mu }_b/\mu =100$, the dashed lines for ${\mu }_b/\mu =200$, the dot-dashed lines for ${\mu }_b/\mu =500$, the dot-dot-dashed lines for ${\mu }_b/\mu =1000$, and the dotted lines for ${\mu }_b/\mu =2000$.

Figure 15

Comparison between the profiles in Ref. [14] and those in the present computation at $M_{-}=1.47$. The figure is a reproduction of a part of Fig. 6 in Ref. [14] (courtesy of S. Taniguchi): The solid lines indicate the result based on the extended thermodynamics, the dashed lines that based on the Navier-Stokes Fourier theory, and the circles in the profile of $\widehat{\rho }$ the experimental result in Ref. [19] (see the caption of Fig. 6 in Ref. [14]). The present result is overdrawn by colored lines: The red line is for $\widehat{\rho }$, the green line for ${\widehat{v}}_1$ (denoted by $\widehat{v}$ in the figure according to Ref. [14]), and the blue line for $\widehat{T}$.

Figure 16

Comparison between the profiles based on the slowly varying solution and those of numerical solution. Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=5$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$ are shown in the figure. (a) Profiles for $-40\le x_1\le 200$; (b) profiles for $-20\le x_1\le 60$. The red line indicates $\check{\rho }$, the green line $\check{v}$, and the blue line $\check{T}$ of the numerical solution. The black dot-dashed line indicates the corresponding profiles obtained on the basis of the slowly varying solution. In panel (b), the numerical solution of the ES model for ${\mu }_b/\mu =\infty$ is also shown by the black dashed line.

Figure 17

Comparison between the profiles based on the slowly varying solution and those of numerical solution. Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.2$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$ are shown in the figure. (a) Profiles for $-100\le x_1\le 1200$; (b) profiles for $-40\le x_1\le 200$. See the caption of Fig. 16.

Figure 18

Comparison between the profiles based on the slowly varying solution and those of numerical solution. Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.138...\left({\tilde{M}}_{-}=1\right)$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$ are shown in the figure. (a) Profiles for $-200\le x_1\le 1800$; (b) profiles for $-160\le x_1\le 160$. The red line indicates $\check{\rho }$, the green line $\check{v}$, and the blue line $\check{T}$ of the numerical solution. The black dot-dashed line indicates the corresponding profiles obtained on the basis of the slowly varying solution.

Figure 19

Comparison between the profiles based on the slowly varying solution and those of numerical solution. Profiles of $\check{\rho },\check{v}$, and $\check{T}$ at $M_{-}=1.05$ for $\delta =4,\mathrm{Pr}=0.761,A_c=\mathrm{const}$, and ${\mu }_b/\mu =100$ are shown in the figure. (a) Profiles for $-2500\le x_1\le 3000$; (b) profiles for $-200\le x_1\le 200$. See the caption of Fig. 18.