Shock-wave structure based on the Navier-Stokes-Fourier equations

We use the Navier-Stokes-Fourier constitutive equations to study plane shock waves in dilute gases. It is shown that the experimental information on the normalized density profiles can be fit by using the so-called soft sphere model, in which the viscosity and thermal conductivity are proportional to a power of the temperature.

Figure 1

Normalized density profiles vs the reduced distance, ${\rho }_n$ vs $s_A$, for Argon at different Mach numbers. (a) M = 8. Solid circles: experiments from Steinhilper [64]; open circles: DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6015$; solid line: solution to Eqs. (23) for $\sigma =0.9$. (b) M = 1.55. Solid circles: experiments from Alsmeyer [65]; open circles: DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$; solid line: solution to Eqs. (23) for $\sigma =1.6$. (c) M = 1.55. Solid circles: experiments from Alsmeyer [65]; open circles: DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6525$, diamonds: DSMC for $\sigma$ (Maxwell model) and ${\alpha }^{-1}=0.6525$, boxes: DSMC for $\sigma =0.0$ (constant transport coefficients) and ${\alpha }^{-1}=0.6525$. The DSMC results using ${\alpha }^{-1}=0.6015$ give practically the same output when ${\alpha }^{-1}=0.6525$ is used. (d) M = 1.2. Solid circles: experiments from Garen et al. [66]; open circles: DSMC for $\sigma =0.81$ and ${\alpha }^{-1}=0.6015$; solid line: solution to Eqs. (23) for $\sigma =3$.

Figure 2

Orbits (solution curves) for Argon at different Mach numbers. (a) Orbit (solution curve) in the $T^{*}-v$ plane at $M=1.2$. Solid line: solution to Eqs. (23); open circles: DSMC for $\sigma =3$ and ${\alpha }^{-1}=0.6015$. (b) Orbit (solution curve) in the $T^{*}-v$ plane at $M=8$. Solid line: Solution to Eqs. (23), recall that the orbits are independent of the viscosity; open circles: DSMC with $\sigma =0.81$ and ${\alpha }^{-1}=0.6015$.

Figure 3

Normalized density or temperature profiles for Argon versus Alsmeyer's reduced distance, ${\rho }_n$ or $T_n$ vs $s_A$. (a)${\rho }_n$ and $T_n$ vs $s_A$ for $M=8$. Experiments [64]: solid circles for ${\rho }_n$. DSMC ($\sigma =0.81, {\alpha }^{-1}=0.6015$): open circles for ${\rho }_n$, diamonds for $T_n$. Solution to Eqs. (23) for $\sigma =0.9$: solid line for ${\rho }_n$, dashed line for $T_n$. (b) $T_n$ vs $s_A$ for $M=8$. DSMC ($\sigma =0.81, {\alpha }^{-1}=0.6015$): diamonds. (c) ${\rho }_n$ and $T_n$ vs $s_A$ for $M=1.2$. Experiments [66]: solid circles for ${\rho }_n$. DSMC ($\sigma =0.81, {\alpha }^{-1}=0.6015$): open circles for ${\rho }_n$, diamonds for $T_n$. Solution to Eqs. (23) for $\sigma =3.0$: solid line for ${\rho }_n$, dashed line for $T_n$. (d) $T_n$ vs $s_A$ for $M=1.2$. DSMC ($\sigma =0.81, {\alpha }^{-1}=0.6015$): diamonds. Solutions to Eqs. (23): dashed line for $\sigma =3.0$; dotted line for $\sigma =0$.

Figure 4

Normalized density profiles vs the reduced distance, ${\rho }_n$ vs $s_A$, for different systems and Mach numbers. (a) $M=6$. Solid diamonds: Experiments for Neon from Steinhilper [64]; open circles: DSMC for Neon with $\sigma =0.66$ [57] and ${\alpha }^{-1}=1.0$; solid line: Solution to Eqs. (23) for $\sigma =0.9$ with initial condition ${\mathbf{P}}_1=(13/48+{10}^{-12},2327/11520)$. (b) $M=8$. Boxes: experiments for Krypton from Steinhilper [64]; asterisks: experiments for Xenon from Steinhilper [64]; solid line: solution to Eqs. (23) for $\sigma =0.9$. (c) M = 6. Solid diamonds: experiments for Neon from Steinhilper [64]; open circles: DSMC for Neon with $\sigma =0.66$ [57] and ${\alpha }^{-1}=1.0$; dashed line: solution to Eqs. (23) for $\sigma =0.9$ with initial condition ${\mathbf{P}}_2=(0.2708333333,0.2019965278)$. (d) $M=8$. Boxes: Experiments for Krypton from Steinhilper [64]; asterisks: experiments for Xenon from Steinhilper [64]; solid line: solution to Eqs. (23) for $\sigma =0.95$.