Question 6.4: “Cold” Normal Shock Wave in Argon In this example, we simula...
“Cold” Normal Shock Wave in Argon
In this example, we simulate a normal shock wave in argon where the freestream temperature is very low. As a result, the attractive force between atoms may be important in the upstream shock region, the shock involves a large variation in temperature, and the shock interior is strongly nonequilibrium; a relevant test case for DSMC using the GHS collision model.
Specifically, the pre-shock conditions (state 1) are that of high Mach number (M_{1} = 10) argon flow at temperature of T_{1} = 20 K and a density of ρ_{1} = 7.6 × 10−5 {kg}/{m^{3}}. Using the jump equations across a normal shock wave, the post-shock conditions (state 2) are M_{2} = 0.45, T_{2} = 642 K, and ρ_{2} = 2.95 × 10^{−4} {kg}/{m^3}. Experiments by Alsmeyer (1976) measured shock wave structure under very similar conditions, but at Mach 7.183. A separate study investigated these conditions using pure MD (LJ 12-6) and DSMC with the VHS model (Valentini and Schwartzentruber 2009b), where no noticeable difference was found between MD and DSMC-VHS solutions. The current Mach 10 conditions induce a larger temperature variation and, in this example, we compare DSMC solutions using GHS and VHS models to see if modeling the attractive portion of the interatomic potential produces noticeable effects.
Since the flow conditions are similar to those used for Example 6.2, the DSMC simulation setup is identical to that used in Example 6.2, and is not repeated here. The VHS model parameters used in this example are also identical those used in Example 6.2, since they were found to give excellent agreement with experiment for the Mach 9 shock wave (where T_{1} = 300 K). Specifically, the VHS parameters are ω = 0.7, d_{ref} = 3.974 × 10^{−10} m, and T_{ref} = 273 K. The GHS parameters are listed in Table 6.2,
Table 6.2 GHS-Weak Model Parameters for Argon
| ∈ ⁄ k(K) | s_{0}[Å] | ψ_{1} | ψ_{2} | α_{1} | α_{2} |
| 124 | 3.418 | 1/6 | 2/3 | 3.85 | 3.10 |
where the GHS-weak assumption is employed. For this example, the GHS parameters were fit to reproduce the LJ 12-6 PES viscosity result over the range 20 K < T < 600 K (the LJ collision integral was plotted in Fig. 5.3(b)).
The viscosity coefficients resulting from the VHS and GHS models are plotted in Fig. 6.12, where they are compared with the result from the LJ PES. All three models produce a viscosity at 273 K in close agreement with the experimentally accepted value of \mu = 2.1 × 10^{−5} kg/m/sec. However, since the VHS model is restricted to a single power-law exponent (ω), it is not able to precisely match the temperature variation of the LJ result, whereas, the GHS model is.
As seen in Fig. 6.13, there is very little difference between VHS and GHS predictions for the density and temperature profiles through the shock. Similar close agreement was observed, for both flow profiles and velocity distribution functions, when DSMC-VHS simulations were compared with pure MD simulations along with the experimental data from Alsmeyer (Valentini and Schwartzentruber, 2009b).
Therefore, for this nonequilibrium problem involving a wide temperature range where both the long-range attractive and short-range repulsive interactions may be important, it is still the repulsive interaction that dominates. As long as the VHS model is properly parameterized, there is only a small difference compared to the more general GHS model. This is due to the fact that only a small region of the shock wave (the upstream portion) involves low temperatures (low collision energies). One would expect larger differences between GHS and VHS solutions for flows that involve a wide temperature range and a substantial portion of the flow field being at low temperature.
