Question 6.3: Normal Shock Wave in a Mixture of Helium and Xenon In this e...

which diffusive transport must be considered. The flow conditions match experiments performed by Gmurczyk, Tarczynski, and Walenta (1979) and further analysis of this problem can be found in Valentini, Tump, Zhang, and Schwartzentruber (2013).
Specifically, the pre-shock conditions (state 1) are that of a moderate Mach number (M_{1} = 3.61) mixture of helium (98.5% by mole) and Xenon (1.5% by mole) at a temperature of T_{1} = 300  K and a mixture density of ρ_{1} = 8.0 × 10^{−5}   {kg}/{m^3}. Using the jump equations across a normal shock wave, the post-shock conditions (state 2) are M_{2} = 0.5, T_2 = 1480  K, and ρ_{2} = 2.6 × 10^{−4}   {kg}/{m^3}.
As in the previous example problem, the length of the simulation domain is 8 cm. Owing to the disparate masses of He and Xe, and also due to the small mole fraction of Xe, this shock wave simulation is more computationally expensive than the previous example problem for argon. The particle weight, W_p, was set to obtain approximately N_p = 4000 particles in each cell in the freestream region. This results in, on average, 3940 He particles and 60 Xe particles in each cell in the freestream. The same boundary condition and iteration procedures are performed as described in the previous example problem. Since the freestream mean free path (λ_1) and mean collision time (τ_1) are that of the mixture, they are not representative of each species individually. Based on mixture properties, a transient period of 1000τ_1 was allowed, after which the solution was sampled during a further simulation time of 100τ_1.
As described in Chapter 5, an interatomic potential can be used to determine viscosity and diffusion coefficients through Eqs. 5.86 and 5.87

\mu _{ij}=\frac{5}{8} \frac{kT}{\Omega ^{(2.2)}}             (5.86)

\mathscr{D}_{ij}=\frac{3}{16} \frac{kT}{nm_{r}\Omega ^{(1,1)}}             (5.87)

which are functions of the collision integrals. The collision integrals resulting from the LJ 12-6 potential were plotted previously in Fig. 5.3(b) and the viscosity and diffusion coefficients resulting from Eqs. 5.86 and 5.87 are now plotted in Fig. 6.10. The LJ potential parameters for He–He, Xe–Xe, and Xe–He pairs, were listed previously in Table 5.1.

Table 5.1 Atomic Parameters

k is Boltzmann’s constant.

As described in Valentini et al.(2013), these LJ parameters result in mixture viscosity and diffusion coefficients that agree well with experimental measurements. Furthermore, pure molecular dynamics (MD) simulations of this normal shock wave were performed Valentini et al. (2013), and are reproduced in Fig. 6.9. For this example, these MD simulation results, using the LJ 12-6 potential, are taken as an accurate baseline solution to which DSMC results using the VHS model are now compared.
The VHS model parameters are listed in Table 6.1. Specifically, the resulting VHS viscosity coefficients μHe−He and μXe−Xe match the viscosity of Helium and Xenon at room temperature, and also match well the trend with temperature. The ω parameter for Xe–He collision pairs is set as the average of the He–He and Xe–Xe values, however, two different values are used for d_{ref}. The first value for d_{ref} is simply the average of the He–He and Xe–Xe values. The second value for d_{ref} is chosen to better match the diffusion coefficient predicted by the LJ potential. Figure 6.10 plots the viscosity and diffusion coefficients for the cross-species pair (He–Xe) determined from the LJ potential, and both sets of VHS parameters. When the VHS parameters for the cross-species pair is taken as the average of the single-species pairs, there is a noticeable discrepancy in both viscosity and diffusion coefficients compared to the LJ result. However, if the value of d_{ref} (specific to He–Xe) is reduced from 4.035 to 3.65, the modified VHS value for the diffusion coefficient comes into close agreement with the LJ result over the full temperature range experienced in the shock wave. Note that this modification to match the diffusion coefficient (D_{He−Xe}(T)) does not necessarily increase the agreement for the viscosity coefficient (μ_{He−Xe}).

DSMC solutions using both VHS models are shown in Fig. 6.9 and compared to the pure MD solution using the LJ potential. In Fig. 6.9(a), the numerical solutions predict a separation in the species densities within the shock. The lighter He atoms “experience” the shock first since their velocities are significantly affected by both collisions with Xe and with other He atoms. Whereas, the heavier Xe atoms are affected significantly only by collisions with other Xe atoms. Such species separation within shock waves has been experimentally documented (for example see Gmurczyk et al. (1979)) and the LJ solution in Fig. 6.9(a) has been verified to agree closely with the experimental result for these shock conditions (Valentini et al., 2013).

As seen in Fig. 6.9(a), the profiles from the modified VHS model (where d_{ref} for He–Xe collisions is set to match the diffusion coefficient) are in much better agreement with the pure MD result. This modification has the effect of increasing the diffusion coefficient and therefore leads to a larger species separation within the shock as predicted by pure MD. To demonstrate improved agreement beyond the density profiles, Fig. 6.9(b) shows that the VHS solution with modified parameters is in much better agreement with MD for the x-translational temperature (T_x) profiles. Since T_x is a measure of the standard deviation of the x-velocity distribution function, it is significantly affected by the bimodal nature of the VDF, and is a more sensitive parameter than the density.

Thus, by matching the cross-species VHS parameters with a known binary diffusion coefficient, D_{He−Xe}(T ) (in this case the LJ result), the VHS model can very accurately simulate this stringent test case involving disparate species masses and concentrations within a strongly nonequilibrium flow involving a wide temperature range. In general, this strategy of choosing VHS parameters for like-species from viscosity data and for unlike-species from diffusion data is quite accurate for a wide range of nonequilibrium flows. It is important to note that while the VHS parameters in Table 6.1 are quite accurate in the temperature range considered, they may be inaccurate at temperatures above or below this range. Researchers using DSMC should carefully determine VHS parameters based on the best available data for viscosity or diffusion coefficients from the appropriate temperature range, on a case-specific basis.