Question 6.2: Normal Shock Wave in Argon In this example, a normal shock w...

The pre-shock conditions (state 1) are that of a high Mach number (M_{1} =9) flow of argon at a temperature of T_{1} = 300   K and a density of ρ_{1} = 1.069 × 10^{-4}   {kg}/{m^{3}}. These conditions match experiments performed by Alsmeyer (1976). Using the jump equations across a normal shock wave, the post-shock conditions (state 2) are M_{2}= 0.456, T_{2} = 7856  K, and ρ_{2} = 4.123 × 10^{-4}    {kg}/{m^{3}} . The length of the simulation domain is 8 cm, which is approximately equal to 100λ_{1} and cells are uniformly sized to be 0.25λ_{1}.
The computational domain was initialized with particles drawn from a Maxwell–Boltzmann velocity distribution function (VDF) corresponding to state 1 for x < 4 cm and corresponding to state 2 for x > 4  cm, with appropriate number densities. The particle weight, W_{p}, was set to obtain approximately N_{p} = 1000 particles in each cell in the freestream region. Before each timestep, any particles residing in either the first 10 cells or the final 10 cells of the domain were deleted and regenerated from Maxwell–Boltzmann distributions corresponding to state 1 and state 2 respectively. This technique enforces pre- and post-shock boundary conditions. The basic DSMC algorithm is iterated with timesteps that are a fraction of the freestream mean collision time, τ_{1}. The steady-state shock wave profile requires approximately 20τ_{1} to develop. After this initial transient period, the solution is sampled during a further simulation time of 5τ_{1}.
It is noted that this simulation is highly resolved, and that accurate solutions can be obtained with fewer particles and larger cell sizes. However, for normal shock wave simulations using this technique, it is desirable to use a large number of particles per cell and sample the solution over as few iterations as possible, to avoid any movement of the shock wave (during sampling) caused by statistical fluctuations. This is discussed in more detail in Bird (1994), in addition to other strategies for simulating normal shock waves.
The technique described previously, however, is accurate and simple to implement for many shock wave conditions. The collision parameters used for the simulation are, d_{ref} = 3.974   Å, T_{ref} = 273  K, and three different values of the viscosity law exponent, ω = 0.5, 0.7, 0.81 are used. Shock wave profiles are typically plotted in normalized variables,

where q is a flow property. The normalized density profiles resulting from the three simulations are shown in Fig. 6.7, where the VHS model using ω = 0.7 best matches the experimental data of Alsmeyer. More comparisons can be found in Valentini and Schwartzentruber (2009b). Note that the simulation using ω = 0.5 corresponds to a hard-sphere collision model and predicts a shock wave that is much thinner than the experimental result. This is expected, since the VHS model more accurately captures the reduction in cross section for higher relative collision velocities (experienced in this temperature range). The reduction in cross section enables both pre- and postshock molecules to be transported further (before colliding), therefore resulting in a thicker shock compared to the hard-sphere result.

The x-velocity distribution functions at four locations within the shock wave (simulated using ω = 0.7) are shown in Fig. 6.8. In the upstream region of the shock, the VDF (Fig. 6.8(a)) is seen to have a narrow peak at high velocity, corresponding to the cold but high-speed freestream flow. However, a low-velocity tail is beginning to appear that results from a finite number of post-shock molecules propagating upstream. The VDFs in the center of the shock (Figs. 6.8(b) and 6.8(c)), exhibit a “bimodal” mixture of the preand post-shock VDFs. Finally, in the downstream portion of the shock (Fig. 6.8(d)), the VDF is approaching a Maxwell–Boltzmann distribution corresponding to state 2.