Question 6.1: Collision Rate in a Simple Gas at Equilibrium In this exampl...

A small volume of argon gas at rest with a density of \rho= 8 × 10^{-5}   {kg}/{m^{3}} and a temperature of T = 500 K is simulated with DSMC using the NTC collision rate algorithm. The VHS collision model is used where d_{ref}= 3.915 ×10^{10}  m , T_{ref} = 273  K, and ω = 0.81. The computational domain consists of a box with dimensions of 5 × 4 × 2.5  mm in x, y, and z coordinate directions, respectively. The domain is divided into 24 collisions cells (4 × 3 × 2 in x, y, and z directions, respectively).

At the beginning of the simulation, the box is filled with simulation particles corresponding to the gas conditions. Specifically, N_{p}=20 or N_{p} =20, 000 particles are randomly positioned within each cell and the particle velocities are initially assigned from a Maxwell–Boltzmann distribution (refer to section A.1.3 in Appendix A) corresponding to zero bulk velocity and temperature T. The particle weight (W_{p}) is set to achieve the desired density ρ. A timestep of \Delta t_{DSMC} = 1.427 × 10^{7} seconds is used and particles that collide with the box walls undergo specular reflection. In this manner, the volume is initialized with a uniform, equilibrium gas which should remain in equilibrium throughout the simulation.

During each timestep, only a fraction of the particles in each cell are
selected to undergo collisions using the NTC algorithm (Eqs. 6.9–6.11 and Eq. 6.13).

N_{coll-max}=\frac{1}{2}N_{p}(N_{p}-1) \frac{ [ \left\langle\sigma (d,g)g\right\rangle]_{max} W_{p} \Delta t_{DSMC} }{V_{DSMC}}           (6.9)

1\leq N_{pairs-tested}\leq floor({N_{p}}/{2})           (6.11)

P_{DSMC}=F_{correction}\times \frac{\sigma (d,g)g}{[\left\langle\sigma (d,g)g\right\rangle ]_{max}}         (6.13)
In this example we define the “collision rate fraction” as the number of collisions performed within the box divided by the number of particles within the box. Simulation results for the case where N_{p} = 20 (and therefore 480 particles within the box) are plotted in Fig. 6.6(a). Here, the discrete and statistical nature of the DSMC simulation is evident. The collision rate fraction, plotted at each iteration, is seen to range between 0.01 and 0.04, which corresponds to between 5 and 20 collisions within the box during any single timestep. However, since the gas is in a steady state, the important result is the average of the collision rate fraction “sampled” over many iterations.
The average value sampled over 5000 iterations (shown by the solid black line in Fig. 6.6(a)) is equal to 0.0251. The simulated collision rate per particle is therefore

\nu _{sim}=\frac{\left\langle \text {Collision Rate Fraction}\right\rangle }{\Delta t_{DSMC}} = 1.759 × 10^{5} \text {collisions/particle/sec}    (6.31)

An exact analytical expression for the collision rate corresponding to the
VHS model can be obtained using the procedure for the HS model previously derived in Eqs. 1.133–1.138.

dZ_{AB}=n_An_B\frac{\left(m_Am_B\right)^{3/2} }{\left(2\pi kT\right)^3 }g\sigma _{AB}\exp \left\{-\frac{1}{2kT}\left[(m_A+m_B)W^2+m^{∗}_{AB}g^2\right]\right\} \ dW_1 \ dW_2 \ dW_3 \ dg_1 \ dg_2 \ dg_3         (1.133)

Z_{AB}=\frac{1}{1+\delta _{AB}}n_An_B\sigma _{AB}\sqrt{\frac{8kT}{\pi M^{∗}_{AB}} }        (1.138)

Specifically for the VHS model, the number of collisions experienced by species A with species B per particle of species A per unit time is

For this example, A = B, the reduced mass becomes m_{r} = m_{Ar}/2, and using the specified argon gas properties the collision rate is ν^{VHS}_{Ar} = 1.752 × 10^{5} collisions per particle per second. Therefore, the simulated collision rate (ν_{sim}) accurately reproduces the analytical collision rate (ν^{VHS}_{Ar}) , within approximately 0.4% for this case.

The instantaneous temperature at each timestep (calculated with Eq. 5.17

T_{tr}=\frac{\left\langle C^{\prime 2}\right\rangle}{3R}= \frac{m}{3k}(\left\langle C^{2}_{x}\right\rangle)+\left\langle C^{2}_{y}\right\rangle+\left\langle C^{2}_{z}\right\rangle -\left\langle C_{x}\right\rangle^{2}-\left\langle C_{y}\right\rangle^{2}-\left\langle C_{z}\right\rangle^{2})       (5.17)

using all particles in the box) is also plotted in Fig. 6.6(a). The temperature
calculated at any single timestep is seen to range significantly from 400 to 600 K. However, the time-averaged “sampled” value for temperature is very close to 500 K as expected.

he results in Fig. 6.6(a) are typical of most steady state DSMC simulations. The collision rate and macroscopic gas properties (such as density, bulk velocity, and temperature) evaluated within each cell at each timestep exhibit large statistical fluctuations. To reduce statistical scatter below a desired tolerance, particle properties must be “sampled” over many iterations to obtain macroscopic properties and molecular distribution functions (refer to Appendix D for more details).

Simulation results for the case where N_{p} = 20, 000 (and therefore 480,000 particles within the box) are plotted in Fig. 6.6(b). Since there are now 1000 times more particles in each cell, in order to obtain the same number of samples in the averaged collision rate, only five iterations are required for time averaging. As seen in Fig. 6.6(b), the instantaneous collision rate and temperature, evaluated at each timestep, now has very little statistical scatter due to the large value of N_{p} . In this case, the average collision rate fraction is found to be 0.0250 which gives a simulated collision rate of ν_{sim}= 1.752 × 10^{5} collisions/particle/sec, in excellent agreement with the analytical result (ν^{VHS}_{Ar} ).
This example demonstrates typical statistical scatter present in a DSMC simulation and how particle properties must be either sampled over many timesteps during steady state, or sampled over large N_{p} . Typically, to limit memory requirements and for overall computational efficiency, it is desirable to minimize N_{p} . However, as evident in the collision rate results in Fig. 6.6(a), the discrete nature of the simulation sets a lower limit on the value of Np. For example, if N_{p} = 4 , either no particle pairs, one pair, or both pairs collide during a given timestep. With such a large percentage variation in the collision rate, even an average taken over many timesteps may not converge to the correct collision rate. If trace species are present or if rare collision events are important, such as certain energy transitions or chemical reactions, then N_{p} must be raised appropriately.

Finally, it is important to note that the results presented in this example could be obtained by using only a single collision cell (with constant N_{p} ) and neglecting particle movement and boundary interactions entirely, since a uniform gas is simulated. Such a zero- dimensional approach is useful for rapidly testing collision algorithms. However, including particle movement within separate collision cells introduces fluctuations in N_{p} and is a more stringent test.