Interdisciplinary Applied Mathematics

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Effects of Statistical Scatter


Figure 3.8(d) aids in better visualization of statistical scatter in the DSMC results, which is insignificant for this case even near the stationary wall (y/L < 0.1). Note that the normalized amplitude does not drop below 1% of the maximum signal for this case. In the simulations, onset of statistical fluctuations is observed when the normalized signal amplitude drops below 1% of the maximum signal. Some of the DSMC results presented in Section 3.3 exhibit statistical fluctuations. In order to explore the statistical fluctuations induced by finite sampling in the presence of thermal fluctuations, we follow (Hadjiconstantinou et al., 2003). In equilibrium statistical mechanics, the ratio of excitation velocity u0 to the thermal fluctuation u’ for an ideal gas is given by (Hadjiconstantinou et al., 2003)


M° = Му^Ур,    (15.1)


J{(uf)


where M is    the    Mach    number, 7 is    the    ratio    of    specific heats,    and    N0    is


the average number of particles per single cell. The velocity fluctuation is defined as u’ = u — {u), where u is the instantaneous velocity and {u) is the average velocity. Since the unsteady DSMC algorithm uses ensemble averages over K repeating runs, the “noise-to-signal” ratio Eu can be expressed as




Based on the above definition, and the typical simulation parameters used in Section    3.3    (u0    = 100    m/s,    K = 5000,    N0    = 100,    and 7 =    5/3),    we


obtain Eu    =    3.4 x    10~3.    Considering    that    the    above    expression    is    obtained


for a medium in equilibrium, the noise level in unsteady computations is expected to be higher due to the presence of strong nonquilibrium effects in high Stokes number rarefied flows.

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