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Statistical constraints

DSMC parameters with respect to total pressure

Scaling behaviour of cell dimension, particle number and time step

For a gas flow simulation, the set up of the simulation domain and the particle statistics requires to decide values for certain parameters, which are listed in the following. For simplicity we assume that all cells have a cubical shape.

  • a = Cell dimension [m]
  • Vc = Cell volume = a3
  • NREAL = Quotient between physical number of particles N and number of simulation particles NS
  • δt = Period of one time cycle

The cell dimension should be less than the mean free path λ which itself reciprocally depends on total pressure p1), which leads to the following considerations:

$a \sim \lambda \sim p^{-1}$,

$V_\textrm{c} \sim a^3 \sim p^{-3}$.

Since the travelling distance of a particle during one time cycle should be also less than the mean free path, a similar relation holds for the time cycle:

$\delta t \sim p^{-1}$.

From the ideal gas equation, we know for the physical number Nc of particles within a cell, that:

$N_\textrm{c} = \dfrac{p\,V_\textrm{c}}{k\,T} \sim p^{-2}$,

provided that the cell dimension a is adjusted reciprocally with total pressure p as mentioned above. It is possible to keep the number Ns of simulation particles per cell constant by adjusting NREAL according to the pressure. In this case NREAL satisfies the relation:

$N_\textrm{s} = \dfrac{N_\textrm{c}}{N_{\scriptscriptstyle\textrm{REAL}}} \stackrel{!}{=} {\rm const.} \Rightarrow N_{\scriptscriptstyle\textrm{REAL}} \sim p^{-2}$.

Especially at higher pressure, the computational effort is almost completely dominated by the collision routine. In the so-called No time counter method 2), the number NCOLL of pre-selected collision pairs is determined as follows:

$N_{\scriptscriptstyle\textrm{COLL}} = \dfrac{N_\textrm{s}^2 N_{\scriptscriptstyle\textrm{REAL}}\,\delta t <\sigma\, c>_{\textrm max}}{V_\textrm{c}}$.

If we put in the above described relations, we find for each cell that NCOLL is independent from the total pressure p. However, given that the number of cells increases with p3 and the time cycle δt behaves reciprocal with p the total computational effort for a gas flow system with constant volume and for a constant total real-time period increases with p4.

Numerical example for Argon

For Argon we know, that for a total pressure of 1 Pa, the mean free path is about 6 mm. Thus, in order to be on the safe side, we chose a cell size of 5 mm at p = 1 Pa.

For the time cycle we consider the mean thermal velocity c of Ar at room temperature, which is approx. 400 m/s. As mean travelling distance at p = 1 Pa, we chose a length of 4 mm which is well below the mean free path resulting in:

$\delta_t(p\textrm{ = 1 Pa}) = 10^{-5}\,s$.

The physical number of particles within a cell sized 5x5x5 mm3 at T = 300 K (room temperature) can be estimated from the ideal gas equation as N = 3.02 x 1013. If we would like to maintain a constant number of simulation particles per cell of Ns = 10, the resulting value for NREAL is 3 x 1012.

Based on these starting considerations and on the scaling relations shown above, we obtain the following set of simulation parameters as a function of total pressure:

Pressure [Pa]Cell dimension a Time cycle [s]NREAL for 10 simulation particles per cell
0.1 50 mm 1 x 10-4 3 x 1014
1.0 5 mm 1 x 10-5 3 x 1012
10.0 0.5 mm 1 x 10-6 3 x 1010
100.0 0.05 mm 1 x 10-7 3 x 108
1000.0 5 µm 1 x 10-8 3 x 106
10000.0 0.5 µm 1 x 10-9 3 x 104
100000.0 (~ 1 bar) 0.05 µm 1 x 10-10 3 x 102
1)
see Mean free path
2)
Bird, G. A.; Monte Carlo Simulation in an engineering context, In: Progr. Astro. Aero. 74 (1981) 239-255.

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